Assignment 12

Assigned 2024-04-10, due 2024-04-17 at noon on Gradescope

Question 1

Consider a financial market consisting of a stock and a money market account. Suppose the money market account has a constant return rate $r$, and the stock price follows a geometric Brownian motion with mean return rate $\alpha$ and volatility $\sigma$. Here $\alpha$, $\sigma$ and $r > 0$ are constants. Let $\beta, K, T > 0$ and consider a derivative security that pays $(S_T^\beta - K)^+$ at maturity $T$. Compute the arbitrage free price of this security at any time $t \in [0, T)$. Your answer may involve $r$, $\sigma$, $K$, $t$, $T$, $S$, and the CDF of the normal distribution, but not any integrals or expectations.

[Hint: The simplest way to solve this problem is to use the risk neutral pricing formula, along with the explicit Black-Scholes formula you already know.]

Question 2

Consider a financial market consisting of a stock and a money market account. Suppose the money market account has a constant return rate $r$ and the stock price is given by a stochastic process $S$ such that \begin{equation} dS_t = \alpha_t S_t \, dt + \sigma_t S_t \, dW_t\,. \end{equation} Here $\alpha = \alpha_t$ is an adapted process, and $\sigma = \sigma(t)$ is a given, non-random, function of~$t$. Let $K, T > 0$ and consider a European call option on $S$ with strike $K$ and maturity~$T$. Given $t \in [0, T)$, find the arbitrage free price of this option at time $t$. Your final answer may involve $r$, $\sigma$, $t$, $T$, $K$, $S_t$, the cumulative distribution function of the standard normal, and Riemann integrals of powers of $\sigma$.

[Hint: Under risk neutral measure find a normally distributed random variable $Y$ such that $S_T = S_t e^Y$, and $Y$ is independent of $S_t$.]

Question 3

A simplified version of the Vasivečk and Ho-Lee model stipulates that the interest rate $R(t)$ is given by \begin{equation} R(t) = r_0 + \theta t + \kappa \tilde B(t)\,, \end{equation} where $r_0$, $\kappa > 0$, $\theta \in \R$ and $\tilde B$ is a Brownian motion under the risk neutral measure $\tilde \P$. Consider a bond that pays $\$1$ at maturity time $T$. Compute the arbitrage free price of this bond at time $0$. Express your answer in terms of $r_0$, $T$, $\theta$ and $\kappa$, without involving expectations or integrals.

Question 4

Let $W$ be a 2-dimensional Brownian motion, $\beta, \theta$ be two processes such that and \begin{equation} dX_t = \beta_t \, dt + \sin(\theta_t) \, dW^1_t + \cos(\theta_t) \, dW^2_t \,. \end{equation} Find infinitely many equivalent measures $\tilde\P$ such that $X$ is a Brownian motion under $\tilde\P$. (You may assume that $\beta$ is nice enough that the Girsanov theorem applies.)