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Abstract： Foreign exchange option， as afinancial derivative， plays an important role in thefinancial market. It is of great theoretical and practical significance to study the foreign exchange options， especially its pricing model. In order to more accurately portray the authenticity of foreign exchange market， this paper applies fractional Brown motion in the fractal market hypothesis and combines with jump diffusion process so as to establish the pricing model of foreign exchange option. Moreover， this paper put forward the pricing formulas of European foreign exchange call and put option， as well as their relationships by using the method of insurance actuary pricing. No matter whether thefinancial market has arbitrage or not， no matter it is complete or not， this conclusion is valid.
Key words： Fractional Brownian motion； Jump-diffusion； Insurance actuary pricing； Foreign exchange option
In 1989， Peters proposed a fractal market hypothesis， and proved that there was fractal structure with non-periodic cycle in different capital market. Fractal market hypothesis does not depend on such assumptions of exchange ratefluctuations as the independent and normal distribution assumptions. What’s more， fractional Brownian motion can better explain many phenomena in foreign exchange market， such as the“thick tail”and long-term correlation， which the efficient market hypothesis can not account for. Therefore， creating a foreign exchange option pricing model by fractional Brownian motion in the fractal market hypothesis， we can more accurately portray the authenticity of the market. In 2000， Hu [1] introduced Wick integral in fractional Brownian motion， and in 2003 [2]， developed Wick integral in fractional Brownian motion when Hurst exponent H > 0.5 through fractional white noise. Bender [3] and Elliott [4] promote Wick integral of fractional Brownian motion to the case of Hurst exponent H∈（0，1）， and set up fractional Black-Scholes model of European contingent claim. What is inadequate is that fractional Brownian motion describes only the continuous change of the price of the underlying asset， that is， the normal changes of asset prices under normal market conditions， but it cannot interpret the abnormal changes of assets prices caused by the unusual circumstances in the market （non-economic factors）， namely the discontinuous and wide“jump”. In order to make up for this deficiency， this paper， adopting the fractional Brownian motion in the fractal market hypothesis， combining with the jump-diffusion process [5，6] and using the method of insurance actuary pricing， discusses the pricing issue of European foreign exchange options. 2. THE CONSTRUCTION OF MATHEMATICAL EVALUATION MODEL
where BH（t） is fractional Brownian motion， Hurst exponent H∈（0，1）；μ（t），σ（t） are continuous functions of the time t； Qt represents the random jump number of foreign exchange price in the period of time [0，t]， assuming that it obeys the Poisson process and its parameter isλ； suppose the random variable J（t） subject to the special normal distribution N（？σ2J/2，σ2J）， eJ（t）？1 represents the relative height of the jump； further assume that J（t）， BH（t） and Qtare independent of each other.
When Hurst exponent H %= 0.5， fractional Brown motion is neither a Markov process， nor a semi-martingale. Therefore， we cannot use normal stochastic integrals to analysis. Wick integral of fractional Brownian motion， a special integral， is used in this paper [2，3]：
Bladt and Rydbergfirstly put forward the method of insurance actuarial pricing in 1998 [7].Compared with the traditional method of martingale pricing， this method’s greatest merit is that it doesn’t make any economic assumptions to the financial market， that is to say it has nothing to do with the basic assumption of market without arbitrage. No matter there is arbitrage or whether it is complete， it is effective. This text is to discuss the pricing of foreign exchange option by insurance actuarial pricing method. Next we will introduce the basic method of insurance actuarial pricing.
Theorem 4.1. Assume that foreign Bond price Pf（t） and domestic Bond price Pd（t） respectively meet Equation （1） and （2）， the price process of foreign exchange rate S（t） meet Equation （3）， K is the exercise price， T is the expiration date， then we obtain the insurance actuarial pricing formulas of foreign exchange option as follows：
Integrated （12）， （13）， （14）， the insurance actuarial pricing formulas of foreign exchange call option are obtained. Similarly， the pricing formula （10） can also be proved. Formula （9） minus formula （10），
5. CONCLUSION
Foreign exchange option， as afinancial derivative， plays an important role in the financial market. Thus， how to accurately price both in theory and practice is extremely important. In order to more accurately portray the authenticity of foreign exchange market， this paper applies fractional Brown motion in the fractal market hypothesis and combines with jump diffusion process so as to establish the pricing model of foreign exchange option under fractional jump-diffusion. Moreover， this paper discusses the pricing issue of European foreign exchange according to the insurance actuary pricing method and put forward the pricing formula of foreign exchange call and put option， as well as their relationships. No matter whether thefinancial market has arbitrage or not， no matter it is complete or not， this conclusion is valid. REFERENCES
[1] Duncan， T. E.， Hu， Y.， & Duncan， B. P. （2000）. Stochastic calculus for factional Brownian motion I： Theory. SIAM Journal on Control and Optimization， 38（2）， 582-612.
[2] Hu， Y.， &？ksendal， B. （2003）. Fraction white noise calculus and application tofinance. Infinite Dimensional Analysis， Quantum Probability and Related Topics， 06（1）， 1-32.
[3] Bender， C. （2003）. An Ito formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter. Stochastic Processes and Their Applications， 104（1）， 81-106.
[4] Elliott， R. J.， & Vander Hoek， J. （2003）. A general fractional White Noise theory and applications tofinance. Mathematical Finance， 13， 71-85.
[5] Chen， L.， & Yang， X. （2007）. Pricing mortgage insurance with house price driven by Poisson jump diffusion process. Chinese Journal of Applied Probability and Statistics， 23（4）， 345-351.
