Higher-order volatility: dynamics and sensitivities
In this addendum to Carey (2005), we draw several more analogies with the Black-Scholes model. We derive the characteristic function of the underlying log process as a function of the volatilities of all orders. Option prices are shown to satisfy an infinite-order version of the Black-Scholes partial differential equation. We find that in the same way that the option sensitivity to the cost of carry is related to delta and vega to gamma in the Black-Scholes model, the option sensitivity to j-th order volatility is related to the j-th order sensitivity to the underlying. Finally, we argue that third-order volatility provides a possible basis for the introduction of a "skew swap" product.
|Date of creation:||24 Aug 2006|
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- Carey, Alexander, 2005. "Higher-order volatility," MPRA Paper 4993, University Library of Munich, Germany.
- 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.
- Jarrow, Robert & Rudd, Andrew, 1982.
"Approximate option valuation for arbitrary stochastic processes,"
Journal of Financial Economics,
Elsevier, vol. 10(3), pages 347-369, November.
- Robert JARROW & Andrew RUDD, 2008. "Approximate Option Valuation For Arbitrary Stochastic Processes," World Scientific Book Chapters, in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 1, pages 9-31 World Scientific Publishing Co. Pte. Ltd..
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