Exponentially fitted block backward differentiation formulas for pricing options

Abstract

A family of Exponentially Fitted Block Backward Differentiation Formulas (EFBBDFs) whose coefficients depend on a parameter and step-size is developed and implemented on the Black–Scholes partial differential equation (PDE) for the valuation of options on a non-dividend-paying stock. Specific EFBBDFs of order 2 and 4 are applied to solve the PDE after reducing it into a system of ordinary differential equations via the method of lines. The methods are shown to be superior to the well-known Crank–Nicolson method since they are $$L$$-stable and do not exhibit oscillations usually triggered by discontinuities inherent in the payoff function of financial contracts. We confirmed the accuracy of the methods by initially applying them to a prototype example based on the one-dimensional time-dependent convection–diffusion equation with a known analytical solution. It is demonstrated that the American put can be exercised early by computing the hedging parameter “delta”, which specifies the condition for early exercise of the put option. Although the methods can be used to price all vanilla options, we elect to focus on the put due to its optimality.

Keywords

• block backward differentiation formula
• exponential fitting
• oscillations
• options
• black–scholes partial differential equation