Overview
The Black Scholes equation
{% \frac{\partial V}{\partial t} + \frac{1}{2} \sigma ^2 \frac{\partial^2 V}{\partial S^2} + rS \frac{\partial V}{\partial S} -rV = 0 %}
can be shown through a change of variables to the 1-dimensional
Heat Equation
{% \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} %}
Derivation
Start with a Change of Variables
{% S = e^x, t = T - \frac{2\tau}{\sigma^2} %}
The {% V %} stated in terms of the new variables becomes
{% V(S,t) = v(x,\tau) = v(ln(S), \frac{\sigma^2}{2}(T-t)) %}
The following partial derivatives follow:
{% \frac{\partial V}{\partial t} = -\frac{\sigma^2}{2} \frac{\partial v}{\partial \tau} %}
{% \frac{\partial V}{\partial S} = -\frac{1}{S} \frac{\partial v}{\partial x} %}
{% \frac{\partial^2 V}{\partial S^2} = - \frac{1}{S^2} \frac{\partial v}{\partial x} + \frac{1}{S^2}\frac{\partial^2 v}{\partial x^2} %}
{% \frac{\partial v}{\partial \tau} =\frac{\partial^2 v}{\partial x^2} + (\frac{2r}{\sigma^2} -1)\frac{\partial v}{\partial x} - \frac{2r}{\sigma^2}v %}
Then, make the following definitions
{% \kappa = \frac{2r}{\sigma^2} %}
{% t=\tau %}
The equation now becomes:
{% \frac{\partial v}{\partial t} = \frac{\partial^2 v}{\partial x^2} + (\kappa - 1)\frac{\partial v}{\partial x} - \kappa v %}
{% 0 <= t <= \frac{\sigma^2}{2}T %}
When the bounary values of the Black Scholes equation are given by {% f(S) %}, then the
boundary values for the new equation are given by
{% v(x,0) = V(e^x, T) = f(e^x) %}
then define a new function {% u(x,t) %} by
{% v(x,t) = e^{\alpha x + \beta t}u(x,t) = \phi u %}
the partial derivatives can be computed as
{% \frac{\partial v}{\partial t} = \beta \phi u + \phi \frac{\partial u}{\partial t} %}
{% \frac{\partial v}{\partial x} = \alpha \phi u + \phi \frac{\partial u}{\partial x} %}
{% \frac{\partial^2 v}{\partial x^2} = \alpha^2 \phi u + 2 \alpha \phi \frac{\partial u}{\partial x} + \phi \frac{\partial^2 u}{\partial x^2} %}
Then defining {% \alpha %} and {% \beta %} as
{% \alpha = - \frac{1}{2}(k-1) = \frac{\sigma^2 - 2r}{2\sigma^2} %}
{% \beta = \frac{1}{4}(k+1)^2 = -(\frac{\sigma^2 + 2r}{2\sigma^2})^2 %}
Then the final equation becomes
{% \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} %}
where the range of {% t %} is given by
{% 0 <= t <= \frac{\sigma^2}{2} T %}
{% u(x,0) = e^{-\alpha x} v(x,0) = e^{-\alpha x}f(e^x) %}