Pricing Basket Options Opt ions With With Skew Skew Dong Qu Quantitative Products Group—Derivatives & Structured Products—Abbey
1.
Introduction
Equity basket options have long been important tools in structuring financial products. They are widely used in portfolio management and as retail products as they are typically more cost-effective than multiple single underlying options. The components in a basket can be single stocks (names), equity indices or funds. Historically, managing basket option books has proved to be problematic and rather difficult for many banks. Quite often, when the markets move, the basket options and their hedges, which are supposed to offset th e basket positions, do not move in the directions as they are supposed to. In these circumstances the basket pricing models used are partially responsible as they fail to capture certain type of fundamental basket risks. Among them, the correlation skew and skew and its structural change triggered by sudden market move are the key contributors. In the absence of a volatility smile/skew, a European option on a basket can be priced using various techniques (ref. 1–4), including moment matching or geometric conditioning. In the presence of a volatility smile/skew, pricing basket options becomes much more complicated. It becomes important to understand how the basket skew behaves and how the skewed correlation structure is related to it. skew has long been a topic of Interestingly,, although the correlation skew has Interestingly study in the context of equity derivatives basket, and the copula technique was one of the proposed methods (ref. 5) to deal with it, it is in the credit derivatives business, the copula technique becomes an industry standard to deal with basket of names. In this paper, we will still focus on equity derivatives basket. The key risks associated with the basket options will be investigated. We will review a technique to extract hi storic basket
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volatility surface and analyse a few examples. We We will develop further t he copula implied basket volatility construction technique (ref. 5) and examine the relationship between the basket skew and correlation skew. The effect of the correlation skew and its structural change on the basket skew will also be examined.
2. Basket Risks A typical basket consists of several underlyings with associated weightings. The basket spot is: n
S B =
w i · Si
i= 1
Where w i is the weightings and Si the individual underlyings. The values of European call and put options on the basket are calculated from: EQ Max (S B − K , 0) ;
Call
EQ Max ( K − S B , 0) ;
Put
A basket option book will generally consist of numerous basket calls and puts, as defined above. In addition to the normal first order risks (e.g. delta, gamma, vega) associated with individual basket underlying, there are several other key basket specific risk exposures. A summary of the basket specific risks is given in the table below. It is evident that a basket (correlation) book is a complex entity involving significant second order effects that needs to be understood and modelled. In the following, we will analyse the basket volatility skew risks and the associated correlation issues. Wilmott
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3. Historic Basket Volatility Surfaces It is possible to construct a historic basket volatility surface from the historic time series of the individual basket components. One such
Basket Specific Risks
Comments
Cross Gamma Risks
The cross Gamma is defined as
∂ 2P
Basket Volatility Skew Risks
FX Correlation Risks
Contractual Risks
Jump Risks
1.0%
0.5%
= j, ,i
∂ Si ∂ S j where P is the basket option price, Si and S j
are the underlying spots. The cross Gamma specifies the delta change of i-th underlying caused by the spot change in the j-th underlying. Basket Correlation Risks
1.5%
When large moves occur in the market, the basket correlation tends to spike and this will impact the pricing as well as risk parameters. This is also referred to as the skewed correlation structure risks. The basket volatility surface is largely driven by the volatility characteristics of the individual components. It is also heavily affected by the skewed correlation structure. In a Quanto or Compo basket, the FX correlation with each individual basket component will impact the price. If merge and acquisition activity happens to the basket member, the basket composition will change and so will its volatility structure. In a large single stock basket, it is more likely that one or some of the basket underlying default. The corresponding underlying price may jump to zero.
