| In 1997,the Nobel Prize in Economics was awarded to Professor Robert·Merton of Harvard University and Professor Myron·Scholes of Stanford University.They conducted creative research on how to evaluate the complexity theory of stock option trading and other financial derivatives,and transformed the risk of option pricing from speculation to science.B-S model not only gives the more reasonable option pricing formula at that time,but also lays a foundation for the future option pricing theory.Since then,the research on option pricing began to develop vigorously.Entering the 21 st century,the rapid development of the financial market has undoubtedly brought broader prospects and greater challenges to the pricing of financial derivatives,therefore,many scholars continue to improve the B-S model.Firstly,the interest rate is no longer assumed to be constant,but follow random process.Secondly,the asset price process is not geometric Brownian motion,but other processes more in line with reality.Thirdly,people are no longer satisfied with the pricing of single asset option and begin to study the pricing of multi asset options.Quotient option is a multi asset options,which takes the ratio of two underlying assets as the underlying option.However,in the existing research on multi asset options,scholars often pay more attention to A Basket of option or Exchange option,and there are few studies on Quotient option at present.From the above perspectives,the constant interest rate is extended to continuous random interest rate and discontinuous random interest rate in this paper.With the assets driven by multi-dimensional Brown motion and O-U process,we study Quotient option with uncertain strike price by measure transformation.The main research contents are as follows:Firstly,assuming that the underlying asset price process follows O-U process driven by multi-dimensional Brown motion and the interest rate obeys multi-dimensional Ho-Lee model,we derive the pricing formula of Quotient option with uncertain strike price by multi-dimensional Girsanov theorem and measure transformation,and analyz the change of price under different volatility by Matlab.Secondly,assuming that the underlying asset price process follows the O-U process driven by multi-dimensional Brown motion and the interest rate follows the multidimensional extended Vasicek model,the pricing formula of Quotient option with uncertain strike price is derived by multi-dimensional Girsanov theorem and measure transformation,and the change of price under different volatility is analyzed by Matlab.Thirdly,assuming that the underlying asset price process follows the O-U process driven by multi-dimensional Brown motion and the interest rate follows the discontinuous random process,the pricing formula of Quotient option with uncertain strike price is derived by multi-dimensional Girsanov theorem and measure transformation under the jump-diffusion model,and the change of price under different volatility is analyzed by Matlab.Finally,we combine Quotient option with Reset option.Assuming that the underlying asset price process follows the O-U process driven by multi-dimensional Brown motion,and the interest rate follows the multi-dimensional extended Vasicek Interest rate model,the pricing formula of Reset quotient option is derived by multi-dimensional Girsanov theorem and measure transformation,and the change of price under different volatility is analyzed by Matlab. |
