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摘 要 假设市场是完备的，在文中使用了计价单位变换，等价鞅测度理论和无套利原理研究了股票价格具有时滞的欧式择好期权，得到了欧式择好期权的定价公式和对冲交易策略.
关键词 股票价格；时滞；择好期权；等价鞅测度
中图分类号 62P05 文献标识码 A
Abstract Assuming that the markets are complete， we use the changes of numeraire， the theory of equivalent martingale measure and noarbitrage property to study the pricing of European betterof options for stock prices with delay. In conclusion， we derive a closedform representation of the option price， and hedging strategy.
Key words stock prices， delay， betterof options， equivalent martingale measure.
1 Introduction
The advent of BlackScholes[1] option pricing model has brought a revolutionary change for the financial industry. Firstly， the return volatility needed to be assessed， when we use the BlackScholes formula to price the stock options. However， options pricing has been questioned under the assumption of the constant volatility， since empirical evidence shows that volatility actually depends on time in a way that is not predictable. For example， Bollerslev[2] found that the volatility was not a constant， besides， the changes of stock return volatility could be more accurately described and predicted by using the GARCH model. So the need for better ways of understanding the behavior of many natural processes has motivated the development of dynamic models of these process. The concept of stochastic volatility model was introduced by Hull and White[3]，and subsequent development includes the work of Wiggns[4]，Johnson and Shannon[5] and Scott[6].On the other hand， some studies found that the rate of change of stock price depends on past prices， such as Christie[7]，Akgiray[8] and Sheinkman and LeBaron[9]. Hobson and Roger[10] suggested a new class of nonconstant volatility models， and the volatility can be regarded as an endogenous factor in the sense that it is defined in terms of the past behavior of the stock price. Furthermore， some scholars began to study the pricing of options for stock price with delay. Kind et al.[11] obtained a diffusion approximation result for processes satisfying some equations with pastdependent coefficients， and they applied the result to a model.of option pricing ， in which the underlying asset price volatility depends on the past evolution to get a generalized Black and Scholes formula. Chang and Yoree[12] studied the pricing of an European contingent claim for the （B，S）securities markets with a hereditary price structure in the sense that the rate of change of the unit price of bond account and rate of change of the stock account S depend not only on the current unit price but also on their historical prices， but the unit price of bond account was not stochastic. Kazmerchuk et al.[13] derived an analogue of BlackScholes formula for vanilla call option value in the market with stock price having timedelay in the diffusion term， and compared the results with the results of classical BlackScholes. Arriojas et al.[14]proposed a model for the diffusion term and drift term with delays， which was sufficiently flexible to fit real market. They also developed an explict formula for pricing European options when the underlying stock price with delay. Basing on the research frameworks of Kazmerchuk et al. and Arriojas et al.， Yaqiong Li and Lihong Huang[15， 16]further expanded to study for option pricing model with dividendpaying stock， and derived option pricing formula for the payment of continuous dividends and discrete dividends. Although many scholars have studied the option pricing problem with delay， their studies were limited to the options pricing on only one risky asset. However， there are many multiasset financial derivatives in real market， such as Exchange options， Better（Worse） of options， Outer performance options， and Quanto options etc. So， in this paper we specially studied the pricing of Betterof options as an application on the basis of establishing the theory of options pricing for two risky assets with delays.
Many studies show that the models of stock prices with delays are more realistic， so the option pricing formula is meaningful with delay. Besides， the pricing ideas and methods that we used to study the Betterof options with delays， have a reference value for studying other multiasset options pricing.
References
[1] F BLACK，M SCHOLES.The pricing of options and corporate liablities[J].Journal of Political Economy， 1973（3）：637-659.
[2] T BOLLERSLEV.Generalized autoregressive conditional heteroskedasticity[J].Journal of Econometrica，1986（31）：307-327.
[3] J HULL，A WHITE.The pricing of options on assets with stochastic volatilities[J].Journal of finance，1987，42（2）：271-301.
[4] J B WIGGINS.Option values under stochastic volatility： theory and empirical estimate[J].Journal of finance，1987，19（2）：351-372.
