The stochastic calculus

Transcription

1 Gdansk

2 A schedule of the lecture Stochastic differential equations Ito calculus, Ito process Ornstein - Uhlenbeck (OU) process Heston model Stopping time for OU process

3 Stochastic differential equations Stochastic differential equations studied in this lecture are typically of the form dx t = a(t, X t )dt + b(t, X t )dw t, (1) where W t is the Wiener process More explicitly the equation (1) has to be understood as X t = X + a(s, X s )ds + b(s, X s )dw s Both integrals exist: the first in the sense of the Lebesgue integration, the second in the sense of the Ito integration when the functions a and b have to fulfill integrability conditions ( a(s, X s ) + b(s, X s ) 2 )ds, as, t

4 Usually we assume that the functions a, b satisfy the Lipschitz condition in the second variable, ie there is constant K such that for all x, y R, t [, T ] a(t, x) a(t, y) + b(t, x) b(t, y) K x y We also assume that there is constant K such for all x R, t [, T ] a(t, x) + b(t, x) K(1 + x ) In that cases the equation (1) has a unique solution X t, t T (so called a strong solution) In spite of the difficulties of the mathematical language there is a numerical procedure which can be helpful to solve (1) and to catch the ideas This procedure is based on a discrete version of the equation (1) In discrete time = t < t 1 < t 2 < < t n = T we consider a recursive formula X tj+1 X tj = a tj (t j+1 t j ) + b tj (W tj+1 W tj )

5 Geometric Brownian motion Symbolically a recursive formula is written as X t = a(t, X t ) t + b(t, X t ) W t In this way we obtain the piecewise linear sample path of the process If sup{ t j t j 1 : j n} then we obtain the solution of (1) We apply this method to the Geometric Brownian motion Samuelson provided the arguments that the price of an asset S t follows the Geometric Brownian motion, ie ds t = S t µdt + S t σdw t (2) Use Program 28 to find out that the price of the asstet S T /S (the solution of (2)) in the moment T has lognormal distribution, where S is an initial value of the asset

6 Lognormal distribution We say that a positive random variable X has the lognormal distribution with parameter m, s, if log X, (log X = log e X = ln X ) has the normal distribution N(m, s 2 ) Moreover, the mean and the variance are given by EX = e m+s2 /2, VarX = (e s2 1)e 2m+s2 Using the Itô calculus we will show that the exact solution of (2) is given by ln(s T /S ) = (µ σ 2 /2)T + σw T Compare the estimated parameters of solution S T /S : s- a shape and m -a scale parameters in Program 28 with the exact value m = µ σ 2, s = σ

7 Itô process Let us define the Ito processes, the important class of stochastic processes, using the differential form: More explicitly dy t = a t dt + b t dw t (3) Y t = Y + a s ds + b s dw s We assume that the processes a, b satisfy the integrability conditions: T a s ds < as and T b s 2 ds < as

8 Itô calculus Theorem (Itô formula) Let Y i t Let, i = 1, 2 be two Itô processes dy i t = a i tdt + b i tdw t F : [, t] R 2 R be twice differentiable Then ξ t = F (t, Y 1 t, Y 2 t ) is the Ito process such that dξ t = F t (t, Y t 1, Yt 2 )dt + F (t, Yt 1, Yt 2 )dyt 1 + F (t, Yt 1, Yt 2 )dyt 2 x 1 x F 2 x1 2 (t, Yt 1, Yt 2 )(bt 1 ) 2 dt F 2 x2 2 (t, Yt 1, Yt 2 )(bt 2 ) 2 dt + 2 F x 1 x 2 (t, Y 1 t, Y 2 t )b 1 t b 2 t dt

9 Geometric Brownian motion We will use the Itô formula to check that the process ln(s t /S ) = (µ σ 2 /2)t + σw t = (µ σ 2 /2)ds + σdw s or equivalently S t = S e (µ σ2 /2)ds+ σdws is the solution of the Geometric Brownian motion, ie solution of the equation ds t = S t µdt + S t σdw t (4)

10 We define Y t = (µ σ 2 /2)t + σw t = Note that Y t is the Ito process and S t = S exp(y t ) (µ σ 2 /2)ds + σdw s The function F (x) = S e x is a smooth function, so twice differentiable and F (x) = S e x, F (x) = S e x By the Itô formula and (4) we get ds t = S e Yt dy t S e Yt σ 2 dt = S e Yt ((µ σ 2 /2)dt + σdw t ) S e Yt σ 2 dt = S t µdt + S t σdw t

11 Ornstein - Uhlenbeck process In the physics, a relaxation means the return of a perturbed system into equilibrium Such processess are modeled by Ornstein - Uhlenbeck process given by dx t = θ(µ X t )dt + σdw t, (5) where θ, σ > and µ R To solve this equation let us consider f (X t, t) = X t e θt, where X t is the solution of (5) From the Itô formula we obtain Since dx t is the solution of (5) then df (X t, t) = θx t e θt dt + e θt dx t df (X t, t) = θx t e θt dt + e θt (θ(µ X t )dt + σdw t ) = θx t e θt dt + e θt θµdt e θt θx t dt + e θt σdw t Thus = e θt θµdt + e θt σdw t X t e θt = X + e θs θµds + e θs σdw s

