COURSE NOTES Financial Mathematics MTH3251 Modelling in Finance and Insurance ETC 3510. Lecturers: Andrea Collevecchio and Fima Klebaner School of Mathematical Sciences Monash University Semester 1, 2016
Contents 1 Introduction. 1.1 Example of mo dels . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Application in Finance . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Application in Insurance . . . . . . . . . . . . . . . . . . . . . . . .
1 1 2 2
2 Review of probability 2.1 Distri tribution tion of Rando dom m Variab riablles. es. Gen Gener eraal. . 2.2 Exp ected value or mean . . . . . . . . . . . . 2.3 Variance Var, and SD . . . . . . . . . . . . . . 2.4 General Properties of Expec pectation . . . . . . 2.5 2.5 Expo Expone nen ntial tial mome momen nts of Norma ormall dist distri ribu buti tion on . 2.6 LogNormal distribution . . . . . . . . . . . . .
4 4 5 6 7 8 9
3 Indep endence. 3.1 Joint and marginal densities . . . . . . . . . . 3.2 Multivariat riatee Normal rmal distribu ributtions . . . . . . . 3.3 3.3 A linea inearr com combina binati tion on of a multi ultiv varia ariate te no norm rmal al 3.4 Independence . . . . . . . . . . . . . . . . . . 3.5 Covariance . . . . . . . . . . . . . . . . . . . . 3.6 Prope ropert rtiies of Covariance nce and Variance nce . . . . 3.7 Covariance function . . . . . . . . . . . . . . . 4 Conditional Expectation 4.1 Conditi dition onaal Distrib ribution tion an and d its mean ean . . . . 4.2 Prope ropert rtiies of Conditi dition onaal Expe Expecctati tation on . . . . 4.3 Expec pectation as bes best predictor . . . . . . . . . 4.4 Conditi dition onaal Expec pectati tation on as Best est Predi edictor tor . . 4.5 4.5 Co Cond ndiition tional al expe expect ctat atiion with with man any y pred predic icto tors rs 5 Random Walk and Martingales 5.1 Simple Random Walk . . . . . . . . . . . . 5.2 Martingales . . . . . . . . . . . . . . . . . 5.3 Martingales in Random Walks . . . . . . . 5.4 Exponent Exponential ial martin martingale gale in in Simple Simple Random Random
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6 Optional Stopping Theorem and Applications 6.1 Stopping Times . . . . . . . . . . . . . . . . . . 6.2 Optional Stopping Theorem . . . . . . . . . . . 6.3 6.3 Hitti itting ng prob probab abiiliti lities es in a simp simplle Ran Rando dom m Walk alk . 6.4 Expec pected duration of a game . . . . . . . . . . . 6.5 Discrete time Risk Mod odeel . . . . . . . . . . . . . 6.6 Ruin Probability . . . . . . . . . . . . . . . . .
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Contents 1 Introduction. 1.1 Example of mo dels . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Application in Finance . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Application in Insurance . . . . . . . . . . . . . . . . . . . . . . . .
1 1 2 2
2 Review of probability 2.1 Distri tribution tion of Rando dom m Variab riablles. es. Gen Gener eraal. . 2.2 Exp ected value or mean . . . . . . . . . . . . 2.3 Variance Var, and SD . . . . . . . . . . . . . . 2.4 General Properties of Expec pectation . . . . . . 2.5 2.5 Expo Expone nen ntial tial mome momen nts of Norma ormall dist distri ribu buti tion on . 2.6 LogNormal distribution . . . . . . . . . . . . .
