The Application Of Principle Of Maximum Entropy On Option Pricing

Since the introduction of option trading on bonds and other underlying assets,it has been one of the most challenging subjects about option pricing.Chicago Board Options Exchange (CBOE)opened in 1973 and at the same time, American scholars Fischer Black and Myron Scholes proposed a model of option pricing,which is the famous Black-Scholes Model (hereinafter referred to as B-S model),and they got the model from a series of hypothesis and rigorous mathematical demonstrations. Binomial Model or Binomial tree proposed by Cox, Ross and Rubinstein in 1979 is also an important method. Compared with B-S Model, it did not ask for better mathematical knowledge and skills and simplified the B-S Model . In general ,Binomial Model is accordance with the B-S Model,so the results from the two method is the same.In the ensuing period, many scholars and researchers are committed to continuous efforts to improve and develop the B-S model and optimize Binomial Model.Meantime there are lots of scholars to work out another way to option pricing.Now the principle of optimality, which lies in Information Theory, has been attached importance in the option pricing.The Maximum Entropy and Minimum Cross-Entropy Distributions are used to the problem of option pricing. In 1996, Stuzer M employed the Minimum Cross-Entropy Distributions to construct the model of option pricing, using the historical return function as a prior distribution.And the same year, Peter W.Buchen and Michael Kelly used the Principle of Maximum Entropy to estimate the neutral-risk probability density, then give the answer to the option price.Assuming risk neutral world , the author selected the problem of maximum entropy subjected to the constraints to deduce and solve the probability density of underlying assets, furthermore get the model of option pricing.First, the constraints are derived from a series of Call and Digital option prices at different strikes and expressed as the form of expectation; Sencond, build up the function according to the combination of maximus problem and the constraints and find the probability density distribution about the underlying assets through the rigorous mathematical deduction and proof.