Chapter 2

Notice that, like the single period ternary model of Chapter 1, the two-step binomial model allows the stock to take on three distinct values at time \(2 .\) Show, however, that every claim can be exactly replicated by a self-financing portfolio, that is, the market is complete. More generally, show that if the market evolves according to a \(k\)-step binomial model then it is complete.

If a stock price is \(S\) just before a dividend \(D\) is paid, what is its value immediately after the payment? Suppose that a stock pays dividends at discrete times, \(T_{0}, T_{1}, \ldots, T_{n}\). Show that it can be optimal to exercise an American call on such a stock prior to expiry.

Suppose that an asset price evolves according to the binomial model. For simplicity suppose that the risk-free interest rate is zero and \(\Delta T\) is 1 . Suppose that under the probability \(\mathbb{P}\), at each time step, stock prices go up with probability \(p\) and down with probability \(1-p\) The conditional expectation $$ M_{n} \triangleq \mathbb{E}\left[S_{N} \mid \mathcal{F}_{n}\right], \quad 1 \leq n \leq N $$ is a stochastic process. Check that it is a P-martingale and find the distribution of the random variable \(M_{n}\).

(a) Find a Markov process that is not a martingale. (b) Find a martingale that is not a Markov process.