Easy Structuring

Derivatives

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NDEX

1

Commerzban Commerzbank k ........................................................................................ ........................................................................................ 4 1.1 1.2 1.3

2

Markets in Corporates & Markets ...................................................... ...................................................... 5 The profit and loss (P&L) ................................................................... ................................................................... 8 The Book Open ................................................................................. ................................................................................. 8

Structured Equity Products and the People who buy them............... them.. .............9 9 2.1 What is a structured product?.. product? ............................................................ .......................................................... 9 2.1.1 Manufacturing aspects of structured products ............................ ............................ 9 2.1.2 Types of structured product ...................................................... ...................................................... 10 2.2 The market for structured products.................................................. products.................................................. 11 2.2.1 Intermediaries in the structured product market ....................... ....................... 12 2.3 Summary Summary ......................................................................................... ......................................................................................... 13

3

Derivatives............................................................................................ Derivatives.. .......................................................................................... 15 3.1 3.2 3.3 3.4

4

Vanilla Vanilla options options ................................................................................. ................................................................................. 15 Barrier Barrier options options ................................................................................. ................................................................................. 17 Combining vanilla products.............................................................. products.............................................................. 18 Put / Call parity ................................................................................ ................................................................................ 19

Black-Scho Black-Scholes les Model ........................................................................... ........................................................................... 20 4.1 4.2 4.3 4.4 4.5

Stochastic Differential Equations (SDEs)......................................... (SDEs)......................................... 20 Lognormal returns for asset prices .................................................. .................................................. 20 The assumption for the Black-Scholes model.................................. model.................................. 20 The Black-Scholes equation ............................................................ ............................................................ 21 Black-Scholes closed formulas ........................................................ ........................................................ 21

5

The Forwa Forward rd ......................................................................................... ......................................................................................... 22

6

Correlation............................................................................................ Correlation............................................................................................ 23 6.1 6.2 6.3

7

How to calculate historical correlation.............................................. correlation.............................................. 23 Implied Implied correlation correlation ........................................................................... ........................................................................... 24 Correlation term structure and correlation skew .............................. .............................. 24

Volatility and Variance......................................................................... Variance......................................................................... 26 7.1 7.2 7.3 7.4 7.5 7.6

How to calculate calculate historical historical volatility volatility ................................................. ................................................. 26 Implied Implied volatility volatility ............................................................................... ............................................................................... 27 Basket volatility ................................................................................ ................................................................................ 27 How to calculate the volatility of a basket ........................................ ........................................ 27 Volatility term structure .................................................................... .................................................................... 28 Skew / Smile.. Smile .................................................................................... .................................................................................. 28

8

Quanto Quanto and Compo Options ............................................................... ............................................................... 30

9

The Gree Greeks ks ........................................................................................... ........................................................................................... 32

10

The The Hedg Hedge e ............................................................................................ ............................................................................................ 34

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Derivatives

11

Sensitivitie Sensitivities s of Exotic Options Options ........................................................... ........................................................... 35

11.1 11.2

The Intrinsic – Time Value Relationship .......................................... .......................................... 35 The pragmatic approach to estimate sensitivities ............................ ............................ 36

12

Being long / short short ................................................................................ ................................................................................ 37

13

Swaps, Notes and Certificates............................................................ Certificates............................................................ 38

13.1 13.2 13.3 13.3 13.4

14

Swaps.............................................................................................. Swaps.............................................................................................. 38 Equity-linked swaps ......................................................................... ......................................................................... 38 Bonds Bonds .............................................................................................. .............................................................................................. 39 Equity Equity Notes .................................................................................... .................................................................................... 39

What What are Models Models .................................................................................. .................................................................................. 41

14.1 The calibration process.................................................................... process.................................................................... 41 14.2 How all derivative pricing models actually work.. work ............................... ............................. 42 14.3 Black Vanilla Vanilla .................................................................................... .................................................................................... 45 14.4 Black Diffusion ................................................................................. ................................................................................. 45 14.5 Vskew Vskew.. .............................................................................................. ............................................................................................ 45 14.6 Pskew Pskew.. .............................................................................................. ............................................................................................ 46 14.7 Local Volatility.. Volatility. ................................................................................. 46 14.8 Stochasti Stochastic c volatility volatility .......................................................................... .......................................................................... 46 14.8.1 14.8.1 The Heston Heston model model .................................................................... .................................................................... 47 14.8.2 The Hagan model ..................................................................... ..................................................................... 48 14.8.3 The Scott-Chesney model ........................................................ ........................................................ 48

15

Cliquets................................................................................................. Cliquets.. ............................................................................................... 49

15.1 Volga (or Convexity) Convexity) ........................................................................ ........................................................................ 49 15.2 Types of Cliquets ............................................................................. ............................................................................. 51 15.2.1 Classic cliquet........................................................................... cliquet........................................................................... 51 15.2.2 15.2.2 Ratchet Ratchet ..................................................................................... ..................................................................................... 52 15.2.3 Reverse cliquet.. cliquet. ........................................................................ 54 15.2.4 15.2.4 Napoleon Napoleon .................................................................................. .................................................................................. 57 15.2.5 15.2.5 Accum Accumulator ulator ............................................................................. ............................................................................. 58 15.2.6 Comparing cliquets.. cliquets ................................................................... ................................................................. 59

16

The Concept Concept of of Risk ............................................................................ ............................................................................ 60

16.1 16.2 16.3 16.4

17

Discontinui Discontinuities ties ..................................................................................... ..................................................................................... 62

17.1 17.2 17.3

18

Delta Risk ........................................................................................ ........................................................................................ 60 Vega risk.......................................................................................... risk.......................................................................................... 60 Correlation Correlation risk ................................................................................ ................................................................................ 61 Second order risks........................................................................... risks.. ......................................................................... 61 Digital Digital call option ............................................................................. ............................................................................. 62 Barrier Barrier shifts .................................................................................... .................................................................................... 63 Evaluatin Evaluating g the discontinu discontinuities ities .......................................................... .......................................................... 64

Pricing Techniques.............................................................................. Techniques.............................................................................. 66

18.1 18.2 18.3 18.4

How to determine determine options options prices ..................................................... ..................................................... 66 Closed formulas............................................................................... formulas.. ............................................................................. 67 Finite difference and Trees .............................................................. .............................................................. 67 The Monte Carlo method ................................................................. ................................................................. 67

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Derivatives

11

Sensitivitie Sensitivities s of Exotic Options Options ........................................................... ........................................................... 35

11.1 11.2

The Intrinsic – Time Value Relationship .......................................... .......................................... 35 The pragmatic approach to estimate sensitivities ............................ ............................ 36

12

Being long / short short ................................................................................ ................................................................................ 37

13

Swaps, Notes and Certificates............................................................ Certificates............................................................ 38

13.1 13.2 13.3 13.3 13.4

14

Swaps.............................................................................................. Swaps.............................................................................................. 38 Equity-linked swaps ......................................................................... ......................................................................... 38 Bonds Bonds .............................................................................................. .............................................................................................. 39 Equity Equity Notes .................................................................................... .................................................................................... 39

What What are Models Models .................................................................................. .................................................................................. 41

14.1 The calibration process.................................................................... process.................................................................... 41 14.2 How all derivative pricing models actually work.. work ............................... ............................. 42 14.3 Black Vanilla Vanilla .................................................................................... .................................................................................... 45 14.4 Black Diffusion ................................................................................. ................................................................................. 45 14.5 Vskew Vskew.. .............................................................................................. ............................................................................................ 45 14.6 Pskew Pskew.. .............................................................................................. ............................................................................................ 46 14.7 Local Volatility.. Volatility. ................................................................................. 46 14.8 Stochasti Stochastic c volatility volatility .......................................................................... .......................................................................... 46 14.8.1 14.8.1 The Heston Heston model model .................................................................... .................................................................... 47 14.8.2 The Hagan model ..................................................................... ..................................................................... 48 14.8.3 The Scott-Chesney model ........................................................ ........................................................ 48

15

Cliquets................................................................................................. Cliquets.. ............................................................................................... 49

15.1 Volga (or Convexity) Convexity) ........................................................................ ........................................................................ 49 15.2 Types of Cliquets ............................................................................. ............................................................................. 51 15.2.1 Classic cliquet........................................................................... cliquet........................................................................... 51 15.2.2 15.2.2 Ratchet Ratchet ..................................................................................... ..................................................................................... 52 15.2.3 Reverse cliquet.. cliquet. ........................................................................ 54 15.2.4 15.2.4 Napoleon Napoleon .................................................................................. .................................................................................. 57 15.2.5 15.2.5 Accum Accumulator ulator ............................................................................. ............................................................................. 58 15.2.6 Comparing cliquets.. cliquets ................................................................... ................................................................. 59

16

The Concept Concept of of Risk ............................................................................ ............................................................................ 60

16.1 16.2 16.3 16.4

17

Discontinui Discontinuities ties ..................................................................................... ..................................................................................... 62

17.1 17.2 17.3

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Delta Risk ........................................................................................ ........................................................................................ 60 Vega risk.......................................................................................... risk.......................................................................................... 60 Correlation Correlation risk ................................................................................ ................................................................................ 61 Second order risks........................................................................... risks.. ......................................................................... 61 Digital Digital call option ............................................................................. ............................................................................. 62 Barrier Barrier shifts .................................................................................... .................................................................................... 63 Evaluatin Evaluating g the discontinu discontinuities ities .......................................................... .......................................................... 64

Pricing Techniques.............................................................................. Techniques.............................................................................. 66

18.1 18.2 18.3 18.4

How to determine determine options options prices ..................................................... ..................................................... 66 Closed formulas............................................................................... formulas.. ............................................................................. 67 Finite difference and Trees .............................................................. .............................................................. 67 The Monte Carlo method ................................................................. ................................................................. 67

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Derivatives

18.4.1 18.4.1 18.4.2

How does it work? .................................................................... .................................................................... 68 Estimate option prices with the MC method ............................. ............................. 70

19

How to Avoid Avoid Errors Errors ............................................................................ ............................................................................ 71

20

Hybrid Products Products.. ................................................................................... ................................................................................. 72

20.1 Overview.......................................................................................... Overview.......................................................................................... 72 20.2 The impact impact of stochastic stochastic IR ............................................................. ............................................................. 73 20.3 Specific case studies ....................................................................... ....................................................................... 76 20.3.1 Equity – Interest Rates linked payoff ........................................ ........................................ 76

21

DIVA DIVA ...................................................................................................... ...................................................................................................... 80

21.1 21.2 21.3 21.4

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Accessing the data in the trading databases ................................... ................................... 80 DIVA Objects Objects ................................................................................... ................................................................................... 80 Pricing complex exotic products products ...................................................... ...................................................... 81 Introduction to EasyDIVA................................................................. EasyDIVA................................................................. 82

Annexe A (Autocallable (Autocallable Products) ..................................................... ..................................................... 84

22.1 22.2

Product Product Description Description ......................................................................... ......................................................................... 84 Product Analysis.. Analysis .............................................................................. ............................................................................ 88

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Derivatives

1 Commerzbank Commerzbank, unlike many other financial institution s, has a relatively simple internal structure. Reporting lines are not as complex as for big US investment banking giants such as Goldman Sachs, Merrill Lynch, JP Morgan for example. The red marked department in Diagram 1.1 is the one involved in the development, sale and hedging stages of derivatives.

Board

Direction

Private client & Asset management

Zentrale Staebe

Commercial Real Estate, Public Finance, Treasury

Asset Management

Fiancial Controlling

ComDirect Private Banking

Mittelstand

Services

Zentrale Service bereiche

Zentrale G eschaeftsfelder

Credit Risk

Group Compliance

Corporate & Investment Banking

Commercial Real Estate

Information Technology

Group Treasury

Transaction Banking

Corporate & Markets Public Finance

Retail

Human Resources

Diagram 1.1 YYYYYYYYYYYYYYYYYYYYYYYYY

The main internal business areas you will be in contact with during this internship are the following: 1. 2. 3. 4.

ZPK (our retail in Germany) : Thomas Timmermann, Andreas Muehleck and Stefan Gotsche ZPB (our private banking in Germany) : Elmar Gehring, Arne Scheehl and David Krentz ComInvest (Our Asset management in Germany) : Thomas Roell, Andreas Muehleck ComDirect (our online broker ) : Philip Lang

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Derivatives

1.1 Markets in Corporates & Markets Within Commerzbank Corporates & Markets, you will find the Investment Banking department. You will be working in the so-called front office which includes six main “teams”: 1. 2. 3. 4. 5. 6.

Trading Team Sales Team Structuring team EMTN Desk Financial Engineering Team Risk Management Team

Diagram 1.2 shows the internal divisions of the Investment Banking part of Commerzbank Corporates & Markets.

Client

Client

Requests

Client

Client

Issuer

New products development

Investment Banking Sales

Issuer

Indicative Prices

New products development

EMTN

Trading price

Structuring

Trading Pricing methodology

New models

Financial Engineering

Risk control

Risk management Model a

roval


The trading team is further segmented into different asset class trading desks as can be seen in diagram 1.3. The desk you will be most in contact with are the following:

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Derivatives 1. 2. 3.

Single stock trading (Thorsten Moos, Carsten Sitter) Exotic stock trading (Bastien Lussault, Nicolas Allano, Frederic Lescaroux) Exotic indices trading (Michel Sibert, Cristoph Hartmann)


On the sales side, the teams can be dived by client type. The complexity of the products sold is another important factor which characterises a given sales team. They are: 1. 2. 3. 4. 5. 6.

Institutional sales ex Germany (Jaime Uribe, Jorge Masalles) Institutional sales Germany (Thomas Roell, Jens Fischer, Frank Mohr) Private Banking sales ex Germany (Guillaume Hellier) Private Banking sales Germany (Elmart Gehring, Arne Scheehl) CBK’s internal retail (Thomas Timmermann, Andreas Muehleck) External retail (Michael Moll, Johannes Neulinger, Fabian Behnke)

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1.2 The profit and loss (P&L) The aim of all companies is to maximize its returns by taking limited level of risks. Banks, like many other financial institutions, offer financial products to their clients, keeping some risk in their trading books. This risk has to be hedged away by the trader using techniques that we’ll analyze later on. At this point a question arises. Where does a bank make its money? The answer to this is relatively simple. When a bank sells a product it has to determine what the fair value for that specific financial instrument is; in other words it has to use some more or less complicated mathematical tool to evaluate the price of a given product. The price determined is the cost for the trading desk to fully hedge the product. That is, if the model is correct and we exactly hedge the product (we’ll see later what this means), once we sell a product at a price X, it will cost the trading desk an amount X to be hedged and at expiry we are left with zero. How can we then explain the trader’s bonuses at the end of the year? Let’s make an example: consider that the price of a financial instrument is € 99. If we proceeded as explained above, we’d need € 99 to fully be hedged. What banks usually do is that they sell that product at a higher price by applying a margin to the fair value: let’s say € 1. The price at which the product is sold is now € 100. The hedging cost is € 99 and the money earned by the bank is € 1. If this is repeated every time a product is sold, the sum of all the margins determines the profit of a bank. But there is more. Traders, usually, don’t fully hedge their books. This means that if they have some specific market exposures they might decide not to cover them. This depends on the level of risk-aversion and on the individual market views of the traders. By not covering a market exposure, they might incurr losses in cases where their views turn out to be wrong. There is obviously a buffer represented by the sales margin made but considering the amount of positions hold by banks it is easy to understand that losses can occur in large extent. The total Profit and Loss, indicated as P&L, can be expressed by the following formula:

P & L

I

J

i =1

j =1

i j = ∑ M arg in Sales + ∑ Positions Trading

i being the total number of products sold and j the total positions held by the trading desk.