4 4 5 6 7 8 9
3 Indep endence. 3.1 Joint and marginal densities . . . . . . . . . . 3.2 Multivariat riatee Normal rmal distribu ributtions . . . . . . . 3.3 3.3 A linea inearr com combina binati tion on of a multi ultiv varia ariate te no norm rmal al 3.4 Independence . . . . . . . . . . . . . . . . . . 3.5 Covariance . . . . . . . . . . . . . . . . . . . . 3.6 Prope ropert rtiies of Covariance nce and Variance nce . . . . 3.7 Covariance function . . . . . . . . . . . . . . . 4 Conditional Expectation 4.1 Conditi dition onaal Distrib ribution tion an and d its mean ean . . . . 4.2 Prope ropert rtiies of Conditi dition onaal Expe Expecctati tation on . . . . 4.3 Expec pectation as bes best predictor . . . . . . . . . 4.4 Conditi dition onaal Expec pectati tation on as Best est Predi edictor tor . . 4.5 4.5 Co Cond ndiition tional al expe expect ctat atiion with with man any y pred predic icto tors rs 5 Random Walk and Martingales 5.1 Simple Random Walk . . . . . . . . . . . . 5.2 Martingales . . . . . . . . . . . . . . . . . 5.3 Martingales in Random Walks . . . . . . . 5.4 Exponent Exponential ial martin martingale gale in in Simple Simple Random Random
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6 Optional Stopping Theorem and Applications 6.1 Stopping Times . . . . . . . . . . . . . . . . . . 6.2 Optional Stopping Theorem . . . . . . . . . . . 6.3 6.3 Hitti itting ng prob probab abiiliti lities es in a simp simplle Ran Rando dom m Walk alk . 6.4 Expec pected duration of a game . . . . . . . . . . . 6.5 Discrete time Risk Mod odeel . . . . . . . . . . . . . 6.6 Ruin Probability . . . . . . . . . . . . . . . . .
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7 Applications in Insurance 7.1 7.1 Th Thee bo boun und d for for the the rui ruin prob probab abil ilit ity y. Cons Consta tan nt R. . . 7.2 R in the Normal mo del . . . . . . . . . . . . . . . . 7.3 Simulations . . . . . . . . . . . . . . . . . . . . . . 7.4 The Acceptance- Rejection method . . . . . . . . .
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8 Brownian Motion 8.1 Definition of Brownian Motion . . . . . . . . . . . . . . . . . . . . . 8.2 Independence of Increments . . . . . . . . . . . . . . . . . . . . . .
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9 Brownian Motion is a Gaussian Pro cess 9.1 9.1 Proo Prooff of Gaus Gaussi sian an prop proper ertty of Brown rownia ian n Moti Motion on 9.2 Proce rocessses obtai btaine ned d from Brownian nian moti otion . . . 9.3 9.3 Co Cond ndiition tional al expe expect ctat atiion with with man any y pred predic icto tors rs . 9.4 Martingales of Brownian Motion . . . . . . . . .
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10 Sto chastic Calculus 10. 10.1 Non on--diffe differe ren ntiab tiabil ilit ity y of Bro Brown wniian motio otion n . . . . . 10.2 Itˆo Integral. . . . . . . . . . . . . . . . . . . . . . 10.3 Distribution Distribution of Itˆ Itˆ o integ integral ral of simpl simplee determ determin inis isti ticc 10.4 Simple Simple stochastic processes processes and their Itˆ o integral 10.5 Itˆo integral for general proce ocesses . . . . . . . . . . 10.6 10.6 Propertie Propertiess of Itˆ o Integral . . . . . . . . . . . . . . 10.7 Rules of Stoch ochastic Calculus . . . . . . . . . . . . 10.8 10.8 Cha Chain in Rule: Ito’s Ito’s formula formula for f ( f (Bt ). . . . . . . . . 10.9 Martingale Martingale property of Itˆ Itoˆ integral . . . . . . . .
