Canonical Representation Of Option Prices and Greeks with Implications for Market Timing

Cadogan, Godfrey (2010): Canonical Representation Of Option Prices and Greeks with Implications for Market Timing.

Abstract

We introduce a canonical representation of call options, and propose a solution to two open problems in option pricing theory. The first problem was posed by (Kassouf, 1969, pg. 694) seeking “theoretical substantiation” for his robust option pricing power law which eschewed assumptions about risk attitudes, rejected risk neutrality, and made no assumptions about stock price distribution. The second problem was posed by (Scott, 1987, pp. 423-424) who could not find a unique solution to the call option price in his option pricing model with stochastic volatility–without appealing to an equilibrium asset pricing model by Hull and White (1987), and concluded: “[w]e cannot determine the price of a call option without knowing the price of another call on the same stock”. First, we show that under certain conditions derivative assets are superstructures of the underlying. Hence any option pricing or derivative pricing model in a given number field, based on an anticipating variable in an extended field, with coefficients in a subfield containing the underlying, is admissible for market timing. For the anticipating variable is an algebraic number that generates the subfield in which it is the root of an equation. Accordingly, any polynomial which satisfies those criteria is admissible for price discovery and or market timing. Therefore, at least for empirical purposes, elaborate models of mathematical physics or otherwise are unnecessary for pricing derivatives because much simpler adaptive polynomials in suitable algebraic numbers are functionally equivalent. Second, we prove, analytically, that Kassouf (1969) power law specification for option pricing is functionally equivalent to Black and Scholes (1973); Merton (1973) in an algebraic number field containing the underlying. In fact, we introduce a canonical polynomial representation theory of call option pricing convex in time to maturity, and algebraic number of the underlying–with coefficients based on observables in a subfield. Thus, paving the way for Wold decomposition of option prices, and subsequently laying a theoretical foundation for a GARCH option pricing model. Third, our canonical representation theory has an inherent regenerative multifactor decomposition of call option price that (1) induces a duality theorem for call option prices, and (2) permits estimation of risk factor exposure for Greeks by standard [polynomial] regression procedures. Thereby providing a theoretical (a) basis for option pricing of Greeks, and (b) solving Scott’s dual call option problem a fortiori with our duality theory in tandem with Riesz representation theory. Fourth, when the Wold decomposition procedure is applied we are able to construct an empirical pricing kernel for call option based on residuals from a model of risk exposure to persistent and transient risk factors.

• Item Type: MPRA Paper
• Language: English
• Keywords: number theory; price discovery; derivatives pricing; asset pricing; canonical representation; Wold decomposition; empirical pricing kernel; option Greeks; dual option pricing
• Subjects: C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C51 - Model Construction and Estimation; G - Financial Economics > G1 - General Financial Markets > G12 - Asset Pricing ; Trading Volume ; Bond Interest Rates; D - Microeconomics > D8 - Information, Knowledge, and Uncertainty > D81 - Criteria for Decision-Making under Risk and Uncertainty; G - Financial Economics > G1 - General Financial Markets > G11 - Portfolio Choice ; Investment Decisions; G - Financial Economics > G1 - General Financial Markets > G13 - Contingent Pricing ; Futures Pricing

References

Ambramowitz, M. and I. A. Stegun (Eds.) (1972). Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (9th Printing ed.). Mineola, N.Y.: Dover Publications, Inc.

Apostol, T. M. (1967). Calculus (2nd ed.), Volume I. New York: John Wiley& Sons, Inc.

Benth, F. E., M. Groth, and O. Wallin (2010). Derivative-free Greeks for the Barndorff-Nielsen and Shephard Stochastic Volatility Model. Stochastics An International Journal of Probability and Stochastic Processes: formerly Stochasticsand Stochastics Reports 82(3), 1744–2516.

Black, F. and M. Scholes (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy 81(3), 637–654.

Bollerslev, T. (1986). Generalized Autoregressive Conditional Heteroskedasticity. Journal of Econometrics 31, 307–327.

