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.

KEYWORDS:

• Variable precision arithmetic
• numerical integration
• adaptive quadrature
• option pricing
• stochastic volatility models

AMS CLASSIFICATIONS:

• 65D30
• 91G60
• 65R10

JEL CLASSIFICATION:

• C63

Acknowledgements

Computational resources were provided by the CESNET LM2015042 and the CERIT Scientific Cloud LM2015085, provided under the programme ‘Projects of Large Research, Development, and Innovations Infrastructure’.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Josef Daněk  http://orcid.org/0000-0002-9744-5107

Jan Pospíšil  http://orcid.org/0000-0002-4288-1614

Notes

1 Available in MATLAB Global Optimization Toolbox, function ga.

2 Available in MATLAB Optimization Toolbox, function lsqnonlin.

Additional information

Funding

This work was partially supported by the Grantová Agentura České Republiky (GACR), grant numbers GA14-11559S Analysis of Fractional Stochastic Volatility Models and their Grid Implementation and GA18-16680S Rough models of fractional stochastic volatility.