Arbitrage is a usual transaction behavior in order to make profits. It is the possibility of riskless profits. If an asset (portfolio) has riskless super profits, there is arbitrage opportunity in the market. Arbitrage is an internal power to make the market efficient, all the prices in the market reflect all the information entirely and immediately. Any price temporarily deviating from its nature will disappear by the arbitrage, that is to say, when capital market reaches balance (supply = demand), there is no arbitrage. Therefore, 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 is 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.However, traditional arbitrage pricing method can only be applied to complete financial market substantially. Completeness is an important condition in noarbitrage pricing. So incompleteness implies that there exist securities which can not be priced using this method. But we can calculate the price of option as the fair premium needed to insure the potential loss from the issuers' 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 establishing partial differential equations, constructing equivalent martingale measure and using insurance actuary pricing method. 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:1) We point out the basic ideas and significance of arbitrage pricing theory and its uses in the pricing of financial products and their derivative products.2) We apply the theory of arbitrage pricing to option pricing. Firstly, by changing basic assumption of BlackScholes option pricing model to the assumption that stock pricing process is a mixed process, give basic option pricing equations with transaction costs whose stock pricing process is a mixed process replicating the payoff of option by portfolio and solving partial differential equations. Secondly, deal with general pricing formulas of European contingent claim by means of martingale method of constituting equivalent martingale measure on the basis of the basic theorem of asset pricing that noarbitrage opportunity is equivalent to the existence of neutralrisk probabilistic measure (equivalent martingale measure). Thirdly, argue about the consistency between arbitrage pricing and neutralrisk pricing.3) We introduce a new method to option pricingan actuarial approach. It turns option pricing into an equivalent insurance or a fair premium determination. The approach is valid even when arbitrage exists and the market is incompleteness and unequilibrium. Under the assumption that the expected rate Î¼(S(t)) , volatilityÏƒ(S(t)) are functions of risk asset S(t), and stock pricing process respectively driven by a general OU process and an exponential of a Levy process, we obtain the accurate pricing formulas and putcall parity of European option. In the end, assume that riskless rate is given, we deal with pricing formulas of European option on foreign currency and apply the approach to the pricing of the convertible bond.
