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Probabilists Develop Improved Stock Pricing Model
Recently, Scholes and Merton won the Nobel prize in Economics for the work they did in the 1970's using modern probability theory (Itô stochastic calculus) to produce a rational method to price an option. [An option is the right to purchase a security at a future time at a given price; one can purchase this right, and the famous Black-Scholes formula then produces a fair price.]
Since the pioneering work of Black, Scholes, and Merton, others have attempted to refine their model. The underlying assumptions have been questioned as researchers have tried to better model the stock market and stock prices. In particular, the Black-Scholes stock price model gives continuous price processes as functions of time, while most researchers believe such price processes are discontinuous, with many little jumps. If one models the market in a discontinuous way, however, the beauty of the model is lost and in particular "market completeness" disappears. Market completeness means that whatever claim one has in the future (such as the right to buy a certain asset at time T at an agreed upon price), there always exists a trading strategy that one can follow to "replicate the claim": that is, by buying and selling according to this strategy, one will have the asset at time T with which to pay off the claim. Such a strategy is called a hedging strategy.
Mathematics Professor Philip Protter, together with Michael Dritschel of the Statistics Department, recently developed a family of models for the stock market that are both complete and discontinuous. Protter used Azéma martingales, a rather esoteric stochastic process whose genesis comes from quantum physics models. This model is only slightly different from the original Black-Scholes-Merton model in the sense that it is indexed by a parameter and converges to it as the parameter tends to 0. Dritschel and Protter tested the model against stock market tick data and found, for example, that for the stock price process of Microsoft stock, the parameter is -0.07.
Protter presented his model first at a meeting in Oberwolfach, Germany and later to the brokerage firm of Morgan-Stanley. Work is under way at Morgan-Stanley to use Protter's model to calculate option prices more precisely than is possible with the Black-Scholes model.
Professor Protter has spent part of the summer in Zurich visiting ETH, where he reported spotting financial gnomes running rampant on the Bahnhofstrasse!
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