Pricing Of Correlation Digital Option Under Time-varying Parameters

With the rapid development of financial markets,many new types of exotic options are designed in order to meet the needs of different investors.Correlation digital option is an exotic options with two assets.Compared with the general options,the market price of the correlation digital options is not easy to be manipulated,and the correlation digital options plays an important role in the foreign exchange market and stock market.Therefore it is of great importance for investors to set a reasonable and effective price for the correlated digital option.In 1973,Black and Scholes assumed that underlying asset price followed the geo-metric Brownian motion,established classic B-S model,and gave the pricing formula of European call option.However,the assumptions of B-S model were too ideal.It could not completely apply to the real financial markets.So many experts and scholars tried to modify the assumptions in the B-S model to better describe changes of the asset price,such as dividend-paying stock,stochastic interest rate,time-varying volatility,the Brow-nian motion to fractional Brownian or mixed fractional Brownian motion,etc..In this paper,we study the price of correlation digital option under three models.The main contents are as follows,Firstly,we suppose the asset price S₁?t?,S₂?t?and the riskless interest rate r?t?follows the stochastic differential equation:here q_i?t?,?_i?t?,a?t?,b?t?are certain functions on t,B_i?t?is the Brownion motion,corre-lation coefficient of B_i?t?and B_j?t?are ?ij,?i,j=1,2,3 and i?j?.We get the pricing formulas of European correlation digital option by the help of multi-dimension Girsanov theory and the change of measure.The results extend the related results.Secondly,we suppose the asset price S₁?t?,S₂?t?and the riskless interest rate r?t?follows the stochastic differential equation:dr?t?=[a?t?-b?t?r?t?]dt+?3?t?dBH3?t?,here q_i?t?,?_i?t?,a?t?,b?t?are certain functions on t,BHi?t?is the fractional Brownion motion with Hurst parameter Hi??1/2,1?,B_H1?t?,B_H2?t?and B_H3?t?are independence.We get the pricing formulas of European correlation digital option by the help of fractional Girsanov theory and the change of measure.Thirdly,we suppose the asset price S₁?t?,S₂?t?follows the stochastic differential equation:???here r?t?,q_i?t?are certain functions on t,?_i,?_i are constants,?_iBH?t?+?_iB?t?is mixed the fractional Brownion motion,H??3/4,1?,BH?t?and B?t?are independence.We get the pricing formulas of European correlation digital option by the help of mixed fractional Ito's formula and the quasi-martingale.Finally,based on the pricing formula,we analyze the sensitivity of the correlation digital option price with respect to parameter,which makes us have a deeper understand-ing of the correlation digital option.