12 Consequently, the solution of (5), ie the Ornstein - Uhlenbeck process is given by X t = X e θt + µ(1 e θt ) + e θ(s t) σdw s From the Itô integral theory we get that the stochastic process U t = e θ(s t) σdw s is the Gaussian process, U t N(, VarU t ), where Hence VarU t = ( σe θt) 2 VarU t = σ 2 e 2θt e2θt 1 2θ e 2θs ds = σ 2 1 e 2θt 2θ

13 Consequetly, EX t = X e θt + µ(1 e θt ), VarX t = VarU t Since θ > we get that lim EX t = µ, (6) t lim VarX t = σ2 t 2θ (7) Use Program 29 to create a sample path of the Ornstein - Uhlenbeck process for large n Note that a trajectories according to (6) return to µ and their oscilations stabilize according to (7)

14 Heston model One of the generalizations of the Samuelson model, is the Heston model We assume that the price of an asset is given by where σ t is the Ornsteina-Uhlenbeck process ds t = µs t dt + σ t S t dz 1 t, (8) dσ t = βσ t dt + δdz 2 t (9) We assume that the Wiener processes (Z 1 t, Z 2 t ) are correlated, ie Cov(Z 1 t, Z 2 t ) = ρt To obtain the equivalent formula of Heston model let us define From the Itô formula ν t = σ 2 t dν t = 2σ t dσ t δ2 dt Taking (9) and puting σ t = ν t we get dν t = 2 ν t ( β ν t dt + δdz 2 t ) δ2 dt

15 Thus dν t = 2δ ν t dz 2 t + (δ 2 βν t )dt = 2δ ν t dz 2 t + β(δ 2 /β ν t )dt We obtain the system of equations equivalent to (8) i (9), ds t = µs t dt + ν t S t dzt 1 dν t = 2β(δ 2 /β ν t )dt + δ ν t dz 2 t A solution of (9) is Ornstein Uhlenbeck process σ t = σ e βt + δ e β(s t) dz 2 s Using similar calculations as in the solution of the Geometric Brownian Motion we get (see (222) Jeanblanc, Yor Chesney) { σ 2 } s S t = S exp µt 2 ds + σ s dzs 1 This form is VERY Complicated!! Use Program 3 to create sample paths of S t and ν t in the Heston model

16 Extension of the Heston model Extension of the Heston model with stochastic interest rates is given in the paper by Grzelak and Oosterlee [211], SIAM J Fin Math pp Assume that the dynamics of an asset is given by following equations corresponding to a risk-neutral measure ds t /S t = r t dt + ν t dw 1 t dν t = κ(ν ν t )dt + γ ν t dw 2 t dr t = λ(θ r t )dt + η r t dw 3 t, with parameters κ, ν, γ, λ, θ, η We assume that the Wiener processes (Wt i, Wt j ) are correlated, ie Cov(Wt i, Wt j ) = ρ ij t It seems reasonable to take ρ 13 = Write a program to value an call option (S T K) +, see Paul Glasserman Monte Carlo Methods in Financial Engineering Springer 23

17 Hitting time- Ornstein -Uhlenbeck processes Recall that X t = θx t dt + dw t, X = x > is OU processes, where µ =, σ = 1 Let T OU = inf{t : X t = } Theorem For any x > the density function of T equals f (t) = A (t)φ( 2x/ x A(t)) A(t), 3/2 where A(t) = and φ is the density function of N(, 1) e 2ks ds

18 Proof The solution OU is given by X t = e θt (x + e ks dw s ) Thus T OU = inf{t : x + e ks dw s = } It is known that for a function f such that the processes is a martingale f 2 (s)ds <, for all t M t = f (s)dw s

19 Martingale =Dubins Schwarz s Theorem A continuous martingale M such that < M > = f 2 (s)ds = is a time changed of Brownian motion In other words, there exists a Brownian motion W such that where M t = W <M>t = W B(t), B(t) = f 2 (s)ds From Dubins Schwarz s Theorem there is BM W such that Consequently T OU = inf{t : W A(t) = x} P(A(T OU ) t) = P(T OU A 1 (t)) = P(T x t)

20 Mixed Processes involving Compound Poisson Processes Let X be a compound Poisson process, ie N(t) X t = Y j, where N(t) is a Poisson process with intensity λ and {Y j } iid and independent of N(t) Let us consider the stochastic equation with constants µ and σ j=1 ds t = S t (µdt + σdw t + dx t ), where S t is left continuous version of S t Theorem The process S t e rt, t is a martingale iff µ + λey 1 = r If Y j 1 then the solution of the stochastic equation is given by N(t) S t = e µt e σwt 1/2σ2t exp( ln(1 + Y j )) j=1