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I am reading Wilmott's book and don't understand why he makes the following step to re-write the PDE. I get equation 8.4, that's just the typical PDE for a dividend yielding stock where r(t), D(t) and sigma(t) are now subbed in to represent they are time dependent.

However, I don't understand the meaning nor the rational of introducing new variables. Why do we re-define the option value? What is time bar? Can anyone explain how to get from 8.4 to 8.5?

Thanks!

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The PDE, in its original form, has got variable coefficients -depend on S and t - e.g., co-efficient of $\frac{\partial^2 V}{\partial S^2}$ has got $\sigma(t)$ and S. They are hard to solve and analyse, and if one can find some transformations of variables, that reduce it to constant coefficient, then it gets a lot easier.

P Wilmott has chosen the three transformations, though these are not the only transformations that would achieve the desired outcome. There are well defined procedures in the PDE world for determining suitable transformations, but that is another topic. For the present purpose, the main aim of these transformations is to get rid of the variable coefficient and simplify the PDE as much as one can, possibly to a form that admits analytical solution. And his transformations achieve that.

You can intuitively interpret his transformations as follows:

$\bar{S}$ is like transforming the stock price to the forward price of the stock.

$\bar{V}$ is transforming the option premium to its forward value.

and $\bar{t}$ is roughly transforming time to total instantaneous variance. Remember there is a very good link between diffusion coefficient and time.

And re-question, how does one go from 8.4 to 8.5, It is very similar to the case when the parameters do not depend on t. The steps would be slightly different but the procedure is the same. You can find the details here:

Transformation from the Black-Scholes differential equation to the diffusion equation - and back

Or in this video:

https://youtu.be/IgMoOcO095U

PS and declaration: I have contributed to the video!

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