Box, G. E. P., G. M. Jenkins, , and G. Riensel (1994). Time Series Forecasting and Control (3rd ed.). New Jersey: Prentice-Hall, Inc.

Brockwell, P. J. and R. A. Davis (1987). Time Series: Theory and Methods. New York, NY: Springer-Verlag.

Brown, D. J. and S. Ross (1988, June). Spanning, Valuation and Options. Cowles Foundation Discussion Paper 873, Yale University, published in Economic Theory, 1991, 1(1):3-12.

Campbell, J. Y., A. W. Lo, and A. C. MacKinlay (1997). The Econometrics of Financial Markets. Princeton, N. J.: Princeton University Press.

Chernov, M. (2003). Empirical Reverse Engineering of The Pricing Kernel. Journal of Econometrics 116, 329–364.

Chow, Y. and H. Robbins. A Martingale System Theorem and Applications. Proc. Fourth Berkeley Symp. on Math. Statist. and Prob. Berkeley CA: University of California Press.

Clark, A. (1971). Elements of Abstract Algebra (Dover reprint 1984 ed.). Belmont CA: Wadsworth Publishing Co.

Davidson, R. and J. G. MacKinnon (2004). Econometric Theory and Methods. New York: Oxford Univ. Press.

Duan, J. (1995). The GARCH Option Pricing Model. Mathematical Finance 5(1), 13–32.

Durrett, R. (2005). Probability: Theory and Examples (3rd ed.). Belmont, CA: Thompson Brooks-Cole.

Easley, D., M. O‘Hara, and P. S. Srinivas (1998, April). Options Volume and Stock Prices: Evidence on Where Informed Traders Trade. Journal of Finance 53(2), 431–465.

Engle, R. (1982, July). Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation. Econometrica 50(4), 987–1008.

Engle, R. F. (2001). GARCH101: The Use of ARCH/GARCH Models in Applied Econometrics. Journal of Economic Perspectives 15(4), 157–168.

Engle, R. F. (2004). Risk and Volatility: Econometric Models and Financial Practice. Amer. Econ. Rev. 94(3), 405–420.

Fama, E. and H. Babiak (1968, Dec.). Dividend Policy: An Empirical Analysis. Journal of American Statistical Society 63(324), 1132–1161.

Flemming, J., B. Ostidek, and R. E. Whaley (1996). Trading Costs and The Relative Rates of Price Discovery in Stocks, Futures, and Options Markets. Journal of Futures Markets 16(4), 353–387.

French, D. W. (1983, Fall). Black-Scholes vs. Kassouf Option Pricing: An Empirical Comparison. Journal of Business Finance and Accounting 10, 395–408.

Gikhman, I. I. and A. V. Skorokhod (1969). Introduction to The Theory of Random Processes (Dover reprint 1996 ed.). Philadelphia, PA: W. B. Saunders.

Gordon, M. J. (1959, May). Dividends, Earnings, and Stock Price. Review of Economics and Statistics 41(2), 99–105.

Grillet, P. A. (2007). Abstract Algebra (2nd ed.). Graduate Text in Mathematics. New York: Springer-Verlag.

Grossman, S. J. and J. E. Stiglitz (1980, June). On The Impossibility Of Informationally Efficient Markets. American Economic Review 70(3), 393–408.

Haug, E. G. and N. N. Taleb (2008). Why We Have Never Used the Black-Scholes-Merton Option Pricing Formula. Wilmott Magazine 1, 72–79.

Heston, S. (1993). A Closed-Form Solution for Options with Stochastic Volatility Application to Bond and Currency Options. Review of Financial Studies 6(2), 327–343.

Hilbert, D. (1998). The Theory of Algebraic Number Fields, Volume English translation of 1898 German edition. New York: Springer-Verlag.

Huang, C. and R. H. Litzenberger (1988). Foundations of Financial Economics. Englewoods Cliffs, New Jersey: Prentice-Hall, Inc.

Hull, J. (2006). Options, Futures, and Other Derivatives (6th ed.). New Jersey:Prentice-Hall, Inc.

