Efficient pricing and hedging under the double Heston stochastic volatility jump-diffusion model

Abstract

The performances of jump-risk mitigation under a double Heston stochastic volatility jump-diffusion (dbHJ) model are examined. The quadratic exponential (QE) scheme is used to simulate the $\mathbb{P}$-measure price paths supposed to follow the dbHJ model, whereas a Fourier-COS-expansion-based scheme (i.e. the COS formula, see [F. Fang and C.W. Oosterlee, A novel pricing method for European based on Fourier-cosine series expansions, SIAM J. Sci. Comput. 31 (2008), pp. 826–848] is employed to price options and to calculate Greeks. Numerical results from extensive dynamic hedging experiments suggest that, when facing a $\mathbb{P}$-measure market with stiff volatility skews and non-trivial jumps, the dbHJ model is better than the plain (double) Heston and Black–Scholes models, and slightly outperforms the Heston stochastic volatility jump-diffusion (HJ) model and the Merton model in mitigating the jump risk of an option. This conclusion holds independent of the model specification of $\mathbb{P}$-measure market.

Keywords:

• option pricing
• Heston model
• COS method
• jump risks
• dynamic hedging

2010 AMS Subject Classifications:

• 91B70
• 91G20
• 91G60
• 60E10

Acknowledgments

The corresponding author is grateful to Prof. Cornelis W. Oosterlee, and two anonymous referees for insightful and detailed comments that have substantially improved the paper. All errors remain my own.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1. Here, I delibrately omit the over-lengthy formulas of $c_4$ that may cover several pages. However, in computational exercise, $c_4$ is needed. According to Fang and Oosterlee [12], including the fourth-order cumulant is sufficient for Equation (21(21) $a=c_1-L\sqrt{c_2+\sqrt{c_4+\cdots }};b=c_1+L\sqrt{c_2+\sqrt{c_4+\cdots }},$(21) ) to obtain a good integration interval for the COS method.

2. The reason for asset's dbHJ model assumption is that this model, with two stochastic volatility processes and a jump feature, is better than its reduced versions in capturing the empirical phenomena (such as skewness, fat tails, and excess kurtosis of return distribution) in financial market.

3. It makes only small difference between the hedging results for ${10}^5$ simulation paths and ${10}^4$ paths. But with more paths, the hedging performance is better.

Additional information

Funding

This work was supported by the Natural Science Foundation of Guangdong Province, China under Grant [No.2014A030313514].