Carey, Alexander (2008): Natural volatility and option pricing.

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Abstract
In this paper we recover the BlackScholes 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 Europeanexercise 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 pathconditional forward volatility, via which the natural volatility can be directly specified in an economically meaningful way.
Item Type:  MPRA Paper 

Original Title:  Natural volatility and option pricing 
Language:  English 
Keywords:  natural filtration, natural volatility, stochastic volatility, local volatility, pathdependent volatility, change of measure, change of filtration, martingale valuation, BlackScholes, pathconditional forward price, pathconditional forward volatility 
Subjects:  G  Financial Economics > G1  General Financial Markets > G13  Contingent Pricing ; Futures Pricing 
Item ID:  6709 
Depositing User:  Alexander Carey 
Date Deposited:  13. Jan 2008 05:16 
Last Modified:  11. Feb 2013 20:40 
References:  ALLEN P., S. EINCHCOMB and N. GRANGER (2006): Conditional Variance Swaps. Product note, J.P. Morgan Securities, London. BILLINGSLEY, P. (1995): Probability and Measure, 3rd edition. New York: John Wiley. BLACK, F. (1976): The Pricing of Commodity Contracts, J. Finan. Econ. 3, 167179. BLACK, F., and M. SCHOLES (1973): The Pricing of Options and Corporate Liabilities, J. Polit. Economy 81, 637654. CARR, P., and K. LEWIS (2004): Corridor Variance Swaps, Risk February, 6772. COX, J., and S. ROSS (1976): The Valuation of Options for Alternative Stochastic Processes, J. Finan. Econ. 3, 145166. DELBAEN, F., and W. SCHACHERMAYER (1998): The Fundamental Theorem of Asset Pricing for Unbounded Stochastic Processes, Math. Ann. 312, 215250. DERMAN, E., and I. KANI (1994): Riding on a Smile, Risk February, 3239. DERMAN, E., I. KANI and M. KAMAL (1997): Trading and Hedging Local Volatility, J. Finan. Eng. 6, 233268. DERMAN, E., and I. KANI (1998): Stochastic Implied Trees: Arbitrage Pricing with Stochastic Term and Strike Structure of Volatility, Int. J. Theoretical Appl. Finance 1, 61110. DOTHAN, M. (1990): Prices in Financial Markets. Oxford: Oxford University Press. DUPIRE, B. (1993): Arbitrage Pricing with Stochastic Volatility. Discussion paper, Paribas Capital Markets, London. DUPIRE, B. (1994): Pricing with a Smile, Risk January, 1820. DUPIRE, B. (1996): A Unified Theory of Volatility. Discussion paper, Paribas Capital Markets, London. DE FINETTI, B. (1937): La Prévision: ses Lois Logiques, ses Sources Subjectives, Ann. Institut Henri Poincaré 7, 168. FOSCHI, P., and A. PASCUCCI (2008): Path Dependent Volatility, Decis. Econ. Finance 31, 120. FOUQUE, J.P., G. PAPANICOLAOU and K.R. SIRCAR (2000): Derivatives in Financial Markets with Stochastic Volatility. Cambridge: Cambridge University Press. GIKHMAN, I., and A. SKOROKHOD (1974): The Theory of Stochastic Processes I. Berlin: Springer Verlag. HAGAN, P., D. KUMAR, A. LESNIEWSKI and D. WOODWARD (2002): Managing Smile Risk, Wilmott September, 84108. HARRISON, J.M., and D. KREPS (1979): Martingales and Arbitrage in Multiperiod Securities Markets, J. Econ. Theory 20, 381408. HARRISON, J.M. and S. PLISKA (1981): Martingales and Stochastic Integrals in the Theory of Continuous Trading, Stoch. Process. Appl. 11, 215260. HESTON, S. (1993): A ClosedForm Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options, Rev. Finan. Stud. 6, 327343. HESTON, S., and S. NANDI (1998): PreferenceFree Option Pricing with PathDependent Volatility: a ClosedForm Approach. Working paper 9820, Federal Reserve Bank of Atlanta. HOBSON, D., and L.C.G. ROGERS (1998): Complete Models with Stochastic Volatility, Math. Finance 8, 2748. HULL, J., and A. WHITE (1987): The Pricing of Options on Assets with Stochastic Volatilities, J. Finance 42, 281300. IBRAGIMOV, I.A., and Y.A. ROZANOV (1978): Gaussian Random Processes. Berlin: SpringerVerlag. KARATZAS, I., and S.E. SHREVE (1991): Brownian Motion and Stochastic Calculus, 2nd edition. Berlin: SpringerVerlag. KAT, H. (1994): Contingent premium options, J. Derivatives 1, 4455. LEDOIT, O., P. SANTACLARA and S. YAN (2002): Relative Pricing of Options with Stochastic Volatility. Working paper, Anderson Graduate School of Management, Los Angeles. LIPSTER, R.S., and A.N. SHIRYAEV (2001): Statistics of Random Processes I. General Theory, 2nd edition. New York: Springer. MERTON, R. (1973): Theory of Rational Option Pricing, Bell J. Econ. Manage. Sci. 4, 141183. MILNE, P. (1997): Bruno de Finetti and the Logic of Conditional Events, Br. J. Philos. Sci. 48, 195 232. NIELSEN, L.T. (1999): Pricing and Hedging of Derivative Securities. Oxford: Oxford University Press RAO, M.M., and R. SWIFT (2006): Probability Theory with Applications, 2nd edition. New York: Springer. REVUZ, D., and M. YOR (1991): Continuous Martingales and Brownian Motion. Berlin: Springer Verlag. RUBINSTEIN, M. (1994): Implied Binomial Trees, J. Finance 49, 771818. SCHÖNBUCHER, P. (1999): A Market Model for Stochastic Implied Volatility, Phil. Trans. R. Soc. A 357, 20712092. STROBL, K. (2001): On the Consistency of the Deterministic Local Volatility Function Model (‘Implied Tree’), Int. J. Theoretical Appl. Finance 4, 545565. KAT, H. (2001): Structured Equity Derivatives. Chichester: John Wiley. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/6709 