Chapter 4 Questions for Chapter 4
Warm-up
Exercise 4.1 Let \(X_1, X_2, X_3\) be a sequence of random variables which can each take three possible values: \(0, 1,\) or \(2\). Construct the filtration generated by \(X_1, X_2,\) and \(X_3\), that is, \(\mathcal{F}_0, \mathcal{F}_1, \mathcal{F}_2,\) and \(\mathcal{F}_3\).
Exercise 4.2 Let \(X_1, X_2, \dots\) be independent identically distributed random variables with mean \(\E[X_i] = 0\) and variance \(\Var[X_i] = 1\). Let \(S_0 = 0\) and \(S_n = X_1 + X_2 + \dots + X_n\). Show that \(M_n = S_n^2 - n\) is a martingale with respect to the filtration \((\mathcal{F}_n)\) generated by \(X_1, X_2, \dots\) (i.e., \(\cF_n = \sigma(X_1,X_2,\dots,X_n)\)).
Main problems
Exercise 4.3 Let \(X_1, X_2, \dots\) and \(S_0, S_1, \dots\) be defined as in Exercise 4.2. Let \(m(\theta) = \E[\e^{\theta X_1}]\) be the moment generating function of \(X_1\). Show that \[ M_n = e^{\theta S_n - n \log m(\theta)} \] is a martingale with respect to the filtration \((\mathcal{F}_n)\) generated by \(X_1, X_2, \dots\).
Exercise 4.4 Suppose that a fair coin is tossed repeatedly, and if we can predict the outcome of a given bet, we will double our stake. Consider the following gambling strategy on the outcomes of the coins. Start at time \(t=1\) by betting £1 on “heads”; if we lose the bet at time \(t\) (the coin comes up “tails”) we double the size of our bet for the next time \(t+1\) and continue betting, if we win at time \(t\) we stop betting. Let \(X_t\) be the winnings in pounds at time \(t\). (So \(X_0 = 0\), before any betting has taken place.) As before we let \(\cF_t\) be the \(\sigma\)-algebra of all events determined by the outcomes of the first \(t\) coin tosses.
- Show that the random variable \(X_t\) depends on the first \(t\) coin tosses, as given by \[ X_t(\omega_1\dotsm\omega_t) = \begin{cases} 1 - 2^t &\text{if $\omega_1=\dotsm=\omega_t = \rT$},\\ 1 &\text{otherwise}, \end{cases} \] for \(t=1,2,\dots\).
- Prove that the sequence \(X_0, X_1, X_2, \dots\) is a martingale under the measure with \(\P[\omega_t = \rH] = \P[\omega_t = \rT] = 1/2\) for all \(t\).
(Note that the above strategy is often called a “martingale gambling strategy”, exactly because the sequence \(X_0, X_1, X_2, \dots\) is a martingale.)
Exercise 4.5 Consider a multi-period Binomial model \(\mathcal{M} = (B_t, S_t)\) with interest rate \(r\). Let \(h_t = (x_t, y_t)\) be a portfolio with value function \(V_t = x_{t+1} B_t + y_{t+1} S_t\). Suppose for every \(t\), \[\begin{equation} V_t - V_0 = \sum_{k=1}^t x_k(B_k - B_{k-1}) + y_k(S_k - S_{k-1}). \tag{4.1} \end{equation}\]
Show that \(h_t\) is a self-financing portfolio, that is, \(V_t = x_t B_t + y_t S_t\).
In Theorem 4.2, the sequence \(\frac{V_t}{(1+r)^t}\) was shown to be a martingale under \(\mathbb{Q}\) using the wealth equation which holds for any self-financing portfolio \(h_t\). Show that Theorem 4.2 also follows from the representation of \(V_t - V_0\) given in (4.1) above.
Exercise 4.6 Let \(h_t = (x_t, y_t)\) be a self-financing portfolio for the multi-period binomial model, with value sequence \(V_t = x_{t+1}B_t + y_{t+1}S_t\), \(t=0,\dots,T\). Show that:
- \(V_t = \dfrac{\E_{\Q}[V_T \mid S_0, \ldots, S_t]}{(1+r)^{T-t}},\)
- \(\E_{\Q}[V_t] = (1+r)^t \E_{\Q}[V_0].\)