[5] H JOHNSON，D SHANNO D.Option pricing when the variance is changing[J].Journal of Finance，1987，22（2）：143-151.
[6] L O SCOTT.Option pricing when the variance changes randomly： theory， estimation and an application[J].Financial and Quantitave. Anal， 987，22（4）：419-438.
[7] A CHRISTIE.The stochastic behaviour of common stocks variances： values leverage and interest rate effects[J].Journal of Financial Economics，1982，10（4）：407-432.
[8] V AKGIRAY.Conditional heteroskedasticity in time series of stock returns： evidence and forecasts[J].The Journal of Business，1989，62（1）：55-80.
[9] J SCHEINKMAN，B LEBARON.Nonlinear dynamics and stock returns[J].The Journal of Business，1989，62（3）：311-337.
[10]D HOBSON，L C G ROGERS.Complete markets with stochastic volatility[J].Mathematical Finance，1998，8（1）：27-48.
[11]P KIND，R LPTSER，W RUNGGALDIER.Diffusion approximation in pastdependent models and applications to option pricing[J].Annals of Applied Probability，1991，1（3）：379-405.
[12]M H CHANG，R K YOUREE.The European option with hereditary price structures： Basic theory[J].Applied Mathematics and Computation，1999，102（2）：279-296.
[13]Y KAZMERCHUK，A SWISHCHUK，J WU.The pricing of option for securities markets with delayed response[J].Mathematics and Computers Simulation，2007，75（3/4）：69-79.
[14]M ARRIOJAS M，Y HU，S E MOHAMMED，et al.A delayed black and scholes formula[J].Stochastic Analysis and Applications，2007，25（2）：471-492.
[15]Yaqiong LI，Lihong HUANG， The pricing of options on a dividendpaying stock with delayed response[J].Journal of Hunan University：Natural Sciences，2009，36（12）：89-92.
[16]Yaqiong LI，HUANG Lihong， Stock option pricing of drift and diffusion terms for with time delay[J].Journal of Quantitative Economics，2011，28（1）：10-13.
[17]K Y KWOK.Mathematical models of financial derivatives[M].Singapore： SpringerVerlag，1998.
关键词 股票价格；时滞；择好期权；等价鞅测度
中图分类号 62P05 文献标识码 A
Abstract Assuming that the markets are complete， we use the changes of numeraire， the theory of equivalent martingale measure and noarbitrage property to study the pricing of European betterof options for stock prices with delay. In conclusion， we derive a closedform representation of the option price， and hedging strategy.
Key words stock prices， delay， betterof options， equivalent martingale measure.
1 Introduction
The advent of BlackScholes[1] option pricing model has brought a revolutionary change for the financial industry. Firstly， the return volatility needed to be assessed， when we use the BlackScholes formula to price the stock options. However， options pricing has been questioned under the assumption of the constant volatility， since empirical evidence shows that volatility actually depends on time in a way that is not predictable. For example， Bollerslev[2] found that the volatility was not a constant， besides， the changes of stock return volatility could be more accurately described and predicted by using the GARCH model. So the need for better ways of understanding the behavior of many natural processes has motivated the development of dynamic models of these process. The concept of stochastic volatility model was introduced by Hull and White[3]，and subsequent development includes the work of Wiggns[4]，Johnson and Shannon[5] and Scott[6].On the other hand， some studies found that the rate of change of stock price depends on past prices， such as Christie[7]，Akgiray[8] and Sheinkman and LeBaron[9]. Hobson and Roger[10] suggested a new class of nonconstant volatility models， and the volatility can be regarded as an endogenous factor in the sense that it is defined in terms of the past behavior of the stock price. Furthermore， some scholars began to study the pricing of options for stock price with delay. Kind et al.[11] obtained a diffusion approximation result for processes satisfying some equations with pastdependent coefficients， and they applied the result to a model.of option pricing ， in which the underlying asset price volatility depends on the past evolution to get a generalized Black and Scholes formula. Chang and Yoree[12] studied the pricing of an European contingent claim for the （B，S）securities markets with a hereditary price structure in the sense that the rate of change of the unit price of bond account and rate of change of the stock account S depend not only on the current unit price but also on their historical prices， but the unit price of bond account was not stochastic. Kazmerchuk et al.[13] derived an analogue of BlackScholes formula for vanilla call option value in the market with stock price having timedelay in the diffusion term， and compared the results with the results of classical BlackScholes. Arriojas et al.[14]proposed a model for the diffusion term and drift term with delays， which was sufficiently flexible to fit real market. They also developed an explict formula for pricing European options when the underlying stock price with delay. Basing on the research frameworks of Kazmerchuk et al. and Arriojas et al.， Yaqiong Li and Lihong Huang[15， 16]further expanded to study for option pricing model with dividendpaying stock， and derived option pricing formula for the payment of continuous dividends and discrete dividends. Although many scholars have studied the option pricing problem with delay， their studies were limited to the options pricing on only one risky asset. However， there are many multiasset financial derivatives in real market， such as Exchange options， Better（Worse） of options， Outer performance options， and Quanto options etc. So， in this paper we specially studied the pricing of Betterof options as an application on the basis of establishing the theory of options pricing for two risky assets with delays.