[6] Li， C.， Chen， L.， & Yang， X. （2009）. Martingale pricing of insurance under the combination of stochastic volatility and jump diffusion. Systems Engineering， 27（3）， 41-45.
[7] Bladt， M. T.， & Rydberg， H. （1998）. An actuarial approach to option pricing under the physical measure and without market assumption. Insurance： Mathematics and Economics， 22（1）， 65-73.
Key words： Fractional Brownian motion； Jump-diffusion； Insurance actuary pricing； Foreign exchange option
In 1989， Peters proposed a fractal market hypothesis， and proved that there was fractal structure with non-periodic cycle in different capital market. Fractal market hypothesis does not depend on such assumptions of exchange ratefluctuations as the independent and normal distribution assumptions. What’s more， fractional Brownian motion can better explain many phenomena in foreign exchange market， such as the“thick tail”and long-term correlation， which the efficient market hypothesis can not account for. Therefore， creating a foreign exchange option pricing model by fractional Brownian motion in the fractal market hypothesis， we can more accurately portray the authenticity of the market. In 2000， Hu [1] introduced Wick integral in fractional Brownian motion， and in 2003 [2]， developed Wick integral in fractional Brownian motion when Hurst exponent H > 0.5 through fractional white noise. Bender [3] and Elliott [4] promote Wick integral of fractional Brownian motion to the case of Hurst exponent H∈（0，1）， and set up fractional Black-Scholes model of European contingent claim. What is inadequate is that fractional Brownian motion describes only the continuous change of the price of the underlying asset， that is， the normal changes of asset prices under normal market conditions， but it cannot interpret the abnormal changes of assets prices caused by the unusual circumstances in the market （non-economic factors）， namely the discontinuous and wide“jump”. In order to make up for this deficiency， this paper， adopting the fractional Brownian motion in the fractal market hypothesis， combining with the jump-diffusion process [5，6] and using the method of insurance actuary pricing， discusses the pricing issue of European foreign exchange options. 2. THE CONSTRUCTION OF MATHEMATICAL EVALUATION MODEL
where BH（t） is fractional Brownian motion， Hurst exponent H∈（0，1）；μ（t），σ（t） are continuous functions of the time t； Qt represents the random jump number of foreign exchange price in the period of time [0，t]， assuming that it obeys the Poisson process and its parameter isλ； suppose the random variable J（t） subject to the special normal distribution N（？σ2J/2，σ2J）， eJ（t）？1 represents the relative height of the jump； further assume that J（t）， BH（t） and Qtare independent of each other.
When Hurst exponent H %= 0.5， fractional Brown motion is neither a Markov process， nor a semi-martingale. Therefore， we cannot use normal stochastic integrals to analysis. Wick integral of fractional Brownian motion， a special integral， is used in this paper [2，3]：
Bladt and Rydbergfirstly put forward the method of insurance actuarial pricing in 1998 [7].Compared with the traditional method of martingale pricing， this method’s greatest merit is that it doesn’t make any economic assumptions to the financial market， that is to say it has nothing to do with the basic assumption of market without arbitrage. No matter there is arbitrage or whether it is complete， it is effective. This text is to discuss the pricing of foreign exchange option by insurance actuarial pricing method. Next we will introduce the basic method of insurance actuarial pricing.
Theorem 4.1. Assume that foreign Bond price Pf（t） and domestic Bond price Pd（t） respectively meet Equation （1） and （2）， the price process of foreign exchange rate S（t） meet Equation （3）， K is the exercise price， T is the expiration date， then we obtain the insurance actuarial pricing formulas of foreign exchange option as follows：
Integrated （12）， （13）， （14）， the insurance actuarial pricing formulas of foreign exchange call option are obtained. Similarly， the pricing formula （10） can also be proved. Formula （9） minus formula （10），
5. CONCLUSION
Foreign exchange option， as afinancial derivative， plays an important role in the financial market. Thus， how to accurately price both in theory and practice is extremely important. In order to more accurately portray the authenticity of foreign exchange market， this paper applies fractional Brown motion in the fractal market hypothesis and combines with jump diffusion process so as to establish the pricing model of foreign exchange option under fractional jump-diffusion. Moreover， this paper discusses the pricing issue of European foreign exchange according to the insurance actuary pricing method and put forward the pricing formula of foreign exchange call and put option， as well as their relationships. No matter whether thefinancial market has arbitrage or not， no matter it is complete or not， this conclusion is valid. REFERENCES
[1] Duncan， T. E.， Hu， Y.， & Duncan， B. P. （2000）. Stochastic calculus for factional Brownian motion I： Theory. SIAM Journal on Control and Optimization， 38（2）， 582-612.
[2] Hu， Y.， &？ksendal， B. （2003）. Fraction white noise calculus and application tofinance. Infinite Dimensional Analysis， Quantum Probability and Related Topics， 06（1）， 1-32.
[3] Bender， C. （2003）. An Ito formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter. Stochastic Processes and Their Applications， 104（1）， 81-106.
[4] Elliott， R. J.， & Vander Hoek， J. （2003）. A general fractional White Noise theory and applications tofinance. Mathematical Finance， 13， 71-85.
[5] Chen， L.， & Yang， X. （2007）. Pricing mortgage insurance with house price driven by Poisson jump diffusion process. Chinese Journal of Applied Probability and Statistics， 23（4）， 345-351.
[6] Li， C.， Chen， L.， & Yang， X. （2009）. Martingale pricing of insurance under the combination of stochastic volatility and jump diffusion. Systems Engineering， 27（3）， 41-45.
[7] Bladt， M. T.， & Rydberg， H. （1998）. An actuarial approach to option pricing under the physical measure and without market assumption. Insurance： Mathematics and Economics， 22（1）， 65-73.