Black-Scholes方程的Legendre有理拟谱方法

Legendre Rational Pseudospectral Method for Black-Scholes Equation

利用变量代换,将带有渐近边界条件的终值Black-Scholes期权定价问题转化为抛物型对流扩散方程的初边值问题,接着构造了该等价问题的弱形式,并建立了相应的半离散Legendre有理拟谱格式.最后,利用Legendre有理正交投影和Legendre-Gauss有理插值逼近结果分析了数值格式的收敛性,并证明了该数值方法在空间方向具有谱精度.本文尽管只考虑了Black-Scholes模型问题,但是构造数值格式和分析收敛性的方法和技巧可以推广到其他线性和非线性问题.

The Black-Scholes option pricing problem with terminal value under asymptotic boundary conditions was transformed to an equivalent parabolic initial boundary value problem of convection diffusion equation by using variable substitution.A weak form of the equivalent problem was constructed,and the corresponding semi-discrete Legendre rational pseudospectral scheme was also proposed.Then by using the approximation results of Legendre rational orthogonal projection and Legendre-Gauss rational interpolation,the convergence analysis of the numerical scheme was derived,the spectral accuracy of which in space was obtained.Although only a simple Black-Scholes model problem was considered,the techniques developed in constructing proper numerical scheme and analysing its convergence could be easily generalized to other linear and nonlinear problems.