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Utility Theory

The utility theory approach is not exactly the most useful of the twelve derivation methods, requiring that we value from the perspective of a particularly unrepresentative investor, an investor with a utility function that is a power law. This idea was introduced by Rubinstein (1976). Even though not the best way to derive the famous formula utility theory is something which deserves better press than it has received so far.

The steps along the way to finding the Black-Scholes formula are as follows. We work within a single-period framework, so that the concept of continuous hedging, or indeed anything continuous at all, is not needed. We assume that the stock price at the terminal time (which will shortly also be an option's expiration) and the consumption are both log-normally distributed with some correlation. We choose a utility function that is a power of the consumption. A valuation expression results. For the market to be in equilibrium requires a relationship between the stock's and consumption's expected growths and volatilities, the above-mentioned correlation and the degree of risk aversion in the utility function. Finally, we use the valuation expression for an option, with the expiration being the terminal date. This valuation expression can be interpreted as an expectation, with the usual and oft-repeated interpretation.

Taylor Series

Taylor series is just a discrete-time version of Ito's lemma. So you should find this derivation of the Black-Scholes partial differential equation very simple.

V (S, t) is the option value as a function of asset S and time t. Set up a portfolio long the option and short A

of the stock:

Now look at the change in this portfolio from time t to t + St, with St being a small time step:

is a discrete-time model for the stock and 0 is a random variable drawn from a normal distribution with zero mean and unit standard deviation. (Aside: Does it matter that 0 is normally distributed? It's a nice little exercise to see what difference it makes if 0 comes from another distribution.)

Now expand STl in Taylor series for small St to get

where all terms are now evaluated at S and t. The variance of this expression is

which is minimized by the choice

Now put this choice for A into the expression for Sn, and set

This is a bit naughty, but I'll come back to it in a second.

Take the resulting equation, divide by St so the leading terms are 0(1) and let St 0. Bingo, you have the Black-Scholes partial differential equation, honest.

The naughty step in this was setting the return on the portfolio equal to the risk-free rate. This is fine as long as the portfolio is itself risk free. Here it is not, not exactly; there is still a little bit of risk. The variance, after choosing the best A,is 0(St3/2). Since there are 0(T/St) rehedges between the start of the option's life and expiration, where T is the time to expiration, the total variance does decay to zero as St 0, thank goodness. And that's the a posteriori justification for ignoring risk.

(In Wilmott, 1994, this analysis goes to higher order in St to find an even better hedge than the classic Black-Scholes -one that is relevant if St is not so small, or if gamma is large, or if you are close to expiration. In that paper there is a small typo, corrected in Wilmott, 2006.)

In our final derivation you will see a less mathsy version of this same argument.

Mellin Transform

This derivation (see Yakovlev & Zhabin, 2003) is another one that I am only going to do in spirit rather than in detail. Again it is in discrete time but with continuous asset price. In that sense it's rather like the previous derivation and also our final derivation, Black-Scholes for accountants, only far, far more complicated.

Vk(S) is the option value when the stock price is S at the kth time step. You set up a portfolio, П, long one option and short a quantity, A, of the underlying asset. The delta is chosen to minimize the variance of this portfolio at the next time step; the resulting expression for A involves the covariance between the stock and the option.

The pricing equation is then

with the obvious notation. There is a slight problem with this, in that there is really no justification for equating value and expectation, at least not until you look at (or check a posteriori) the total variance at expiration and show that it is small enough to ignore (if the time steps are small enough). Anyway...

This equation can be rewritten just in terms of V as

And since we know the option value at expiration, this is Vo(S) then we can in principle find Vk(S)for all k. Next you go over to the Mellin transform domain (this is not the place to explain transform theory!)

So far none of this has required the stock to be lognormally distributed, it is more general than that. But if it is lognormal then the above iteration will result in the class formula for calls and puts.

 
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