Sep

7

Thanks to a helpful hint from my colleague Vince Fulco I have recently become acquainted with an academic paper that I do not think I had seen previously, and would like to remark on:

Michael Stutzer: A Simple Nonparametric Approach to Derivative Security Valuation, Journal of Finance, Vol 51 #5, December 1996, pp1633-1652

As my friend Kris Falstaff often points out, the Black-Scholes framework for option valuation is based on an erroneous assumption, that stock price changes are lognormal. Of course alternative models can be and have been developed, such as those that incorporate jumps in prices and fluctuations in volatility, to get around this limitation. But then Kris could reply "that is not the real stock price process either."

A more radical approach is to make no assumptions about the distribution of stock price changes but just use the actual changes that have been observed in the past. This would amount to using a histogram of price changes instead of an analytical form for the distribution (for example the lognormal form). If the observation period is sufficiently long this should give an accurate representation of real life stock price changes. This can be called a 'nonparametric' approach or a 'historical' or 'empirical' approach to option valuation. ('nonparametric' in this context simply means "without assuming a distribution"). The Stutzer paper gives a simple procedure to implement this approach.

In brief there are three steps:

1. Using a large amount of historical price data, compute the empirical distribution of stock price changes over the time horizon T of interest (T= the maturity of the options we are trying to value). This gives a vector RH of all the possible price changes that have occurred over intervals of length T, and a vector PIHAT that assigns a probability to each. Since we have no reason to assume any one outcome is more likely than any other to occur in the future, all the entries in PIHAT should be the same, i.e. an equiprobable distribution. For example if we have 1000 different entries in RH, we should set PIHAT(i)= 1/1000 for i=1 to 1000.

2. We transform the empirical distribution found in (1) into a risk neutral distribution. Stutzer argues this should be done using the Kullback-Leibler Information Criterion. The vector of possible outcomes RH remains the same, but the probabilities PIHAT associated with these outcomes are replaced by a different set of probabilities PISTAR. The beauty of the Kullback-Leibler Criterion is that it gives an explicit, relatively simple way to compute PISTAR:

PISTAR(i) = \frac{exp[\gamma RH(i) / r^T}{\sum_j exp[\gamma RH(j) / r^T}

where \gamma is a constant given by another relatively simple expression, and r is the interest rate.

3. We can now compute the value of any option (or other European-style derivative) by taking the expectation of the payoff under the risk neutral distribution. For example to value a call option we would compute the expectation of Max[S-E,0] over all scenarios contained in the RH/PISTAR vectors.

It is a very interesting algorithm. The part that I am not completely convinced about is the idea that the Kullback-Leibler criterion is the correct one to use to find the risk neutral distribution; Stutzer has an explanation that makes it sound plausible, but somehow it was not completely persuasive (or rigorous) to me.

This is the best published paper on empirical option pricing in my opinion (although there are not many published), and it forms the basis for Emanuel Derman's Strike-Adjusted-Spread concept, that we can talk about next time.

Laurence Glazier comments:

This is very interesting and it would be good to see a worked example. It does rest upon an assumption that previous stock price movement is to some extent predictive of the future. Can we test if this is so? Also if Black-Scholes or similar is universally believed in by options traders does that not make it effectively true in a cultural context? I would be most interested in pricing theory to see an account made of the latent energy of an option, i.e. as the stock drifts slowly up, the option is gearing up, tensing to jump to the next level, and we want to identify this point so we can buy just beforehand. I am thinking here of a spiral motion up from a kind of Argand plane — when a full revolution is made the real option price moves up.

I think the weakness of Black-Scholes is the use of Vega, which is like the god of the gaps. It is a truly useful piece of social engineering, however, which enables the industry to run.


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