Actuarial Methods In Option Pricing Applications And Research

Arbitrage is a usual transaction behavior in order to make profits.It is the possibilitity of riskless profits.Arbitrage is an internal power to make the market efficient,all the prices in the market reflect all the information entirely and immediately.Arbitrage pricing theory,which is based on the efficiency of the market,uses equilibrium methodology.The theory can be applied to the pricing of financial products and their derivative products widely.The problem of option pricing id one of key problems in financial mathematics.Under the condition of efficient,equilibrium markets,options can be priced using arbitrage pricing method. According to arbitrage theory,when there is no arbitrage,a riskless portfolio can only obtain normal riskless profits.Therefore we can make use of constructing a risk free portfolio or separating and compounding securities for option pricing.There are three traditional methods of option pricing:the Black-Scholes model;binomial tree methods;Martingale.All the three methods are based on the assumption that the financial market is arbitrage free,equilibrium,and complete.They are the application of arbitrage free principle to the pricing of financial products and derivatives.When these methods are used,the option pricing is obtained by replicating.However,if the market is with arbitrage,or non-equilibrium,or incomplete, it is difficult to price options with these traditional methods.In this case,we can calculate the price of option as the fair premium needed to insure the potential loss from the issurer's point of view, using the principle of fair premium and physical probabilistic measure of price process.We call this method insurance actuary pricing.This paper deals with some option pricing problems in finance with all kinds of factors,by using insurance actuary pricing method,and setting up some mathematics models,it calculates equations of the price of options and rational price of options,trying to obtain some outcomes to guide financial practice and make it easy to be operated.The essay mainly answers the following problems:①We point out the basic ideas and significance of arbitrage pricing theory and its uses in the pricing of financial products and their derivative products.②We introduces the actuarial approach to option pricing in an all-round way.Then it generalizes the result of Mogens Bladt and Tina Hviid Rydberg on the European option pricing,and deals with the general Black-Scholes model.That is,the model of stock pricing process may be changed,for example,the stock pricing processis driven by exponential Ornstein-Uhlenback process or by exponential Levy process.The analysis of the consistent between the actuarial approach and non-arbitrage pricing is also given.③We apply the insurance actuarial approach to pricing of other derivatives and other exotic options,such as Bi-direction European options,warrents and convertible bond.Among these,the pricing of Bi-direction European options and warrents are what we focus on.According to the property of Bi-direction European options,under the assumptions that stocks price process driven by non-homogeneous Poisson jump-diffusion process and geometric fractional Brownian motion,and the expected rate,volatility and risk-less rate are functions of time,using the method of insurance actuary pricing by physical probabilistic measure of pricing process and the principle of fair premium,pricing formula of Bi-direction European options were obtained considering the price of stock dividends-payment.④we give the pricing of Bi-direction European options with the exercise price is uncertain when the stock price submitting to geometry Brown walk and geometric fractional Brownian motion. Then according to the property of warrents,we further give the pricing formulae by using insurance actuary pricing method.