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Abstract： This paper discusses the problem of pricing on some multiasset option European exchange option in jump-diffusion model by martingale method. Supposing that risk assets pay continuous dividend regarded as the function of time. By changing basic assumption of William Margrabe exchange option pricing model to the assumption that jump process is count process that more general than Poisson process. It is established that the behavior model of the stock pricing process is jump-diffusion process. With risk-neutral martingale measure， pricing formula and put-call parity of European exchange options with continuous dividends are obtained by stochastic analysis method. The results of Margrabe are generalized.
Key words： Dividend； European exchange options； Jump-diffusion； Dividends； Count process
1. INTRODUCTION
Option pricing theory is always one of the kernel problems onfinancial mathematics. The domestic and foreign scholars have done a great deal of researches on BlackScholes model and obtained many results which is instructive tofinancial practice. The option pricing model is options to exchange one asset to another. William Margrabe [1] studied an equation for the value of the option to exchange one risky asset for another. His paper discusses the option pricing model when exercise price is random variable. However the appearance of important information will cause the stock price to a kind of not continual jumps [2–4]. In this paper， an equation for the value of the option to exchange one risky asset for another is developed. The option-pricing model with continuous dividends is established when exercise price is a random variable. The option-pricing model is options to exchange one asset to another. Pricing formula of European option is also given. The results of Margrabe [1]， Yang and Hao [5] are generalized.
Proposition 2 Assume that the dynamics of two risky assets S1t， S2tare given by （2）. Ifγi+ 1 have lognormal distribution with mean parameterμγ1，μγ2and varianceσγ12，σγ22. Then the price of a European-type option is given by the expression following when exercise price is S1（t） and expiry date is T.
We can use put-call parity tofind the price of a European put option on a stock with the same parameters as earlier.
4. SUMMARY
In this paper， we establish the option-pricing model when exercise price is random variable. Supposing that risk assets pay continuous dividend regarded as the function of time. Assume that jump process is count process which is more general than Poisson process， it is established that the model of the stock pricing process is jump-diffusion process with continuous dividends. European option pricing formula and their parity are obtained when the jump distribution is lognormal.
REFERENCES
[1] Margrabe， W. （1978）. The value of an option to exchange one asset for another. Journal of Finance， 33（1）， 177-186.
[2] Ball， C.， & Roma， A. （1993）.A jump diffusion model for the European monetary system. Journal of International Money and Finance， 12（6）， 475-492.
[3] Harold， J.， & Kushner （2000）. Jump-diffusions with controlled jumps： existence and numerical methods. Journal of Mathematical Analysis and Applications， 249（1）， 179-198.
[4] Yumiharu Nakano （2004）. Minimization of shortfall risk in a jump-diusion model. Statistics & Probability Letters， 67（1）， 87-95.
[5] Yang， Y. F.， & Hao， J. （2012）. The value of European exchange option. Internationl Conference on Engineering Technology and Economic Management， 1（1）， 449-452.
Key words： Dividend； European exchange options； Jump-diffusion； Dividends； Count process
1. INTRODUCTION
Option pricing theory is always one of the kernel problems onfinancial mathematics. The domestic and foreign scholars have done a great deal of researches on BlackScholes model and obtained many results which is instructive tofinancial practice. The option pricing model is options to exchange one asset to another. William Margrabe [1] studied an equation for the value of the option to exchange one risky asset for another. His paper discusses the option pricing model when exercise price is random variable. However the appearance of important information will cause the stock price to a kind of not continual jumps [2–4]. In this paper， an equation for the value of the option to exchange one risky asset for another is developed. The option-pricing model with continuous dividends is established when exercise price is a random variable. The option-pricing model is options to exchange one asset to another. Pricing formula of European option is also given. The results of Margrabe [1]， Yang and Hao [5] are generalized.
Proposition 2 Assume that the dynamics of two risky assets S1t， S2tare given by （2）. Ifγi+ 1 have lognormal distribution with mean parameterμγ1，μγ2and varianceσγ12，σγ22. Then the price of a European-type option is given by the expression following when exercise price is S1（t） and expiry date is T.
We can use put-call parity tofind the price of a European put option on a stock with the same parameters as earlier.
4. SUMMARY
In this paper， we establish the option-pricing model when exercise price is random variable. Supposing that risk assets pay continuous dividend regarded as the function of time. Assume that jump process is count process which is more general than Poisson process， it is established that the model of the stock pricing process is jump-diffusion process with continuous dividends. European option pricing formula and their parity are obtained when the jump distribution is lognormal.
REFERENCES
[1] Margrabe， W. （1978）. The value of an option to exchange one asset for another. Journal of Finance， 33（1）， 177-186.
[2] Ball， C.， & Roma， A. （1993）.A jump diffusion model for the European monetary system. Journal of International Money and Finance， 12（6）， 475-492.
[3] Harold， J.， & Kushner （2000）. Jump-diffusions with controlled jumps： existence and numerical methods. Journal of Mathematical Analysis and Applications， 249（1）， 179-198.
[4] Yumiharu Nakano （2004）. Minimization of shortfall risk in a jump-diusion model. Statistics & Probability Letters， 67（1）， 87-95.
[5] Yang， Y. F.， & Hao， J. （2012）. The value of European exchange option. Internationl Conference on Engineering Technology and Economic Management， 1（1）， 449-452.