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PS 1

Problem 1.

Use Itˆo’s rule to find the Stochastic Differential Equation for these processes:

(a) Brownian motion to the fourth power, W4 (t).

(b) S 4 (t) where S(t) = e (µ−σ 2/2)t+σW(t) is the stock price in the Merton-Black-Scholes model.

(c) The product W(t)S(t)

(d) the ratio S(t)/W(t).

Problem 2.

Show that the process X(t) := W2 (t) − t is a martingale, i.e., that E[X(t)|X(s)] = X(s) for s < t. You can do this directly, or you can use Ito’s rule on the process X(t).

Problem 3.

The Black-Scholes-Merton model assumes that the stock price S(T) at time T > 0 is a random variable given by

S(T) = S(0)e aT +σ √TZ

where S(0) > 0, a and σ > 0 are constants, and Z is a standard normal random variable. Given a constant K > 0, consider the financial derivative that pays $1.00 at time T if S(T) ≥ K, and zero otherwise. In other words, its payoff at time T is the indicator random variable given by

X = 1{S(T)≥K}.

- a) Find the formula for the probability that the payoff at time T will not be zero, in terms of the standard normal cumulative distribution function N(x) = P(Z ≤ x).

- b) Suppose the price of the derivative is defined as E[X]. Compute the value of E[X] for the case

1 = S(0) = K = T , σ = 0.2, a = −0.02.

You may use any quantitative software to compute N(x), or you can use normal distribution calculators that can be found on the internet.

Problem 4.

Find the values of daily prices of a stock or a stock index for the last 20 trading days, where you can choose the stock yourself. For example, you can go to Yahoo Finance, and use ”open” prices. Assume that the stock follows the Black-Scholes model and estimate its drift µ and its volatility squared σ2, in annual terms. For µ, you can do this by computing the sample mean of the ratios S(tk)/S(tk+1) of the stock prices on consecutive days to estimate its expected value, and use the fact that this value satisfies

E [S(tk)/S(tk + 1)] = e (tk+1−tk)×µ

For σ2, use the expression

V ar [log (S(tk + 1)) − log(S(tk))] = (tk+1 − tk) × σ2

and compute the corresponding sample variance from the data.

Assume that one day is equal to 252/1 years, because there are, on average, 252 working days on Wall Street in a year. You may use software or internet to compute the sample mean and variance.