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46 . . . . . . . . . . 46 . . . . . . . . . . 46 proces processes ses . . . . 47 . . . . . . . . . . 48 . . . . . . . . . . 49 . . . . . . . . . . 49 . . . . . . . . . . 50 . . . . . . . . . . 51 . . . . . . . . . . 52
11 Sto chastic Differential Equations 54 11. 11.1 Ordi Ordina nary ry Differ ifferen enti tial al equa equati tion on for for gro growth wth . . . . . . . . . . . . . . . 54 11.2 11.2 Blac Blackk-Sc Schol holes es stocha stochasti sticc diff differe erent ntia iall equ equati ation on for for stoc stocks ks . . . . . . . 54 11.3 11.3 Solv Solvin ingg SDE SDEss by by Ito Ito’s ’s form formul ula. a. Black Black-S -Sch chol oles es equat equatio ion. n. . . . . . . . 55 11.4 Itˆo’s o ’s form formul ulaa for for func functi tion onss of two varia ariabl bles es . . . . . . . . . . . . . . 56 11.5 11.5 Stoc Stocha hast stic ic Produ Product ct Rule Rule or Inte Integr grat atio ion n by part partss . . . . . . . . . . . 57 11.6 Ornstein-Uhlenbec beck proce ocess. . . . . . . . . . . . . . . . . . . . . . . 57 11.7 1.7 Vasicek’ ek’s mod odeel for intere teresst rates ates . . . . . . . . . . . . . . . . . . . 58 11.8 Solution to the Vasicek’s SDE . . . . . . . . . . . . . . . . . . . . . 59 11.9 11.9 Stochast Stochastic ic calculu calculuss for processes processes driven driven by two or more Brownian Brownian motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 11.10S 0Su ummary of stoch ochastic calculus . . . . . . . . . . . . . . . . . . . . 60 12 Options 12.1 Financial Concepts . . . . . 12.2 Functions x unctions x + and x− . . . . . 12.3 The problem of Option price 12.4 One-step Binomial Mod odeel . 12.5 2.5 One-peri period od Bino nom mial Pric ricing
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12.6 Replicating Portfolio . . . . . . . . . . . . 12.7 Option Price as expec pected payoff . . . . . . 12.8 Martingale Martingale property of the stock under p . 12.9 2.9 Binomi omial Mod odel el for Opti ption prici ricing ng.. . . . . 12.10Bl 0Black-Scholes formula . . . . . . . . . . . .
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13 Options pricing in the Black-Scholes Mo del 13.1 Self-financing Portfolios . . . . . . . . . . . . . . 13.2 13.2 Repl Replic icat atio ion n of Opti Option on by self self-fi -fina nanc ncin ingg portf portfol olio io 13.3 3.3 Repli plicati ation in Black-Scholes oles mod odeel . . . . . . . 13.4 13.4 Black Black-S -Scchole holess Parti artial al Diffe Differe ren ntial tial Equa Equati tion on . . . 13. 13.5 Opti Option on Pric Pricee as as dis disco coun unte ted d exp expec ecte ted d pa payoff . . . 13.6 13.6 Stock Stock price price S T T under EMM Q . . . . . . . . . .
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14 Fundamental Theorems of Asset Pricing 14.1 I nt ntroduction . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Ar Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 14.3 Fun unda dame men ntal tal the theor orem emss of Math Mathem emat atic ical al Fina Financ ncee . . . . . 14.4 14.4 Co Comp mple leten teness ess of Blac Blackk-Sc Schol holes es and Binom Binomia iall mode models ls . . . 14.5 4.5 A gene eneral ral form ormula for for opti ption price rice . . . . . . . . . . . . . 14.6 Su S ummary . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Mo dels for Interest Rates 15.1 Term Structure of Interest Rates . . . . . 15.2 Bonds and the Yield Curve . . . . . . . . . 15.3 General bo bon nd pricing formula . . . . . . . 15.4 Mo dels for the sp ot rate . . . . . . . . . . 15.5 Fo Forward rates . . . . . . . . . . . . . . . . 15.6 Bonds in Vasicek’s mod odeel . . . . . . . . . . 15. 15.7 Bon onds ds in Co Coxx-In Inge gers rsol olll-Ro Ross ss (CIR (CIR)) mode modell . 15.8 Options on b onds . . . . . . . . . . . . . . 15.9 Caplet as a Put Option on Bond . . . . .
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Intro
1
Introduction.
In order to study Finance and Insurance, we need mathematical tools. We start with a review of probability theory: random variables, their expected values, variance, independence etc. We introduce Random Walks, Martingales and Brownian motion, stochastic differential equations. These are sophisticated mathematical tools. We compromise: we are going to learn how to use these tools, which are useful in other areas, such as Engineering and Biology.
1.1
Example of models
Let xt the amount of money in a savings account. Suppose the interest rate is r, and x0 > 0. The evolution of x t is described by the differential equation dxt = rxt . dt We solve this equation as follows. Divide by x t to get x ′t /xt = r. We know that the derivative of ln xt equals x′t/xt . Hence, by integration, we have ln xt = rt + C , where C is a constant. Finally, xt = eC ert . In order to find the value of eC we need to know x 0 . In fact, by plugging t = 0, we have x0 = eC . Hence, we get xt = x 0 ert . What is it for? It allows to predict xt at a future time t. Or it allows to find rate r if both x t and x0 are known.
