A reduced PDE method for European option pricing under multi-scale, multi-factor stochastic volatility

Abstract

The number of tailor-made hybrid structured products has risen more prominently to fit each investor’s preferences and requirements as they become more diversified. The structured products entail synthetic derivatives such as combinations of bonds and/or stocks conditional on how they are backed up by underlying securities, stochastic volatility, stochastic interest rates or exchanges rates. The complexity of these multi-asset structures yields lots of difficulties of pricing the products. Because of the complexity, Monte-Carlo simulation is a possible choice to price them but it may not produce stable Greeks leading to a trouble in hedging against risks. In this light, it is desirable to use partial differential equations with relevant analytic and numerical techniques. Even if the partial differential equation method would generate stable security prices and Greeks for single asset options, however, it may result in the curse of dimensionality when pricing multi-asset derivatives. In this study, we make the best use of multi-scale nature of stochastic volatility to lift the curse of dimensionality for up to three asset cases. Also, we present a transformation formula by which the pricing group parameters required for the multi-asset options in illiquid market can be calculated from the underlying market parameters.

Keywords:

• Multi-asset
• Multi-scale
• Stochastic volatility
• Asymptotics
• Dimension reduction
• Illiquid market

AMS Subject Classifications:

• G13

Acknowledgements

We appreciate valuable comments and suggestions given by the reviewers which have made this paper improved significantly. The authors are responsible for any errors.

Notes

No potential conflict of interest was reported by the authors.

1 We opt for the LL2VI model in Chernov et al. (2003) and the SV2-EUL model in Durham (2006).

2 http://www.statsmodels.org

Additional information

Funding

The work of J.-H. Kim was supported by the National Research Foundation of Korea [grant number NRF- 2017R1A2B4003226] and the work of J. Huh was partially supported by the Graduate School of Yonsei University Research Scholarship Grants in 2017.