Research On Singular Perturbation Analysis Of Nonlinear Black-scholes Equation With Transaction Cost

It is well known that the Black-Scholes model has limitations when describing real market behavior.Especially when transaction costs arise,there is market friction and the market is incomplete,which may lead to two different prices for option holders and option sellers.It can be seen that the theory and practice of the Black-Scholes model are seriously disjointed.Due to the importance of transaction costs in option trading,research on option pricing with transaction costs can play a crucial role and has significant practical significance.This thesis aims to study the pricing of options with transaction costs under weak volatility and fast reverting process volatility,analyze the impact of transaction costs and random volatility on option prices,and focus on the pricing of European options with transaction costs under weak volatility,European options with transaction costs under fast mean regression volatility,and permanent American options with transaction costs under fast reverting process volatility.As any relaxation of the basic assumptions in the Black-Scholes model can lead to a more complex solution,we need to overcome the difficulties of nonlinearity caused by transaction costs,multivariable systems caused by fast reverting process,and the optimal implementation boundary caused by permanent American options.Therefore,we use singular perturbation methods and functional analysis to solve problems such as nonlinear and multivariable systems,and obtain an asymptotic approximate solution of the pricing model.We also provide a uniformly valid error estimate for this approximate solution,in an attempt to obtain better or guiding results for financial practice and easy to operate and calculate.The main content of the paper is as follows:1.This thesis mainly studies the pricing of European options with transaction costs under weak volatility.Assuming that weak volatility is a small parameter,the singular perturbation method is used to transform the nonlinear problem into a series of linear models.The model is solved using differential equations and first integration methods.A uniformly valid estimate of error is obtained using the parabolic maximum principle and De Giorgi iteration technique.2.This thesis mainly studies the pricing of European options with transaction costs under fast reverting process volatility.The nonlinear model is expanded by singular perturbation to obtain a second order linear equation,and then the multivariable system is transformed into a single variable using an adjoint operator.A series of standard Black-Scholes models are obtained and their asymptotic approximate solutions are obtained,giving a uniformly valid error estimate for this approximate solution.3.This thesis mainly studies the pricing of permanent American options with transaction costs under fast reverting process volatility,and obtains a linear equation by singularly perturbing the nonlinear model.The adjoint operator is used to simplify the linear equation and convert the multivariable system into a single variable.At the same time,it is necessary to use the Black-Scholes expression to obtain the optimal implementation boundary.The singular perturbation expansion is performed on the overall optimal implementation boundary.The constant coefficient linear ordinary differential equation method is used to obtain the optimal implementation boundary solution for each expansion,and then the overall optimal implementation boundary and asymptotic approximate solution are formed.Finally,we give a consistent and effective error estimate.