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1 Stochastic Calculus Week 3 Topics: Towards Black-Scholes Stochastic Processes Brownian Motion Conditional Expectations Continuous-time Martingales Towards Black Scholes Suppose again that S t+δt equals S t u with probabililty p and S t d with probability (1 p), where u = exp(σ δt), d = exp( σ δt), p = 1 ( 1+ δt µ ), 2 σ where µ is some constant representing the drift in S t. As before we have δt = T/n, and we look at the behaviour of the stock price as n, and hence δt 0. Let Y i = log(s iδt /S (i 1)δt ) be the ith δt-period continuously compounded return. Then E P (Y i ) = σ δt(p [1 p]) = µδt, var P (Y i ) = σ 2 δt(p +[1 p]) µ 2 δt 2 σ 2 δt, where the approximation gets better as δt 0. Now log S T = log S 0 + Y i = log S 0 + µt + σ T 1 n ( ) Yi µδt σ. δt 1

2 Each of the terms in parentheses has mean zero and variance (approximately) 1; furthermore they are independent. This implies a Central Limit Theorem: 1 ( ) Yi µδt n σ d Z, δt where Z N(0, 1), the standard normal distribution: P (Z x) = x 1 2π e 1 2 z2 dz = x φ(z)dz, with φ( ) the standard normal density function. The symbol d means that the distribution of the left-hand side converges to the distribution of the right-hand side. All this implies, as n, log S T d log S 0 + µt + σ TZ N(log S 0 + µt, σ 2 T ), that is, stock-prices have a log-normal distribution. So far distributions were under P. It can be shown that q = erδt e σ δt e σ δt e 1 (1+ δt r 1 ) 2 σ2. σ δt 2 σ Following the same derivations with µ replaced by r 1 2 σ2, it follows that under Q, Remarks The distribution under Q implies the Black-Scholes formula for a European call option via ( C 0 = E Q e rt [S T K] +). Using similar methods, we can show that under P, log S t log S s N(µ(t s),σ 2 (t s)), for all 0 < s < t < T. This defines log S t to be a Brownian motion process, and S t itself is a geometric Brownian motion process. When Z has a standard normal distribution, it can be derived that E ( e µ+σz) = e µ+ 1 2 σ2. This implies that under Q, E Q (e rt S T )=S 0. log S T d N(log S 0 +(r 1 2 σ2 )T,σ 2 T ). 2

3 Stochastic Processes Random variable: X :Ω R, and hence X(ω),ω Ω; Random vector: X :Ω R k, and hence X(ω) =[X 1 (ω),...,x k (ω)],ω Ω; Stochastic process: X :Ω T R, and hence X t (ω),t T,ω Ω. For fixed t: X t (ω) is random variable. For fixed ω, X t (ω) is a sample path, ortrajectory. See Figure 1.2.1, p.24 of Mikosch. Some further classes: Gaussian processes have Gaussian (normal) fidis; they are completely characterized by µ X (t) and c X (t, s); Stationary: (X t1,...,x tn ) has same distribution as (X t1 +h,...,x tn +h) Stationary increments: (X t2 X t1,...,x tn X tn 1 ) has same distribution as (X t2 +h X t1 +h,...,x tn +h X tn 1 +h); Independent increments: (X t2 X t1 ),...,(X tn X tn 1 ) are independent. Discrete time: T = {0, 1, 2,...}; Continuous time: T =[0,T] or T =[0, ). Fidis: finite-dimensional distributions of vectors (X t1,...,x tn ),t 1...,t n T. Process is partly characterized by first two moments: µ X (t) = E(X t ), σ 2 X (t) = var(x t) and c X (t, s) = cov(x t,x s ). 3

