The Research Of Exchange Option Under Various Models

Options are one of the most important risk management tools in financial markets,reasonable pricing of options is an important aspect of studying options.Exchange options belong to singular options,and their pricing complexity lies in the two underlying assets.Convertible bonds are another type of corporate bonds,which have the characteristics of converting bonds into stock assets.In order to accurately describe the movement of asset prices,the classical Black-Scholes option pricing model is modified.The pricing formulas of exchange options and convertible bonds under several modified models are obtained by using Mellin transform and finite difference method,the numerical simulation and empirical analysis verify the usefulness of the model,which enriches the option pricing theory.Firstly,the exchange option pricing with dividend factor is considered.Under the condition that the underlying asset price model obeys the geometric Brownian motion,the partial differential equation satisfied by the exchange option price is solved by Mellin transformation method,and the closed-form solution of the option pricing formula is obtained.Secondly,the geometric Brownian motion is improved to sub-fractional Brownian motion,and the exchange option pricing formula under sub-fractional Brownian motion is obtained by solving the partial differential equation.The default risk model based on company value is introduced to the model,constructing the portfolio and using the double Mellin transformation method to solve the partial differential equation,then the closed-form solution of the exchange option price with default risk in the sub-fractional Brownian motion environment is obtained.The influence of model parameters on option price is analyzed by numerical simulation.Finally,the assumption that the daily return of assets in Black-Scholes model obeys normal distribution is improved to Tsallis entropy distribution with long-term memory effect and long-range interaction.The numerical solution of two-factor convertible bond price based on Tsallis entropy distribution is obtained by finite difference method.Selecting three market convertible bond data for analysis,the optimal parameters of Tsallis entropy distribution are estimated by goodness-of-fit method,and then the comparison figure between theoretical price and actual price of convertible bonds is drawn.The results show that the theoretical price of convertible bonds based on Tsallis entropy distribution of stock price model is closer to the actual price than that based on normal distribution,which shows that Tsallis entropy distribution has a good fitting effect on daily return distribution of stock price.