Fast and Simple Method for Pricing Exotic Options Using Gauss-Hermite Quadrature on a Cubic Spline Interpolation

Abstract

There is a vast literature on numerical valuation of exotic options using Monte Carlo, binomial and trinomial trees, and finite difference methods. When transition density of the underlying asset or its moments are known in closed form, it can be convenient and more efficient to utilize direct integration methods to calculate the required option price expectations in a backward time-stepping algorithm. This paper presents a simple, robust and efficient algorithm that can be applied for pricing many exotic options by computing the expectations using Gauss-Hermite integration quadrature applied on a cubic spline interpolation. The algorithm is fully explicit but does not suffer the inherent instability of the explicit finite difference counterpart. A 'free' bonus of the algorithm is that it already contains the function for fast and accurate interpolation of multiple solutions required by many discretely monitored path dependent options. For illustrations, we present examples of pricing a series of American options with either Bermudan or continuous exercise features, and a series of exotic pathdependent options of target accumulation redemption note (TARN). Results of the new method are compared with Monte Carlo and finite difference methods, including some of the most advanced or best known finite difference algorithms in the literature. The comparison shows that, despite its simplicity, the new method can rival with some of the best finite difference algorithms in accuracy and at the same time it is significantly faster. Virtually the same algorithm can be applied to price other path-dependent financial contracts such as Asian options and variable annuities.

Valuation of Variable Annuities with Guaranteed Minimum Withdrawal and Death Benefits via Stochastic Control Optimization

“…The accuracy and efficiency of GHQC method in pricing GMWB under optimal withdraw is well demonstrated in Luo & Shevchenko (2014a). From numerical point of view, once the problem is correctly formulated, pricing GMWDB (with combined GMWB and GMDB features) uses the same key algorithm components as those for pricing GMWB, such as numerical quadrature integration for the expectations and cubic spline interpolation for applying the jump conditions.…”

“…This method can be applied when transition density of the underlying asset between withdrawal dates or it's moments are known in closed form and required expectations are 1d integrals. It has also been successfully used to price exotic options such as American, Asian, barrier, etc; see Luo & Shevchenko (2014a).…”

Fast numerical method for pricing of variable annuities with guaranteed minimum withdrawal benefit under optimal withdrawal strategy

“…In general, the abscissas and the weights for the Gauss-Hermite quadrature for a given order q can be readily computed, e.g. using functions in Press et al (1992); also as a reference, the abscissas and the weights for q = 5, 6, 16 are presented in Luo and Shevchenko (2014). Applying a change of variable x = y/ √ 2 and use the Gauss-Hermite quadrature to (14), we obtain…”

“…For convenience, hereafter we refer this new algorithm as GHQC (Gauss-Hermite quadrature on cubic spline). We adopt the method developed in Luo and Shevchenko (2014) for pricing American options and extend it to solving optimal stochastic control problem for pricing GMWB variable annuity. This allows to get virtually instant results for typical GMWB annuity prices on the standard desktop PC.…”