approach is to minimise the relative entropy (ref. 6) in which a risk neutral probability density function (PDF) for the basket can be built. Subject to constraints of the forward and ATM implied volatility, the risk neutral PDF is used to construct a historic volatility surface for the basket. Specifically, given the historic time series of the basket spot Si the following steps are taken to build a historic basket volatility surface:
Step 1: Recover the historic PDF for the basket at time interval T by calculating the log-return at appropriate time interval:
0.0% 70%
90%
110%
130%
150%
Figure 1: Historic PDF (Mean Drift = 9%). where N defines the time interval. The bucketed log-returns RT create the historic PDF denoted as P . An example of a historic PDF is shown in Fig. 1;
Step 2: Recover the risk neutral PDF (Q ) from the historic PDF (P ). This is a key step and a numerical optimisation technique is used for this. The penalty function for optimisation is based on a relative entropy, which is a mathematical measure of the uncertainty of a distribution. The relative entropy of Q to P is defined as follows: S( P , Q ) = EQ (ln Q − ln P )
By minimising relative entropy S( P , Q ), one can solve for Q , subject to the following constraints,
Q (ST ) · ST · dST = Forward Q (ST ) · dST = 1
Additional optimisation constraint can also be applied, such as imposing the ATM volatility. The optimisation solution for the problem posed above is: Q (ST ) =
P (ST )
P (S) · exp(−λS) · dS
exp (−λST )
Where λ is a constant that can be obtained by iterating constraints during the numerical optimisation. Note that given P (ST ) is always positive, Q (ST ) is guaranteed to be positive. In Fig. 2, an example of risk neutral PDF is shown and it is derived from the historic PDF shown in Fig. 1. The mean drift is the expected drift of the forward given the historic yield curve and dividends.
Step 3: Use the risk neutral PDF Q to obtain option prices with different strikes K i : ∞
Call ( K i , T ) = DF ·
(ST − K i ) · Q (ST ) · dS T
K
RT = ln
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St + N St
K
Put ( K i , T ) = DF ·
( K i − ST ) · Q (ST ) · dS T
^
0
59
1.5% 50% 1.0% 40%
30%
0.5%
20% 0.0% 70%
90%
110%
130%
150%
10% 7.0
Figure 2: Risk Neutral PDF (Mean Drift = 6.75%).
0%
Where DF is the discount factor. The option prices can then be used to recover the implied volatility points at K i . By repeating the above steps at different time interval T , one can build an historic volatility surface for the basket. In the following, we examine a few examples of the historic volatility surfaces. Fig. 3 shows a historic volatility surface for FTSE-100. The short end skew is quite strong as expected. The historic volatility surface shown in Fig. 4 is that of the FTSE-100 banking sector basket. Clearly there is a smile at the short end and skew at the longer end. In Fig. 5, the historic volatility surface for an equally weighted index basket of FTSE-100, S&P and EUROSTOCK 50 is shown. Again the smile/skew is apparent in the figure. While the historic basket volatility surfaces provide valuable smile/skew information of the basket, it is relatively static. When the market moves, the historic information changes little given that the recent contribution constitutes a relatively small proportion of the time series. For day-to-day trading and hedging purposes, one requires different approaches to cater for rapidly changing market conditions and the basket volatility surfaces should incorporate the latest market information.
% 0 5
% 0 8
2.0
% 0 1 1
% 0 4 1
% 0 7 1
0.08
% 0 0 2
Figure 4: FT-Banks Historic Vol Surface.
25%
20%
15%
10%
5% 3.0 0.75
0%
50%
% 0 7
40%
30%
% 0 9
0.08
% 0 1 1
% 0 3 1
Figure 5: Index Basket Historic Vol Surface. 20%
4. Implied Basket Volatility Surfaces
10% 7.0 0%
% 0 5
2.0
% 0 8
% 0 1 1
% 0 4 1
% 0 7 1
Figure 3: FTSE Historic Vol Surface.
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0.08
% 0 0 2
It is conceivable that one can construct an implied basket volatility surface from the implied volatility surfaces of individual basket components. One potential approach is to use the local volatility surfaces, in which the stripped local volatility surfaces of the basket components can be integrated with appropriate correlation structures to build an implied basket volatility surface. The local volatility approach is a small step operation and can be computationally expensive. A more efficient Wilmott
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technique (ref. 5) of constructing an implied basket volatility surface is to use a copula alike technique, in large steps dealing with the terminal distributions.
4.1.
