Assignment 14
Assigned 2024-04-24. Not due
Question 1
Let $X$ be an It\^o process such that $dX_t = t^2 \, dt + t \, dW_t$. Fix $T > 0$, and let $Z_T$ be an $\mathcal F_T$-measurable random variable such that $Z_T > 0$ and $\E Z_T = 1$. Define a new measure $\tilde\P$ by $d\tilde\P = Z_T \, d\P$. Find a formula for $Z_T$ so that the process $X$ is a martingale under $\tilde\P$.
Question 2
Let $B$ and $W$ be two independent, standard, one dimensional Brownian motions. Compute \begin{equation} \E \int_0^{B(t)^2} W(s)^2 \, ds \,. \end{equation} Express your answer as a function of $t$ without involving $W$, $B$, integrals, expectations or probabilities.
Question 3
Let $S$ be a geometric Brownian motion with mean return rate $\alpha$ and volatility $\sigma$. Given $T > 0$ and a non-random function $f$, the Markov property guarantees that there exists a non-random function $g$ such that for any $t \leq T$ we have \begin{equation} \E \paren[\big]{ f(S(T)) \given \mathcal F_t } = g(t, S(t))\,. \end{equation} Find non-random functions $h_1$, $h_2$, $h_3$ (that may depend on $x$, $t$, $\alpha$, and $\sigma$, but not on $S$, $T$, $f$ or $g$) such that \begin{equation*} \partial_t g(t, x) = h_1(t, x) \partial_x g(t, x) + h_2(t, x) \partial_x^2 g(t, x) + h_3(t, x) \end{equation*} Note: You are not required to find a formula for $g$ itself.
Question 4
Find an equivalent measure $\tilde\P$ under which $2W_t - t^2$ is a martingale. If $0 \leq s \leq t$, then compute $\tilde\E_s W_t$, and express your answer without using expectations or integrals.
Question 5
Let $c(t, x)$ be given by the Black–Scholes formula. Compute $\partial_x c(t, x)$ by differentiating the risk neutral pricing formula, and use this to provide a (shorter) proof that $\partial_x c = N(d_+)$.
Question 6
Find the forward price of a dividend paying stock. Also find the trading strategy to replicate the corresponding forward contract.
[Recall the forward price at time $t$ is the price at which the forward contract is worth nothing at time $t$.]