Equivalence Formula Of Option Pricing Under The Jump-diffusion Model

In all financial derivative products pricing，the option pricing theory has always beenconcerned，and is one of the main problems of financial engineering. B-S option pricingmodel was found not suitable for the reality of the financial markets，so researchers haveproposed some option pricing models which have broader scope，jump-diffusion optionpricing model is one of them.There are many method of pricing options. In this paper，the method which the paperused is no-arbitrage pricing and Fourier transform. No-arbitrage pricing is based on theprinciple of risk-neutral pricing，the present value of all future expected loss of Financialderivative securities represented in the form of mathematical expectation, then by theequivalent martingale theory simplify or calculate the mathematical expectation. Fouriertransform method is to represent the price of the option price with continuous stateArrow-Debreu securities portfolio by Fourier transform.In this paper, we first recount the development of option pricing theory and the optionsconditions，in addition，we briefly review the basics of option pricing theory andmathematical knowledge used in the paper，such as the martingale，Ito processes and Itoformula，Girsanov theorem，characteristic functions，and the Fourier transformation.The Second part of this paper，we used two methods to price European call optionwhich obey B-S option pricing model. One method is based on no-arbitrage pricing，theother is based on the Fourier transform，then we prove that the option pricing formula of thetwo pricing methods are equivalent.The third part of this paper，we discuss the option pricing problem under thecircumstances of which the underlying asset price is jump-diffusion process，we use twomethods to obtain the European call option pricing formula，and prove that these twopricing formulas are equivalent. Moreover，we assume that distribution of the jump size ofthe jump-jump process is certain，and obtain the specific pricing formula of European calloption，then we prove that the two pricing formula are equivalent.