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

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

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.

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  • [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.