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### 4.3.1 Transformation to log-normal variables

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

 (4.3.1#eq.1)

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

 (4.3.1#eq.2)

showing that only two dimensionless parameters in fact characterize the problem

 (4.3.1#eq.3)

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

 (4.3.1#eq.4)

transforms Black-Scholes into a normalized diffusion equation

 (4.3.1#eq.5)

which bears a strong resemblance with the heat-equation from engineering sciences. This equation has to be solved for , using boundary , and initial conditions that have to be derived from no-arbitrage arguments with financial variable via the transformations (4.3.1#eq.1 and 4.3.1#eq.4). .

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