1.3 The Book Open The term Book Open (BO) defines the price of a security (swap, bond, certificate etc) without taking into account any kind of margin, neither the one of the bank nor the one of t he client (if sold to third parties). It represents the fair price of that specific security and is defined as:

SecurityOf fer Pr ice − M arg in Bank − M arg inClient = BookOpen ( BO)

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Derivatives

2

Structured Equity Products and the People who buy them

Commerzbank’s equity derivative business is mainly focussed on what are called “structured products” (specifically those linked to equities – there are also interest rate, FX, credit, and other types of structured product but we are not concerned with them here). These products are manufactured and bought by a variety of types of investors for a variety of reasons.

2.1 What is a structured product? A structured product is a simple concept – it is essentially a bond, which has a coupon and / or a redemption value which, instead of being fixed like a typical bond, is linked to equity prices. A very simple example would be a bond which paid no coupons during its lifetime, but at maturity redeemed at an amount equal to 100, plus the greater of zero and 70% of the percentage change in the value of an equity index measured over the bond’s lifetime. This very simple product can be decomposed into two financial instruments –a zero coupon bond, which provides the return of 100 at maturity, plus a call option on the equity index, with a notional amount of 70% of the bond notional. This illustrates how a structured product is a “manufactured” product, consisting of an equity derivative plus a fixed income instrument, all sold as a single legal security. Strucutured products do not always take the legal form of a bond (in industry parlance, they are “wrapped” as something else), but the economics are the same.

2.1.1 Manufacturing aspects of structured products A structured product is a single security, which is “made” of a bond plus an equity derivative. The entity selling the structured product effectively assembles the product by buying (or replicating) the individual pieces. The issuer then brings the product into existence through the legal document which defines the exact obligation that the bond issuer has to bond holders. The holder of the bond will almost certainly not retain any physical proof of ownerhsip – their holding will be recorded in some elecronic system such as Clearstream or Euroclear which record securities ownership in accounts in the same way as a bank account records cash ownership. A structured product can also take the form of an investment fund, where a fund is set up which buys all the bits of t he structured product. An investor in the shares of the fund then has an exposure which is identical to having bought all the pieces independently. This is often done for regulatory reasons – in the UK, for example, it is legal to sell investment funds, but not legal to sell many types of bond directly to the public. There are a number of different legal forms that these bonds can take, with the most common being the Medium Term Note (MTN, or sometimes Euro MTN or EMTN), and the certificate. Whilst these are slightly different legally, this is not particularly important from the point of view of pricing the products, and for all practical purposes they are just bonds. Often a certificate is a non-principal protected product (i.e. redemption could be below par) but this is not necessarily the case.

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Derivatives The issuer of a structured product is the instution which actually issues the bond which constitues the product. In many cases this is Commerzbank, but in a large number of cases Commerzbank will structure and sell a bond which is issued by an external party. The motivation for the issuer of any bond is to finance their business activities. Generally the coupon that they pay on the bond reflects both the market’s required level of compensation to take the risk that the issuer of the bond can repay the principal, but the issuer will have preferences on how it pays the coupon – for example some issuers will have costs linked to interest rates, and prefer to pay a coupon linked to interest rates, rather than a fixed coupon. In order to be able to use as wide a range of issuers as possible, most structured products are manufactured in the following way:

Commerzbank ZCM Bond issuer pays Commerzbank Coupon stream in format of Issuers choice (frequency, Currency etc.)

Commerzbank pays Equity-linked amount to Bond issuer at maturity / Over bond lifetime

Bond Issuer

Bond issuer sells bond With desired equityLinked payoff to bond investor

Bond investor pays bond issuer 100 EUR for the bond at Inception

Bond investor


This way, the net position of the bond issuer is that they have issued a bond, and pay the coupon stream of their choice (almost always an amount linked to short-term interest rates), the investor has a bond, issued by the counterparty of their choice, and with the payoff of their choice, and Commerzbank has a net postion in the equity-linked derivative element of the bond which is then managed in the equity derivatives trading portfolio. Often the issuer will be Commerzbank’s own treasury department (this is the department of the bank which is responsible for balancing the cashflows arising from the bank’s many different financing, and deposit-taking activities). However in many cases it is an external institution. In some cases it is actually this external instution who is Commerzbank’s client rather than the bond investor; this will be explained in a later section.

2.1.2 Types of structured product There are many diffferent types of structured product, but almost all of them fall into one of three broad categories:

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Derivatives

Delta-1 products These are the simplest structured products of all, and simply replicate the exposure that is gained from holding a security, basket of securities, or index. Often they only replicate the price return of the security, ie the investor does not receive the benefit of dividends. In this case the cashflow stream that comes fr om the dividends is used to price These products are used for what is known as “market access”. For example, a particular type of investor may not be able to invest in a particular market, but once the product is legally transformed through a certificate “wrapper” its new legal nature makes it a permissible investment, or a more efficient investment from the point of view of regulation or tax. Alternatively, an investor may not have the time or money to actually invest in the underlying (for example the S&P 500 index cannot be tracked by an investor who does not have several hundred thousand euros, plus the time and infrastructure to trade in five hundred stocks at once).

Yield enhancement products Financial instruments, in particularly bonds, offer a higher coupon where there is a greater risk of the principal not being repayed in full. Typically this is because there is credit risk on the issuer of the bond. However it is also possible to create a bond whose repayment at maturity is linked to equity prices, where the redemption is r educed if equity prices decline. The simplest such instrument would reduce the principal repayment by 1% for every 1% that an equity index dropped over the lifetime of the bond, in return for an increased coupon compared to the risk-free coupon. Such a bond can be “manufactured” by combining a long position in a simple fixed coupon bond, and a short position in an equity put option. The premium on the equity put option is then added to the coupon so that the structured product pays a higher coupon. There are a great variety of yield enhancement products which involve the investor taking some sort of downside risk on equities in return for an increased coupon. Some of the names that you will come across are reverse convertible, autocallable, premium certificate, bonus certificate, discount certificate, sidestep note, and many others. Yield enhancement products are widely used by investors who have a weakly positive view of the market, and want to generate an income stream from taking some equity risk on the downside.

Capital protected products The example given in the first section, of a bond which redeems at par (in the absence of bankruptcy of the issuer), plus a participation in the positive performance of an equity underlying, is what is generally known as a capital (or principal) protected (or guaranteed, although there are sometimes regulatory implications of the use of the word “guaranteed”) products. They are widely used by investors who have a positive view of equity, but who for some reason do not want to take any risk on the equity downside.

2.2 The market for structured products Most structured products which are sold by Commerzbank are ultimately destined for retail investors. Whilst there is no single definition of what exactly a retail investor is, it is

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Derivatives approximately true to say that if somebody has thousands, or tens of thousands of euros to invest, they are a retail investor. In most developed markets, the relationship between banks (and other financial companies) and retail investors is very closely regulated. One step up (in terms of wealth, and also regulation) from retail investors are the so-called “high net worth” or “HNW” individuals, who in simple terms have hundreds of thousands, or millions, of euros to invest. There is generally less regulation covering this market, and they can buy much risker and illiquid products. At the top, are the so-called “instutional investors”. These are professional money managers, who spend all their time looking for investments. They include insurance companies (who need to invest the premium that they receive from selling insurance policies), pension funds (both private and state), family offices (who act on behalf of the super-rich with billions to invest, often called “ultra” high net worth.), fund managers who are allowed to invest in structured products rather than directly into equities and bonds, and sometimes large corporations, who invest some of the cash that they hold for liquidity purposes to try and get a higher return than they could from simply putting the cash on deposit. Institutional investors are often driven by very specific regulatory, tax, or accounting concerns as much as they are by the actual economics of the product, so the legal form that the product takes may well be its most important feature. In fact many structurers are devoted solely to addressing these types of issues. Different groups of clients are covered by different sales teams at Commerzbank, often with some regional segmentation as well. Coverage is also a little confused by the fact that the same company may be covered by more than one sales team, depending on the capacity in which they are a client.

2.2.1 Intermediaries in the structured product market Obviously Commerzbank’s equity derivatives division cannot directly service thousands upon thousands of individual retail clients (nor do the regulators allow it to). Instead, the retail market is covered by instutions such as banks, insurance companies, fund management companies, brokers, and financial advisers, who have large networks of salespeople and complex admnistration systems which allow them to deal directly with the public. One such intermediary is Commerzbank’s own retail network of high-street branches in Germany, through which a large number of structured products are sold via the internal organisation known as ZPK. So Commerzbank does actually sell products directly, although for the purposes of ZCM, this is done through an internal counterparty, which is treated as though it were an external client. Such intermediaries generally earn their living from fees which they earn in two ways: from the investor directly by selling a bond for 102 EUR when its par value is 100 EUR, for example; or indirectly via the product manufacturer. The manufacturer in such a case will structure a bond which has an additional coupon of say 0.25% per annum which is paid to the intermediary rather than the investor. Obviously the level of the fee is related to the level of service offered by the intermediary. From Commerzbank’s point of view, the fee makes no difference to us. If a fee is paid to the distributor, then the present value of this fee is simply deducted from the amount that the investor has to spend on the equity derivative element of the structured product. Some of these intermediaries do not buy completed structured products. Instead they will buy the raw derivative pieces of the product, and actually manufacture the product themselves. Many banks with large retail client-bases but no derivative trading capabililty will do this, and issue the necessary structured bond themselves. Some bank products are sold in the form of a “structured deposit”, which is economically the same as a bond, but has the legal structure of a bank account rather than a bond.

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Derivatives

Insurance companies also do this, however instead of issuing the product as a bond (although they often do call it a bond) it is issued in the form of a life insurance contract. The insurance company then holds a bond, or a derivative, as a hedge for the liability of the insurance contract. The contract buyer pays a regular premuim to the insurance company. In return, the contract buyer will get back an amount when the policy matures which is linked to equity performance over the policy lifetime. (Additionally, they will get back something if they die during the policy lifetime, which is why it is a life insurance contract. However this is not a significant element of the product, these are almost invariably bought as investment products). Large fund management companies also sell “structured funds” which are investment funds which have a defined payoff on a specific future date. As mentioned above, these funds are engineered by buying the derivatives which give the specific payoff. The product manufacturer has to take account of the fees paid for fund admnistration, custodians etc. when assembling such a product. Large manufacturer-distributors like large banks, some asset management companies, and insurance companies are the easiest to service, as they only want to trade derivatives, not manufactured products. They are professional market counterparties so the regulatory burden of dealing with them is very low. However, as a consequence of this, they are highly competitive and although they will deal in large size, they will invariably get several competing quotes from rival derivative houses which makes it difficult to make large profits fr om them. The retail market is also serviced by brokers and financial advisers, who do not manufacture products themselves. They need to have a product which is in a suitable form to be sold directly to retail investors, often including marketing material and other things. They simply offer a network of sales people, and administration systems, which enable product manufacturers to sell their products directly to the public without actually having to deal with the public. However, because they sell “finished” products, the products that they sell have to be manufactured to a high standard in terms of things like being listed on stock exchanges, having the documentation produced to certain standards, etc. They may also need the ability to execute small trades (for a few thousand euros) efficiently and quickly, and legally binding commitments from product providers to offer secondary market liquidity with defined bid-offer spreads. Moving up the wealth chain, HNWI are generally serviced by a sector of the intermediary market called Private Banks. These are effectively high-powered advisers and brokers, who may have some limited capacity to manufacture products (but normally not), but who can be serviced much more easily than retail brokers because HNWI can be sold products with very much less regulation than retail investors. Some very large financial institutions may well be a client in more than one way. They will hedge equity retail products which they manufacture themselves, they will offfer Commerzbank manufactured products to their wealthiest clients, and they may trade equity derivatives directly to hedge, for example, employee share schemes. This situation occasionally results in confusion when talking about “institutional” business. In this case it is important to identify exactly who the end-investor is going to be.

2.3 Summary There are many types of structured product, and many types of client / investor. Structured products fall into one of three broad classes:

• • •

Simple market-access products Yield enhancement products Capital protected products

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Derivatives

Commerzbank sells these to various classes of investor, both directly and indirectly.

• • • •

Retail investors directly through Commerzbank braches Retail and HNW investors indirectly via intermediaries who sell Commerzbank products Retail investors indirectly via intermediaries who sell their own products which are manufactured partly from derivatives which they buy from Commerzbank, sometimes known as the “retail hedge” business. Institutional investors who manage assets professionally (who in some cases are the same companies as the intermediaries mentioned above)

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Derivatives

3 Derivatives The word Derivatives is a general term describing investment products, which derive their payoff – therefore called Derivatives - from an underlying asset. The relationship Asset – Derivative is rarely linear, meaning that if the value of the asset increases by a given amount, the value of the derivative doesn’t vary by the same amount. There is therefore some form of convexity which depends on the volatility of the underlying (this can be seen by analyzing Jensen’s inequality). We therefore deduce that the concept of volatility is fundamental when dealing with these products.

3.1 Vanilla options The most common options are named calls and puts. They are therefore defined as being vanilla options to express their simplicity compared to exotic options, which are more complex from a valuation point of view. A call is a buying option, it gives the holder (the person who bought the option for an amount X1) the right but not the obligation to buy the asset underlying the option at a given price. A put is a selling option, it gives the holder (the person who bought the option for an amount X 2) the right but not the obligation to sell the asset underlying the option at a given price. Their value is usually a function of various parameters. Let’s define the value of a general option (Vanilla or exotic) with the symbol V, then the following relation is true:

V (t ) = V ( S (t , σ , ); K , T ; r ) The semicolons are used here to distinguish between different types of parameters: underlying-dependent parameters, product-dependent parameters and market-dependent parameters. The underlying dependent parameters can’t be modified (unless we replace the underlying with another one) because they are part of the asset and define its behavior over time. The second ones can be changed at our discretion in order to fulfill the investor’s needs. The last ones can’t be modified since they are implied market parameters. The parameters are defined like follows: S: the spot of the asset σ: the volatility of the asset μ: the drift of the asset K: the strike of the option T: the maturity of the option r: interest rates

The payout of a vanilla call at maturity T is:

Max (0, S T − K ) Where St is the value of the underlying on maturity date and K is the prefixed strike level. Its graphical representation is:

15

Derivatives

Payoff

S

K


The payout of a vanilla put at maturity T is:

Max (0, K − S T ) And its graphical representation is:

Payoff

S K


As can be seen on the graphs and in the formulation of the payouts, these options always have a non-negative value for their holders. Call holders believe in the increase of the underlying prices, put holders in their decrease. It is important to notice that the maximum gain a call holder can make is unlimited, whereas the maximum gain a put holder can make is equivalent to the strike value.

