Assignment 3

Exercise 3.9 Consider a European call option with strike price \(K=£ 8\) and maturity \(T=2\) on a two-period binomial model. The price at time 0 of the stock is \(£ 10\). Over the first period, the stock market is expected to be very variable, and the underlying stock’s price is expected to increase or decrease by 20%. Over the second period, a more stable market is expected and the stock price is expected to increase or decrease by 10%. Assume that the risk-free rate is 5% over each period. What is the value of this call option? Describe the self-financing hedging portfolio that replicates the call option.

Exercise 3.10 Consider a 4-period binomial model, with \(u=1.1\), \(d=0.8\) and \(r=0.05\), and suppose the share price at time 0 is \(S_0 = 40\). What is the Cox–Ross–Rubinstein price at time 0 of a European call option with strike price 40 and expiry time 4? And what is the CRR price of the option at time 2, if the share price at time 2 is \(S_2 = 35.2\)?

Exercise 3.11 Let \(Y \sim \Bin(T,q)\) be a Binomial random variable.

1. Using the fact that \(Y = \sum_{i=1}^T X_i\) where \(X_1,\dots,X_T\) are i.i.d. Bernoulli random variables, or otherwise, show that the moment generating function \(m_Y(\theta)\) of \(Y\) equals \((1-q+q\e^\theta)^T\).
2. Suppose \(S_T = S_0 u^Yd^{T-Y}\), where \(S_0\), \(u\) and \(d\) are positive constants. Calculate \(\E[S_T]\) using part (a) and find the value of \(q\) that solves \[ S_0 = \frac{1}{(1+r)^T}\E[S_T]. \]