Abstract
We present a methodology for extracting information from option prices when the market is viewed as knowledgeable. By expanding the information filtration judiciously and determining conditional characteristic functions for the log of the stock price, we obtain option pricing formulae which when fit to market data may reveal this information. In particular, we consider probing option prices for knowledge of the future stock price, instantaneous volatility, and the asymptotic dividend stream. Additionally the bridge laws developed are also useful for simulation based on stratified sampling that conditions on the terminal values of paths.
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References
G. Bakshi, and D. Madan, “Spanning and derivative-security valuation,” Journal of Financial Economics vol. 55 pp. 205–238, 2000.
C.A. Ball, and W.N. Torous, “Bond price dynamics and options,” Journal of Financial and Quantitative Analysis vol. 18 pp. 517–531, 1983.
Barndorff-Nielsen, and N. Shephard, “Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics (with discussion),” Journal of the Royal Statistical Society. Series B, Statistical Methodology vol. 63 pp. 167–241, 2001.
F. Black, and M. Scholes, “The pricing of options and corporate liabilities,” Journal of Political Economy vol. 81 pp. 637–654, 1973.
L. Canina, and S. Figlewski, “The informational content of implied volatility,” The Review of Financial Studies vol. 6 pp. 659–681, 1993.
P. Carr, and D. Madan, “Option valuation using the fast fourier transform,” Journal of Computational Finance vol. 2 pp. 61–73, 1998.
P. Carr, and D. Madan, “A note on sufficient conditions for no arbitrage,” Finance Research Letters vol. 2 pp. 125–130, 2005.
P. Carr, H. Geman, D. Madan, and M. Yor, “Stochastic volatility for Lévy processes,” Mathematical Finance vol. 13(3) pp. 345–382, 2003.
S. Cheng, “On the feasibility of arbitrage based option pricing when stochastic bond prices are involved,” Journal of Economic Theory vol. 53 pp. 185–198, 1991.
J.C. Cox, J. Ingersoll, and S. Ross, “A theory of the term structure of interest rates,” Econometrica vol. 53 pp. 385–408, 1985.
L. Campi, S. Polbennikov, and A. Sbuelz, “Assessing credit with equity: a CEV model with jump to default,” Discussion Paper 27, Tilburg University Center for Research, 1985.
M. Davis, and D. Hobson, “The range of traded option prices,” Mathematical Finance, forthcoming, 2007.
D. Duffie, J. Pan, and K. Singleton, “Transform analysis and asset pricing for affine jump-diffusions,” Econometrica vol. 68 pp. 1343–1376, 2000.
D. Dufresne, “The distribution of a perpetuity with applications to risk theory and pension funding,” Scandinavian Actuarial Journal pp. 39–79, 1990.
B. Dumas, J. Fleming, and R. Whaley, “Implied volatility functions: empirical tests,” Journal of Finance vol. 53 pp. 2059–2106, 1998.
I.S. Gradshteyn, and I.M. Ryzhik, Table of Integrals, Series and Products, Academic: New York, 1994.
S. Heston, “A closed-form solution for options with stochastic volatility with applications to bond and currency options,” Review of Financial Studies vol. 6 pp. 327–343, 1993.
J. Lamperti, “Semi stable markov processes,” Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete vol. 22 pp. 205–225, 1972.
R.C. Merton, “Theory of rational option pricing,” Bell Journal of Economics and Management Science vol. 4 pp. 141–183, 1973.
J. Noh, R.F. Engle, and A. Kane, “Forecasting volatility and option prices on the S&P 500 Index,” Journal of Derivatives vol. 2 pp. 17–30, 1994.
E. Niccolato, and E. Venardos, “Option pricing in stochastic volatility models of the Ornstein–Uhlenbeck type with a leverage effect,” Finance vol. 13 pp. 445–466, 2003.
J. Pitman, and M. Yor, “A decomposition of Bessel bridges,” Zeitschrift für Wahrscheinlichkeitstheorie verw. Gebiete vol. 59 pp. 425–458, 1982a.
J. Pitman, and M. Yor, “Sur une décomposition des ponts de Bessel.” In M. Fukushima (ed), Functional Analysis in Markov Processes, pp. 276–285, Lecture Notes in Mathematics, No. 923, Springer: Berlin Heidelberg New York, 1982b.
J. Pitman, and M. Yor, “Bessel processes and infinitely divisible laws,” In D. Williams (ed), Stochastic Integrals, pp. 285–370, Lecture Notes in Mathematics, No. 851, Springer: Berlin Heidelberg New York, 1981.
C. Ribeiro, and N. Webber, “Valuing path dependent options in the variance gamma model by Monte Carlo with a gamma bridge,” working paper, University of Warwick Business School, 2002.
L. Scott, “Pricing stock options in a jump-diffusion model with stochastic volatility and interest rates: applications of fourier inversion methods,” Mathematical Finance vol. 7 pp. 413–426, 1997.
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Geman, H., Madan, D.B. & Yor, M. Probing Option Prices for Information. Methodol Comput Appl Probab 9, 115–131 (2007). https://doi.org/10.1007/s11009-006-9005-3
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DOI: https://doi.org/10.1007/s11009-006-9005-3