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4.3.1 Transformation to log-normal variables $ \spadesuit$

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The log-normal distribution of the price increments ($ \kappa=1$ in 3.3.1#eq.1) chosen to derive the Black-Scholes equation (3.4#eq.4) shows that the asset price $ S$ and the time $ t$ are in fact not a natural choice of variables for the price of an option that expires at a time $ t=T$ . This motivates a transformation from financial variables $ V(S,t)$ to log-normal variables $ v(x,\tau)$ defined by

$\displaystyle S = K\exp(x), \hspace{1cm} t=T-2\tau/\sigma^2, \hspace{1cm} V=K v(x,\tau)$ (4.3.1#eq.1)


Substitute these in the Black-Scholes equation (be careful with the second derivative)

$\displaystyle \frac{\partial v}{\partial \tau} -\frac{\partial^2 v}{\partial x^2} -(k_2-1)\frac{\partial v}{\partial x} +k_1 v = 0$ (4.3.1#eq.2)


showing that only two dimensionless parameters in fact characterize the problem

$\displaystyle k_1=\frac{2r}{\sigma^2} , \hspace{1cm} k_2=\frac{2(r-D)}{\sigma^2}$ (4.3.1#eq.3)


With a little more algebra, you can verify that further scaling by

$\displaystyle V=Kv=K\exp\left[-\frac{1}{2}(k_2-1)x -\left(\frac{1}{4}(k_2-1)^2 +k_1\right)\tau\right] u(x,\tau)$ (4.3.1#eq.4)


transforms Black-Scholes into a normalized diffusion equation

$\displaystyle \frac{\partial u}{\partial \tau} -\frac{\partial^2 u}{\partial x^2} = 0$ (4.3.1#eq.5)


which bears a strong resemblance with the heat-equation from engineering sciences. This equation has to be solved for $ x \in [-\infty; \infty]$ , $ \tau\ge 0$ using boundary $ u(-\infty, \tau)$ , $ u(+\infty, \tau)$ and initial conditions $ u(x,0)$ that have to be derived from no-arbitrage arguments with financial variable $ V(S,t)$ via the transformations (4.3.1#eq.1 and 4.3.1#eq.4). .

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