Stochastic Calculus, Fall 2002 (http://www.math.nyu.edu/faculty/goodman/teaching/StochCalc/)

Assignment 8.

Given November 7, due November 14 Last revised November 8.

Focus: Ito calculus

1. We will calculate the first four Picard iterates for the SDE dX_{t} = X_{t}dW_{t}, with X_{0} = 1. These are given by X_{t}^{(0)} = 1 for all t ≥ 0, and

X_{t}^{(k+1)} = 1 + ∫_{0}^{t} X_{s}^{(k)}dW_{s} .

This is partly to provide practice using the Ito calculus on the various stochastic integrals that come up.

a. Compute X_{t}^{(1)}.

b. Compute X_{t}^{(2)}. It will be simpler to compute

Y_{t}^{(2)} = X_{t}^{(2)} − X_{t}^{(1)} = ∫_{0}^{t}(X_{s}^{(1)} − X_{s}^{(0)})dW_{s} = ∫_{0}^{t} Y_{s}^{(1)}dW_{s} .

c. The exact solution of the SDE is X_{t} = e^{W_{t}}e^{−t/2}. Expand e^{W_{t}} and e^{−t/2} is Taylor series in W_{t} and t respectively, keeping terms up to and including O(t). Multiply these to get a short time approximation to X_{t} up to O(t). Show that this agrees with your answer to part b.

d. Calculate Y_{t}^{(3)} = ∫_{0}^{t} Y_{s}^{(2)}dW_{s}. To simplify the answer, you need a relationship between ∫_{0}^{t} W_{s}ds and ∫_{0}^{t} sdW_{s}.

e. Calculate Y_{t}^{(4)} = ∫_{0}^{t} Y_{s}^{(3)}dW_{s}. Combine with previous results to get X_{t}^{(4)}.

f. Extend the calculation of part c to get an approximation containing all terms up to and including O(t^{2}). Check that this agrees with your answer to part e.

2. a. Using our bounds for X^{(k+1)} − X^{k} from Lecture 7, paragraph 1.5 with the constant in (3), show that X_{t} − X_{t}^{(k)} = O(t^{(k+1)/2}). For this you will need to know that

∑_{j=k}^{∞} z^{j} = \frac{z^{k}}{1 − z} = O(z^{k}) , as z → 0.

b. Justify the approximation X_{t+Δt} = X_{t} + σ(X_{t})ΔW_{Δt} + O(Δt). Here, ΔW_{Δt} = W_{t+Δt} − W_{t}. This uses part a and facts about Ito integrals. Hint: it might be easier notationally if you replace t by 0 and Δt by t. c. Justify the approximation

X_{t+Δt} = X_{t}+σ(X_{t}, t)ΔW_{Δt}+a(X_{t}, t)Δt+\frac{1}{2}σ(X, t)∂_{X}σ(X_{t}, t)(ΔW_{Δt}^{2}−t)+O(t^{3/2}) .

This requires the approximation in part b and the technique behind it.

3. Suppose dX_{t} = a(X_{t}, t)dt + σ(X_{t}, t)dW_{t} and f(x, t) = E_{x,t}[V (X_{T})]. Use the fact that f_{t} = f(X_{t}, t) is a martingale and Ito’s lemma for solutions of SDEs to find a partial differential equation ∂_{t}f + ⋯ = 0. This is another in our collection of useful backward equations.

4. Derive the Black Scholes formula for European option pricing. For this, we need the cumulative distribution of a standard normal random variable. This is, for Z ∼ N (0, 1),

N(z) = P (Z ≤ z) = \frac{1}{\sqrt{2π}} ∫_{y=−∞}^{z} e^{−y^{2}/2}dy .

Many integrals involving gaussians can be expressed in terms of N. For example, if X ∼ N (μ, σ^{2}), then P (X ≤ μ + zσ) = N(z). In earlier assignments, we have used the fact that E[e^{X}] can be calculated by writing the integral, completing the square in the exponent, and recognizing this as another gaussian integral.

a. A geometric Brownian motion satisfies the SDE dS_{t} = rSdt + σS_{t}dW_{t}. Note that what we usually call σ is now σS_{t}. Compute Y_{t}, the ordinary calculus guess at the solution (if the solution if W_{t} were a differentiable function of t). Use the Ito calculus to find and verify the correct solution in the form S_{t} = A(t)Y_{t}.

b. Find an expression for S_{T} in terms of S_{t} and W_{T} −W_{t}. This is just a line of algebra. c. For f(S, t) = E[V (S_{T}) | F_{t}], write the backward equation for f. This asks you

to translate the general result in question 3 into a more specific equation that holds for geometric Brownian motion. The resulting equation is called the “Black Scholes” equation.

d. For any final payout function V (s), write a formula for f(S, t) as the expected value of some function of a gaussian random variable whose mean and variance depend on t in a simple way. This is an application of the result of part b and the definition of f. It does not use the backward equation.

e. For V (s) = max(s − K, 0) = (s − K)_{+}, compute the integral in part d explicitly in

terms of the N function. This is the “Black Scholes formula”. f. Verify by direct differentiation that the formula satisfies the backward equation.

5. Use Ito’s lemma to show that each of the following is a martingale. Comment on the difficulty of doing it this way or directly, as in an earlier assignment.

a. X_{t} = e^{k^{2}t/2} sin(kW_{t})

b. Y_{t} = W_{t}^{4} − 6 ∫_{0}^{t} W_{s}^{2}ds.