16

Derivatives

3.2 Barrier options Very common options are the time-dependent versions of vanilla calls and vanilla puts. These products are built with barriers that can be t ouched by the underlying asset at any time during their lifetime. The product can either activated or deactivated when the barrier is touched, and the barrier can be above or below the strike level. There are 4 types of barrier calls and 4 types of barrier puts:

Call option

Put option

European/American

European/American

UP & IN

UP & IN

UP & OUT

UP & OUT

DOWN & IN

DOWN & IN

DOWN & OUT

DOWN & OUT


To illustrate these barrier options, we will consider the example of an Up & Out call with strike K and barrier B. Its payoff is the same as that of a vanilla call, but the payment is conditioned by the fact the underlying asset has never traded at or above its barrier level during the lifetime of the product. The following graph shows the payoff of the Up & Out call. Payoff

S K

B


Let’s now consider a Down & In put with strike K and with barrier B. The payoff formula of this put is the same as that of the vanilla put. The put will be activated if the underlying asset trades at or below the barrier level B. The following graph illustrates the product payoff:

17

Derivatives

Payoff

S B

K


3.3 Combining vanilla products Several investment products, which at a first glance might seem to be exotic, are effectively composed of vanilla options and are therefore vanilla themselves. Thre are various ways to combine vanilla products and the aim of this chapter is to explain the basic principle behind them. Lets consider, for example, a zero strike call (a call with a strike equal to zero) and a put D & O, strike K, with barrier at B. Diagram XX shows these two basic structures in dotted lines and their sum in blue.

Payoff

S B

K


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Derivatives

3.4 Put / Call parity There is a very important relationship in finance known as Put / Call parity. It establishes a linear relationship between the value of a call option, a put option, the underlying asset and a cash position. We will deduce the Put / Call parity in two steps. First we consider the long / short position of an investor at time T and we then discount the portfolio with the risk free rate in order to get the actualized value of the portfolio which will express the Put / Call parity. The investor has the following positions in his portfolio Π: 1. 2. 3.

he is long a cash position K he is long a call strike K with expiry T he is short a put strike K with expiry T

At time T, the value of the portfolio is therefore:

Π T = K + C T − PT It is easy to see that the portfolio position corresponds to the value of the underlying asset at time T, and therefore:

Π T = ST If we calculate the present value of the portfolio Π at time 0, we get:

Π 0 = PV (Π T ) = PV ( K ) + C 0 − P0 = S0 And therefore:

PV ( K ) + C 0

= S 0 + P0

This last relation is called the Put / Call parity.

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Derivatives

4

Black-Scholes Model

The Black-Scholes model (BS) is the central in model finance theory. It still remains the most widely used model and represents the starting point for more complex and complete models such as the Local Volatility Model (LVM).

4.1 Stochastic Differential Equations (SDEs) The basic form of a stochastic differential equation for a process X is composed of two parts: the Newtonian term (the deterministic term) and the Brownian term (the random or stochastic term). It can usually be written in the following form

dX = __ dt + __ dW In finance we find SDE in the form of an I to process, which can be written as

dX = a( X , t )dt + b( X , t )dW

4.2 Lognormal returns for asset prices It can be shown that asset returns are lognormally distributed and the Ito process assumes therefore the form:

dS S

= μ dt + σ dW t

Where

a ( X , t ) = S and

b( X , t ) = σ S

4.3 The assumption for the Black-Scholes model The key assumptions of the Black–Scholes model are:

• • • • • • •

The price return of the underlying asset follows a lognormal distribution with constant drift μ and constant volatility σ It is possible to short sell the underlying stock. There are no arbitrage opportunities. Trading in the stock is continuous. There are no transaction costs or taxes. All securities are perfectly divisible (e.g. it is possible to buy 1/100th of a share). It is possible to borrow and lend cash at a constant risk-free interest rate.

20

Derivatives

4.4 The Black-Scholes equation The Black-Scholes equation can be seen as a diffusion-convection-absorption equation. It determines how the price of a derivative, written on the asset S, varies over time. Its evolution can be described by the partial differential equation (PDE)

∂V 1 2 2 ∂ 2V ∂V rS = σ S + − rV ∂t 2 ∂S ∂S 2

It is notable that the equation does not contain μ, the drift of the stock.

4.5 Black-Scholes closed formulas It can be shown from the Black-Scholes equation that the price of a call, denoted by C, and the price of a put option, denoted by P, option written on the asset S with strike K, expiry T, volatility σ and interest rates r can be expressed by means of the following closed formula: − rT C ( S , T ) = Sφ ( d 1 ) − Ke φ (d 2 )

P( S , T ) = Ke

− rT

φ ( −d 2 ) − Sφ ( −d 1 )

Where

d 1

=

2 S ⎞ ⎛ σ ⎞ ⎛ ⎟⎟T ln⎜ ⎟ + ⎜⎜ r + 2 K ⎝ ⎠ ⎝ ⎠

σ T

And

d 2

= d 1 − σ

T

Here Φ is the standard normal cumulative distribution function.

21

Derivatives

5

The Forward

A very important concept when pricing an exotic product is the forward of an asset. The forward is the expected value of the underlying at a point in the future. Here we will explain how to evaluate the forward and how it varies over time. Let’s consider the lognormal distributes process previously described:

dS S

= μ dt + σ dW t

If we don’t take into account the Brownian term, this equation can be rewritten as:

dS S

= μ dt

And the solution of this simple first order differential equation, where the initial condition is given by S0 for t=0, is:

S (T ) = S 0 e μ

T

= S 0 e ( r −q )T

Or expressed as a percentage of the initial spot S 0

F % (T ) =

S (T ) S0

= e ( r −q )T

We see that the forward increases as interest rates increases and decreases as dividends increases. This is a very important observation that will be very useful when we’ll have to deal with the optimization problem.

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Derivatives

6 Correlation The concept of correlation is widely used in finance and statistics to indicate the linear relationship between two random variables or between two time series. In other words a correlation ρ between a random variable X and a random variable Y indicates the “probability” of X changing in a given direction and in which direction for a given change in Y.

Definition: The correlation can be seen as a strength vector between X and Y, which expresses the intensity and the direction of their linear relationship. It is important to notice that the correlation is a static value, meaning that it doesn’t tell us anything about the future variation of the relationship. The general definition of the correlation ρ between two random variables X and Y, with mean μX and μY and standard deviation σX and σY is:

ρ =

cov( X , Y ) σ X σ Y

=

E (( X − μ X )(Y −

Y

))

σ X σ Y

E ( XY ) − E ( X ) E (Y )

=

2 2 2 2 E ( X ) − E ( X ) E (Y ) − E (Y )

2

Where E is the expected value of the variable and cov is its covariance. Since μX = E(X), σX 2 2 = E(X ) − E (X) and likewise for Y, in order for the correlation to have a mathematical sense the standard deviations have to be different from zero.

It can be shown (Cauchy-Schwarz inequality corollary) that the maximum value that the correlation can assume is 1, meaning that for each up movement of X, Y moves up as well and vice versa. Notice that the correlation is not telling us anything about the size of the individual movements. Notice that the correlation ρ is not telling us anything about the intensity of the changes, since this is expressed by the volatility.

6.1 How to calculate historical correlation Let’s consider a time series of N elements on two normal distributed processes X and Y, defined by the two vectors X = [x 1, x 2, x 3, ..., xN] and Y = [y1, y 2, y 3, …, yN], then the historical correlation can be estimated as:

N

∑

N xi yi

ρ =

i =1

N

∑

N xi i =1

2

⎛ N ⎞ − ⎜ ∑ xi ⎟ ⎝ i=1 ⎠

N

N

i =1

i =1

− ∑ xi ∑ yi 2

N

∑ y

N

i =1

2 i

⎛ N ⎞ − ⎜ ∑ yi ⎟ ⎝ i=1 ⎠

2

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Derivatives

6.2 Implied correlation Let’s consider an index I having N members as constituents and a variance implied correlation of the Index I is defined as:

ρ I ,t =

σ2, then the

N

∑∑ λ

ρ ij ,t

ij ,t

i =1 j >i

Where

λ ij ,t =

wi ,t w j ,t σ i ,t σ j ,t N

∑∑ w

w j ,t σ i ,t σ j ,t

i ,t

i =1 j >i

The implied correlation is an average correlation value that, if used to calculate the variance 2 of the index I would give the same variance σ .

6.3 Correlation term structure and correlation skew As we have seen previously, correlation expresses a linear relationship between two random variables which is constant in time and strikes. In practice we know that correlation has a more complex behaviour and has to be expressed as a function of time and strike. We know, for example, that two assets which have a given correlation today might evolve more or less correlated in the future. More specifically, it has been shown that correlation tends to be mean reverting over time. That is, it can be expressed as a function of time t:

ρ = ρ (t ) If actual correlation is high compared to historical correlation then it will lower over time resulting in a downward sloping correlation curve; if it low compared to historical correlation then it will be upward sloping, as shown in diagramm XXX.

CORRELATION TERM-STRUCTURE Correlation

Historical Correlation

Maturity


24

Derivatives

Similarly, correlation is not constant for different market levels. In a bearish market, assets move with a correlation close to one, whereas in a bullish market assets move with lower correlation. This is generally defined as correlation skew and therefore the following equation holds:

ρ = ρ ( K ) Diagram XX shows how the correlation evolves with different market levels. The correlation is asymptotic to one in a bearish market and to zero in a bulish market. This is true for assets within the same asset class. If assets from different asset classes are taken, corelation might behave in a different way. CORRELATION SKEW Correlation

Strike


Similar to the volatility, we can represent correlation as a surface function of time and strike and defined as follows:

ρ = ρ (t , K )

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Derivatives

7

Volatility and Variance

The standard deviation σ of a lognormal distributed variable X r efers to the average change of its value from the mean μ. The standard deviation of a random variable X is usually defined as the square root of the variance, where

Var ( X ) = E (( X − E ( X )) 2 ) = E ( X 2 ) − ( E ( X )) 2 And therefore

σ X

=

2 2 E ( X ) − ( E ( X ))

The T-period volatility σT is defined as

σ T

= σ

T

Therefore the wider the data is spread around the mean the higher will be the volatility.

7.1 How to calculate historical volatility Let’s consider a time series of N elements in a normally distributed process X, defined by the vectors X = [x1, x2, x3, …, xN] , then the historical volatility can be estimated as:

⎛ 1 N ⎞ xi ⎟ σ hist , N Days = ∑ ⎜ xi − N ∑ N i =1 ⎝ i =1 ⎠ 1

N

2

Example

26

Derivatives

7.2 Implied volatility The implied volatility is the market view on the future volatility of a given asset. To calculate the implied volatility we use the inverse form of the Black-Scholes formula. In other words we observe at which price options are trading in the market and we calculate the volatility that has to be used in the BS formula in order to match that price. It is important to notice that the BS formula is highly dependent on the dividends of the asset used to estimate the price of the option. There are several ways to estimate dividends and the following is a non exhaustive list of methodologies used by traders, in order to get the most accurate evaluation: 1.

2.

3.

Calculate the implied dividends form a synthetic forward (call minus put). Inconvenient: the liquidity of options is usually limited to a few years, for long term dividends alternatives estimates have therefore to be used. Forecast future dividends form more recent dividends Inconvenient: this methodology is risky in the sense that past dividends not necessarily imply the same amount in the future Rely on analysts forecasts Inconvenient: the analysts view can be wrong and therefore the estimate bring to losses

Another important point is represented by the taxable amount on the dividends. European banks, for example, are taxed by 20% on the dividends of US companies. It is therefore important to take this effect into account when dealing with foreign shares.

7.3 Basket volatility 2

2

2 X+Y

Let’s consider two random variables X and Y with variance σX and σY . The variance σ the combined process X+Y is defined as:

of

Var ( X + Y ) = Cov( X + Y , X + Y ) = Var ( X ) + Var (Y ) + 2Cov( X , Y ) And therefore

Var ( X + Y ) = Var ( X ) + Var (Y ) + 2 ρ X ,Y Var ( X )Var (Y )

7.4 How to calculate the volatility of a basket Let’s consider N assets of a basket B with weights W = [w1, w2, w3, …, wN] and implied Tperiod volatility given by σ = [σ1, σ2, σ3, ... σ Ν]. The volatility of the basket for maturity T is given by: N −1

σ B ,T

= ∑ w σ + 2∑∑ ρ ij wi w jσ i ,T σ j ,T 2 i

2 i ,T

i =1 j >i

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Derivatives

7.5 Volatility term structure The volatility term structure describes the non-constant nature of the volatility with respect to time. If we draw the implied volatilities calculated via the Black-Scholes model from options written on the same underlying but with different expiries, we generally obtain an upwards sloping curve like shown in diagram XXX.

VOLATILITY TERM-STRUCTURE Implied volatility

Maturity


This means that, usually, the market quotes longer term options with a higher volatility than shorter term options. This can be explained if we think of the risk associated to longer maturities compared to shorter maturities.

7.6 Skew / Smile Skew (or smile) is a phenomenon which describes the non constant nature of volatility with respect to the strike level. If we plot the log-returns of the equity market we can see that they slightly deviate from a lognormal distribution. The distribution shows signs of leptokurtosis (or fat tails), particularly on the downside in stock returns.

VOLATILITY SKEW

VOLATILITY SMILE

Implied volatility

Implied volatility

Strike

Strike


28

Derivatives This has important repercussions on option prices. Options with different strikes and different expiries tend to trade at different implied volatilities. When implied volatilities for options with the same expiry are plotted, the graph looks like a smile (or a skew), with at-the-money volatility in the middle and in-the-money volatilities gently rising on either side (smile) or only on the downside (skew) There are various explanations for why volatilities exhibit skew. First, let’s consider a firm. We know that a firm’s value is the sum of its debt and its equity. This value is relatively constant over time and thus if the firm’s equity declines, its debt and consequently risk and volatility will increase. On the other hand, if equity increases, debt will decrease along with risk and volatility. This argument shows that we can expect the volatility of equity to be a decreasing function of price.

Equity

Asset = Equity + Debt

Debt


Another explanation refers to behavior of equity in market crashes. In a bearish market investors suffer from fears of large losses and, therefore, increase their trading activity. In a bullish market on the other hand investors are confident and tend to hold for longer time their equity position reducing therefore the trading activity.

29

Derivatives

8

Quanto and Compo Options

Investors reserve their main interests for “domestic” investments. In other words they prefer to invest in known domestic equity instead of less known and, therefore, more risky foreign equity. For diversification reasons, investors are willing to spread the risk in their portfolio over different countries. Some of these countries can’t even be considered as “risky” since their economy has shown a big stability over the past decade, like the United States and Japan. Nevertheless a direct investment in US or Japanese equity would not only have an associated Equity-risk but also an implied FX-risk. A European investor should, in fact, convert its euro notional into the foreign currency, buy foreign equity and, at the end of the investment, convert the gain (or loss) back at an unknown exchange rate (which could have risen or fallen). If investing in derivatives the situation is even more complicated since the payoff should be discounted at the domestic interest rates. Diagram XXX shows the cashflows involved in a foreign equity investment.

