| In the application of financial mathematics,the pricing problem of options has always been one of the most complex research topics.In option trading,the most important thing is the reasonable pricing of the premium.Option pricing is the core and key of option research.In option contracts,the price of options changes according to the changes in supply and demand in the market,which directly affects the profits or losses of buyers and sellers,and since the last century,the issue of option pricing has made breakthroughs and is still the core of the option problem.Options pricing methods mainly include:Black-Scholes formula method,binary tree method,risk-neutral pricing method,martingale pricing method,partial differential equation method,finite difference method,trident tree method method,etc.The traditional BlackScholes option pricing model has strict assumptions and is not in line with the actual market situation,as it has an important assumption that risk-free interest rates and volatility are both constant.This article will study option pricing under the uncertainty of interest rate and volatility.Avellaneda,Levy and Paras once proposed an uncertain volatility model to study the pricing and hedging issues of European options.Based on this model,this article combine the uncertainty of risk-free interest rate and volatility,assuming that the risk-free interest rate and volatility are not exactly known,but between the two known extreme values,i.e.satisfiedσmin≤σt≤σmax,rmin≤rt≤rmax,to study the pricing problems of European options and Asian options.Firstly,the binary tree model under the uncertainty of risk-free interest rate is discussed,and the option pricing formulas of single-step binary tree and two-step binary tree are given respectively.And the risk-free interest rate between two steps in the two-step binary tree is discussed with the same value and the different value points that can be taken.Then,we study the option pricing model under the uncertainty of interest rate and volatility.By constructing the portfolio and using Ito formula,we derive the nonlinear partial differential equation satisfied by European options and Asian options,that is,the BSB equation under the uncertainty of interest rate.Through the equation,we can get the best price and the worst price.Then we introduce the second best price and the second worst price as the reference interval of option pricing through the equation.Finally,we solve the BSB equation under the uncertainty of interest rate by discretization the price process through the triple tree method to solve the pricing problem of European options and Asian options,We conduct numerical simulation on European call option,European put option,floating strike price call arithmetic average Asian option,fixed strike price call arithmetic average Asian option,and give examples to verify the method. |
