Canonical Representation Of Option Prices and Greeks with Implications for Market Timing
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.
|Date of creation:||21 Jun 2010|
|Date of revision:|
|Contact details of provider:|| Postal: |
Web page: http://mpra.ub.uni-muenchen.de
More information through EDIRC
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- Sanford J Grossman & Joseph E Stiglitz, 1997.
"On the Impossibility of Informationally Efficient Markets,"
Levine's Working Paper Archive
1908, David K. Levine.
- Grossman, Sanford J & Stiglitz, Joseph E, 1980. "On the Impossibility of Informationally Efficient Markets," American Economic Review, American Economic Association, vol. 70(3), pages 393-408, June.
- Jun Pan & Allen Poteshman, 2004.
"The Information of Option Volume for Future Stock Prices,"
NBER Working Papers
10925, National Bureau of Economic Research, Inc.
- Jun Pan & Allen M. Poteshman, 2006. "The Information in Option Volume for Future Stock Prices," Review of Financial Studies, Society for Financial Studies, vol. 19(3), pages 871-908.
- Kassouf, S. T., 1976. "The lag structure of option price," Journal of Econometrics, Elsevier, vol. 4(4), pages 303-310, November.
- Tim Bollerslev, 1986.
"Generalized autoregressive conditional heteroskedasticity,"
EERI Research Paper Series
EERI RP 1986/01, Economics and Econometrics Research Institute (EERI), Brussels.
- Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
- David Easley & Maureen O'Hara & P.S. Srinivas, 1998. "Option Volume and Stock Prices: Evidence on Where Informed Traders Trade," Journal of Finance, American Finance Association, vol. 53(2), pages 431-465, 04.
- Engle III, Robert F., 2003.
"Risk and Volatility: Econometric Models and Financial Practice,"
Nobel Prize in Economics documents
2003-4, Nobel Prize Committee.
- Robert Engle, 2004. "Risk and Volatility: Econometric Models and Financial Practice," American Economic Review, American Economic Association, vol. 94(3), pages 405-420, June.
- Frank Milne & Dilip Madan, 1991.
"Option Pricing With V. G. Martingale Components,"
1159, Queen's University, Department of Economics.
- 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.
- Chernov, Mikhail, 2003. "Empirical reverse engineering of the pricing kernel," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 329-364.
- Madan, Dilip B & Seneta, Eugene, 1990. "The Variance Gamma (V.G.) Model for Share Market Returns," The Journal of Business, University of Chicago Press, vol. 63(4), pages 511-24, October.
- Jin-Chuan Duan, 1995. "The Garch Option Pricing Model," Mathematical Finance, Wiley Blackwell, vol. 5(1), pages 13-32.
- 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.
When requesting a correction, please mention this item's handle: RePEc:pra:mprapa:23426. See general information about how to correct material in RePEc.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Ekkehart Schlicht)
If references are entirely missing, you can add them using this form.