Assignment 9

Assigned 2024-03-20, due 2024-03-27 at noon on Gradescope

Question 1

(Asian options) Consider a market with a bank (with interest rate $r$) and a stock. Let $S$ be a geometric Brownian motion with mean return rate $\alpha$ and volatility $\sigma$, modelling the price of a stock. Let $Y_t = \int_0^t S_s \, ds$.

  1. Let $f = f(t, x, y)$ be any function that is $C^2$ in $x, y$ and $C^1$ in $t$. Find a condition on $f$ such that $X_t = f(t, S_t, Y_t)$ represents the wealth of an self financing portfolio.

Let $g = g(x, y)$ be a function and consider a security that pays $g(S_T, Y_T)$ at time $T$. Note, if $g(x, y) = (y/T - K)^+$ then this is exactly an Asian option with strike price $K$.

  1. Suppose this security can be replicated and $f = f(t, x, y)$ is a function such that $f(t, S_t, Y_t)$ is the wealth of the replicating portfolio of this security at time $t$. Assuming $c$ is $C^1$ in $t$ and $C^2$ in $x, y$ when $t < T$, find a PDE and boundary conditions satisfied by $c$.

    [The PDE you obtain will be similar to the Black-Scholes PDE, but will also involve derivatives with respect to the new variable $y$. Unlike the case of European options, the PDE you obtain here will not have an explicit solution.]

  2. Conversely, if $f$ is the solution to the PDE you found in the previous part then show that the security can be replicated, and $f(t, S_t, Y_t)$ is the wealth of the replicating portfolio at time $t$.

Question 2

Let $Y_t = \int_0^t e^{-r} W_r \, dr$, and $X_t = W_t^2 Y_t$. Find $\E_s X_t$ and $\E X_t$. Express $\E_s X_t$ in without involving expectations or conditional expectations (you may have unsimplified It\^o or Riemann integrals). Express $\E X_t = h(t)$ for some (non-random) function $h$ that you compute explicitly.

Question 3

Let $W$ and $B$ be two independent (one dimensional) Brownian motions. Let $M$, $N$ be defined by \begin{equation*} M(t) = \int_0^t W(s) \, dB(s) \qquad\text{and}\qquad N(t) = \int_0^t B(s) \, dW(s)\,. \end{equation*} Show $\jqv{M,N} =0$. Also verify $\E M(t)^2 \E N(t)^2 \neq \E M(t)^2 N(t)^2$, and show that $M$, $N$ are not independent even though $\jqv{M,N} = 0$.

Question 4

Consider a market with one stock (whose price is denoted by $S$), and a money market account. The price of one share of the money market account is given by $C_t = e^{rt}$. At time $t$ a self-financing portfolio holds $\Delta_t$ shares of stock and $\Gamma_t$ shares of the money market account. If $X_t$ is the wealth of this portfolio, then derive the self-financing condition \begin{equation} S_t \, d\Delta_t + d\jqv{S, \Delta}_t + e^{rt} \, d\Gamma_t = 0\,. \end{equation}