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Derivatives on volatility: some simple solutions based on observables

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  • Steven L. Heston
  • Saikat Nandi

Abstract

Proposals to introduce derivatives whose payouts are explicitly linked to the volatility of an underlying asset have been around for some time. In response to these proposals, a few papers have tried to develop valuation formulae for volatility derivatives—derivatives that essentially help investors hedge the unpredictable volatility risk. This paper contributes to this nascent literature by developing closed-form/analytical formulae for prices of options and futures on volatility as well as volatility swaps. The primary contribution of this paper is that, unlike all other models, our model is empirically viable and can be easily implemented. ; More specifically, our model distinguishes itself from other proposed solutions/models in the following respects: (1) Although volatility is stochastic, it is an exact function of the observed path of asset prices. This is crucial in practice because nonobservability of volatility makes it very difficult (in fact, impossible) to arrive at prices and hedge ratios of volatility derivatives in an internally consistent fashion, as it is akin to not knowing the stock price when trying to price an equity derivative. (2) The model does not require an unobserved volatility risk premium, nor is it predicated on the strong assumption of the existence of a continuum of options of all strikes and maturities as in some papers. (3) We show how it is possible to replicate (delta hedge) volatility derivatives by trading only in the underlying asset (on whose volatility the derivative exists) and a risk-free asset. This bypasses the problem of having to trade numerously many options on the underlying asset, a hedging strategy proposed in some other models.

Suggested Citation

  • Steven L. Heston & Saikat Nandi, 2000. "Derivatives on volatility: some simple solutions based on observables," FRB Atlanta Working Paper 2000-20, Federal Reserve Bank of Atlanta.
  • Handle: RePEc:fip:fedawp:2000-20
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    References listed on IDEAS

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    1. Breeden, Douglas T & Litzenberger, Robert H, 1978. "Prices of State-contingent Claims Implicit in Option Prices," The Journal of Business, University of Chicago Press, vol. 51(4), pages 621-651, October.
    2. Harrison, J. Michael & Pliska, Stanley R., 1981. "Martingales and stochastic integrals in the theory of continuous trading," Stochastic Processes and their Applications, Elsevier, vol. 11(3), pages 215-260, August.
    3. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, pages 307-327.
    4. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters,in: Theory Of Valuation, chapter 5, pages 129-164 World Scientific Publishing Co. Pte. Ltd..
    5. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters,in: Theory Of Valuation, chapter 8, pages 229-288 World Scientific Publishing Co. Pte. Ltd..
    6. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    7. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, pages 327-343.
    8. Rubinstein, Mark, 1994. " Implied Binomial Trees," Journal of Finance, American Finance Association, vol. 49(3), pages 771-818, July.
    9. Leland, Hayne E, 1994. " Corporate Debt Value, Bond Covenants, and Optimal Capital Structure," Journal of Finance, American Finance Association, vol. 49(4), pages 1213-1252, September.
    10. Grunbichler, Andreas & Longstaff, Francis A., 1996. "Valuing futures and options on volatility," Journal of Banking & Finance, Elsevier, vol. 20(6), pages 985-1001, July.
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    Cited by:

    1. Windcliff, H. & Forsyth, P.A. & Vetzal, K.R., 2006. "Pricing methods and hedging strategies for volatility derivatives," Journal of Banking & Finance, Elsevier, vol. 30(2), pages 409-431, February.
    2. Dimitris Psychoyios & George Dotsis & Raphael Markellos, 2010. "A jump diffusion model for VIX volatility options and futures," Review of Quantitative Finance and Accounting, Springer, vol. 35(3), pages 245-269, October.
    3. Wang, Xingchun & Fu, Jianping & Wang, Guanying & Wang, Yongjin, 2015. "Quadratic hedging strategies for volatility swaps," Finance Research Letters, Elsevier, vol. 15(C), pages 125-132.
    4. Teh Raihana Nazirah Roslan & Wenjun Zhang & Jiling Cao, 2016. "Pricing variance swaps with stochastic volatility and stochastic interest rate under full correlation structure," Papers 1610.09714, arXiv.org.
    5. Ben-zhang Yang & Jia Yue & Nan-jing Huang, 2017. "Variance swaps under L\'{e}vy process with stochastic volatility and stochastic interest rate in incomplete market," Papers 1712.10105, arXiv.org, revised Jan 2018.
    6. Gonzalez-Perez, Maria T., 2015. "Model-free volatility indexes in the financial literature: A review," International Review of Economics & Finance, Elsevier, vol. 40(C), pages 141-159.
    7. repec:eee:apmaco:v:277:y:2016:i:c:p:72-81 is not listed on IDEAS

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    Keywords

    Derivative securities ; Hedging (Finance) ; Options (Finance);

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