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Martingales

The martingale pricing methodology was formalized by Harrison & Kreps (1979) and Harrison & Pliska (1981).[1]

We start again with

The Wt is Brownian motion with measure P. Now introduce a new equivalent martingale measure Q such that

Under Q we have

Introduce

The quantity er(T t)Gt is a Q-martingale and so

for some process at. Applying Ito's lemma,

This stochastic differential equation can be rewritten as one representing a strategy in which a quantity aGt/oS of the stock and a quantity (G — aGt/o)er(T-t) of a zero-coupon bond maturing at time T are bought:

Such a strategy is self financing because the values of the stock and bond positions add up to G. Because of the existence of such a self-financing strategy and because at time t = T we have that Gt is the call payoff we must have that Gt is the value of the call before expiration. The role of the self-financing strategy is to ensure that there are no arbitrage opportunities.

Thus the price of a call option is

The interpretation is simply that the option value is the present value of the expected payoff under a risk-neutral random walk.

For other options simply put the payoff function inside the expectation.

This derivation is most useful for showing the link between option values and expectations, as it is the theoretical foundation for valuation by Monte Carlo simulation.

Now that we have a representation of the option value in terms of an expectation we can formally calculate this quantity and hence the Black-Scholes formulae. Under Q the logarithm of the stock price at expiration is normally distributed with mean m = Xn(Stj + (r — <r2) (T — t) and variance v2 = a2(T t). Therefore the call option value is

A simplification of this using the cumulative distribution function for the standardized normal distribution results in the well-known call option formula.

Change of Numeraire

The following is a derivation of the Black-Scholes call (or put) formula, not the equation, and is really just a trick for simplifying some of the integration.

It starts from the result that the option value is

This can also be written as

where H(S — K) is the Heaviside function, which is zero for S < K and 1 for S > K.

Now define another equivalent martingale measure Q' such

that

The option value can then be written as

It can also be written as a combination of the two expressions,

Notice that the same calculation is to be performed, an expectation of H(S — K), but under two different measures. The end result is the Black-Scholes formula for a call option.

This method is most useful for simplifying valuation problems, perhaps even finding closed-form solutions, by using the most suitable traded contract to use for the numeraire.

The relationship between the change of numeraire result and the partial differential equation approach is very simple, and informative.

First let us make the comparison between the risk-neutral expectation and the Black-Scholes equation as transparent as possible. When we write

we are saying that the option value is the present value of the expected payoff under the risk-neutral random walk

The partial differential equation

means exactly the same because of the relationship between it and the Fokker-Planck equation. In this equation the diffusion coefficient is always just one half of the square of the randomness in dS. The coefficient of d V/dS is always the risk-neutral drift rS and the coefficient of V is always minus the interest rate, — r, and represents the present valuing from expiration to now.

If we write the option value as V = SV then we can think of V as the number of shares the option is equivalent to, in value terms. It is like using the stock as the unit of currency. But if we rewrite the Black-Scholes equation in terms of V using

then we have

The function V can now be interpreted, using the same comparison with the Fokker-Planck equation, as an expectation, but this time with respect to the random walk

And there is no present valuing to be done. Since at expiration we have for the call option

we can write the option value as

Change of numeraire is no more than a change of dependent variable.

  • [1] If my notation changes, it is because I am using the notation most common to a particular field. Even then the changes are minor, often just a matter of whether one puts a subscript t on a dW for example.
 
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