Numerical aspects of integration in semi-closed option pricing formulas for stochastic volatility jump diffusion models

Abstract

In mathematical finance, a process of calibrating stochastic volatility (SV) option pricing models to real market data involves a numerical calculation of integrals that depend on several model parameters. This optimization task consists of large number of integral evaluations with high precision and low computational time requirements. However, for some model parameters, many numerical quadrature algorithms fail to meet these requirements. We can observe an enormous increase in function evaluations, serious precision problems and a significant increase of computational time.In this paper we numerically analyse these problems and show that they are especially caused by inaccurately evaluated integrands. We propose a fast regime switching algorithm that tells if it is sufficient to evaluate the integrand in standard double arithmetic or if a higher precision arithmetic has to be used. We compare and recommend numerical quadratures for typical SV models and different parameter values, especially for problematic cases.

Solution of option pricing equations using orthogonal polynomial expansion

“…By the numerical implementation of Galerkin's method in weighted Sobolev spaces we found an alternative representation of the solution to both BS and Heston models. The obtained representation is a smooth approximation of the solution that does not share the serious numerical difficulties of existing semi-closed formulas as they were presented by Daněk and Pospíšil (2019).…”

“…Both in BS and Heston model, one can derive the pricing partial differential equation (PDE) in several different ways, for example using the Fokker-Planck equation for the transition probability density function. Although semi-closed formulas have been widely used in practice for a long time, only recently Daněk and Pospíšil (2019) showed that for certain values of model parameters these formulas can bring serious numerical difficulties especially in evaluation of the integrands in these formulas and their implementation therefore sometimes requires a demanding high precision arithmetic to be adopted.…”

Isogeometric analysis in option pricing

“…If there exists a semi-closed form solution for studied stochastic volatility jump diffusion model, it is of course superior to any other means of numerical solution of corresponding pricing equations. However, even these formulas can have serious numerical problems as was shown, for example, by Daněk and Pospíšil (2017). Let us mention at least one minor advantage of using finite elements over semi-closed formulas, namely one finite element solution give us prices of options for one strike and all maturities at once.…”

Solution of option pricing equations using orthogonal polynomial expansion

“…Both in BS and Heston model, one can derive the pricing partial differential equation (PDE) in several different ways, for example (Wilmott, 1998;Rouah, 2013;Hull, 2018) using arbitrage arguments with self-financing trading strategies, approaches with martingale measures or the Fokker-Planck equation for the transition probability density function. Although semi-closed formulas have been widely used in practice for a long time, only recently Daněk and Pospíšil (2020) showed that for certain values of model parameters these formulas can bring serious numerical difficulties especially in evaluation of the integrands in these formulas and their implementation therefore sometimes requires a demanding high precision arithmetic to be adopted.…”