Option pricing with Legendre polynomials

TL;DR

This paper introduces a novel option pricing method using Legendre series expansion of the density function, enabling rapid and accurate computation by leveraging the characteristic function and addressing numerical stability issues.

Contribution

The paper develops a new Legendre polynomial-based approach for option pricing that improves accuracy and stability, especially for non-smooth payoff functions, with proven exponential convergence.

Findings

Achieves rapid and accurate density function recovery

Provides exponential convergence in numerical experiments

Addresses numerical stability through a difference equation approach

Abstract

Here we develop an option pricing method based on Legendre series expansion of the density function. The key insight, relying on the close relation of the characteristic function with the series coefficients, allows to recover the density function rapidly and accurately. Based on this representation for the density function, approximations formulas for pricing European type options are derived. To obtain highly accurate result for European call option, the implementation involves integrating high degree Legendre polynomials against exponential function. Some numerical instabilities arise because of serious subtractive cancellations in its formulation (96) in proposition 7.1. To overcome this difficulty, we rewrite this quantity as solution of a second-order linear difference equation and solve it using a robust and stable algorithm from Olver. Derivation of the pricing method has been accompanied by an error analysis. Errors bounds have been derived and the study relies more on smoothness properties which are not provided by the payoff? functions, but rather by the density function of the underlying stochastic models. This is particularly relevant for options pricing where the payoff of the contract are generally not smooth functions. The numerical experiments on a class of models widely used in quantitative finance show exponential convergence.