91Ó°ÊÓ

In a binomial model. give an example of a stochastic process that is a martingale but is not Markov.

We consider a binomial asset pricing model as in Chapter 1, except that, after each movement in the stock price, a dividend is paid and the stock price is reduced accordingly. To describe this in equations, we define $$ Y_{n+1}\left(\omega_{1} \ldots \omega_{n} \omega_{n+1}\right)=\left\\{\begin{array}{l} u, \text { if } \omega_{n+1}=H \\ d, \text { if } \omega_{n+1}=T \end{array}\right. $$ Note that \(Y_{n+1}\) depends only on the \((n+1)\) st coin toss. In the binomial model of Chapter \(1, Y_{n+1} S_{n}\) was the stock price at time \(n+1 .\) In the dividend-paying model considered here, we have a random variable \(A_{n+1}\left(\omega_{1} \ldots \omega_{n} \omega_{n+1}\right)\), taking values in \((0,1)\), and the dividend paid at time \(n+1\) is \(A_{n+1} Y_{n+1} S_{n} .\) After the dividend is paid, the stock price at time \(n+1\) is $$ S_{n+1}=\left(1-A_{n+1}\right) Y_{n+1} S_{n} $$ An agent who begins with initial capital \(X_{0}\) and at each time \(n\) takes a position of \(\Delta_{n}\) shares of stock, where \(\Delta_{n}\) depends only on the first \(n\) coin tosses, has a portfolio value governed by the wealth equation (see (2.4.6)) $$ \begin{aligned} X_{n+1} &=\Delta_{n} S_{n+1}+(1+r)\left(X_{n}-\Delta_{n} S_{n}\right)+\Delta_{n} A_{n+1} Y_{n+1} S_{n} \\ &=\Delta_{n} Y_{n+1} S_{n}+(1+r)\left(X_{n}-\Delta_{n} S_{n}\right) \end{aligned} $$ (i) Show that the discounted wealth process is a martingale under the riskneutral measure (i.e., Theorem \(2.4 .5\) still holds for the wealth process (2.8.2)). As usual, the risk-neutral measure is still defined by the equations $$ \tilde{p}=\frac{1+r-d}{u-d}, \quad \tilde{q}=\frac{u-1-r}{u-d} $$ (ii) Show that the risk-neutral pricing formula still applies (i.e., Theorem 2.4.7 holds for the dividend-paying model). (iii) Show that the discounted stock price is not a martingale under the riskneutral measure (i.e., Theorem 2.4.4 no longer holds). However, if \(A_{n+1}\) is a constant \(a \in(0,1)\), regardless of the value of \(n\) and the outcome of the coin tossing \(\omega_{1} \ldots \omega_{n+1}\), then \(\frac{S_{n}}{(1-a)^{n}(1+r)^{n}}\) is a martingale under the risk-neutral measure.