Diffusion Equations: Convergence of the Functional Scheme Derived from the Binomial Tree with Local Volatility for Non Smooth Payoff Functions

The function solution to the functional scheme derived from the binomial tree financial model with local volatility converges to the solution of a diffusion equation of type $${h_t}(t,x) + {{{x^2}{\sigma ^2}(t,x)} \over 2}{h_{xx}}(t,x) = 0$$ht(t,x)+x2σ2(t,x)2hxx(t,x)=0 as the number of discrete dates $$n \to \infty $$n→∞ . Contrarily to classical numerical methods, in particular finite difference methods, the principle behind the functional scheme is only based on a discretization in time. We establish the uniform convergence in time of the scheme and provide the rate of convergence when the payoff function is not necessarily smooth as in finance. We illustrate the convergence result and compare its performance to the finite difference and finite element methods by numerical examples.