Write down the Black-Scholes partial differential equation (PDE), and explain briefly its relevance to option pricing.

Show how a change of variables reduces the Black-Scholes PDE to the heat equation:

$$\begin{aligned} &\frac{\partial f}{\partial t}+\frac{1}{2} \frac{\partial^{2} f}{\partial x^{2}}=0 \text { for all }(t, x) \in[0, T) \times \mathbb{R}, \\ &f(T, x)=\varphi(x) \text { for all } x \in \mathbb{R} \end{aligned}$$

where $\varphi$ is a given boundary function.

Consider the following numerical scheme for solving the heat equation on the equally spaced grid $\left(t_{n}, x_{k}\right) \in[0, T] \times \mathbb{R}$ where $t_{n}=n \Delta t$ and $x_{k}=k \Delta x, n=0,1, \ldots, N$ and $k \in \mathbb{Z}$, and $\Delta t=T / N$. We approximate $f\left(t_{n}, x_{k}\right)$ by $f_{k}^{n}$ where

$$0=\frac{f^{n+1}-f^{n}}{\Delta t}+\theta L f^{n+1}+(1-\theta) L f^{n}, \quad f_{k}^{N}=\varphi\left(x_{k}\right)$$

and $\theta \in[0,1]$ is a constant and the operator $L$ is the matrix with non-zero entries $L_{k k}=-\frac{1}{(\Delta x)^{2}}$ and $L_{k, k+1}=L_{k, k-1}=\frac{1}{2(\Delta x)^{2}}$. By considering what happens when $\varphi(x)=\exp (i \omega x)$, show that the finite-difference scheme $(*)$ is stable if and only if

$$1 \geqslant \lambda(2 \theta-1),$$

where $\lambda \equiv \Delta t /(\Delta x)^{2}$. For what values of $\theta$ is the scheme $(*)$ unconditionally stable?