CHAPTER 5 ELEMENTARY STOCHASTIC CALCULUS. In all of these X(t) is Brownian motion. 1. By considering X 2 (t), show that

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1 CHAPTER 5 ELEMENTARY STOCHASTIC CALCULUS In all of these X(t is Brownian motion. 1. By considering X (t, show that X(τdX(τ = 1 X (t 1 t. We use Itô s Lemma for a function F(X(t: Note that df = df dx dx + 1 d F dx dt. df = [F ] t, and let F = X (t. Then df = X (t X ( = X (t, as X( =. Applying Itô s Lemma to F gives us that df = XdX + dt, (X(τdX(τ + dτ = X (t.. Show that X(τdX(τ = 1 X 1 t. τdx(τ = tx(t X(τdτ.

2 4 ELEMENTARY STOCHASTIC CALCULUS We use Itô s Lemma for a function F(X(t,t: df = df ( df dx dx d F dt dx dt. Let F = tx(t. Then df = tx(t. Applying Itô s Lemma to F gives us that df = tdx + Xdt, (τdx(τ + X(τdτ = tx(t. τdx(τ = tx(t X(τdτ. 3. Show that X (τ dx(τ = 1 3 X3 (t X(τdτ. Let F = X 3 (t. Then df = X 3 (t X 3 ( = X 3 (t, as X( =. Applying Itô s Lemma to F gives us that df = 3X dx + 3Xdt, (3X (τ dx(τ + 3X(τdτ = X 3 (t. X (τ dx(τ = 1 3 X3 X(τdτ. 4. Consider a function f(t which is continuous and bounded on [,t]. Prove integration by parts, i.e. f(τdx(τ= f(tx(t X(τ df (τ.

3 INSTRUCTOR S MANUAL 5 Let F = f(tx(t. Then df = f(tx(t f(x( = f(tx(t. Applying Itô s Lemma to F gives us that df = fdx+ Xdf, (f (τ dx(τ + X(τ df (τ = f(tx(t. f(τdx(τ= f(tx(t 5. Find u(w, t and v(w,t where dw(t = udt+ vdx(t X(τ df (τ. and (a W(t = X (t, (b W(t = 1 + t + e X(t, (c W(t = f(tx(t, where f is a bounded, continuous function. We use Itô s Lemma for a function W(X(t,t: dw = dw ( dw dx dx d W dt dx dt. (a (b Therefore dw = XdX+ dt = WdX+ dt. u(w, t = W and v(w,t = 1. dw = e X(t dx + ( 1 + e X(t dt. Rearranging the formula for W(t,we find that e X(t = W(t 1 t,

4 6 ELEMENTARY STOCHASTIC CALCULUS (c and so Therefore dw = (W(t 1 tdx + (W(t tdt. u(w, t = W(t 1 t and v(w,t = W(t t. dw = fdx+ X df dt dt. Therefore u(w, t = f(t and v(w,t = W(t df f(t dt. 6. If S follows a lognormal random walk, Use Itô s lemma to find the differential equations satisfied by (a f(s= AS + B, (b g(s = S n, (c h(s, t = S n e mt, where A, B and n are constants. Itô s lemma for a function f(s is df = σs df ds dx + ( µs df ds + 1 σ S d f ds dt. Then (a df = Aσ SdX + AµSdt = AdS. (b dg = nσ S n dx + ns n ( µ + 1 (n 1σ dt. (c dh = nσ S n e mt dx + S n e mt ( m + nµ + 1 n(n 1σ dt. 7. If ds = µs dt + σs dx, use Itô s lemma to find the stochastic differential equation satisfied by f(s= log(s. Itô s lemma for a function f(s is df = σs df ( ds dx + µs df ds + 1 σ S d f ds dt. Now df ds = 1/S and d f ds = 1/S, so df = σdx+ ( µ 1 σ dt.

5 INSTRUCTOR S MANUAL 7 Note that this stochastic differential equation for log(s has constant coefficients. For this reason, S is described as satisfying a lognormal random walk. 8. The change in a share price satisfies ds = A(S, tdx + B(S,tdt, for some functions A, B, what is the stochastic differential equation satisfied by f(s,t? Can A, B be chosen so that a function g(s has a zero drift, but non-zero variance? We could use Itô s Lemma directly to answer this and the following question, but as a teaching aid, will derive the results informally from first principles. We apply Taylor s theorem to find the change in f over a small time step, f(s+ δs,t + δt: f(s+ δs,t + δt = f(s,t+ δs + S t δt + 1 f S δs + f S t δsδt + 1 f t δt +... Substitute for δs = σsδx+ µsδt to get δf = (AδX + Bδt + δt S t + 1 (A δx + B δt + ABδXδt f S δs + (AδXδt + Bδt f S t + 1 δt f t +... Discarding terms of O ( δt 3/ and smaller, δf = A S δx + ( B S + t δt + 1 A f S δx + O ( δt 3/. As δt, replace δt by dt, δx by dx and δx by dt to find the stochastic differential equation satisfied by f(s,t: df = A ( S dx + B S + 1 A f S + dt. t A function g(s will therefore satisfy the equation dg = A dg ( ds dx + B dg ds + 1 A d g ds dt.

6 8 ELEMENTARY STOCHASTIC CALCULUS For g(s to have a zero drift but non-zero variance, we require that B dg ds + 1 A d g ds =. We can find a solution to this problem if A /B is independent of time. 9. Two shares follow geometric Brownian motions, i.e. ds 1 = µ 1 S 1 dt + σ 1 S 1 dx 1, ds = µ S dt + σ S dx, The share price changes are correlated with correlation coefficient ρ. Find the stochastic differential equation satisfied by a function f(s 1,S. We apply Taylor s theorem to find the change in f over a small time step - f(s 1 + δs 1,S + δs : f(s 1 + δs 1,S + δs = f(s+ δs 1 + δs + 1 f S 1 S δs 1 S 1 f + δs 1 δs + 1 f S 1 S δs S +... Substituting for δs 1 and δs, and discarding terms of O ( δt 3/ and smaller, we find that δf = σ 1 S 1 δx 1 + σ S δx + µ 1 S 1 δt + µ S δt S 1 S S 1 S + 1 σ 1 f S 1 S δx σ f S 1 S δx f + σ 1 σ S 1 S δx 1 δx + O ( δt 3/. S 1 S As δt, replace δt by dt, δx 1 by dx1, δx by dx, δx1 by dt, δx by dt and δx 1δX by ρdt to find the stochastic differential equation satisfied by f(s 1,S : df = σ 1 S 1 dx 1 + σ S dx S 1 S ( + µ 1 S 1 + µ S + 1 S 1 S σ f 1 S 1 S + 1 σ f S 1 S + ρσ 1 σ S 1 S f S 1 S dt.

5. Itô Calculus. Partial derivative are abstractions. Usually they are called multipliers or marginal effects (cf. the Greeks in option theory).

5. Itô Calculus. Partial derivative are abstractions. Usually they are called multipliers or marginal effects (cf. the Greeks in option theory). 5. Itô Calculus Types of derivatives Consider a function F (S t,t) depending on two variables S t (say, price) time t, where variable S t itself varies with time t. In stard calculus there are three types