European investor (€ domestic Ccy)

€ 100 FXini

€ ZZ Buy a J-stock

Sell the J-stock

¥ XX

FXfin ¥ YY

Japanese Market (¥ foreign Ccy)


It is easy to see that the investor incurs two different risks: 1.

2.

The risk associated with the equity performance between the buying and the selling date of the J-stock. In case the equity performed negatively (any other parameter remaining the same), the investor would have incurred in a loss. The risk associated with the exchange rate at maturity. In case the yen weakened against the euro, meaning that you need more yen in order to buy euros (any other parameter remaining the same), the investor would have incurred in a loss.

Derivatives where the payoff (expressed in foreign currency) is converted back into the domestic currency with the exchange rate at maturity and discounted with the domestic discount factor are called compo options or compo derivatives.

30

Derivatives Not all the investors are ready to take the FX-risk associated with compo derivatives. This problem can be overturned by selling to the bank the embedded foreign exchange risk and conserving only the equity risk. These kinds of options are called quanto options or quanto derivatives. Here the final payoff is first discounted using the foreign discount factor and is then converted into the domestic currency using a fixed exchange rate. The pricing of quanto options is relatively simple and requires the forward to be redefined. Let FXt be the exchange rate at time t, which corresponds to the domestic currency amount for one unit of foreign currency. Let’s define r d and r f as the riskless domestic and foreign interest rates. As we know, the exchange rate follows a lognormal distributed stochastic process:

dFX t FX t

= (r d − r f )dt + σ FX dW FX

Where σFX is the volatility of the foreign exchange and dW FX is the Wiener process. The equity price follows the similar stochastic process:

dS t S t

= (r f − d )dt + σ S dW S

Where σS is the volatility of the stock price and dW S is the Wiener process. It is possible to construct a portfolio Π with the quanto derivative, an amount of foreign currency and a given amount of shares: Π = V(FX, S, t) – ΔFX FX – ΔS S FX Where ΔFX FX is the Euro amount of the Yen amount being short and ΔS S FX is the Euro amount of the shares being short. By repeating the same procedure as shown in the chapter 3, it is easy to see that quanto options have to satisfy a similar equation to the Black-Scholes one, where the dividends have to be replaced by:

d’ = r f + d + ρ σS σFX

and the forward can, therefore be rewritten as

S (T ) = S 0 e μ T

= S 0 e ( r −d ')T = S 0 e ( r −r −d − ρ ⋅σ d

d

f

FX

⋅σ S )T

Or expressed as a percentage of the initial spot S 0

F % (T ) =

S (T ) S0

= e ( r −r − d − ρ ⋅σ d

f

FX

⋅σ S )T

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Derivatives

9 •

The Greeks The delta measures sensitivity of the option price V to spot price S.

Δ= •

∂V ∂S

The gamma measures second order sensitivity of the option price V to spot price S.

∂ 2V Γ= 2 ∂S . •

The speed measures third order sensitivity of the option price V to spot price S.

∂ 3V speed = ∂S 3 . •

The vega measures sensitivity of the option price V to volatility σ.

vega

•

=

The theta measures sensitivity of the option price V to the passage of time. With T the amount of time to expiry:

θ = −

•

∂V ∂T .

The rho measures sensitivity of the option price V to the applicable interest rate r.

ρ =

•

∂V ∂σ .

∂V ∂r .

The vega gamma or volga or convexity measures second order sensitivity of the option price V to implied volatility σ.

∂ 2V volga = ∂σ 2 . •

The vanna measures cross-sensitivity of option value with respect to change in underlying level and the underlying volatility, which can also be interpreted as the sensitivity of delta to a unit change in volatility.

∂ 2V vanna = ∂S ∂σ . 32

Derivatives

•

The sensitivity to the skew is also considered like a greek:

skew

= slope ⋅110 − slope

Where

slope =

σ 105%

− σ 95%

10%

is the curvature of the volatility smile.

The underlined greeks are the most important for a structuring position. In general, we define a GreekX as the sensitivity of the option price V with respect to the parameter X:

Greek X

=

∂V ∂ X

We always approximate the derivative by a small variation:

Greek X

where

=

∂V ΔV V X − V X ≈ = ∂ X Δ X X 2 − X 1 2

1

X 2 − X 1 is small.

33

Derivatives

10 The Hedge

34

Derivatives

11 Sensitivities of Exotic Options The aim of this chapter is to evaluate the sensitivities of vanilla options with respect to market parameters. There are two ways to do this: 1.

2.

Mathematical approach: we can differentiate the option with respect to the parameter we would like to analyze and therefore evaluate the impact on the price for a small change in that parameter from its actual value Pragmatic approach: we can proceed in a more practical (and efficient) way by estimating how the probabilities for being more in the money (respectively out of the money) are going to change.

N.B. The two methodologies are not exclusive and crosschecks should always been done in order to minimize errors in the pricing stage.

11.1 The Intrinsic – Time Value Relationship We’ll start by considering an at the money (ATM) European call option with strike K and expiry T. We know that at maturity T, the value of the call option is equal to its payout (the dark line in diagram XX):

⎛ ST

C(T ) = max⎜⎜

⎝ S0

⎞ − K ,0 ⎟⎟ ⎠

In any previous instant t the value of a call option can be decomposed into two components: 1. 2.

The intrinsic value It Its time value θt

The intrinsic value I t is the value the option would have if exercised at time t; it corresponds therefore to the payout of the same option but with expiry t. The time value θt is the value that takes into account the probability for the call option to be in the money at maturity T. The Intrinsic – Time value relationship for a call C at time t expresses the value of the option at that time and can be written as:

Ct

⎛ S ⎞ = I t + θ t = max⎜⎜ t − K ,0 ⎟⎟ + θ(t ) ⎠ ⎝ S0

Diagram XX schematises the Intrinsic – Time value relationship

35

Derivatives

Price

θt It K

S


Generally speaking we can say that, for vanilla options, the intrinsic value is directly a function of the forward and therefore will be sensitive to interest rates and dividends and of the strike of the option. The time value on the other side is a function of time and of the volatility. In case of a vanilla call it grows asymptotically to volatility multiplied by square root of time.

11.2 The pragmatic approach to estimate sensitivities Based on the previous discussion we can now analyze the sensitivities of an ATM European call option to different market parameters such as: 1. interest rates 2. dividends 3. volatility 4. strike 5. maturity

The intrinsic value is directly proportional to the forward and therefore as interest rates increase the intrinsic value tends to increase, whereas the opposite effect is observed if dividends increase. On the other hand the intrinsic value will increase if the strike decreases since the option would be in the money already since its issue. Higher volatility and longer maturity have both a positive effect on the time value for an ATM European option since the probability for being in the money at expiration increases.

36

Derivatives

12 Being long / short Short and long are general terms used in finance to specify what the position of an investor or a trader is with respect to a market parameter X if a derivative Y is exchanged between the two parties. Let’s suppose that an investor A buys a call option Y from a bank B as shown in the diagram below; B is then said to be short Y and A is said to be long Y. A long position is therefore associated with buying Y and a short position is associated with selling Y. It is common in finance to switch from one definition to the other: buying Y or being long Y are therefore completely equivalent and are used indistinctively.

A

Y(t) Y(0)

B time


The terms are used even in a more general and wider sense. Y in the previous example was a call option but could in effect also be a market parameter such as volatility, skew, dividends, correlation etc.

Definition: Counterparty A is said to be long (respectively short) a derivative or a generic market parameter Y if the increase of Y, at any time t after he first established his long (respectively short) position at time 0, represents a gain (respectively a loss) for A. In the case of the call for example we said that A bought the call and is therefore long the option. A is said also to be long vega since an increase in volatility during the lifetime would increase the time value of the derivative itself. Analyzing further, A is said to be short dividends since the increase in dividends would decrease the intrinsic value of the call option.

37

Derivatives

13 Swaps, Notes and Certificates There are different ways a bank can issue a derivative instrument to its clients. The most well known are of three types: swaps, notes and certificates.

13.1 Swaps A T-year swap is a derivative contract written between two counterparties where they agree to swap a stream of cash flows at prefixed dates in the future expiring at time T. The simplest form of swap is the interest rate swap where the counterparties agree to exchange a fixed against a floating amount. The floating amount is usually linked to Libor (London Inter Bank Offer Rate).

13.2 Equity-linked swaps If we replace the fixed leg stream of a swap with an option we obtain an equity swap. Diagram XX shows the basic structure of an equity swap.

A option Libor i - X

B T

time


Counterparty B pays a floating amount, which corresponds to Libor minus a small fixed amount X called funding, at prefixed intervals to counterparty A. The total amount received from counterparty A is therefore

N

∑ ( Libor − X ) i

i

i =1

38

Derivatives Counterparty A on the other hand pays the equity option to counterparty B at maturity T. in order for the payment to be consistent the value of the floating amount leg has to be equal to the value of the equity option (plus the sales margin if any). This means that N

∑ ( Libor − X ) = Option + M arg in i

i

Bank

i =1

13.3 Bonds Bonds are special financial instruments which are used to raise capital. It is a debt instrument with maturity T issued by a company or by governments in exchange of a principal amount. The bond issuer has then the obligation to pay back its debt at maturity T by repaying the principal amount. The simplest form of bond doesn’t pay any coupon during its lifetime and is called a zero coupon bond. It is easy to understand that the value of a bond varies over time since its value is linked to interest rates. The value of a zero coupon bond which pays X at maturity T is worth today the net present value of X, evaluated with the current levels of interest rates.

13.4 Equity Notes In the derivative world bonds are used to construct specific products which are called capital guaranteed notes. A capital protected note ensures the investor to receive back at maturity at least the amount initially invested (this is of course less than what he could have done by earning interest on the same amount). Notes are simply the combination of a zero coupon bond plus an equity option as shown in Diagram XX.

option

Bond

X

X

B T

time


If counterparty B wishes to invest an amount X into a 100% capital protected note then he would pay X to counterparty A. We know that in order to have X at maturity T we need to

39

Derivatives invest the net present value of X into the money market, earning interest on it. The remaining amount (i.e. the difference between A and its net present value) is then used to buy an equity option with maturity T plus the margin for counterparty A (and the margin for counterparty B if it will be sold further to a third party). The following equality has therefore to be satisfied:

ZCBond + Option + M arg in Bank + M arg inClient = 100%

40

Derivatives

14 What are Models There are various models that can be and have to be used in finance to price derivatives. They can vary from very simple Black-Scholes model to more complex (and complete ones) such as the local volatility model and the stochastic volatility models. There are, at this point, two basic questions to which we should try to give an answer: 1. 2.

Why are there different models in finance? How can we choose the right model when pricing derivatives?

In finance, like in physics, different models have been developed depending on the complexity of the derivative which has to be analyzed. A model could generally be defined as follows:

Definition of model A model is a series of laws which have been derived from empirical observations and which have to describe and predict a given process in the best possible way. Options are characterized by a non-linear behavior. In other words if we evaluate first derivatives with respect to a given parameter, this will generally be different from zero. When pricing options one has to take into account the effects of the non-linearity. For vanilla products most of the second order effects can be neglected and therefore a simple model (for example the B-S model) is good enough in order to get a reliable price. There are cases when second order effects cannot be neglected because they have a significant impact on the price of the option. As previously discussed, in the B-S model the volatility of the underlying option is considered to be constant. We have seen that this is not exactly the case since in the market we observe a skewed behavior of lognormal returns. Skew doesn’t affect ATM European option and they can be therefore correctly priced with the B-S model. This is not the case when dealing with OTM or ITM options; a model where the skew is correctly modeled is necessary. Before pricing a derivative we have therefore to analyze which effects have to be taken into account and which one be neglected and this decides the model which has to be used in order to correctly evaluate the price of an option.

14.1 The calibration process The calibration proces of a model is crucial in order to fairly evaluate a derivative. The word “fairly” means that the model has to be able to correctly price securities which are available in the market. Let’s give an example to better understand this concept. Suppose that we need to price an exotic barrier option linked on a basket of two shares. This product can’t be priced with the B-S model because the skew has a significant impact on the price. A more “exotic” model has, therefore, to be used. This model, which is different from the B-S model, has to be able to correctly evaluate vanilla options on the individual shares, giving exactly the same price as the B-S model. In addition to this, it has to correctly price fixed income securities such as bonds. The process of “adapting” the parameters of the model in order to get the same price as the B-S model is called the “calibration process”.

41

Derivatives

B-S Model

OK

Exotic option

OK

OK

X

OK Vanilla option on share A and on share B

Exotic option on a basket of shares A + B


14.2 How all derivative pricing models actually work All derivative pricing models have at their heart one very important principle: that the price of a derivative is the cost of removing the risk of having sold (or bought) the derivative in the first place. The Black and Scholes approach assumes that the way to remove the risk is to trade the share underlying the option. The price that one gets from the Black and Scholes equation is simply the cost of running a trading strategy that will replicate the derivative payoff. Consider a very simple derivative – a forward. This contract is simply an agreement for two counterparties to trade a share on a future date at a price to be agreed on the deal date. Both counterparties are obliged to trade the share, and neither can walk away from the deal once it has been struck. So it is not like an option, where the buyer can abandon the contract with no penalties. The price at which the share will be traded in the future is agreed so that there is no initial value to the contract i.e. no money need change hands up-front in order to make the contract “fair”. What can the seller to do remove the risk? If you know for absolute certain that somebody will come and buy a specific thing from you for a known price on some defined date in the future, the obvious strategy is to own the thing now. Once you have bought the thing, you no longer care about its price in the market. So if you have entered into a forward contract where you have to sell a share, the obvious hedge is to buy the share as soon as possible. Then there is no more risk, and the outcome is known with certainty. So what is the fair price to agree to buy / sell the share in the future, so that no money need change hands up-front as part of the contract? The hedging strategy is to buy the share, which will have at any time a known market price. However this is not the full cost of the strategy. The share has to be “financed”, i.e. the money has to be found to buy the share.

