Relative Entropy Option Pricing And Positive Definite Analysis Of Discounted Factor

In the development of international financial markets, investors were bothered with finding a reasonable method for option pricing. During the past 40 years, investors made a lot of wealth with Black-Scholes-Merton option pricing model. But Black-Scholes-Merton model is not perfect. In this paper, we get the optimal risk-neutral measure through minimizing the relative entropy between risk-neutral measure and prior measure. The functional optimal problem was solved by the dual problem. We supposed the form of discounted process according to the solution of the functional optimal problem. Using the discounted process, we get the option pricing formula based on the minimization of relative entropy.We do a large amount of numerical experiments. At first, we research the relationship between two parameters of discounted process, discounted factor and risk aversion factor, and other parameters in the problem. It's easy to know the risk aversion factor is always less than zero. Moreover, we find that the discounted factor is always greater than zero through numerical experiments. We show the fact under reasonable conditions. Then, we study the relationship between the Black-Scholes-Merton option pricing model and option pricing model by mini-mizing the relative entropy. We find that the price from the traditional Black-Scholes-Merton model is always greater than or equal to the price from our model.