This paper aims to develop a new free stochastic volatility model, joint with jumps. By freeing the power parameter of instantaneous variance, this paper takes Heston model and 3/2 model for special examples, and extends the generalizability. This model is named after free stochastic volatility model, and it owns two distinctive features. First of all, the power parameter is not constrained, so as to enable the data to voice its authentic direction. The Generalized Methods of Moments suggest that the purpose of this newly-added parameter is to create various volatility fluctuations observed in financial market. Secondly, even upward and downward jumps are separately modeled to accommodate the market data, this paper still provides the quasi-closed-form solutions for futures and option prices. Consequently, the model is novel and highly tractable. Here, it should be noted that the data on VIX futures and corresponding option contracts is employed to evaluate the model, in terms of its pricing and implied volatility features capturing performance. To sum up, the free stochastic volatility model with asymmetric jumps is capable of adequately capturing the implied volatility dynamics. Thus, it can be regarded as a model advantageous in pricing VIX derivatives with fixed power volatility models.
Free stochastic volatility Jumps VIX derivatives
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This work is supported by the National Natural Science Foundation of China (No. 11571310A011402) and Jin E. Zhang has been supported by an establishment grant from the University of Otago and the National Natural Science Foundation of China (Project No. 71771199).
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
Baldeaux, J., & Badran, A. (2014). Consistent modelling of VIX and equity derivatives using a 3/2 plus jumps model. Applied Mathematical Finance, 21(4), 299–312.CrossRefGoogle Scholar
Bluman, G., & Kumei, S. (1989). Symmetries and differential equations. Applied Mathematical Sciences, Vol. 81. New York: Springer.Google Scholar
Breeden, D. T., & Litzenberger, R. H. (1978). Prices of state-contingent claims implicit in option prices. Journal of Business, 51(4), 621–651.CrossRefGoogle Scholar
Chan, K. C., Karolyi, G. A., Longstaff, F. A., & Sanders, A. B. (1992). An empirical comparison of alternative models of the short-term interest rate. The Journal of Finance, 47(3), 1209–1227.CrossRefGoogle Scholar
Craddock, M., & Lennox, K. A. (2009). The calculation of expectations for classes of diffusion processes by Lie symmetry methods. The Annals of Applied Probability, 19(1), 127–157.CrossRefGoogle Scholar
Drimus, G. G. (2012). Options on realized variance by transform methods: A non-affine stochastic volatility model. Quantitative Finance, 12(11), 1679–1694.CrossRefGoogle Scholar
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
Goard, J., & Mazur, M. (2013). Stochastic volatility models and the pricing of VIX options. Mathematical Finance, 23(3), 439–458.CrossRefGoogle Scholar
Grasselli, M. (2017). The 4/2 stochastic volatility model: A unified approach for the Heston and the 3/2 model. Mathematical Finance, 27(4), 1013–1034.CrossRefGoogle Scholar
Grunbichler, A., & Longstaff, F. A. (1996). Valuing futures and options on volatility. Journal of Banking & Finance, 20(6), 985–1001.CrossRefGoogle Scholar
Hansen, L. P. (1982). Large sample properties of generalized method of moments estimators. Econometrica, 50(4), 1029–1054.CrossRefGoogle Scholar
Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies, 6(2), 327–343.CrossRefGoogle Scholar
Lewis, A. L. (2000). Option valuation under stochastic volatility. Newport Beach, CA: Finance Press.Google Scholar
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
Lin, W., Li, S. H., Luo, X. G., & Chern, S. (2017). Consistent pricing of VIX and equity derivatives with the 4/2 stochastic volatility plus jumps model. Journal of Mathematical Analysis and Applications, 447(2), 778–797.CrossRefGoogle Scholar
Olver, P. J. (1993). Applications of Lie groups to differential equations, 2nd Edn. Graduate Texts in Mathematics, 107. New York: Springer.Google Scholar
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
Park, Y. H. (2016). The Effects of asymmetric volatility and jumps on the pricing of VIX derivatives. Journal of Econometrics, 192(1), 313–328.CrossRefGoogle Scholar