Chapter 3 Questions for Chapter 3

Exercise 3.1 Consider a binomial price tree two periods long with \(S_0 = 100\), \(u = 2\), \(d = 1/2\). Find the price of the call option with expiry 2 and strike price 150 as a function of \(r\), the interest rate per period.

Exercise 3.2 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 in the situation when the stock price increases over the first period.

Exercise 3.3 Suppose that the stock price \(S_t\) behaviour over two periods is given by the following tree: and suppose that \(r=0.1\). Find the martingale probabilities at each node of the tree. Using these probabilities, find the fair prices of the following digital options:

  1. a cash-or-nothing call option, with payoff 66 if \(S_2 \geq 50\) and 0 if \(S_2 < 50\),
  2. an asset-or-nothing call option, with payoff \(S_2\) if \(S_2 \geq 50\) and 0 if \(S_2 < 50\),

Exercise 3.4 Suppose the initial stock price is \(S_0=80\) and that for at each time \(t=1,2,3\) the stock price either increases by 10, or decreases by 10. (So, at time \(t=3\) the stock price can be 50, 70, 90, or 110.) Assume the interest rate is always zero. Can we use the Cox–Ross–Rubinstein formula on this market? What is the fair price of a European call option with strike price 80 and expiry time 3?

Exercise 3.5 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 3.6 Suppose we want to use a multi-period binomial model to model the behaviour of the share price of a risky asset over the course of one year (from \(t=0\) to \(t=1\)). We believe that over this time the most that the share price will increase by is 40% and the most it will decrease by is 20%. We assume that the annual interest rate is 4%, compounded monthly. What values should we take for \(u, d\) and \(r\) if we wish to model this with a 12-period model, so that the prices change at monthly intervals (\(t=0, 1/12, 2/12, \dots, 1\))? Is this model arbitrage free? Supposing the risky asset price at time 0 is 60, what is the price at time 0 of a European put option with strike price 70 and expiry at the end of the year (\(T=1\))?

Exercise 3.7 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 \[ S_0 = \frac{1}{(1+r)^T}\E[S_T]. \]

Exercise 3.8 Find an expression for the price at time \(T-t\) of a European put option with strike price \(K\) and expiry date \(T\), analogous to the Cox–Ross–Rubinstein formula for the price of a call option. Confirm that the CRR prices at time \(T-t\) for a call and put with the same strike price \(K\) and expiry date \(T\) satisfy the put-call parity formula \(P + S_{T-t} = C + K(1+r)^{-t}\).