Many studies show that the models of stock prices with delays are more realistic， so the option pricing formula is meaningful with delay. Besides， the pricing ideas and methods that we used to study the Betterof options with delays， have a reference value for studying other multiasset options pricing.
References
[1] F BLACK，M SCHOLES.The pricing of options and corporate liablities[J].Journal of Political Economy， 1973（3）：637-659.
[2] T BOLLERSLEV.Generalized autoregressive conditional heteroskedasticity[J].Journal of Econometrica，1986（31）：307-327.
[3] J HULL，A WHITE.The pricing of options on assets with stochastic volatilities[J].Journal of finance，1987，42（2）：271-301.
[4] J B WIGGINS.Option values under stochastic volatility： theory and empirical estimate[J].Journal of finance，1987，19（2）：351-372.
[5] H JOHNSON，D SHANNO D.Option pricing when the variance is changing[J].Journal of Finance，1987，22（2）：143-151.
[6] L O SCOTT.Option pricing when the variance changes randomly： theory， estimation and an application[J].Financial and Quantitave. Anal， 987，22（4）：419-438.
[7] A CHRISTIE.The stochastic behaviour of common stocks variances： values leverage and interest rate effects[J].Journal of Financial Economics，1982，10（4）：407-432.
[8] V AKGIRAY.Conditional heteroskedasticity in time series of stock returns： evidence and forecasts[J].The Journal of Business，1989，62（1）：55-80.
[9] J SCHEINKMAN，B LEBARON.Nonlinear dynamics and stock returns[J].The Journal of Business，1989，62（3）：311-337.
[10]D HOBSON，L C G ROGERS.Complete markets with stochastic volatility[J].Mathematical Finance，1998，8（1）：27-48.
[11]P KIND，R LPTSER，W RUNGGALDIER.Diffusion approximation in pastdependent models and applications to option pricing[J].Annals of Applied Probability，1991，1（3）：379-405.
[12]M H CHANG，R K YOUREE.The European option with hereditary price structures： Basic theory[J].Applied Mathematics and Computation，1999，102（2）：279-296.
[13]Y KAZMERCHUK，A SWISHCHUK，J WU.The pricing of option for securities markets with delayed response[J].Mathematics and Computers Simulation，2007，75（3/4）：69-79.
[14]M ARRIOJAS M，Y HU，S E MOHAMMED，et al.A delayed black and scholes formula[J].Stochastic Analysis and Applications，2007，25（2）：471-492.
[15]Yaqiong LI，Lihong HUANG， The pricing of options on a dividendpaying stock with delayed response[J].Journal of Hunan University：Natural Sciences，2009，36（12）：89-92.
[16]Yaqiong LI，HUANG Lihong， Stock option pricing of drift and diffusion terms for with time delay[J].Journal of Quantitative Economics，2011，28（1）：10-13.
[17]K Y KWOK.Mathematical models of financial derivatives[M].Singapore： SpringerVerlag，1998.