A novel term-structure-based Heston model for implied volatility surface

Abstract

It remains a challenge for existing financial models to accurately capture the shapes of implied volatility (IV) of options with all maturities simultaneously. Inspired by Chen et al.’s empirical finding, ‘the shorter the option expiry, the higher the IV is’, we propose a novel and simple approach to solve this challenge, by introducing a term-structure-based correction (i.e. an exponential increase function of the options expiry) to the volatility of volatility (vol–vol) term of the classical Heston stochastic volatility model. We derive an approximate formula for the IV under the corrected model with the perturbation method and further apply the formula to predict the IVs of options written on the Shanghai Stock Exchange 50 ETF. Numerical experiments and empirical results show that the introduction of a term-structure-based correction function surely overcomes the deficiency of the classical Heston model in capturing the short-term IVs, thus improving notably its performance of IV forecasting.

KEYWORDS:

• Implied volatility
• term structure
• stochastic volatility
• Heston model
• perturbation method

Mathematics Subject Classifications:

• 91B70
• 91G20
• 91G60
• 60E10

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 We have tried another form, the power-law function, i.e., $H(\tau )=A{\tau }^{\alpha }$, which also shows an explosive effect in strengthening the IVs of very short-term options. However, the discontinuity of $H(\tau )=A{\tau }^{\alpha }$ at $\tau =0$ and the sharp drop when $\tau \to 0$ bring big challenges to well calibrate the H-DA-EEG model when facing the in-samples with very short terms. Hence, we abandon the power-law setting for the correction function.

2 Actually, we have developed the third-order expansion to the characteristic functions as well as the associated IV formula. Please see the Appendix D for details. However, compared with their second-order counterparts, the higher-order expansion does not bring a better result but increases the computation complexity geometrically. This result is consistent with Takahashi [45], Friz & Gatheral et al. [23].

Additional information

Funding

This research is supported by the National Natural Science Foundation of China under grant number 72271064 and the Natural Science Foundation of Guangdong Province of China under grant number 2022A1515011125.