1.0
The Copula Technique to Construct Basket Volatility Surfaces
0.5
In order to build an implied basket volatility surface, one needs to obtain the joint distribution of the basket components. In general, however, knowing the marginal distributions of the individual component and the correlation structure does not guarantee an unique joint distribution. The copula technique allows us to specify a joint distribution function for known marginal distributions with a dependence (correlation) structure. Specifically, for given univariate marginal distributions F i (Si ), the Copula links the univariate marginal distributions to their multi variate joint distribution F (S1 , S2 , · · ·, Sn ) : F (S1 , S2 , · · ·, Sn ) = C ( F 1 (S1 ), F 2 (S2 ), · · ·, F n (Sn ))
Given the implied volatility surfaces of each basket component, the Cumulative Distribution Functions (CDF) at a chosen time slice for the individual component can be built and these CDFs are the univariate marginal distributions. If we impose correlation structures onto these CDFs and link them up using a Gaussian copula, the joint (basket) distribution can be created as follows: F (S1 , S2 , · · ·, Sn ) = C (CDF 1 (S1 ), CDF 2 (S2 ), · · ·, CDF n (Sn ))
The joint distribution given above allows us to simulate the basket spot path by sampling the correlated CDFs. Assuming there are n individual components in the basket, at a given time slice T , the following steps can be taken to construct an implied basket volatility surface: • Build CDFs for all basket components at a chosen time slice T . This
can be achieved by calculating digital calls or puts at the appropriate strikes since a undiscounted digital option specifies the terminal probability for the underlying spot to be above (call) or under (put) the strike. The CDFs are obviously distributed between 0 and 1 and a typical CDF is shown in Fig. 6. The CDFs can be used later on to sample the individual component spot path given a uniformly distributed random number; • Generate n independent random Gaussian numbers (g i ) and impose a
correlation structure among them. This is a key step in which one can either use a simple correlation number or skewed correlation structure. If Cholesky decomposition technique is used, given the decomposed correlation matrix C ij , the correlated Gaussian number is given by Gi = j C ij · g j ;
• Convert the correlated Gussian numbers back to uniformly distrib-
uted numbers by reversing the Wiener process U i = W − 1 (Gi ) ; • Use the correlated uniform numbers (U i )to sample individual CDFs
to obtain underlying spots (Si ), and in turn calculate the basket spot n S B = i= 1 w i · Si ;
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0.0 0.0
0.5
1.0
1.5
2.0
2.5
3.0
Figure 6: Sampling CDF.
• Calculate either a call or put on the basket for a given strike K :
Call = E Max (S B − K , 0) Put = E Max ( K − S B , 0) • Calculate the basket implied volatility from either the call or put by reversing the Black-Scholes formula (σ B ( K , T ) = BS−1 ( K , T )) .
The above steps can be repeated for different strikes (K ) and maturities (T ) to build an entire basket implied volatility surface. It is important to note that in order to obtain a high quality basket implied volatility surface using this technique, some numerical smoothing techniques are needed in the process. The constructed basket implied volatility surface could be used in a variety of ways. Firstly, it allows one to quantify the basket skew and hence analyse the skew exposures. Secondly, the basket can be treated as a new single underlying and other basket payoffs may be priced and risk managed efficiently in a consistent manner using the same basket implied volatility surface.
4.2.
Basket Skew Under A Constant Correlation
In the following, we examine a basket of three underlyings. Their volatility surfaces are shown in Fig. 7, Fig. 8 and Fig. 9 respectively. The volatility surface for the underlying 1 has a relatively low ATM volatility but strong skew. The volatility surface for the underlying 2 has a medium ATM volatility with a medium skew. The volatility surface for the underlying 3 has a high ATM volatility with medium skew. The underlying volatility surfaces are chosen to be representative to cover a wide range of possible market implied volatility surfaces. Fig. 10 shows the constructed implied basket volatility surface using a constant correlation coefficient of 40%. As can be seen in figure, the basket skew is quite pronounced. The implied volatility points in the near right corner of the basket volatility surface for the short maturities and high strikes are not fully recovered. In practice, this is not a problem given that the missing points are in the very low probability region and extrapolation into this region is relatively straightforward.
^
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50%
75%
40% 50% 30%
20% 25% 10% 7.0
7.0 0%
y r i p x E
2.0
% % 0 0 % % 5 7 0 0 9 1 1
% 0 3 1
Strike
% 0 5 1
0.1
% 0 7 1
0%
% 0 9 1
y r i p x E
2.0
% % 0 0 % % 5 7 0 0 % 9 1 0 1 3 1 Strike
% 0 5 1
0.1
% 0 7 1
% 0 9 1
Figure 9: Implied Vol Surface (Underlying 3).