4 Brownian Motion/Wiener Process Stochastic process B t,t [0, ), with 1. B 0 =0; 2. Stationary & independent increments; 3. B t N(0,t) for all t; 4. Continuous sample paths. Alternatively: Gaussian process with µ B (t) =0and c B (t, s) = min(t, s): e.g., when s<t, c B (t, s) = E(B t B s )=E([B s +(B t B s )]B s ) = E(Bs)+E[(B 2 t B s )(B s B 0 )] = s +0, because of independent increments. Further properties of sample paths: Continuity: by assumption, but intuitively: B t+h B t N(0,h) 0 as h 0; Nowhere differentiability: intuitively: ( B t B t h N 0, 1 ) B t+h B t, h h h independently; N Unbounded variation: for 0=t 0 <...<t n = T : sup Bti B ti 1 =. ( 0, 1 ), h All this implies that integrals like T 0 f(t)db(t) cannot be defined as T 0 f(t)db(t) dt dt. (Unlike Riemann- Stieltjes integrals such as xdf (x) = (x) xdf dx dx = xf(x)dx).) Extensions: Brownian motion with drift: X t = X 0 + µt + σb t,so µ X (t) =X 0 + µt and c X (t, s) =σ 2 min(t, s). Geometric Brownian motion X t = X 0 exp(µt + σb t ). In this case µ X (t) =X 0 exp(µt σ2 ). 4

5 Representation of B t as limit of sequence Let Y i be i.i.d. with mean zero and unit variance, and let S n (t) = 1 [nt] n where [nt] is the largest integer smaller than or equal to nt. This process does not have continuous sample paths (step function). However, its fidis converge as n to the fidis of B t on [0, 1]. Y i, We can make it continuous by linear interpolation: S n (t) = 1 [nt] n Y i + 1 n Y [nt]+1 (nt [nt]). Again, the fidis of S n converge as n to those of B t on [0, 1]. Conditional Expectations Probability space (Ω, F, P). Indicator function of B F: 1 if ω B, I B (ω) = 0 if ω/ B. Note that E(I B )=1 P(B)+0 [1 P(B)] = P(B). A discrete random variable X n can be defined as follows X n (ω) = x i I Ai (ω), where {A i } is partition of Ω: A i A j = and n A i =Ω. Conditional expectation of X n given event B with P(B) > 0: E(X n B) = X n (ω)dp(ω B) Ω P(A i B) = x i P(B) 1 = X n (ω)i B (ω)dp(ω) P(B) Ω 1 = P(B) E(X ni B ). 5

6 Any random variable can be approximated arbitrarily well by X n, choosing partitioning finer and finer. Hence E(X B) = lim E(X n B) = 1 n P(B) E(XI B). Conditional expectation of X given general σ-field F: random variable satisfying: 1. E(X F) is a function of F; 2. E[E(X F)I A ]=E(XI A ) for all A in F. Definition is not very constructive (doesn t tell us how to find the conditional expectation). Often in practice, F = σ(y 1,...,Y i ) or σ(y s,s t). In that case the defining property is that X E(X F) is uncorrelated with any function of (Y 1,...,Y i ) or (Y s,s t). Example: Ω = {ω 1,ω 2,ω 3,ω 4 }, F = {, Ω, {ω 1,ω 2 }, {ω 3,ω 4 }}, P(ω i )=0.25, and X(ω 1 )=2, X(ω 2 )=X(ω 3 )=1and X(ω 4 )=0. Then define 3 2 if ω {ω 1,ω 2 }, E(X F)(ω) = 1 2 if ω {ω 3,ω 4 }. This satisfies the defining properties of the conditional expectation (check!). Rules for working with conditional expectations (same as discrete): E(X F) =X if X is F-measurable E(XY F) =XE(Y F) if X is F-measurable E(X F 0 )=E(X), where F 0 = {, Ω} E(aX + by F) =ae(x F)+bE(Y F) E[E(X F)] = E(X) E[E(X F) G] =E[E(X G) F] =E(X F) if F G 6

