Numerical Methods For Option Pricing Under Hard-to-borrow Models

Option pricing is the central issue in option contract,however an important assumption implicit in the classical Black-Scholes formula is to allow unrestricted short selling,thus one need to study the hard-to-borrow stock model when considering the option pricing under short selling.In general,one cannot obtain the analytic pricing formulas under the hard-to-borrow model,this enables the thesis to explore the numerical methods for the value partial differential equations.This thesis firstly considers the European option pricing under the hard-to-borrow model,then extends to the cases of regime-switching and jump-diffusion,and mainly focuses on the pricing algorithms and convergence problems.For the European option pricing,since the spatial delay term in partial differential equations has an important influence on the convergence and accuracy of the numerical methods,a mesh-dependent Taylor expansion approach without fixed truncation error is introduced to approximate the spatial delay term,then the Crank-Nicolson difference scheme and the operator splitting alternative direction difference scheme on the expansion are studied respectively.Moreover the convergence of the Crank-Nicolson scheme is theoretically proved.Numerical results show that the proposed methods not only improve the computational accuracy,but also achieve the optimal convergence rates.For the regime-switching model,there are three distinct characteristics of the partial differential equations: firstly,the system of equations are coupled with respect to the regime states;secondly,the system of partial differential equations have delay terms in two spatial directions;moreover one boundary condition is determined by another initial-boundary value problem of the system of partial differential equations,which bring some difficulties in algorithms designing and convergence analysis.The Crank-Nicolson difference scheme and the operator splitting alternative direction difference scheme for the system of partial differential equations are studied,then the rigorous framework of convergence analysis is established for the Crank-Nicolson scheme,moreover the second-order convergence rates are rigorously obtained.Finally,numerical examples are given to demonstrate the theoretical results.For the jump-diffusion model,since the integral term of the partial integrodifferential equation cannot be discretized by the trapezoidal formula directly,then the mesh-dependent Taylor expansion and the trapezoidal rule are combined to discretize the integral term,further a implicit-explicit difference method on the scheme for the partial integro-differential equation is studied where the integral term handled explicitly and the rest terms implicitly.This treatment saves the CPU memory and improves the computation efficiency.Moreover the first-order convergence rates for time and the second-order convergence rates for space are rigorously proved.Besides,this thesis also studies the operator splitting alternative direction difference scheme for the partial integro-differential equation.Finally,the thesis studies the Laplace transform and finite difference methods for European option pricing,more specifically: the partial differential equation is firstly semi-discretized by the finite difference method;then taking Laplace transform to the semi-discretized system with respect to time variable obtains parameterized linear system,finally the linear system is iteratively solved on the trajectory of the numerical hyperbolic contour to invert the Laplace transform and to recover the option prices.Moreover the rigorous convergence analysis of the method is carried out to prove that the scheme achieves the second-order convergence rates for space and the exponential-order convergence rates for the quadrature nodes.Besides,the method is extended to the case of regime-switching option pricing.