Assignment 7

Assigned 2024-02-28, due 2024-03-13 at noon on Gradescope

Question 1

This problem outlines how you would go about solving the Black-Scholes-Merton PDE. Suppose $f = f(t, x)$ solves $\partial_t f + r x \partial_x f + \frac{\sigma^2 x^2}{2} \partial_x^2 f = r f$, with boundary conditions $f(t, 0) = 0$, linear growth as $x \to \infty$, and terminal condition $f(T, x) = g(x)$ for some given function $g$.

  1. Set $y = \ln x$ and compute $\partial_x f$, $\partial_x^2 f$ in terms of $y$, $\partial_y f$ and $\partial_y^2 f$. Use this to find constants $\beta_1$, $\beta_2 \in \R$ such that $\partial_t f + \beta_1 \partial_y f + \beta_2 \partial_y^2 f = r f$.

  2. Let $\tau = T - t$, $z = y + \gamma_2 \tau$ and $v(\tau, z) = e^{\gamma_1 \tau}f(t, y)$. Find $\gamma_1$ and $\gamma_2$ so that $\partial_\tau v = \kappa \partial_z^2 v$ for some constant $\kappa > 0$. Express $\gamma_1$, $\gamma_2$ and $\kappa$ in terms of $\sigma^2$ and $r$.

The equation you obtained for $v$ above is called the heat equation, whose solution formula can be found in any standard PDE book. Namely, if we set $h(y) = v(0, y)$, then at times $\tau > 0$ the function $v$ is given by \begin{equation} v(\tau, y) = \frac{1}{\sqrt{4 \pi \kappa \tau}} \int_\R h(y-z) \exp\paren[\Big]{\frac{-z^2}{4\kappa \tau}} \, dz \end{equation} This is very similar to the formula you should have obtained in a previous homework question.

  1. (Optional) Using the above formula for $v$, substitute back and derive the solution formula given in Proposition 6.8 in the notes.

    [Note: While this is good practice, it is a little tedious. We will derive the formula in class using risk neutral measures.]

Question 2

Use the solution formula in Proposition 6.8 to derive the Black-Scholes formula for the price of an European call with strike $K$ and maturity $T$ (as stated in Corollary 6.9 in the notes).

Question 3

Consider a market a bank and a stock. The interest rate in the bank is $r$, and the price of the stock is modelled by a geometric Brownian motion with mean return rate $\alpha$ and volatility $\sigma$. Let $S_t$ denote the spot price of the stock at time $t$. Consider a derivative security that pays $g(S_T)$ at maturity time $T$, for some increasing function $g$. True or false: The replicating portfolio for this security is always long on the stock. Prove it, or find a counter example.