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Twelve Different Ways to Derive Black-Scholes

The twelve different ways of deriving the Black-Scholes " equation or formula that follow use different types of mathematics, with different amounts of complexity and mathematical baggage. Some derivations are useful in that they can be generalized, and some are very specific to this one problem. Naturally we will spend more time on those derivations that are most useful or give the most insight. The first eight ways of deriving the Black-Scholes equation/formula are taken from the excellent paper by Jesper Andreason, Bjarke Jensen and Rolf Poulsen (1998).

Note that the title of this chapter doesn't explicitly refer to the Black-Scholes equation or the Black-Scholes formula. That s because some of the derivations result in the famous partial differential equation and some result in the famous formula for calls and puts.

In most cases we work within a framework in which the stock path is continuous, the returns are normally distributed, there aren't any dividends, or transaction costs, etc. To get the closed-form formula (the Black-Scholes formula we need to assume that volatility is constant, or perhaps time dependent, but for the derivations of the equation relating the greeks (the Black-Scholes equation) the assumptions can be weaker, if we don't mind not finding a closed-form solution.

In many cases, some assumptions can be dropped. The final derivation, Black-Scholes for accountants, uses perhaps the least amount of formal mathematics and is easy to generalize. It also has the advantage that it highlights one of the main reasons why the Black-Scholes model is less than perfect in real life. I will spend more time on that derivation than on most of the others.

I am curious to know which derivation(s) readers prefer. Please mail your comments to This email address is being protected from spam bots, you need Javascript enabled to view it if you are aware of other derivations please let me know.

Hedging and the Partial Differential Equation

The original derivation of the Black-Scholes partial differential equation was via stochastic calculus, Ito's lemma and a simple hedging argument (Black & Scholes, 1973).

Assume that the underlying follows a lognormal random walk

Use n to denote the value of a portfolio of one long option position and a short position in some quantity A of the underlying:

The first term on the right is the option and the second term is the short asset position.

Ask how the value of the portfolio changes from time t to t + dt. The change in the portfolio value is due partly to the change in the option value and partly to the change in the underlying:

From Itô's lemma we have

The right-hand side of this contains two types of terms, the deterministic and the random. The deterministic terms are those with the dt, and the random terms are those with the dS. Pretending for the moment that we know V and its derivatives then we know everything about the right-hand side except for the value of dS, because this is random.

These random terms can be eliminated by choosing

After choosing the quantity A, we hold a portfolio whose value changes by the amount

This change is completely riskless. If we have a completely risk-free change dTl in the portfolio value n then it must be the same as the growth we would get if we put the equivalent amount of cash in a risk-free interest-bearing account:

This is an example of the no-arbitrage principle.

Putting all of the above together to eliminate n and A in favour of partial derivatives of V gives

the Black-Scholes equation.

Solve this quite simple linear diffusion equation with the final condition

and you will get the Black-Scholes call option formula.

This derivation of the Black-Scholes equation is perhaps the most useful since it is readily generalizable (if not necessarily still analytically tractable) to different underlyings, more complicated models, and exotic contracts.

 
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