Figure 7: Implied Vol Surface (Underlying 1).
60%
40%
30% 40% 20%
20%
10% 7.0 0%
0%
y r i p x E
2.0
% % 0 0 % % 5 7 0 0 % 9 1 0 1 3 1 Strike
% 0 5 1
% 0 7 1
0.1
% 0 9 1
Figure 8: Implied Vol Surface (Underlying 2).
7.0
% 0 % 5 0 8
i r y p E x
2.0
% 0 1 1 Strike
% 0 4 1
% 0 7 1
0.1
% 0 0 2
Figure 10: Basket Implied Vol Surface (Cor = 40%). The correlation effect on the basket volatility is shown in Fig.11. The implied basket volatility curves (volatility versus strike) at the time slice year 1 for different correlation coefficients (−20%, 0%, 50% and 90%) are plotted in the figure. It is apparent that the higher the correlation, the higher the basket volatility level. This is a well understood fact as the higher correlations indicate that the basket components move more coherently, resulting in the higher basket volatility. Fig. 11 also shows that the slope of the volatility curve changes with the correlation. This indicates that the skew is also a function of the correlation. Fig. 12 plots the basket skew versus correlation at the time slice year 1. The basket skew is defined as the volatility slope at the strike of 100%. As can be seen in Fig. 12, the basket skew increases (slo pe becomes
62
more negative) as the correlation increases. This is because the larger correlation makes the constituents of the basket move together more often and increases the probability of the basket spot reaching large extreme values. The resulting probability density function can only be matched by that from an implied volatility curve with a larger skew. When correlation approaches 100%, the basket skew reaches maximum of −9.8% . This is substantially less skewed than the most skewed underlying in the basket which has a skew of −23.4% . Actually the basket skew is even less than the least skewed underlying which has a skew of −11.6% . Let us now examine the full skew term structures. Fig. 13 plots the skew term structures of two basket underlyings and those of basket for Wilmott
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TECHNICAL ARTICLE 2
55%
Cor = −20% Cor = 0% Cor = 50% Cor = 90%
45%
−40%
Underlying 1 Underlying 2 Basket (Cor = 0%)
−30%
Basket (Cor = 80%)
35%
−20%
25%
−10%
0%
15% 60%
80%
100% Strike
120%
140%
0
1
2
3
4
5
6
7
Figure 13: Skew Term Structure.
Figure 11: Basket Implied Volatility (1Y).
−25%
−20% Basket
−15%
Und 1 Und 2
−10%
Und 3
−5%
0%
−20%
0%
20%
40%
60%
80%
100%
Correlation
Figure 12: Skew (1Y) vs Correlation.
different correlation coefficients (0% and 80%). Overall the basket skew tends to be reduced compared with those of the basket underlyings. This can be qualitatively understood from the central limit theorem. Given the skew indicates a non-log-normality, and when there are more basket underlyings contributing towards the joint distribution, the log-normality tends to be preserved and t his reduces the skew.
4.3. Basket Skew Under A Skewed Correlation Structure In contrast to a constant correlation coefficient, a skewed correlation structure illustrates a more complex but realistic relationship among the basket components. Fig. 14 contains 10 years worth of historic spot time series for FTSE-100, ESTX (EUROSTOCK 50) and S&P. As can be seen in the Wilmott
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figure, typically, when markets fall dramatically, they tend to fall together, indicating a much more correlated market move. A skewed correlation structure incorporates the fact that the correlation becomes larger when basket component spots are falling. The skewed correlation structures can be built into the basket implied volatility surface during the construction process outlined in section 4.1. One such approach is to impose a linear correlation structure depending on where the average basket underlying spot is in a simulated path. Fig. 15 plots three basket volatility curves versus strike for a basket of three underlyings. All curves are subtracted by the ATM volatility to highlight the basket skew. The first curve is generated using a flat correlation coefficient of 44%, which is chosen such that the basket ATM volatility equals to that in the second and third curve illustrated below. The second curve is constructed with a skewed linear correlation structure in the local volatility space in small steps. For each small step, the initial correlation of 40% is scaled up linearly to a maximum of 95% when the average of the basket underlying spots falls below the prevailing average at the start of that small time step. It is apparent from the f igure that the skewed correlation structure increases the slope of the basket volatility curve and the basket skew. The third curve is created also with a skewed linear correlation structure but using one big step, in which the linear correlation coefficient (between 40% and 95%) is dependent on half of the average of the terminal basket spot drop. The large step approach is effectively an approximation to t he small step local volatility approach. This can be clearly seen in the figure as the third curve is very close to the second curve. In practice, different practitioners may prefer to impose different correlation structures depending on different market views and hedging needs. The copula large step approach is flexible enough to allow various correlation structures to be incorporated into the basket. The correlation structure impact on the basket skew and basket pricing and risk exposures could be significant.
