Transform the Black-Scholes PDE to the Heat Equation

Given the Black-Schole partial differential equation $$- r V_t + \frac{\partial V}{\partial t} + r S \frac{\partial V}{\partial S} + \frac{1}{2} S_t^2 \sigma^2 \frac{\partial^2 V}{\partial S^2}= 0,$$ the variables are non-dimensionalized as follows: $\hat{t}=\frac{t}{T}$, $\hat{S}=\frac{S}{E}$, and $\hat{V}=\frac{V}{E}$.

The partial derivatives of $\hat{V}$ are calculated as follows: $$\begin{array}{c} \frac{\partial V}{\partial t}= \frac{E}{T} \frac{\partial \hat{V}}{\partial \hat{t}} \\ \frac{\partial V}{\partial t}=\frac{\partial \hat{V}}{\partial \hat{S}}\\ \frac{\partial^2 V}{\partial S^2}=\frac{1}{E}~\frac{\partial^2 \hat{V}}{\partial \hat{S}^2}. \end{array}$$ We then get $$- \beta \hat{V}_{\hat{t}} + \frac{\partial \hat{V}}{\partial \hat{t}} + \beta \hat{S} \frac{\partial \hat{V}}{\partial \hat{S}} + \alpha \hat{S}^2 \frac{\partial^2 \hat{V}}{\partial \hat{S}^2}= 0,$$ where $\alpha=\frac{1}{2} T \sigma^2$ and $\beta=rT$.

Let $x=\mathrm{log} \hat{S}$, $\tau=\alpha-\alpha \hat{t}$ and $\hat{V}(\hat{S},\hat{t})=v(x,\tau)$ and compute the following partial derivatives $$\begin{array}{c} \frac{\partial \hat{V}}{\partial \hat{t}}= -\alpha \frac{\partial v}{\partial \tau}\\ \frac{\partial \hat{V}}{\partial \hat{S}}=\frac{1}{\hat{S}} \frac{\partial v}{\partial x}\\ \frac{\partial^2 \hat{V}}{\partial \hat{S}^2}=\frac{1}{\hat{S}^2} \left( \frac{\partial^2 v}{\partial x^2} - \frac{\partial v}{\partial x} \right). \end{array}$$ This gives $$v_t - v_{xx}+(1-w)~v_x+w~v= 0,$$ where $w=\frac{\beta}{\alpha}=\frac{2r}{\sigma^2}$.

Let $v(x,\tau)=e^{\lambda x + \mu \tau} u(x,\tau)$ and by choosing $\lambda=\frac{1-w}{2}$ and $\mu=-\frac{(1+w)^2}{4}$, we get the heat equation $$u_t - u_{xx}= 0.$$ Making the relevant substitutions for a Put option, we get the initial condition of $$u(x,0)=\mathrm{max}\left(e^{\frac{w-1}{2}x}-e^{\frac{w+1}{2}x},0 \right).$$ Similarly, the following boundary conditions are obtained, $$\begin{array}{cc} \lim_{x \to -\infty}u(x,\tau)=&e^{\frac{\tau}{4} (w+1)^2};\\ \lim_{x \to \infty}u(x,\tau)=&0. \end{array}$$

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References

1. Tang, Q. Mathematical Models in Industry and Finance, Lecture Notes University of Sussex (2008).
   
2. Wilmott P, Howison S and Dewunne J, The Mathematics of Financial Derivatives: A Student Introduction, (1995).