Risk Latte - Understanding the Black-Scholes Partial Differential Equation (PDE)

Understanding the Black-Scholes Partial Differential Equation (PDE)

Team latte
August 9, 2009

The other day we had a little discussion about the Black-Scholes partial differential equation (PDE) with a couple of trainee traders in a bank. Most of these guys had a very good understanding of the Black-Scholes model and all its pitfalls as well as how to adapt the model to real life trading (if ever they were to use it). However, one thing that became clear was that there was very little intuitive understanding of the underlying partial differential equation (PDE) – the governing equation, if you will – for the Call option, or for that matter, any other financial derivative for which a closed for solution can be derived.

One of the best and perhaps the most intuitive explanations of Black-Scholes partial differential equation (PDE) is given by Paul Wilmott*.

A Black-Scholes partial differential equation is a generalized diffusion equation that we find in Physics. The generalized PDE is given by:

...................................... (1)

Where, , and are constants and is the value of the financial derivative (e.g. a call option) and is the stock (equity) price. The key to understanding the Black-Scholes PDE is to understand what the constants , and are and how they drive the above PDE.

The two critical, and brilliant, observations that Wilmott makes is that both cash (money in the bank) and equity (stock) have to be solutions of the above PDE if the above PDE has to hold for all derivatives.

Cash is the simplest of all financial instruments and hence it is the smallest decomposable fragment (SDF) of all derivative contracts. Therefore, cash has to be a solution of the above. If that is the case then is a solution of the above PDE, where is the initial value of the cash in the bank (this is constant) and is the interest rate in the economy. Therefore, differentiating we have,

Substituting the above in the PDE gives:

Therefore, the value of the last coefficient, is .

Equity (stock), , can itself be considered as an option (a financial derivative). Equity can be thought of as a call option with zero strike. Therefore, the stock price itself has to be a solution of the PDE. If this is the case, i.e. if then we get:

Substituting the above in the PDE gives:

Thus, the value of the second coefficient, is .

Therefore, even without going into any detailed finance theory or stochastic calculus we have estimated the value of two out of three coefficients of the Black-Scholes PDE by making two simple, yet fundamental assumptions. The Black-Scholes PDE now becomes:

...................................... (2)

The only remaining coefficient is the first one, . Actually, it can also be shown implicitly (read Wilmott for more on this) that this coefficient is the coefficient of diffusion and is related to the term, .

The real intuition behind the Black-Scholes PDE can be had if one tries to understand how a synthetic portfolio can be created with cash and equity to replicate a call option and how the rate of return of this portfolio will be the risk free rate. We get a glimpse of that from Wilmott’s analysis.

References:   *Paul Wilmott Introduces Quantitative Finance
                    FAQ in Quantitative Finance by Paul Wilmott

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