On pricing variance swaps in discretely-sampled with High Volatility model

Year 2021, , 105 - 115, 30.06.2021

Youssef El-khatib , Mariam Alshamsi , Jun Fan

https://doi.org/10.53006/rna.908113

Abstract

In this paper, the valuation of discretely sampled variance swaps is investigated in a financial asset price model with an increase in volatility. More precisely, we consider a
stochastic differential equation model with an additional parameter that augments
volatility. This is to cover the impact of financial crunches on the prices of a given asset.
Under these settings, the calculation of the annualized delivery price of a variance swap is not
sure in a closed-form. Following the literature, the delivery price can be written as a finite sum of conditional expectations. We focus our attention on the computations
of these expectations and we obtain some interesting results. This leads to a semi-
analytical solution to the variance swaps pricing problems. Some illustrations showing
the goodness of our model are provided.

Keywords

Stochastic Dierential Equations, discretely-sampled variance swaps, high volatility model

Supporting Institution

United Arab Emirates University

Project Number

UPAR Grant No.31S369.

Thanks

The authors would like to express their sincere appreciation to the United Arab Emirates University Research Office for the financial support UPAR Grant No.31S369.

References

• [1] M. Broadie and A. Jain, The effect of jumps and discrete sampling on volatility and variance swaps, International Journal of Theoretical and Applied Finance, 11(8)(2008) 761-797.
• [2] K. Demeter, E. Derman, M. Kamal, and J. Zou, More than you ever wanted to know about volatility swaps, Goldman Sachs Quantitative Strategies Research Notes, (1999).
• [3] G. Dibeh, H-M. Harmanani, Option pricing during post-crash relaxation times, Physica A., 380(2007), 357-365.
• [4] Y. El-Khatib, and Hatemi-J, A., Computations of Price Sensitivities After a Financial Market Crash, In Ao SI., Gelman L. (eds) Electrical Engineering and Intelligent Systems., Lecture Notes in Electrical Engineering, (2013), vol 130. Springer, New York, NY.
• [5] Y. El-Khatib, and Hatemi-J, A., Option valuation and hedging in markets with a crunch, Journal of Economic Studies, 44(5)(2017) 801-815.
• [6] Y. El-Khatib, and Hatemi-J, A., Option pricing in high volatile markets with illiquidity, AIP Conference Proceedings 2019 Jul 24, 2116(1), AIP Publishing LLC.
• [7] Hatemi-J, A and Y. El-Khatib, Stochastic optimal hedge ratio: Theory and evidence, Applied Economics Letters, 19(8)2012 699-703.
• [8] Hatemi-J, A and Y. El-Khatib, Portfolio selection: An alternative approach, Economics Letters, 135(2015) 141--143.
• [9] S. Heston, A closed-form solution for option pricing with stochastic volatility with application to bond and currency options, Review of Financial Studies, 6(2)(1993), 327-343.
• [10] A. Javaheri, P. Wilmott, and E. Haug, GARCH and volatility swaps, Quantitative Finance, 4(5)(2004), 589--595.
• [11] T. Little and V. Pant, A finite-difference method for the valuation of variance swaps, The Journal of Computational Finance, 5(1)(2001), 81--101.
• [12] N. Privault, Understanding Markov Chains Examples, and Applications, (2013) Springer Singapore, 2nd edition.
• [13] S. Rujivan and S. Zhu, A simplified analytical approach for pricing discretely-sampled variance swaps with stochastic volatility, Applied Mathematics Letters, 25(11)(2012), 1644-1650.
• [14] S. Zhu and G. Lian, A closed-form exact solution for pricing variance swaps with stochastic volatility, Mathematical Finance, 21(2)(2011) 233-256.
• [15] S.P. Zhu, A. Badran, and X. Lu, A new exact solution for pricing European options in a two-state regime-switching economy, Computers, and Mathematics with Applications, 64(8)(2012), 2744--2755.