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Natural volatility and option pricing

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  • Carey, Alexander

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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Bibliographic Info

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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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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  1. Rubinstein, Mark, 1994. " Implied Binomial Trees," Journal of Finance, American Finance Association, vol. 49(3), pages 771-818, July.
  2. Ledoit, Olivier & Santa-Clara, Pedro & Yan, Shu, 2002. "Relative Pricing of Options with Stochastic Volatility," University of California at Los Angeles, Anderson Graduate School of Management qt7jp8f42t, Anderson Graduate School of Management, UCLA.
  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.
  4. 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.
  5. Paolo Foschi & Andrea Pascucci, 2008. "Path dependent volatility," Decisions in Economics and Finance, Springer, vol. 31(1), pages 13-32, May.
  6. Mark Rubinstein., 1994. "Implied Binomial Trees," Research Program in Finance Working Papers RPF-232, University of California at Berkeley.
  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, vol. 6(2), pages 327-43.
  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.
  9. Robert C. Merton, 1973. "Theory of Rational Option Pricing," Bell Journal of Economics, The RAND Corporation, vol. 4(1), pages 141-183, Spring.
  10. 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.
  11. 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.
  12. David G. Hobson & L. C. G. Rogers, 1998. "Complete Models with Stochastic Volatility," Mathematical Finance, Wiley Blackwell, vol. 8(1), pages 27-48.
  13. Black, Fischer, 1976. "The pricing of commodity contracts," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 167-179.
  14. 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.
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