# Intuition behind the change of variable of BS into Heat Equation

$$\tau=T-t$$ $$u=Ce^{r\tau}$$ $$x=\ln(\frac{S}{K})+(r-\frac{1}{\sigma^2})\tau$$ The first transform is reverse in time. The second is reversely discounted call price. But I find it hard to see the intuition behind the third transformation. And I am also wondering why this is not a one-to-one transform of time, call price, and price of the underlying. Why is 'multiplying by a Jacobian' doesn't work in this case?

the third one is really two transformations:

$$y = \log (S/K)$$ $$x= y + (r-0.5\sigma^2)\tau$$

Move to log coordinates -- the stock followed geometric Brownian motion so it's log follows Brownian motion. So the equation should be simpler in log coordinates.

The second one is remove the drift. The log has drift $r-0.5\sigma^2$ so removing that changes to coords in which the log stock is driftless.

(there are actually two ways to do the reduction to the heat equation, see Concepts and Practice etc by me)

• Can you add in more details? e.g. how to compose the first and the second together. How do I see Concepts and Practice etc? – ZHU Apr 5 '17 at 4:23
• Re the "Concepts and Practice" book see here markjoshi.com/concepts – noob2 Apr 5 '17 at 18:30