The derivation of the formulae for calls in Wilmott on QF (7.2 page 100-101) is excellent, but I would like more explanation on one step.To go from the BS EDP to the heat equation, there are some changes in variables. These changes are explained not only from the technical point of vue, but also from the financial point of vue. They are1) change from present value to future value2) reversing time : tau = T-t3) replace stock S by returns xi = log S4) 'Using the forward price of the asset instead of the spot price as a variable': x = xi + (r-1/2 vol^2)*tauCan anyone give more explanation on step 4?

The forward price is a Martingale ... fancy words, but it means that the forward price has no drift (in other words, the expected value of the stock price on the exercise date is the forward price). If one visualizes what this means, one sees that the probability distribution of the stock price on the exercise date will be centered around the forward price. This is conceptually (and computationally) simpler than using the stock price directly, since today's stock price is not in the center of the distribution of stock prices on the exercise date.If one wants the operation reason, one changes the variable to get rid of the drift (first derivative) term, and any time one gets rid of a term in a PDE, one is a step closer to solving it.

Pat, You should be teaching

pat, I am glad to see that you believe that the probabilistic risk-neutral evaluation approach can help us understand the BS PDE better.MJ

I can't help it; it's a disease which affects all fixed income type guys ... risk neutral evalutions/change of numeraire tricks to set up the problem, and then PDEs to solve it. Besides with appropriatley obscure language, even we can appear deep.

Guys,I am very glad that my question triggered all these remarks, but I would like someone to explain me how this x=xi+(r-vol^2)*tau equation is related with the forward. I thought that the forward price is related to the spot in a way that is independent of vol?Thanks