Hull, J. and A. White (1987, June). The Pricing of Options on Assets with Stochastic Volatility. Journal of Finance 42(2), 281–300.

Jacobson, N. (1951). Lectures in Abstract Algebra, Volume I of Graduate Text in Mathematics. New York: Springer-Verlag.

Karatzas, I. and S. E. Shreve (1991). Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics. New York, N.Y.: Springer-Verlag.

Kassouf, S. T. (1968, May). Stock Price Random Walks: Some Supporting Evidence. Review of Economics and Statistics 50(2), 275–278.

Kassouf, S. T. (1969, Oct.). An Econometric Model for Option Price with Implications for Investor Audacity. Econometrica 37(4), 685–694.

Kassouf, S. T. (1976). The Lag Structure of Option Price. Journal of Econometrics 4, 303–310.

Kolmogorov, A. N. (1956). Grundbegriffe der Wahrscheinlichkeitrechnung (“Foundations of The Theory of Probability”) (2nd English ed.), Volume English translation of 1933 German edition. New York: Chelsea Publishing Co.

Koverlachuk, B. and E. Vitayev (2002). Data Mining in Finance: Advances in Relational and Hybrid Methods. The Kluwer International Series in Engineering and Computer Science. Moscow: Kluwer Academic Publishers.

Madan, D. B. and F. Milne (1991). Options Pricing With V. G. Martingale Components. Mathematical Finance 1(4), 39–55.

Madan, D. B. and E. Seneta (1990). The Variance Gamma Model for Share Market Returns. Journal of Business 63(4), 511–524.

McCullagh, P. and J. A. Nelder (1989). Generalized Linear Model (2nd ed.). Number 37 in Monographs in Statistics and Applied Probability. New York:Chapman & Hall.

Merton, R. C. (1973). A Theory of Rational Options Pricing. Bell Journal of Economics and Management Science 4(1), 141–183.

Mills, T. C. and T. N. Markellos (2008). The Econometric Modeling of Financial Time Series (3rd ed.). New York: Cambridge University Press.

Pan, J. and A. M. Poteshman (2006, Feb.). The Information in Option Volume for Future Stock Prices. Review of Financial Studies 19(3), 871–908.

Passarelli, D. (2008). Trading Option Greeks. New York: Bloomberg Press.

Pollard, H. and H. G. Diamond (1975). The Theory of Algebraic Numbers (2nd ed.). Number 9 in Carus Mathematical Monograph. Washington, D.C.: Mathematical Association of America.

Rao, C. R. (1973). Linear Statistical Inference and Its Applications (2nd ed.). Wiley Series in Probability and Statistics. New York, N. Y.: John Wiley & Sons, Inc.

Reed, M. and B. Simon (1980). Modern Methods of Mathematical Physics (Revised Enlarged ed.), Volume Functional Analysis. New York: Academic Press, Inc.

Scott, L. O. (1987, Dec). Option Pricing When the Variance Change Randomly: Theory, Estimation and Application. Journal of Financial and Quantitative Analysis 22(4), 419–438.

Shephard, N. (1996). Time Series Models in Econometrics, Finance and Other Fields, Chapter Statistical Aspects of ARCH and Stochastic Volatility, pp. 1– 67. London: Chapman & Hall.

Snell, J. L. (1952, Sept.). Applications To Martingale System Theorems. Trans. Amer. Math. Soc. 73(2), 293–312.

Wang, Y. and R. Caflish (2010, May). Fast Computation of Upper Bounds for American-style Options without Nested Simulation. Dep‘t of Mathematics, Univ. California-Los Angeles, available at SSRN http://ssrn.com/abstract=1612336.

Weisberg, S. (2005). Applied Linear Regression (3rd ed.). Wiley Series in Probability and Statistics. New York: John Wiley & Sons, Inc.

Whaley, R. E. (2000). The Investor Fear Gauge: Explication of the CBOE VIX. Journal of Portfolio Management 26(3), 12–17.