7 Continuous-time Martingales Filtration F t : collection of σ-fields with F s F t if s<t. Example: F t = σ(b s, 0 s t). This σ-field contains all the possible events involving B s,s t. It is also the σ-field generated by the random variables (B t1,...,b tn ), for arbitrary t 1... t n t. Martingale:{X t, F t } satisfying: 1. E( X t ) < ; 2. X t is F t -measurable (adapted); 3. E(X t F s )=X s, s<t. Examples: X t = E(X F t ). Brownian motion: when F t = σ(b s, 0 s t), then E(B t F s ) = E(B s +[B t B s ] F s ) = B s +0 because of independent increments. Note that E( B t ) < because B t N(0,t); Geometric Brownian motion X t = X 0 exp(µt + σb t ): E(X t F s ) = E(X s exp(µ[t s]+σ[b t B s ]) F s ) = X s exp(µ[t s])e(exp(σ[b t B s ])) = X s exp(µ[t s]) exp( 1 2 σ2 (t s)). Hence this is a martingale only if µ = 1 2 σ2. X t = Bt 2 t: X t = Bs 2 +(B t B s ) 2 +2B s (B t B s ) t. Because E((B t B s ) 2 F s )=(t s) and E(2B s (B t B s ) F s )=2B s E((B t B s ) F s )=0,wehave E(X t F s )=Bs 2 +(t s) t = X s. 7

8 Martingale transform in discrete time: {B ti,i = 1,...,n}, increments i B = B ti B ti 1, filtration F i = σ(b t1,...,b ti ) and previsible sequence {C i }, such that C i is in F i 1. Then the martingale transform C B is k (C B) k = C i i B, which is a discrete-time martingale with respect to F i. Interpretation: financial gain from a trading strategy; see exercise. Exercises 1. Let (Z 1,Z 2 ) be a pair of independent standard normal random variables. Define the vector ( ) ( ) X1 X = = µ 1 + σ 1 1 ρ2 Z 1 + ρz 2. X 2 µ 2 + σ 2 Z 2 Show that X has a bivariate normal distribution with mean vector µ =(µ 1,µ 2 ) and covariance matrix ( ) σ 2 Σ X = 1 ρσ 1 σ 2. ρσ 1 σ 2 σ Show that the Brownian motion process is 0.5-selfsimilar, as follows. Let B t be a Brownian motion, and define, for T>0, X t = T 1/2 B tt Show that X t satisfies the defining properties of a Brownian motion. 8

9 3. Let Y i,i = 1,...,n be independent random variables with P(Y i =1)=P(Y i = 1)=0.5. Let S n (t) be a continuous process on t [0, 1], defined by ( ) i S n = 1 i Y j. n n j=1 Show that as n, this process has the same finitedimensional distributions as a Brownian motion proces B t at the points t i = i/n. 4. Consider a multiplicative tree in which the stock price S satifies S i = S i 1 u or S i = S i 1 d, both with probability 0.5. Let u = exp(n 1/2 ) and d =1/u, and define the continuous process X n (t) on t [0, 1], such that ( ) i X n = S i. n Show that as n, this process has the same finitedimensional distributions as a geometric Brownian motion at the points t i = i/n. What are the values of µ and σ 2? 5. Let S t and B t be a continuous-time stock price and bond processes, such that the discounted stock price process Bt 1 S t is a martingale under Q. Let V t = φ t S t + ψ t B t be the value of a self-financing portfolio, where the weights φ t and ψ t only change at discrete points in time t i,i =0, 1,...,n. More precisely, the weights are constant over de intervals [t i 1,t i ), and are previsible in the sense that their value in the period [t i 1,t i ) depends only on {S t,b t,t t i 1 }. Show that the self-financing property implies, with t [t m t<t m+1 ), m m V t = V 0 + φ ti 1 (S ti S ti 1 )+ ψ ti 1 (B ti B ti 1 ) +φ tm (S t S tm )+ψ tm (B t B tm ). Show also that under Q, Bt 1 V t is a martingale with respect to F t = σ(s s,s t). 9

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