42

Derivatives Let’s assume that this money has to be borrowed. There is a cost to this borrowing equal to the interest rate paid. The total cost of the hedging strategy is therefore the cost of the share, plus the cost of borrowing the money to buy the share, between the deal inception date and the date on which the share is going to be sold. As an example, assume that the share price is 100 EUR, the term of the contract is 1 year, and the interest rate is 5%. The fair price to sell the share in 1 year will be 105 EUR, which is the current price of the share, plus the cost of borrowing 100 EUR for 1 year. The seller, on the deal date, borrows 100 EUR, and buys the share. In 1 year, they sell the share for 105 EUR, and pay back the borrowing which will have grown to 105 EUR. So their position at the end of the contract is economically identical to their position at the start, irrespective of any changes in the price of the share. If the agreed price for the trade in the future (commonly known as the “strike”price) is different to 105 EUR, some money will need to change hands up front to compensate one counterparty or the other. The important thing here is that the forward price of the share is not defined using some arbitrary formula. The formula comes from somewhere – it is the cost of removing the risk on the contract. When it comes to a contract where the party who is buying the share can walk away from the deal with no penalty, the situation is different. The simple strategy described above will not work in the case that the share price is below 105 EUR on the expiry date, because the buyer would be better off buying the share in the market. So they will abandon their side of the contract, leaving the seller holding a share which is worth less than 105 EUR, but with a loan of 105 EUR to repay. So the seller needs to do something else. One possible strategy would be to sell the share immediately should the price fall below the strike price of the contract, and buy it again as soon as the price goes above the strike price. Another would be to “smooth” the trading, by buying half a share initially, assuming that it is about equally likely that the share is going to go up as down, and buying or selling another half share at some point in the future depending on the way in which the price moves. Consider a very simple world where it is known that the share price is either going to be 90 EUR or 110 EUR at maturity. To further simplify the analysis in this case, assume that interest rates are zero. The strike price of the option is 100 EUR. The seller of the contract buys half a share on the deal date. At expiration, if the price is 110 EUR, the seller will hold half a share worth 55 EUR for which they paid 50 EUR, and will need to buy another half share, at a price of 55 EUR, and deliver the lot to the counterparty for a total price of 100 EUR. The total cost for the seller was 105 EUR, so they will lose 5 EUR in this case. If the share price drops, the seller of the option is left holding half a share, worth 45 EUR, for which they paid 50 EUR. The buyer is not interested in the option because they can buy the share more cheaply in the market than they can from the option seller. So the seller can do nothing other than sell the half-share, for a loss of 5 EUR. The critical point here is that the seller loses 5 EUR in both cases. So if they charge 5 EUR up front for the option, they will break even under all possible scenarios. So the fair value of the option in this very simple model universe is 5 EUR. It is not necessary for the seller to know in advance whether it is more likely that the share go up or down – they are completely indifferent to the final price of the share. In the real world, things are clearly a little more complicated and the above model is inadequate. In reality a much wider range of outcomes is possible, and a more complicated model of the world is needed to accurately cover the possible outcomes. An important point about the hedging strategy in the simple world model, is that it involves buying shares at a higher price than they are sold. This is what actually costs money and why the premium for an option is not zero. This, in simple terms, is what all derivative hedging is about, and what all models are about. They will, given a model of how share prices move

43

Derivatives about in the short term, tell you how much you will lose from this process of buying high and selling low. Another important point is that the hedger does not care whether the price of the share goes up or down, they just care that the model covers the potential outcomes regardless of their probabilities. Black and Scholes option pricing is therefore just a complicated way of working out the loss from running a hedging strategy like the one described above, which systematically buys shares at a higher price than they are sold. The model makes assumptions about how the share price behaves in the very short term, and then adds up the effect of these very short term moves, combined with the hedging strategy which specifies how many shares are actually being held at any time, to calculate the losses that result. Also included in the pricing model is the cost of financing the hedge. The Black and Scholes breakthrough was that it represented a simple model of how the market behaves which is reasonably accurate, and also that it incorporates a hedging strategy that, if the market behaves in the way the model describes, will result in an exact outcome for the hedger i.e. the hedger is entirely indifferent to the performance of the share during the option’s lifetime. Remember that the Black and Scholes model of a share price only describes how the price varies over short periods of time, and does not make any predictions about its long term performance. For shares whose price jumps around a lot, and by large amounts, the losses that result from the buy high / sell low strategy are larger than they will be for a less volatile share. This is why vanilla option prices are linked to volatility. As described above, basic derivative pricing assumes that the price of the underlying jumps around continuously, and the model describes the way that the price moves in the very short term. Approximately speaking, the standard Black and Scholes model assumes that the size of the change of the price of the stock is related to two things: the square root of the length of time over which the change is being moved; and the “volatility” of the stock, which is a fixed number. Another important aspect of the model is that, on average, the stock price moves down as often, and by as much, as it moves up. The model also assumes that the universe is static, in that the volatility does not change, and there is no time-dependency of the process. This standard model leads to the well-known lognormal distribution of returns, when observed over some long time period (remember that the “model” of the stock price only describes what happens in the very short term). However when the Black and Scholes equation is solved, it is seen that the price of the option is also equal to the sum, over the possible prices at maturity, of the payoff of the option at each price, multiplied by the probability of ending up at that price. This gives us an alternative way of calculating the price of the option. So the option price that you arrive at, by calculating the loss that results from following the hedging strategy over the lifetime of the option, is the same as the price that you get from calculating the average value of the option, given the final distribution of share prices that you get from the short-term model of how the price of the underlying evolves. Monte-carlo simulation uses this fact of option pricing, to avoid having to simulate the entire path followed by a share price during the lifetime of the option. It simulates a number of single jumps from start to option expiration, so that the distribution of the prices at the end of the jump is the same as the distribution of prices that would result if the share price followed the exact short-term process described in the Black and Scholes equation. For each simulated jump, it calculates the option payoff that would result. Then the option price is simply the average of all of these simulated payoffs (discounted by the appropriate amount). However when option prices are observed in the market, one observes skew and smile. The conclusion is that the simple Black and Scholes model of the short-term evolution of the share price is not quite right. Therefore the distribution of returns over time is not log-normal either. We can acalculate what the implied distribution of long-term returns must look like, using a relatively simple mathematical formula. Note that calculating this implied distribution does not

44

Derivatives actually tell us anything about the short-term process that generates this distribution – there are in fact infintely many short-term processes that could in theory produce the same distribution of returns over the longer term. But for Monte-Carlo simulation, we don’t have to worry about the short-term process, just the eventual distribution of returns. If you can work out what the eventual distribution of returns is, you can run a Monte-Carlo simulation that takes account of the skew. However, this approach can lead to difficulties when a path-dependent option is priced (that is, an option whose value at expiration is linked to the value of the underlying during the lifetime of the option as well as just the value of the underlying at expiration, for example a barrier option or an asian option).

14.3 Black Vanilla Black vanilla is the simplest model, and can only be used for analytic pricing of vanilla options (and products which are made up of vanilla options, like variance swaps) The model includes a complete volatility surface for each underlying, but cannot be used in conjunction with either Monte Carlo simulation or finite difference pricing. Because it can only be used with analytic pricing (i.e. not splife products) most things cannot be priced with this model.

14.4 Black Diffusion This is the simplest possible model which can be used with Monte-Carlo simulation or finite difference pricing. It has a term-structure of volatility for each underlying, but no skew. Splife products can be priced with Black Diffusion, but the effect of skew is not taken into account. For many products this effect is quite significant, so black diffusion is not often used.

14.5 Vskew This model takes account of skew, but suffers from some drawbacks linked to the way it prices path dependent options. As described above, this model takes account of skew by calculating the implied distribution of returns of an option underlying, by examining vanilla option prices. It is simple to price non path-dependent options by simulating returns of the underlying according to the implied distribution of returns at maturity. If we do this, we do not have to worry about what is supposed to be happening to the share price between the start date of the option and the expiry. Imagine now that we have an “path-dependent” option; i.e. one whose value at maturity is linked to the price on some intermediate date as well as at expiration. We have to run a Monte-Carlo simulation that uses the correct distribution of returns for both of these fixing dates. In a world with no skew, this is not too difficult. We just simulate one jump from the start date to the first fixing date, and another jump from the first fixing date to the second fixing date. This is possible because we know that the same short-term process governing share prices applies equally over both periods of time. The final price of the option for any given path can be calculated by examining the share price on the two fixing dates. Note that with a path-dependent option, we cannot treat the price of the underlying on the two fixing dates as independent things. This is because they are not independent – a large drop in the share price between the start date and the first fixing date makes it likely that the share price will also be relatively low on the second fixing date, compared to the situation where the share price is high on the first fixing date. The price on the two fixing dates is correlated.

45

Derivatives

However, as mentioned earlier, the distribution of returns on each of these dates might be known, but nothing is known about the process followed by the share price in the short-term. So we know the distribution of returns between today and the first fixing date, and we know the distribution of returns between today and the second fixing date. We do not know anything about the distribution of returns between the two fixing dates.

14.6 Pskew Pskew attempts to overcome the problems that Vskew has with pricing forward starting options. It does this by generating simulated Monte-Carlo paths by taking each jump in turn, according to a distribution that is calculated by assuming that skew stays the same over each jump. Another way of looking at this is to say that this model assumes that the 1 year skew in 1 year’s time will be exactly the same as the 1 year skew now. The problem with this model is that when a long-dated vanilla option is priced using this model, where intermediate points in the path are generated (these points do not impact the pricing directly, but the fact that they are calculated will influence the distribution of returns on the expiration date of the option), it does not give the right price. What one can observe is that the model does not produce enough skew in the long-term. This is a direct result of the Central Limit Theorem, that if one takes successive draws from any distribution at all, the overall distribution tends towards a normal distribution. This leads us to the conclusion that there is no simple process of share prices that can lead to the skew that is observed in the market, as all processes tend to produce the normal distribution in the long-term. This lead market practitioners to develop models of the share price process where there is a correlation between the share price and the volatility, and models where the volatility is itself a stochastic process. These are explained later. So Pskew can be used to price simple cliquet options, but not anything else.

14.7 Local Volatility Local volatility is a simple concept which say that the instantaneous volatility of the process which drives share price changes is a fixed function of time and the actual share price. So in a local volatility world, one should be able to say, for any day in the future and level of share price, what the volatility will be on that day. By examining vanilla option prices, it is possible to produce a matrix which is populated with short-term volatilities for each share price and future date. Furthermore, this model leads itself to Monte-Carlo simulation, as the volatility for each short-term jump can be calculated easily from the matrix. However, long-term jumps cannot be calculated so easily and the model is therefore quite slow, because of the calibration step, and the need to calculate paths with many intermediate points and not just the fixing dates which are needed. However local volatility is a widely used model for pricing many types of option, both spot starting and forward starting. There are some known problems with the so-called “dynamics” of the implied volatility with this model, but it is a very powerful and widely used model which is the benchmark for most pricing.

14.8 Stochastic volatility

46

Derivatives Stochastic volatilty attemps to address the “dynamics” problem mentioned in the previous paragraph, and also the problem that local vol assumes that the only thing which causes volatility to change is changes in share prices and the passage of time. This is clearly not true, one can observe that there is a random element to volatility. This can be very important in pricing certain types of option which have “vega convexity”. This means that their sensitivity to implied volatility is not constant, and as implied volatility goes up and down, so will the sensitivity to this parameter. Imagine a situation where an exotic option which has a positive sensitivity to implied volatility is vega-hedged with a vanilla option. If the exotic option has “positive” convexity of vega, then its sensitivity to implied volatility increases as implied volatility increases. So the total position will no longer be hedged if implied volatility goes up, as the exotic option will have become more sensitive. The hedger will need to sell some more vanilla options in order to have a hedged position again. If implied volatility then drops, the hedger will have to unwind the vanilla option trade. They will make money on this unwind, as they are buying back the vanilla option at a lower level of implied vol compared to the level at which they bought it. So a positive vega convexity position will consistently make the holder money when implied volatility varies itself. The problem arises where an exotic option position has negative vega convexity. The holder of such a position will systematically lose money if implied volatility changes, and it is clear that it does. So they need a model to calculate the cost of these changes, in just the same way that the basic Black and Scholes model calculates the cost of re-heding the delta of an option. Stochastic volatility models generally have five components. The average level of the basic volatility of the share price, the volatility of this volatility, the correlation between the basic volatility and the “vol of vol”, and the speed to which the basic volatility reverts to its mean level. (This mean reversion is necessary as it is clear that, unlike a share price, implied volatiltiy does not increase without limit. A share price can double, treble, quadruple or even more, whereas volatility cannot feasibly increase above a certain level, except for very short periods.) So stochastic volatility models are used to price options with vega convexity. The classic example of an options with vega convexity are certain types of cliquet option, often called “best-of ratchets” or “reverse cliquets” or “napoleons”. Note that there is a common misconception that it is the forward-starting nature of these options than means they have to be priced with stochastic volatility models. This is not true, it is the fact that they have strong vega convexity. Other types of options, for example simple barrier options, can also have strong vega convexity. It is even possible to construct a portfolio of vanilla options (the “butterfly” strategy”) which has strong convexity. However with the vanilla options, the convexity cost of the strategy is accounted for in the shape of the volatility surface. The mathematical characteristics of stochastic volatility models are the following:

14.8.1

The Heston model

The dynamics of the calibrated Heston model predict that:

• • •

volatility can reach zero stay at zero for some time or stay extremely low or very high for long periods of time.

47

Derivatives

⎧ dS = μ Sdt + v SdW S ⎪ ⎨dv = κ (θ − v)dt + α v dW v ⎪ E [dW dW ] = ρ dt S v ⎩

14.8.2

The Hagan model

The dynamics of the Hagan model predict that:

• • •

the expectation of volatility is constant over time variance of instantaneous volatility grows without limit the most likely value of instantaneous volatility converges to zero.

⎧dS = μ Sdt + σ SdW S ⎪ ⎨ d σ = ασ dW σ ⎪ E [dW dW ] = ρ dt S σ ⎩

14.8.3

The Scott-Chesney model

The main drawback of the Scott-Chesney model is that:

• •

it requires very high correlation between the spot and the volatility process to calibrate to a pronounced skew the skew is fully deterministic

These features are also shared by all of the above discussed models.

⎧ dS = μ Sdt + e y SdW S ⎪ ⎨dy = κ (θ − y )dt + α dW y ⎪ E [dW dW ] = ρ dt S y ⎩

48

Derivatives

15 Cliquets A cliquet, ratchet option or strip of forward start options is a derivative where the strike is reset each observation date at the then current spot level. The profit can be accumulated until final maturity, or paid out at each reset date. Cliquets are complex exotic products where second order effects can significantly affect the pricing. There are two main effects which have to be considered: 1. 2.

Volatility of volatility effects (vega convexity) Forward skew effects

Not all cliquets are sensitive to volatility of volatility and forward skew. We’ll see what the impact is on the main types of re-striking options.

15.1 Volga (or Convexity) Let’s consider an ATM European call option. We have seen that its price can be written as

C ≈ 0.4 * σ * T If we draw the price of this option with respect to the volatility we can see that it is a straight line with positive slope, as shown in Diagram XX.

Price

Sigma

Diagram XX A call option price function of volatility

It is easy to verify that the vega in this case is equal to a constant value; it is therefore independent from the level of the volatility. The vega is a constant line as shown in diagram XX.

49

Derivatives

Price

Sigma

Diagram XX A call option vega function of volatility

So for an ATM call

∂ 2V =0 Volga = ∂σ 2 Consequently, vanilla options don’t have price convexity. Some exotic derivatives can have a non zero convexity which needs to be hedged like in the case of cliquets. Let’s suppose that the price a generic derivative follows a parabolic curve like shown in diagram XXX.

Price

Sigma

Diagram XX Exotic option price function of volatility showing convexity

50

Derivatives The vega is a linear function of volatility and will therefore change sign around a value σ* (like shown in diagram 8) Vega

Sigma

Diagram XX Exotic option vega function of volatility.

As you can see the vega is here not constant with respect to volatility but is a linear function of volatility. We call σ* the volatility where the vega is equal to zero and changes of sign. If we are buying volatility, the more σ increases, the shorter the vega is. In order to hedge our position, we need to buy the volatility, therefore we buy the volatility when it increases and we sell it when it decreases. This hedging cost has therefore to be included in the price of the derivative. Finally when the Volga is non zero, we are dealing with the volatility of the volatility, so we need to consider a stochastic volatility.

