Assignment 13

Assigned 2024-04-17. Due 2024-04-24 at noon on Gradescope

Question 1

Let $\alpha \in \R^m$ and $\sigma$ be a $m \times d$ matrix, and $W$ be a $d$-dimensional Brownian motion. Suppose \begin{equation} dS^i_t = \alpha_i S^i_t \, dt + S^i_t \sum_j \sigma_{i,j} dW^j_t \,. \end{equation}

  1. Show that each $S^i$ is a geometric Brownian motion. Find the mean return rate and the volatility.

  2. Find $d\jqv{S^i_t, S^j_t}$.

  3. Find $\E (S^i_t S^j_t)$.

Question 2

Consider a market with a bank and one stock. The bank has interest rate $r$, and the stock price is modelled by \begin{equation} dS^1_t = \alpha S^1_t \, dt + \sigma_{1,1} S^1_t dW^1_t + \sigma_{1,2} S^1_t dW^2_t \,. \end{equation} Here $\alpha \in \R$ and $\sigma_1, \sigma_2 > 0$.

  1. Given $\alpha_2 \in \R$, and $\sigma_{2,1}$ and $\sigma_{2,2}$, let $S^2$ be a process that satisfies \begin{equation} dS^2_t = \alpha S^2_t \, dt + \sigma_{2,1} S^2_t dW^1_t + \sigma_{2,2} S^2_t dW^2_t \,. \end{equation} Find $\sigma_{2,1}$ and $\sigma_{2,2}$ so that $S^2$ is independent of $S^1$.

  2. Find infinitely many risk neutral measures. (Note the only traded stock in the market is $S^1$. The process $S^2$ is simply a process we constructed for our convenience, and does not represent the price of a traded asset.)

  3. Explicitly find a security with maturity $T$ and an $\mathcal F_T$ measurable payoff $V_T$ so that this security can not be replicated.

Question 3

Consider a market with 3 stocks and a bank. The bank has interest rate $r$, and the stock prices are modelled by \begin{equation} dS^i = \alpha_i S^i \, dt + \sum_{j=1}^2 \sigma_{i,j} S^i_t \, dW^j_t \,, \end{equation} where \begin{equation} \alpha = \begin{pmatrix} 1 \\ 2 \\3 \end{pmatrix} \qquad \sigma = \begin{pmatrix} 1 & 1\\ 0 & 1\\ 1 & 0 \end{pmatrix}\,. \end{equation}

  1. Find all $r \in \R$ under which the market has no arbitrage.

  2. Find all $r \in \R$ under which the market is complete and arbitrage free.

  3. When the market has arbitrage, find an explicit arbitrage opportunity.