Transforming the Time-Dependent Black–Scholes PDE
Summary
The document explains why a Black–Scholes partial differential equation with time-varying interest rates, dividends, and volatility may be rewritten using new variables. The original equation has coefficients that vary with the underlying price and time, which can make it harder to solve and analyze. Suitable variable transformations can simplify the equation, potentially turning it into one with constant coefficients or a form that admits an analytical solution.
The answer gives an intuitive interpretation of the transformed variables: the stock price becomes similar to a forward price, the option value becomes similar to its forward value, and transformed time roughly tracks accumulated instantaneous variance. It says the derivation follows a procedure similar to the constant-parameter case, while acknowledging that the detailed steps differ. The response offers intuition rather than a derivation, and does not specify the exact transformations or demonstrate the algebra from one equation to the next.
Key ideas
- Variable transformations can simplify a Black–Scholes PDE with time-dependent parameters.
- The original PDE has coefficients that vary with the underlying price and time.
- The transformed underlying and option values can be interpreted in forward-value terms.
- Transformed time roughly corresponds to accumulated instantaneous variance.
- The document explains the purpose of the transformation but does not show its derivation.
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# Modifying Basic Black Scholes Equation For Time Dependent Variables - Per Wilmott?
# Modifying Basic Black Scholes Equation For Time Dependent Variables - Per Wilmott?
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!
## Answer by Magic is in the chain (score 2, accepted)
https://quant.stackexchange.com/a/46753
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!Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.