The Actuarial Approach To The Problem Of Option Pricing

The problem of option pricing is one of key problems in financial mathematics. There are 3 traditional methods of option pricing : the Black-Schoels model; binomial tree methods; Martingale. All the 3 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 finanical products and derivatives. When these methods are used, the option pricing is obtained by replicating. However, if the market is with arbitrage , or non-eqilibrium, or in-complete, it is difficult to price options with these traditional methods.But we can calculate the price of option as the fair premium needed to insure the potential loss from the issuer'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 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 according to the references [3][11][15]. That is, the model of stock pricing process may be changed , for example, the stock pricing process is driven by exponential Ornstein-Uhlenback process or by exponential Levy process. The comparison is also given between the actuarial approach and non-arbitrage pricing.Last,we apply the insurance actuary approach to pricing of other derivatives and other exotic options, such as warrents, convertible bond, Asian option, quanto option, as well as compound option. Among these, the first one and the last one are what we focus on. This paper makes the mathmatical models according to the property of warrents and compound options, and further give the pricing formulae. In particular, in our results, when the investor'sexpected return rate equals the riskless interest rate , the pricing is just the neutral price. As for Asian option and quanto option ,this paper makes some changes to the mathematical models introduced by references [26][27], then gets the reasonable pricing formulae.