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STOCHASTIC VOLATILITY MODEL WITH CORRELATED JUMP SIZES AND INDEPENDENT ARRIVALS

Published online by Cambridge University Press:  12 February 2020

Pengzhan Chen Open the ORCID record for Pengzhan Chen [Opens in a new window]  and
Wuyi Ye
Pengzhan Chen
Affiliation:
School of Management, University of Science and Technology of China, Hefei 230026, PR China E-mail: [email protected]; [email protected]
Wuyi Ye
Affiliation:
School of Management, University of Science and Technology of China, Hefei 230026, PR China E-mail: [email protected]; [email protected]
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Abstract

In light of recent empirical research on jump activity, this article study the calibration of a new class of stochastic volatility models that include both jumps in return and volatility. Specifically, we consider correlated jump sizes and both contemporaneous and independent arrival of jumps in return and volatility. Based on the specifications of this model, we derive a closed-form relationship between the VIX index and latent volatility. Also, we propose a closed-form logarithmic likelihood formula by using the link to the VIX index. By estimating alternative models, we find that the general counting processes setting lead to better capturing of return jump behaviors. That is, the part where the return and volatility jump simultaneously and the part that jump independently can both be captured. In addition, the size of the jumps in volatility is, on average, positive for both contemporaneous and independent arrivals. However, contemporaneous jumps in the return are negative, but independent return jumps are positive. The sub-period analysis further supports above insight, and we find that the jumps in return and volatility increased significantly during the two recent economic crises.

Keywords

jump riskmodel specificationstochastic volatilityVIX
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 35 , Issue 3 , July 2021 , pp. 513 - 531
Copyright
© Cambridge University Press 2020

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References

1.Aït-Sahalia, Y., Cacho-Diaz, J., & Laeven, R.J. (2015). Modeling financial contagion using mutually exciting jump processes. Journal of Financial Economics 117(3): 585–606.CrossRefGoogle Scholar
2.Amengual, D. & Xiu, D. (2018). Resolution of policy uncertainty and sudden declines in volatility. Journal of Econometrics 203(2): 297–315.CrossRefGoogle Scholar
3.Bardgett, C., Gourier, E., & Leippold, M. (2019). Inferring volatility dynamics and risk premia from the S&P 500 and VIX markets. Journal of Financial Economics 131(3): 593–618.CrossRefGoogle Scholar
4.Bates, D.S. (2000). Post-'87 crash fears in the S&P 500 futures option market. Journal of Econometrics 94(1–2): 181–238.CrossRefGoogle Scholar
5.Black, F. & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy 81(3): 637–654.CrossRefGoogle Scholar
6.Bollerslev, T., Tauchen, G., & Zhou, H. (2009). Expected stock returns and variance risk premia. The Review of Financial Studies 22(11): 4463–4492.CrossRefGoogle Scholar
7.Britten-Jones, M. & Neuberger, A. (2000). Option prices, implied price processes, and stochastic volatility. The Journal of Finance 55(2): 839–866.Google Scholar
8.Carr, P. & Madan, D. (2001). Optimal positioning in derivative securities. Quantitative Finance 1(1): 19–37.CrossRefGoogle Scholar
9.Da Fonseca, J. & Ignatieva, K. (2019). Jump activity analysis for affine jump-diffusion models: evidence from the commodity market. Journal of Banking & Finance 99: 45–62.CrossRefGoogle Scholar
10.Dotsis, G., Psychoyios, D., & Skiadopoulos, G. (2007). An empirical comparison of continuous-time models of implied volatility indices. Journal of Banking & Finance 31(12): 3584–3603.Google Scholar
11.Duan, J.-C. (1994). Maximum likelihood estimation using price data of the derivative contract. Mathematical Finance 4(2): 155–167.CrossRefGoogle Scholar
12.Duan, J.-C. & Yeh, C.-Y. (2010). Jump and volatility risk premiums implied by VIX. Journal of Economic Dynamics and Control 34(11): 2232–2244.CrossRefGoogle Scholar
13.Duffie, D., Pan, J., & Singleton, K. (2000). Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68(6): 1343–1376.CrossRefGoogle Scholar
14.Eraker, B. (2004). Do stock prices and volatility jump? Reconciling evidence from spot and option prices. The Journal of Finance 59(3): 1367–1403.CrossRefGoogle Scholar
15.Eraker, B., Johannes, M., & Polson, N. (2003). The impact of jumps in volatility and returns. The Journal of Finance 58(3): 1269–1300.CrossRefGoogle Scholar
16.Han, X. (2019). Valuation of vulnerable options under the double exponential jump model with stochastic volatility. Probability in the Engineering and Informational Sciences 33: 81–104.CrossRefGoogle Scholar
17.Heston, S.L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies 6(2): 327–343.CrossRefGoogle Scholar
18.Hull, J. & White, A. (1987). The pricing of options on assets with stochastic volatilities. The Journal of Finance 42(2): 281–300.CrossRefGoogle Scholar
19.Jiang, G.J. & Tian, Y.S. (2005). The model-free implied volatility and its information content. The Review of Financial Studies 18(4): 1305–1342.CrossRefGoogle Scholar
20.Jones, C.S. (2003). The dynamics of stochastic volatility: evidence from underlying and options markets. Journal of Econometrics 116(1–2): 181–224.CrossRefGoogle Scholar
21.Kou, S.G. (2002). A jump-diffusion model for option pricing. Management Science 48(8): 1086–1101.CrossRefGoogle Scholar
22.Kou, S., Yu, C., & Zhong, H. (2017). Jumps in equity index returns before and during the recent financial crisis: a Bayesian analysis. Management Science 63(4): 988–1010.CrossRefGoogle Scholar
23.Li, J. & Zinna, G. (2018). The variance risk premium: components, term structures, and stock return predictability. Journal of Business & Economic Statistics 36(3): 411–425.CrossRefGoogle Scholar
24.Li, H., Wells, M.T., & Yu, C.L. (2008). A Bayesian analysis of return dynamics with Lévy jumps. The Review of Financial Studies 21(5): 2345–2378.CrossRefGoogle Scholar
25.Lian, G.-H. & Zhu, S.-P. (2013). Pricing VIX options with stochastic volatility and random jumps. Decisions in Economics and Finance 36(1): 71–88.CrossRefGoogle Scholar
26.Lin, Y.-N. & Chang, C.-H. (2010). Consistent modeling of S&P 500 and VIX derivatives. Journal of Economic Dynamics and Control 34(11): 2302–2319.CrossRefGoogle Scholar
27.Pan, J. (2002). The jump-risk premia implicit in options: evidence from an integrated time-series study. Journal of Financial Economics 63(1): 3–50.CrossRefGoogle Scholar
28.Pyun, S. (2019). Variance risk in aggregate stock returns and time-varying return predictability. Journal of Financial Economics 132(1): 150–174.CrossRefGoogle Scholar
29.Tankov, P. (2003). Financial modelling with jump processes. CRC Press. https://www.crcpress.com/Financial-Modelling-with-Jump-Processes/Tankov-Cont/p/book/9781584884132.CrossRefGoogle Scholar
30.Todorov, V. & Tauchen, G. (2011). Volatility jumps. Journal of Business & Economic Statistics 29(3): 356–371.CrossRefGoogle Scholar