15.2

Types of Cliquets

There are various types of cliquets options and an extensive list would not be possible. We’ll present only the main typologies since the effects which have to be taken into account are common to all of them.

15.2.1

Classic cliquet

The classic cliquet is a forward starting option, which fixes its strike at time t (from today) and expires at time T (from today). For a cliquet call option the payoff would be:

⎛

ST

⎝

S t

Payoff T = Max⎜⎜ 0,

⎞ − 1⎟⎟ ⎠

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Derivatives

15.2.1.1

Cliquet behavior

Let’s consider an ATM forward starting call option. This derivative would pay at time T the performance of the underlying asset over the time T – t if positive, zero otherwise. As we know an ATM call option is not sensitive to skew effect. Similarly a forward starting call option will not exhibit any sensitivity in change in volatility with respect to the strike, meaning that the sensitivity to forward skew is equal to zero. The only parameter that has to been taken into account is the forward starting volatility with maturity T – t. This can be evaluated with the simple variance equality:

σ 1 T 1 + σ 12 T 12 2

2

= σ 2 2T 2

and therefore

σ 2T 2

=

σ 12

− σ 1T 1

T 12

Diagram XX shows the legs and the volatility considered in the formulas above. S0

St

ST

σ1, T1

σ12, T12 σ2, T2 σ1 , T12

Diagram XX YYYYYYYYYYYYYYYYYYYYYYYY.

the price of a european ATM call starting at time t with maturity T 12 is therefore the same as the one for a european ATM call starting today and with maturity T 12 adjusted by the change * in volatility (from σ1 to σ12) times the volatility sensitivity (because the vega convexity is equal to zero).

C (σ 12 , T 12 ) = C (σ 1 , T 12 ) + Vega * (σ 12 *

15.2.2

− σ 1 )

Ratchet

A ratchet option is a strip of forward starting options, which fixes their strikes at time i and are evaluated at time i+1. The performances evaluated are thereafter summed together and paid at maturity T. For a call ratchet r atchet option the payoff would be: be:

Payoff T

N ⎛ S ⎞ = ∑ Max⎜⎜ 0, i − 1⎟⎟ i =1 ⎝ Si−1 ⎠

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Derivatives

15.2.2.1

Ratchet behavior

We have seen that classic cliquets are insensitive to forward starting skew. Ratchets are therefore insensitive as well by definition, being the sum of forward starting cliquets. But there are other interesting effects which have to be analyzed. Let’s consider a strip of two ATM forward starting call options. This derivative pays at time T the sum of the positive performance between each subsequent period, T 1 and T12. As seen before, the only parameter that has to been taken into account is the forward starting volatility between the two fixing date t 1 and T. This can be evaluated with the simple variance formula:

σ 12

=

σ 2T 2

− σ 1T 1

T 12

In the case of a ratchet the effect of volatility of volatility has to be taken into account. To understand this let’s evaluate how the vega position would be at an instant t, between the initial strike date t0 and the first fixing date t 1 like shown in Diagram XX.

t t0

t1

T time

σ1, T1

σ12, T12 σ2, T2


At an instant t, the call with expiry t 1 would have a positive vega exposure with respect to the volatility σ1. The call with expiry T would have as well a positive vega exposure with respect to the volatility σ12, but this volatility decreases if σ1 increases, meaning that even if the overall exposure of this call is vega positive, it is in effect vega negative with respect to the volatility corresponding to the previous fixing. This means that at a future instant t the price of the ratchet varies in a non linear way since the first call tends to increase in value if σ1 increases, whereas the second call will tend to decrease if σ1 increases (assuming that σ2 hasn’t changed). Diagram XX shows how σ12 varies for a 1% increase in σ1 considering that σ2 is not changing.

53

Derivatives

Δσ12

σ1

Diagram XX YYYYYYYYYYYYYYYYYYYYYYYYYY XX YYYYYYYYYYYYYYYYYYYYYYYYYY

It is easy to understand that the vega is highly sensitive to the evolution of the volatility term structure: change in the volatility of the volatility have therefore to be t aken into account. It is important to highlight that in the case of a volatility surface moving all by the same amount, the vega would have shown a linear behavior.

15.2.3

Reverse cliquet

A reverse cliquet can be generally defined as a globally floored option where the payoff depends on locally capped performances. Usually a reverse cliquet has a maximum payout starting at X which decreases as the sum of forward starting put options increases in value. The payoff would therefore be:

N ⎛ ⎛ S ⎞ ⎞ ⎜ Payoff = Max 0, X + ∑ Min⎜⎜ 0, i − 1⎟⎟ ⎟ ⎜ i =1 ⎝ S i −1 ⎠ ⎠⎟ ⎝

The reverse cliquet is sensitive to both the volatility of volatility and the forward skew. In the following section we will show why this is the case.

15.2.3.1

Reverse cliquet behavior

Let’s consider how the vega of the reverse cliquet behaves when the volatility varies from small to significant values. If volatility is low the sum of the values of the ATM put options will be very sensitive in volatility changes since a small shift in the surface will be reflected in the change in value of all the single put options and the total vega will therefore be the same as the sum of the individual vegas. If volatility is high, on the other hand, a small change in the volatility surface will not affect the vega since the probability for the puts to be in the money is very high, which in turn is very likely to quickly exceed X. The vega profile can therefore be represented as follows:

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Derivatives

Vega

Sigma


The reverse cliquet, like several other similar cliquets, can therefore be considered as a put option on volatility as shown in diagram XX.

Vega

Sigma


As we know a put option has a positive sensitivity to vega. It is, therefore, easy to see that if we are using a model which doesn’t take into account the volatility of volatility we are in fact pricing a put option without its time value and the price is therefore incorrect. A model which simulates the stochastic behavior of volatility is here needed in order to correctly price the additional optionality.

The reverse cliquet is equally sensitive to forward skew, even if its sensitivity is here less important if compared to that of volatility of volatility. To see why this is the case, we’ll analyze * the skew exposure is when we fix the last strike (meaning the fixing before maturity) maturity) at time t .

55

Derivatives

t0 time N-1

N


Let’s suppose that the following relation is valid at time t

X ′ = X +

N −1

⎛

Si

∑ Min⎝ ⎜⎜ 0, S i =1

i −1

*

⎞ − 1⎟⎟ ⎠

[ZZZ]

*

The payoff at time t for the maturity T is therefore:

⎛ ⎜ ⎝

⎛

S N

⎝

S N −1

Payoff = Max⎜ 0, X ′ + Min⎜⎜ 0,

⎞ ⎞ − 1⎟⎟ ⎟⎟ ⎠ ⎠

This can be seen as being short an ATM call and long a call strike done by 1-X’, both fixing the strike at N-1 and with expiry at N, as shown in figure XXX. Payoff

1-X’

1

S


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Derivatives *

The dotted line shows the total position of the holder of the reverse cliquet at time t , which results in a call spread. It is easy to see that the call one is long is always in-the-money and this results in a positive sensitivity to skew for this call. The call one is short, on the other hand, is always at-the-money and is consequently non sensitive to skew. The total exposure is therefore a positive exposure to skew.

15.2.4

Napoleon

Let’s now consider a Napoleon with monthly resets, which pays a yearly coupon expressed by the following formula:

⎛ ⎜ ⎝

⎛ S i

Payoff = Max⎜ 0, X + Min⎜⎜ i

⎝ S i −1

⎞ ⎞ − 1⎟⎟ ⎟⎟ ⎠ ⎠

The client receives, therefore, a coupon of X plus an amount which corresponds to the smallest monthly return. The Napoleon as well, like the reverse cliquet, is sensitive to volatility of volatility and to forward skew.

15.2.4.1

Napoleon behavior

The vega is, in this case again, a non-linear function of volatility. Let’s consider the case where the volatility of volatility is close to zero. This means that the volatility won't vary significantly from a given fixed value. The individual monthly puts will therefore have a given probability to be in the money and the more valuable of them will have a given probability Π1 to assume a value X1. If we increase now the volatility of volatility the probability associated to each monthly put will in general be different and the more valuable of them will have a probability Π2 to assume the same a value X, which will be greater than Π1. In other words, if we consider stochastic volatility the most valuable monthly put will assume higher values compared to the case where there is no volatility of volatility. If volatility is very high, small shifts of the volatility surface will in general leave unaffected the price of the Napoleon. Increasing the probability Π for the put being in the money will have no effect on the price since the payoff has an overall floor at zero. The vega is therefore close to zero. This is not the case if volatility is low. In this case increasing the probability Π will have a significant impact on the options price and this corresponds to a significant vega. The vega profile as a function of volatility is here again similar to the one shown in diagram XXXX. Let’s now analyze the sensitivity of the Napoleon with respect to the forward skew. The forward skew has less significant impact if compared to the reverse cliquet. To understand this let’s see what the payoff would be at the last strike date (the one before maturity):

⎛ ⎜ ⎝

⎛ S N ⎛ S ⎞ ⎞ ⎞ 1, Min ⎜⎜ i − 1⎟⎟ ⎟ ⎟ − ⎜ S N −1 i =[ 0 ,..., N −1] S ⎝ i −1 ⎠ ⎠⎟ ⎠⎟ ⎝

Payoff = Max⎜ 0, X + Min⎜

We can easily see that the payoff is worth zero if

57

Derivatives

⎛ S i ⎞ ⎜ − 1⎟⎟ < X i =[ 0 ,..., N −1]⎜ S ⎝ i−1 ⎠ Min

And corresponds again to a call spread otherwise. The lower strike corresponds to

S N S N −1

− 1 = − X

Whereas for the upper strike corresponds to

⎛ S i ⎞ ⎜ − 1⎟⎟ i =[ 0 ,..., N −1]⎜ S ⎝ i −1 ⎠ Min

But we have to distinguish two cases:

⎧ ⎛ S i ⎞ − 1⎟⎟ > 0 Min ⎪i =[0,..., N −1]⎜⎜ S ⎪ ⎝ i −1 ⎠ ⎨ ⎞ S ⎪ Min ⎛ ⎜⎜ i − 1⎟⎟ < 0 ⎪⎩i =[0,..., N −1]⎝ S i −1 ⎠ In the two cases the sensitivity to skew will be different. If the first condition is true then the lower strike is below the ATM and the upper strike is above the ATM. If the second condition is verified then the two strikes lie below the ATM. The probability that the second condition is verified is higher than the probability associated to the fir st condition. This is true since we are taking the smallest return over the period. The two strikes tend, therefore, to be closer if compared to the reverse cliquet. Since the holder of the option is long the call spread he is effectively buying the call with the lower strike and selling the call with higher strike. The contract is in general long skew where the overall sensitivity is lower than the one of the reverse cliquet since the two strikes lie closer in this case.

15.2.5

Accumulator

⎛ N ⎛ ⎛ ⎞ ⎞ ⎞ S Payoff = Max⎜ 0, ∑ Min⎜ Cap, Max⎜⎜ Floor , i − 1⎟⎟ ⎟ ⎟ ⎜ ⎜ i=1 S i −1 ⎠ ⎠⎟ ⎠⎟ ⎝ ⎝ ⎝

15.2.5.1

Accumulator behavior

XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX

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Derivatives

15.2.6

Comparing cliquets

Table for comparison

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Derivatives

16 The Concept of Risk When a bank issues a structured product to a client, it is in fact selling a contract (under form of swap, note or certificate) that ensures the holder receives a given percentage of the notional invested back, depending on the performance of the underlying asset(s). The role of the trader is to ensure that the right amount of risk is always hedged away in order to be able to fulfill the conditions stated in the contract. Due to the complexity of the financial world, and especially that of exotic products, the hedges are far from being completely accurate. This is often not related to the ability of the trader but rather to the nature of risk which has to be hedged away. It is possible to categorize the main risks into the following sub-categories: 1. 2. 3. 4.

delta risk vega risk correlation risk second order risks

The value of an option can vary over time because of several market parameters. The trader has always to assume an opposite position in the market (with respect to the issued products) in order to reflect the change in value of the derivative instruments. The main components which have to be hedged away are the so-called first-order risk indicated above (delta, vega and correlation). Once these have been hedged, the trader has still to verify the presence of second order risk like volga or vanna. Usually their effect is negligible for vanilla options and most of the more common exotic options. This is not the case for cliquets and other unusual exotic payoffs.

16.1 Delta Risk The delta represents for the trader, without any doubt, the most important risk to be hedged. This is achieved by buying (or selling) the right amount of shares (futures in case of indices) in order to have at all time (or as many time as possible) a flat delta position. In other words the trader has to be long (or be short) at any time an amount of the underlying asset so that if added to its delta position, resulting from the products issued, equals zero. Liquidity is in this case a very important aspect which has to be analyzed when checking if a given underlying can be hedged or not. It is important to verify that the amount of shares traded per day corresponds to the delta of the product which has been sold since the trader, as stated previously, has to buy (or sell) that amount of shares.

16.2 Vega risk Once the delta component is hedged away, the trader still has the risk associated with the volatility of the underlying. Suppose that a bank sells an at-the-money call option on one underlying today to one of its clients. This option has a value X 1 which can be estimated with t he Black-Scholes equation as previously seen. Suppose that after a period t, the market has not moved since the day we sold the option, meaning that all the market parameters have remained constant over time. The value of the option has therefore not changed (if we neglect the time decay) and the call is still worth X 1.

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Derivatives Suppose now that after the period t, all the market parameters have not seen any change but the volatility of the underlying (on which the option is written), which has increased. The option is now worth X 2 (where X2 is greater than X1 since a call is long vega). In order to be hedged, the trader has, therefore, to buy volatility.

16.3 Correlation risk Correlation risk is one of the most important and dangerous component to which banks are exposed. Due to the nature of the products sold over the past decade, banks are generally short on correlation. This is because of the attractiveness nature of low correlation. Suppose to have a generic products “worst of”, meaning that the payoff, usually represented by a big coupon or the capital protection, depends on event that one (or more) of the N underlyings has touched a barrier or not. The lower the correlation the more attractive the final payoff will be for the investor. Consider a reverse convertible worst of where, at maturity, the investor is long a bond and a coupon X and is short a put down and in (with barrier B) on the worst performing stock. He receives therefore his notional back plus the coupon X if none of the underlying stocks ever traded below B. In case the condition is not verified, and therefore one or more stocks did trade below the barrier B, the client still receives the coupon X, but the notional invested is reduced by an amount corresponding to the highest drop among the stocks at maturity. In order to have an attractive coupon it is in the investor’s interest to choose stocks with as low correlation as possible. By reducing the correlation we increase the probability of one of the shares touching the barrier B. This will, in turn, increase the probability of losing the capital protection at maturity, leaving more to spend for the coupon X.

16.4 Second order risks Modeling risk refers to the model used to evaluate the price of the derivative instrument. Due to the complexity of exotic products it is crucial to take into account all the effects that could affect their value. There are products for which second order effects (second order derivatives with respect to a given market parameter) do not have to be taken into account. This is especially the case for simple products like vanilla options or even simple exotic options. Consider, for example, an at-the-money call option on a given underlying asset. This option can be valued with the Black-Scholes model or with the more “sophisticated” Local Volatility Model and the price we would get would be exactly the same. Skew effects have, in fact, no influence when valuating an at-the-money vanilla call option. Consider now a barrier option, like a down-and-out put option with barrier at 60%. The model used in this case assumes a very important role. The Black-Scholes model would use the same volatility for the strike and for the barrier, whereas the Local Volatility Model would consider two different volatilities because of presence of skew.

