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Natural volatility and option pricing

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Author Info
Carey, Alexander

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Abstract

In this paper we recover the Black-Scholes and local volatility pricing engines in the presence of an unspecified, fully stochastic volatility. The input volatility functions are allowed to fluctuate randomly and to depend on time to expiration in a systematic way, bringing the underlying theory in line with industry experience and practice. More generally we show that to price a European-exercise path-(in)dependent option, it is enough to model the evolution of the variance of instantaneous returns over the natural filtration of the underlying security. We call the square root of this new process natural volatility. We develop the associated concept of path-conditional forward volatility, via which the natural volatility can be directly specified in an economically meaningful way.

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Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 6709.

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Date of creation: 12 Jan 2008
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Handle: RePEc:pra:mprapa:6709

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Related research
Keywords: natural filtration natural volatility stochastic volatility local volatility path-dependent volatility change of measure change of filtration martingale valuation Black-Scholes path-conditional forward price path-conditional forward volatility

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G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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  1. Rubinstein, Mark, 1994. " Implied Binomial Trees," Journal of Finance, American Finance Association, vol. 49(3), pages 771-818, July. [Downloadable!] (restricted)
  2. Black, Fischer, 1976. "The pricing of commodity contracts," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 167-179. [Downloadable!] (restricted)
  3. Hull, John C & White, Alan D, 1987. " The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June. [Downloadable!] (restricted)
  4. Pascucci, Andrea & Foschi, Paolo, 2006. "Path dependent volatility," MPRA Paper 973, University Library of Munich, Germany. [Downloadable!]
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  5. Robert C. Merton, 1973. "Theory of Rational Option Pricing," Bell Journal of Economics, The RAND Corporation, vol. 4(1), pages 141-183, Spring. [Downloadable!] (restricted)
  6. Harrison, J. Michael & Kreps, David M., 1979. "Martingales and arbitrage in multiperiod securities markets," Journal of Economic Theory, Elsevier, vol. 20(3), pages 381-408, June. [Downloadable!] (restricted)
  7. Cox, John C. & Ross, Stephen A., 1976. "The valuation of options for alternative stochastic processes," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 145-166. [Downloadable!] (restricted)
  8. 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 637-54, May-June. [Downloadable!] (restricted)
  9. Mark Rubinstein., 1994. "Implied Binomial Trees," Research Program in Finance Working Papers RPF-232, University of California at Berkeley. [Downloadable!]
  10. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Oxford University Press for Society for Financial Studies, vol. 6(2), pages 327-43. [Downloadable!] (restricted)
  11. Steven L. Heston & Saikat Nandi, 1998. "Preference-free option pricing with path-dependent volatility: A closed-form approach," Working Paper 98-20, Federal Reserve Bank of Atlanta. [Downloadable!]
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