Finite difference solutions of the CEV PDE

This work studies the valuation of European options under the constant elasticity of variance model. The model generalises the Black-Scholes framework for option pricing by incorporating a local instantaneous volatility term which is a function of the stock price. The model has the ability to fit certain implied volatility structures exhibited by market option prices, but the computation of the closed-form European option formula is not always stable and can be largely inaccurate for some parameter ranges because of the difficulties associated with the computation of the non-central chi-square distribution in the valuation formula. As an alternative to one line of research which aims at accelerating and stabilising the analytical price computation, we study finite difference techniques to obtain European option prices and associated hedging parameters. It is numerically demonstrated that a direct discretisation of the pricing equation in combination with an exponential integrator in time performs better than other schemes based on Crank-Nicolson discretisations of two transformed problems, one posed on an infinite domain and the other on a finite domain.