61

Derivatives

17 Discontinuities Discontinuities in the payoff are very common and they need to be analyzed in detail when pricing exotic derivatives. In the following section we describe the dynamic of a digital call option and the issues related to hedging.

17.1 Digital call option A Digital Call option with strike K and paying an amount N at maturity T has the following Payoff:

Payoff = N ⋅ I ( St > K ) Diagram 9 shows the payoff profile of a digital call option at maturity. The option pays N if the underlying is above the strike K, it pays 0 otherwise.

Payoff

K

St

Diagram 18.1 Digital Call option.

The black shape in diagram 10 represents the value of the digital call option at time T 1 and its corresponding delta profile. As you can see, at time T 2 > T1 (and closer to maturity), the value of the option steepens around K and the delta tends to increase consequently.

62

Derivatives

Payoff

Δ

t = T1

t = T1

St

St K

K

Payoff

Δ

t = T2>T1

t = T2>T1

St

St

K

K

Diagram 18.2 Value and delta of a digital option as a function of strike K. a) Shows the value of the digital option with the corresponding delta profile at time T1. b) Shows the value of the digital option with the corresponding delta profile at time T2 > T1. We therefore observe a discontinuity in the payoff at maturity around K. By discontinuity we mean a point where the first derivative of the value of the option tends towards infinity. This is the case of digital options where for the limit of t -> Maturity the delta tends to assume infinite values; in other words close to K and close to maturity the delta, is given by:

Δ=

∂P ∂S

is a Dirac function.

17.2 Barrier shifts In order to keep the Δ finite, we need to modify the payoff in order to take into account the effects of hedging a discontinuous payoff. The most natural solution for a digital call option is to represent the payoff as a call spread. The other solution is the so-called barrier shift. If we shift the strike K of the call to the left we notice that we decrease the value of the delta on the original strike. We could therefore sell to the client a digital call option with strike K at a cost of a digital with strike K* (which is more valuable then a digital call with strike K) and hedge this option instead in order to avoid a delta tending to infinity.

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Derivatives

Payoff

K

St

Payoff

St Δ

K

St

Diagram 18.3 Barrier shift and the effect on the delta of the option.

17.3 Evaluating the discontinuities Evaluating discontinuities in the payoff (or value of the product) could be misleading when dealing with exotic products. It is therefore necessary to give a formal definition of how a discontinuity can be estimated. The formal definition of a discontinuity can be explained as follows. Let’s consider a product with maturity T, with interim payment cash flows at time t i with 0 < t1 < t2 < t3, .. , T where the payment depends on the condition C i .Ci is in this case a Boolean variable which can assume two value (True or False). In other words the CashFlowTrue will be paid at time if C i is True otherwise CashFlowFalse will be paid.

Discontinuity (i ) = χ (i) C i =true

+ θ (i) C =true − χ (i) C = false − θ (i) C = false i

i

i

Where:

= is the intrinsic value of the product if the condition C i is true θ (i) C =true = is the time value of the product if the condition C i is true χ (i) C = false = is the intrinsic value of the product if the condition C i is false θ (i) C = false = is the time value of the product if the condition C i is false

χ (i) C i =true i

i

i

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Derivatives

Example:

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Derivatives

18 Pricing Techniques Once the model with which one will determine the value of the option has been chosen, it is still necessary to select a pricing method. For sufficiently simple products it is possible to determine closed formulas for option evaluation. This is not the case when dealing with exotic products. In this case more sophisticated techniques have to be used as will be explained in the following sections.

18.1 How to determine options prices There are various methods to estimate the value of derivative products. Usually it is the complexity of the derivative that determines whether an evaluation methodology is appropriate or not. They can be listed as follows:

• • •

Closed formula (analytical method) Finite difference and Trees (numerical methods) Monte Carlo

Generically we could say that as the dimensionality of the problem increases (and therefore as its complexity increases) the use of a Monte Carlo approach becomes necessary. Schematically this can be seen like in diagram XX

Dimensionality

Monte Carlo

Finite Difference Trees

Closed formula

Diagram 19.1 YYYYYYYYYYYYYYYYYYYYYYYY

66

Derivatives

18.2 Closed formulas The analytical approach is the classical, and most natural, method to evaluate the solution of a mathematical problem. It enables us to find the exact solution to a well posed problem. Let’s consider the following equation:

x 2

− 3x + 2 = 0

If we want to find the roots of this equation, we can decompose the equation into the product of two members and subsequently evaluate the values for the variable x that satisfies the equation.

( x − 1) ( x − 2) = 0 It is easy to see that the equation is equal to zero if and only if one of the two members is equal to zero this means:

x − 1 = 0 x − 2 = 0 The solutions of the equation are therefore x = 1 and x = 2. We used an analytical approach to solve the equation.

Closed formula definition: Under some condition a value V of an option with payoff f can be determined via a closed formula. A closed formula is a function f depending from some specific parameters of the underlying like the volatility, the dividends, the interest rates, maturity etc. V can therefore be written as:

V = g(sigma, d, r, T, … )

The closed formula solution to the Black and Scholes differential equation has been determined by using an analytical approach. Nevertheless it is very rare that the value of a derivative can be determinate by following this methodology. This is especially the case with more “exotic” structures where the complexity of the product requires a more powerful evaluation tool. Let’s consider, for example, an American call option where the holder has the right to early exercise the option. It is not possible to estimate an American call via a closed formula. The use of Finite difference or Trees becomes, therefore, necessary

18.3 Finite difference and Trees XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX

18.4 The Monte Carlo method The Monte Carlo (MC) method allows us to analyze more complex problems (basket options, exotic payoffs etc). The theory behind the MC approach is to use a random number generator to estimate the stochastic evolution of an asset, calculate the payoff of the option for all the

67

Derivatives stochastic paths, calculate the average of the payoffs and discount it in order to obtain today’s value.

Monte Carlo method definition: The Monte Carlo methodology (also called Monte Carlo Integration) is a stochastic technique for obtaining solutions of complex problems by means of random numbers. When we determine the value V of an option with payoff f (and discounted payoff f d ) with the MC approach we are effectively calculating the average of the function f on a set of randomly sampled points, each one having a given probability of occurring.

V=

18.4.1

1 M

M

∑ f ( x ) d

i

i =1

How does it work?

Let’s consider a circle C circumscribed by a square S as illustrated in figure XX. We want to calculate the value of the surface of C via the MC methodology.

Diagram XX YYYYYYYYYYYYYYYYYYYYYYYY

Considering the symmetry of the problem we can also analyze only one quarter of Figure XX. Figure XX shows the arc of circumference and the square S’ resulting from this operation. Let’s discretize S’ in a finite number of smaller squares by dividing the x and the y axis in M intervals.

y

68 i

i

y

i

(x , y )

Derivatives


First we randomly extract M pairs of uncorrelated numbers and to associate them with the i i point (x , y ) on S’ by means of a mapping function F: j j i i F : ( x , y ) ⇒ ( x , y ) i

i

Let’s count the number of times N the point (x , y ) falls inside the arc of circle. We can then denote the surface of the arc of circle with A and the surface of the square S’ with Q. the ratio of these two surfaces will be proportional to the ratio of random extracted points which fell inside and outside the area A:

AreaArc AreaSquare

A

=

Q

≈

Po int sArc TotalPo int s

=

N M

For the limit of M going to infinity the ratios converge to each other and the expression above becomes an equality. It is interesting to point out that the MC method applied to this simple geometric problem allows us to obtain an estimate of the constant π. We have seen that for a given set of M of sampling points

A Q

≈

N M

The ratio on the right hand side tends towards the left ratio for a sufficiently high population of sampling points. We also know that

π l 2

A =

4

And that

Q And therefore an estimate for the constant

= l2

π is

N π ≈ 4 M

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Derivatives

18.4.2

Estimating option prices with the MC method

The procedure described in the previous section is very similar to that used to evaluate the price of an option. Let’s consider an exotic option with payoff at maturity described by the function f and having only one underlying S, which is lognormally distributed. We want to calculate the value V of this option using the MC method. The procedure can be decomposed as follows: 1. 2. 3. 4. 5.

we extract a vector U = [x1, x2, x3, … , xN] of N random numbers following a normal probability distribution function we evaluate the values assumed by S for all the different xi, obtaining a vector S = [S1, S2, S3, …, SN] we calculate the payoff f corresponding to each Si, which will be as well a vector f = [f 1, f 2, f 3, …, f N] we calculate the discounted value of f, called f d = [f d,1, f d,2, f d,3, …, f d,N] we calculate the average of all the components of the vector f d, which corresponds to the value V of the option

S S

100%

f

t


70

Derivatives

19 How to Avoid Errors

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Derivatives

20 Hybrid Products The term Hybrid usually refers to structured products where the payoff is contingent to more than one asset class, such as equity, foreign exchange, commodities, interest rates, inflation etc. these kinds of products are often, but not always, treated by a specific team within the bank. The reason behind this is that even if some of these underlyings follow the same stochastic process from a pricing point of view – which should in theory simplify the pricing – the risks involved from a trading point of view are quite different. Risk management tools are very sophisticated for single asset classes but are less flexible and manageable for multi asset classes. Financial institutions are making big efforts to minimise these weaknesses by putting in places dedicated hybrid traders.

20.1 Overview It is a general mistake to define Hybrids as products where several market parameters have to be taken into account when estimating their fair value. This is obviously not the case since, as we have seen in the initial chapters, the value V of a vanilla option depends on several market variables:

V (t ) = V ( S (t , σ , ); K , T ; r ) Some of these are pure equity variables, for example the volatility and the forward. Others are non-equity variables such as interest rates and foreign exchange (if quanto or compo). Vanilla options, like many other exotic options, are not “pure” hybrid products since all the non-equity variables can be hedged away in a static way, whereas equity variable have to be hedged in a dynamic way. Hybrid products can usually categorized into two sub-categories: 1. 2.

Payoff involving explicitly more asset classes Payoff involving one asset class which is highly affected by stochastic behavior of other asset classes

In both cases a specific model has to be used where each individual asset class is correctly priced (e.g. the vanilla market values have to be matched) and where a correlation between different asset classes has to be taken into account. The steps involved to build a proper hybrid multi-asset model can be listed as follows: 1. 2. 3. 4.

Choice of the underlyings (e.g. Equity, FX, Interest Rates commodity etc.) Construction of a single-asset model for each asset class involved Definition of a correlation factor between members of different asset classes Construction of a global multi-asset hybrid model

Schematically this can be represented as in Diagram XXXX.

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Derivatives

Asset Class 1 (AC1)

Asset Class 2 (AC2)

Market Model (AC1)

Market Model (AC2)

Correlation AC1 – AC2

Hybrid Model

Market Values


20.2 The impact of stochastic IR When we consider a model where Interest Rates have a stochastic behaviour, there are two major effects which have to be taken into account: 1. 2.

The actualization of each individual payment will be stochastic The correlation between IR and Equity affects Equity-Volatility surface convexity

The convexity may, or may not, have a significant impact on the value of the option. This depends, of course, on the sensitivity of the product itself to the shape of the volatility surface. The shape of the volatility surface is modified as follows: 1. 2.

the volatility above the ATM level decreases the volatility below the ATM level decreases

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Derivatives The dotted line in diagram XXX shows how the surface is afffected.

ATM


The reason for this is that the model calibrates to vanilla options. The Black-Scholes volatility (the volatility implied by the Black-Scholes model) can’t therefore to be deduced from the local volatility with the known formula

Σ BS ( K ) =

1 K − S

K

∫ σ (κ )d κ S

There is an additional stochastic term coming from the Interest Rates which has to be taken into account. Basically the hybrid model is calibrated in order to match, at each local point, the Black-Scholes volatility for vanilla options if we consider the stochastic behaviour of the rates but the local volatility will in general be different from a non-hybrid model. Let’s consider a 10Y lookback certificate, for example, where the client gets at maturity the highest value of the index observed on a monthly basis. This product has the following sensitivities (values are taken as an example)

Vega

1.1%

Skew

-0.20% 1

Convexity

0.30%

Table 21.1 YYYYYYYYYYYYYYYYYYYYYYYY

1

The sensitivity to the convexity is calculated by bumping the Hagan alpha parameter by 5% in relative value.

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Derivatives As you can see the sensitivity to the convexity of the volatility surface is quite important. If we increase the convexity the price of the certificate increases by 30 Bps. By pricing this option with the hybrid model and with a non-hybrid equity model, we obtain a substantial difference in the price because of different local volatilities implied by the two models. As we have seen, the hybrid model reduces the convexity of the vol surface. We’d expect, therefore, that the hybrid model will show a cheaper price for a lookback certificate. This is confirmed by the results shown in Table XXX

Hybrid Model

96.80%

Equity Model

98.30%


One could suppose that the difference in the price arises from the high vega of the lookback (and therefore from a different deformation of the volatility surface like a parallel shift for example). Let’s therefore consider a product which shows no sensitivity (or very small sensitivity) to convexity and skew but with a significant vega. This is the case of a barrier risk reversal 90 – 110 (the client is short a put strike 90% down and out at 70% and long a call strike 110% up and out at 130%). The sensitivities are shown below:

Vega

0.70%

Skew

-0.02%

Convexity

-0.004%


If we price this product with a hybrid model and with a non-hybrid equity model, we obtain the following results:

Hybrid Model

92.69%

Equity Model

92.61%


The change in price here is very small because of the low sensitvity to the convexity. It is therefore easy to see that the volatility surface is effectively deformed as described above.

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Derivatives

20.3 Specific case studies Covering all types of issues related to hybrid products would be beyond the scope of this book. It is, therefore, more suitable to proceed by analyzing some of the most common products and determine the effects of multi-asset pricing. We will consider the following combination of asset classes: 1. 2. 3.

