Pricing Options Based On Esscher Transform

As an important part of the financial derivatives,the theory of option pricing is one of the core issues of the financial mathematics.How to give the option pricing has become a hot topic for many scholars including mathematician,economist and so on.We can obtain an equivalent martingale measure by the Esscher transform and get some pricing formulas of options by Girsanov theorem and the theory of change of numeraire etc.The main contents of this thesis are as follows:1.Assume that the underlying risk asset price follows the differential equation of the continuous diffusion process under the risk neutral measure.We introduce the definition of the digital power option to extend the application of power option.And then,we use the Esscher transform and make it as Radon-Nikodym derivative to determine an equivalent martingale measure.We obtain the pricing formulas of the Bi-direction European option,the geometric average Asian option and the digital power option respectively by applying Girsanov theorem.2.Suppose that the underlying asset follows the differential equation with the jump diffusion process under the real probability measure.Using the Esscher transform,we give an equivalent martingale measure—the risk neutral measure equivalent to the original measure by choosing some suitable parameters.Under the risk neutral measure,the intensity and the amplitude of the jump diffusion process change and the pricing formula of the digital power option based on the jump diffusion process is obtained to extend the pricing model of the above digital power option.3.With the help of the Esscher transform theory,we redefine the underlying asset model with the jump diffusion process under the risk neutral measure.Next,we introduce an underlying asset which follows the continuous diffusion process.Two types of exchange options pricing formulas are obtained:one taken the underlying asset with the jump diffusion process as the strike price and the other taken the underlying asset without the jump diffusion process as the strike price respectively by applying change of numeraire.4.Studying the pricing models of the digital power option,the exchange option and the power exchange option,we propose the definition of the digital power exchange option and obtain its pricing formula by applying the Esscher transform theory.It extends the application scope of the power exchange option model.It also extends the application scope of the digital power option and the exchange option.5.Under the domestic real probability measure,suppose that the exchange rate follows the differential equation with the jump diffusion process.By the Esscher transform theory,we obtain a new martingale measure equivalent to the domestic real probability measure.The new measure is the risk neutral measure under which we redefine the differential equation of the exchange rate.Further,we get two types of option pricing formulas:one taken the price of the domestic currency₁at maturity time as the strike price and the other taken the domestic risk asset as the strike price.Based on these option pricing formulas,we take some numerical experiments to simulate and analyze these pricing models in Matlab R2013.