^
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Both volatility and skew are positive functions of the correlation. Skewed correlation structures can be imposed onto the basket during the construction process. The correlation structure impacts the basket skew and basket option prices. A realistic and consistent basket skew framework built by incorporating a skewed correlation structure should be able to capture some risk exposures of dramatic market falls. It is possible to extend the basket volatility surface construction technique to other multi-asset basket structures. For example, in the following multi-asset option structures:
7500 FTSE ESTX S&P 5000
2500
0 0
530
1060
1590
2120
Figure 14: Historic Spot (FTSE, ESTX, S&P). 2.0% Flat Cor (44%) Skewed Cor (Local Vol) Skewed Cor (Large Step)
1.0%
0.0% 80%
90%
100%
110%
120%
−1.0%
−2.0%
Call On Worst Of :
max (min(S1 , S2 , · · · , Sn ) − K , 0)
Put On Worst Of :
max ( K − min(S1 , S2 , · · · , Sn ), 0)
Call On Best Of :
max (max (S1 , S2 , · · · , , Sn ) − K , 0)
Put On Best Of :
max ( K − max (S1 , S2 , · · · , Sn ), 0)
2650
the worst-of Smin = min(S1 , S2 , · · · , Sn ) and the best of Smax = max (S1 , S2, · · · , Sn ) can be treated as the new single underlying similar to the basket underlying S B . The new single underlying’s terminal spots can be simulated using the copula technique which links the individual CDFs with an appropriate dependence (correlation) structure. The simulated spots can then be used to calculate the option prices at different strikes. The implied volatility surfaces for the new single underlying can be backed out from option prices, and the skew risk measurements of these multi-asset option structures can be conducted similarly. Finally, although this paper deals with the copula and correlation skew in the context of equity basket option, the techniques and the thought process can potentially benefit credit derivatives professionals too. It will be very interesting to see cross fertilization between equity and credit derivatives professionals.
Strike
Figure 15: Basket Implied Volatility (1Y)
5. Conclusions
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A basket volatility surface construction technique is developed using the copula methodology. It captures the information in the implied volatility surfaces of the individual basket components. The copula basket volatility surface construction technique is computational efficient as it is in large steps at finite time slices. Once the basket volatility surface is built, the basket can be treated as a new single underlying and the surfaces could be interpolated or extrapolated for pricing and risk managing basket products in a consi stent manner. The risk parameters relevant to individual volatility skew as well as the correlation effects can be generated within the same framework.
REFERENCES D. Gentle, “Basket Weaving”, Risk, Vol. 6, No. 6, p. 51 (1993) M. A. Milevsky and S. E. Posner, “A Closed-Form Approximation for Valuing Basket Options”, The Journal of Derivatives, Summer 1998, p. 54 I C. B. Huynh, “Back to Baskets”, Risk, Vol. 7, No. 5, p. 59 (1994) I M. Curran, “Valuing Asian and Portfolio Options by Conditioning on the Geometric Mean Price”, Management Science (1994), No. 12, p. 1705–1711. I D. Qu, “Basket Implied Volatility Surface”, Derivatives Week, 4 June 2001. I J. Zou and E. Derman, “Strike-Adjusted Spread: A New Metric For Estimating The Value Of Equity Options”, Goldman Sachs Quantitative Strategies Research Notes, July 1999. I I
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