Equity – Interest Rates linked payoff Pure Equity (but with hybrid effects on Interest Rates) linked payoff Equity – Foreign Exchange linked payoff

20.3.1

Equity – Interest Rates linked payoff

In this section we’ll discuss hybrid products, where the payoff depends on the Equity and on the Interest Rates performance. Usually the interest which is t aken into account when dealing with Interest Rates is the CMS rate (Constant Maturity Swap rate). It corresponds to the rate associated to a swap where the maturity remains unchanged throughout the lifetime of the swap. The hybrid options we will analyze in this section are: 1. 2. 3.

a strip of CMS10Y digitals on the Eurostoxx50 a strip of CMS10Y digitals on the Eurostoxx50 (autocallable if above a barrier) “best of” between Eurostoxx50 and CMS10Y

20.3.1.1

Strip of CMS10Y digitals on the Eurostoxx50

Let’s consider a 3 years product where the client receives, each year, the CMS10Y rate if the Eurostoxx50 performed positively (is above the strike) at the end of that year (since inception); the client receives nothing otherwise. Diagramm XXX shows the CMS10Y linked coupon (red line) if the Eurostoxx50 peforms positively (yellow area) at the end of each year (dottet line), wheras the coupon would be zero if the Eurostoxx 50 performed negatively (grey area). Eurostox50

100%

time


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Derivatives

The conditional floating payment can be expressed as

E [1 X 1 ⋅ (1 − P(T 0 , T 1 ))] + E [1 X 2 ⋅ P(T 0 , T 1 )(1 − P(T 1 , T 2 ))] + E [1 X 3 ⋅ P (T 0 , T 2 )(1 − P (T 2 , T 3 ))] Where

1 Xi

⎧1 if S (i) > S (0) =⎨ ⎩ 0 otherwise

And it is called the indicator function for Xi. Let’s now analyze each individual expectation in the espression showing the floating payment. In general, they are

E [1 Xi ⋅ P(T i − 2 , T i −1 )(1 − P(T i −1 , T i ))] for i=[2,3] The three terms in the expectation are going to determine if the presence of correlation between Interest Rates and Equity is going to increase the value of the option. Let’s assume that the correlation is positive (the same reasoning can be done of course if a negative correlation is taken into account). If the whole Interest Rates curve shifts by X% (without considering convexity effects involved) we can assume that the discount factor decreases by an amount which is smaller than the amount by which the CMS rate increases. This is shown in table XXX

1 − P(T i −1 , T i ) ↑↑

if Equity ↑

P(T i − 2 , T i −1 ) ↓

if Equity ↑


The amount of arrows indicates the intensity of the changes if Equity (and therefore Interest Rates) moves up. The product of the discount factor and the individual floating amount increases if equity goes up.

P(T i − 2 , T i −1 )(1 − P(T i −1 , T i )) ↑

if Equity ↑


We have now to analyze what happens to the indicator function if Equity moves up. As we have seen previously, when we increase the correlation between Equity and Interest Rates

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Derivatives we are effectively reducing the equity-skew and therefore we are reducing the probability for the equity being above strike at any observation date. This effects tends to reduce the value of the expections and, therefore, of the whole floating amount. It has, nevetherless, a negligeable effect on the final price. The sensitivity of the individual expectations is summirized in table XXX.

E [

o

]↑

if Equity ↑


The overall sensitivity of the sum of the expectations is directly proportional to the correlation between the two asset classes, as shown in Diagram XXX.

Price

ρ


20.3.1.2

Strip of CMS10Y digitals on the Eurostoxx50 (autocallable)

Let’s consider the same product as in the previous section. The product can be also called, this time, if Equity perofrms particularly well and trades, on any annual observation date, above a barrier B (above the spot price). As we have seen before if we increase the correlation we are effectively increasing the value of the Interest Rate linked coupon whereas the change of the indicator function had no significant impact on the price. In this example the floating coupon is still subject to the same changes if the correlation increases, whereas the effect on the indicator function is not negligeable anymore. As we said before, introducing a correlation between IR and Equity has a significant impact on the Equity-skew. Any autocallable product has a given sensitivity to Equity-skew. If the autocallable barrier is above the spot level then the product has a negative sensitvity (if we increase the Equity-skew, then the price decreases).

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Derivatives

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Derivatives

21 DIVA DIVA is a library of C++ implemented code developed by Commerzbank Corporates & Markets in order to price complex exotic products, which uses Excel as a graphical interface (for more information please refer to Annex 1)

21.1 Accessing the data in the trading databases Market data is sourced from the trading databases such as Murex, Diva/Avid, Xenomorph and the MDR. The data is used for live pricing, but this is not a guarantee that it is completely upto-date. The data can be accessed by functions which s tart with the following syntax:

1. 2. 3.

= av__( ) for AVID, = feMarket__( ) for Murex = X__( ) for Xenomorph.

Diagram 1 shows how DIVA is linked to the trading database systems and the functions that are used to access the data.

Diva Excel

= av__( )

Database

AVID

= feMarket__( )

Murex

= X__( )

Xenomorph

MDR


21.2 DIVA Objects There are additional Financial Engineering functions which enable us to create specific objects containing market data and store them in the computer memory. The principle is similar to the C++ object-oriented language (for more information please refer to Annex 1). For example, if we want to create an object containing the volatility of a given underlying, we will proceed in the following way:

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Derivatives

1.

take its value from a database (here Murex) = feMarketVol(Index, Strike, Date)

2.

create an object containing the previous value = feCreateEquityBlackVolatilityMatrixMarket(Index, Value, Date)

3.

repeat this process for all the data in the Market (IR, vols, dividends, …)

4.

create an object combining all the data called a MarketDataCollection = feCreateMarketDataCollection( )

Diagram 2 shows how market data are withdrawn form the databases and how different Objects, associated to each Underlying (Indices or Socks) are created in the computer memory as a MarketDataCollection.

MarketDataCollection

Indices, Stocks

rates

volatilities

dividends

…

Database


21.3 Pricing complex exotic products To price a product in Diva, you need three elements: a model, a product and an implementation. The model contains all the market data information on the underlyings, and also information on how prices evolve with time (for example, stochastic volatility, local volatility etc.). The product contains details of the payoff, for example fixing dates and cashflows. The implementation specifies the numerical method that should be used to evauate the price of the derivative (MonteCarlo, FiniteDifference and Trees). The product, the model and the implementation are built in DIVA and stored in computer memory as an object. Combining these three objects together we can evaluate the price of the product. The main steps to calculate the price for an option are: 1.

create 3 different objects: a.

a Product object with the function

=feCreateSplifeProduct( )

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Derivatives which uses the SPLiFE language to describe the product in details (currency, maturity, underlying, payoff,…) b.

a Model object with the function

=feCreateEquityModel( ) using the object MarketDataCollection as an input for the parameters of the underlyings c.

an Implementation object with the function

=feCreateImplementation( ) which gives the valuation method 2.

use these objects as parameters to get the price with the function

= feValueProduct(Product, Model, MonteCarloImplementation)

Diagram 3 is a graphical representation of the steps 1.a, 1.b, 1.c and 2.

Product = feCreateSplifeProduct( )

Model = feCreateEquityModel( )


Implementation = feCreateImplementation( )

Price = feValueProduct(Product , Model, Implementation)


21.4 Introduction to EasyDIVA EasyDIVA is a VBA based pricing tool which uses DIVA libraries functions to create models and implementation obects. There are two components to Easy Diva – the Model Builder spreadsheet, and the product template spreadsheets. The inputs for the Model Builder spreadsheet are the Indices and/or

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Derivatives Stocks tickers, the type and name of the Model. After the model has been build the user needs to create a SPLiFE product in a separate spreadsheet and use the

= priceproduct( ) function to evaluate the price. Diagram 4 is a schematic representation of EasyDIVA’s structure (for more information please refer to annex B).

Product = feCreateSplifeProduct( )

Index/Stock tickers

EasyDivaModelBuilder Model = feCreateEquityModel( )


MonteCarloImplementation = feCreateImplementationMonteCarlo( )

Price = priceproduct(Product , “No”, Model, MonteCarloImplementation )


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22 Annexe A (Autocallable Products)

22.1 Product Description The autocallable product is a structure which pays the investor a fixed annualized amount if the underlying reaches a given strike level at discrete observation dates. If the condition is satisfied the product redeems at 100%, meaning that the investor gets the notional invested back regardless from any previous performance. If the product is never called the investor is exposed to the equity downside contingent to the past equity performance. If, at maturity, the product is not called and the a barrier has never been touched the client gets his notional back, otherwise the client is long the underlying (meaning that he’s short a put Down and In on the underlying)

Example of Payoff Maturity: 3 years Currency: EUR Underlying: Eurostoxx50 (BBG Code: SX5E Index) Observation dates: yearly Barrier: 65% (of initial spot) Annualized amount: 7% p.a. Strike level: 100% (of initial spot)

At the end of the first year We observe the underlying on a yearly basis. If the Eurostoxx50 is above issue level (above 100%) then the client gets 7% of the notional invested and the prducts redeems at 100%. This means that if the client invested 100EUR, he would get 107EUR at the end of the first year and the contract between the seller and the buyer is now closed.

At the end of the second year If the Eurostoxx50 is below issue level (below 100%), the product continues and we observe the level of the underlying at the end of the second year. If it is above issue level (above 100%), the client gets 14% of the notional invested plus and the product redeems at 100%. This means that if the client invested 100EUR, he would get 114EUR at the end of the second year and the contract between the seller and the buyer is now closed.

At the end of the third year If the Eurostoxx50 is below issue level (below 100%), the product continues and we observe the level of the underlying at the end of the third year. If it is above issue level (above 100%), the client gets 21% of the notional invested plus and the product redeems at 100%. In case the Eurostoxx50 is below the strike level (below 100%) but the barrier ( 65% of initial spot) has never been touched, the product redeems at 100%, otherwise the client is long the underlying at maturity (let’s suppose 76% of initial spot as example). This means that if the client invested 100EUR, he would get 121EUR at the end of the third year if the condition is satisfied, 100EUR if the conditin is not satisfied and the barrier never touched, 76EUR otherwise.

Advantages and disadvantages Thus, this structure has some downside risk with potential loss of a part of the investment, but it is no more risky than a direct investment in the underlying. The downside has even a protection at maturity up to t he trigger

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Derivatives Besides, if this structure is autocalled, the bonus earned is fixed at the agreement of the deal, so the return could be either more important than a direct investment in the underlying, and the client receives an high bonus coupon, or the underlying could increase highly, and then the return may be less important than a direct investment (which is more risky).

Payoff Examples For all the examples, we take the following structures: Maturity: 3 Years Initial Stock Price: €100 Strike Price: At the Money Autocall Bonus: 8% after the first year 16% after the second year 24% after the third year Barrier at maturity: 60% of the Strike First example: at the end of the first year, the underlying stock closed equal to €105 (so to 105% of the initial strike). Then, the product is autocalled and the client will receive €108 which is the notional + the autocall Bonus. So, we see here that the return of this product is higher than a direct investment in the underlying.


Second example: at the end of the first year, the underlying stock closed equal to €110. So, the product is autocalled, and the client will receive €108 (i.e. the notional + the autocall bonus). This time, the return is less than for a direct investment in the underlying.


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Derivatives

Third example: at the end of the first year, the underlying stock closed equal to €95. The product is not autocalled the first year. Then, the deal continues until the next observation at the end of the second year. Different cases may happen:

• • •

Closing price at the end of the 2nd year: €120 => Client receives €116 Closing price at the end of the 2nd year: €105 => Client receives €116 Closing price at the end of the 2nd year: €95 => Product is not called after the 2nd year


Fourth example: the product is called neither in the first year, nor in the second year. The payoff of this derivative will now depend on the closing price at maturity.

• • •

Closing date at maturity: €105 => Client receives €124 (notional + call bonus at maturity) Closing date at maturity: €80 => Client is protected at maturity and he will receive €100 which is his initial investment Closing date at maturity: €55 => The barrier has been passed at maturity; the client is no more protected, so he will receive 1 share with a €55 notional.


We can see in those examples that the client is protected on the downside, and that his investment is not more risky than a direct investment in the underlying.

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Derivatives We must point out here that we have consider for our example an european barrier, but the 100% redemption at maturity if the stock closes below 100% could also be conditioned by an american barrier: If at maturity, so without early redemption, the underlying stock has never quoted below 60% of the strike, the client receives his 100% investment back. Otherwise, the clients receives shares.

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Derivatives

22.2 Product Analysis Marketing points A salesperson can underline the following points to sell this product: On the upside, the autocall gives a high bonus coupon, sometimes with a return greater than in a direct investment. On the downside, the initial investment is protected as long as a barrier is not reached. This derivative is autocallable early. If the product ends before the final maturity, the client received back their notional early.

Discontinuities In this product, at each yearly observation there is a discontinuity that we have to estimate in order to properly shift the autocall triggers and barrier. To do this, we can to use the formula:

Discontinuity (i ) = χ (i) C i =true

+ θ (i) C =true − χ (i) C = false − θ (i) C = false i

i

i

Where:

= is the intrinsic value of the product if the condition C i is true θ (i ) C =true = is the time value of the product if the condition C i is true χ (i) C = false = is the intrinsic value of the product if the condition C i is false θ (i ) C = false = is the time value of the product if the condition C i is false

χ (i) C i =true i

i

i

On the first year, the condition Ci is whether the closing price of the underlying is below (false) or above (true) the strike price. If we only consider the payoff we expect to receive the day after the first observation, we get the following graph.

108%

100%

100%


On the right side, the product is in the money. So it will pay 108 after this first observation and then the deal is over whereas, on the left side, the product is not called and we have to wait the second observation.

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Derivatives So, on the right side, the condition Ci is true, the intrinsic value is 108 and the time value is 0 because the deal will be over after this payment. On the other hand, the product on the left side possesses both intrinsic and time value. In our example, if the closing price is above 60, the intrinsic value is the value of receiving the protection (€100) within two year s, whereas the time value takes into account the probabilities of all possible outcomes of the product life – e.g. being autocalled after the second year, or being in the money at maturity, etc. The same reasoning can be applied when the closing price is below the 60% protection, with another intrinsic and temporal value. Thus, if the closing level of the underlying is below 100%, (so the condition Ci is false), we

χ (i)

+ θ (i )

C i = false C i = false have a total worth equal to . Usually we consider that this value is close to 100%, because the probability to receive at least this value whenever the product terminates is quite high.

This calculation can be computed every year, and the following graph represents the discontinuity each year:

124% 116% 108% 100%

100%


So, each year, this effectively results in an ATM digital, and the discontinuity will increase with the ascending coupon (discontinuity = 8% the first year, 16% the second year and 24% the third year) As a general rule we need to shift the barrier according to the rule: 1% shift per 1 million discontinuity in terms of product notional. So, the shift in the 100% trigger will change every year, and will be ascending like the ascending coupon. However, the value on the left side is not always 100% and depends on the ascending coupon. If this coupon is quite high, this value increases and it can happen that it is more convenient to not be autocalled the first years in order to receive the final high coupon at maturity. This case means that the lefthand value is worth more than the righthand value due to the high coupon we could receive at maturity. In that case, the trigger should be shifted towards the right in order to make the product more expensive.

Discontinuity at maturity

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Derivatives Besides the European discontinuity at 100%, we are also short a put down and in at maturity. This put could be valued according to European observation at maturity, like in our example with the 60% trigger. But a put D&I with a continuous (American) barrier is more frequent. The following graph presents this discontinuity:

100% 60%

60%


If the observation at maturity is European, then the discontinuity at the 60% european barrier is worth 40%, whereas this discontinuity is smaller if the observation is continuous. Sensitivities: Let’s try to understand the sensitivities on the most common autocall we have to price: an autocall on the worst performer of a basket with an American barrier at maturity for the 100% redemption. This product is usually: Short Vega: the higher the volatilities, the higher the chance to touch the barrier, and to make the put kick in, so the lower the price. Long correlation: The lower the correlation, the higher the probability for one underlying to touch the down barrier, so the lower the price. Short skew: To understand that, let’s consider the following graph w hich is the payoff at maturity:

124% 100% 60%

60%


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Easy Structuring

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