What if we introduce a random perturbation? dX t = rX t dt + dξ t , where ξ t is a random process. This is a strong generalization. We will introduce and study how to solve some cases of this class of equations. They are called Stochastic Differential Equations.
1
1.2
Application in Finance
Prices 6 . 3
0 . 5 1
0 . 3
0 . 3 1
0
50
150
250
0
50
BHP
250
Boral
5 . 7 1
0 . 5 1
150
0 2
0 1
0
50
150
250
0
50
LLC
150
250
NCP
Figure 1: Prices of stocks Observed prices of stocks as functions of time
Plot price at time t, S t of the y-axis and time t on the x-axis. A model for such functions. Notice that simulated functions of time that look like stock prices. Simulations These are random functions, continuous but not smooth, (not differentiable). Using models we solve the problem of Options Pricing in Finance. Option is a financial contract that allows to buy assets in the future for the agreed price at present. This is modern approach to risk management in markets used by Banks and other large Financial Companies.
1.3
Application in Insurance
Consider a sequence of independent games, and suppose that your payoff at the end of each game is X i , which is a random variable. We assume that X i are identically distributed. The Random Walk is simply ni=1 X i for n N. This is the discrete counterpart of Brownian motion. Using Random Walk to model Insurance surplus we can calculate the ruin probability.
∑
2
∈
2 . 1 4 . 1 8 . 0
0 . 1
6 . 0
4 . 0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
mu = -1
0.6
0.8
1.0
0.8
1.0
mu = 0
6 5 2
5 4
5 1
3 2
5
1
0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
mu = 1
0.6
mu = 2
Figure 2: Computer simulations The equation for surplus at the end of year n is n
U n = U 0 + cn
� −
X k ,
k=1
where U 0 the initial funds, c is the premium collected in each year and X k is the amount of claims paid out in year k. The insurance company wants to compute the probability of ruin, i.e. the probability that soon or later the process (U n, n 1) to hit zero or become negative. This model allows to find sufficient initial funds to control the probability of ruin.
≥
3
2
Review of probability
2.1
Distribution of Random Variables. General.
A random variable refers to a quantity that takes different values with some probabilities. A random variable is completely defined by its cumulative probability distribution function. Cumulative probability distribution function
F (x) = Pr(X
≤ x),
x
∈ IR.
cdf
The probability of observing an outcome in an interval A = (a, b] is
∈ A) = F (b) − F (a).
Pr(X
Sometimes it is more convenient to describe the distribution by the probability density function. Probability density function for continuous random variables
f (x) =
d F (x) dx
pdf
Using the relation between the integral and the derivative we can calculate probabilities of outcomes by using the pdf. The probability of observing an outcome in the range (a,b] (or (a, b)) is
�
b
Pr(a < X
≤ b) = F (b) − F (a) =
f (x)dx.
a
Any probability density is a non-negative function, f (x)
�
≥ 0 that integrates to 1
f (x)dx = 1.
Conversely, any such f corresponds to some probability distribution. Uniform(0,1) have density f (x) =
�
1 0
if x (0, 1) otherwise.
∈
4
The cumulative function in this case is F (x) =
if x if x if x
0 x 1
≤0 ∈ (0, 1) ≥ 1.
Exponential with parameter λ have density f (x) =
�
λe−λx if x > 0 0 otherwise.
The cumulative function in this case is F (x) =
�
if x
0 1
λx
− e−
≤0
Standard Normal Distribution N (0, 1)
f (x) =
√ 12π e−
x2 2
General Normal ¸ Distribution involves two numbers (parameters) µ and σ
The density of normal N (µ, σ 2 ) distribution is given by f (x) =
1 √ 2πσ e−
(x−µ)2 2σ2
The cumulative probability function of Standard Normal is denoted by Φ(x).
�
x
Φ(x) =
f (u)du.
−∞
It cannot be expressed in terms of other elementary functions. It is available in Excel and Tables.
2.2
Expected value or mean
The expected value or the mean is defined as E (X ) =
�
xf (x)dx.
Interpretation, if f (x) is the mass density then EX is the centre of gravity. 5
