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, $$ \begin{array}{cc} \lim_{x \to -\infty}u(x,\tau)=&e^{\frac{\tau}{4} (w+1)^2};\\ \lim_{x \to \infty}u(x,\tau)=&0. \end{array} $$
References
- Tang, Q.
Mathematical Models in Industry and Finance
, Lecture Notes University of Sussex (2008).
- Wilmott P, Howison S and Dewunne J,
The Mathematics of Financial Derivatives: A Student Introduction
, (1995).
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