Canonical Option Pricing and Greeks with Implications for Market Timing
44 Pages Posted: 23 Jun 2010 Last revised: 29 Jan 2011
Date Written: January 27, 2011
We introduce canonical representation of a call option premium which solves two open problems in option pricing theory. The ﬁrst problem was posed by (Kassouf, 1969, pg. 694) who sought “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 posed 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 option pricing and or market timing. This explains the taxonomy of variegated option pricing formulae presented in Haug and Taleb (2008) critique of the Black and Scholes (1973); Merton (1973) formula. To wit, we claim that investor sentiment is an algebraic number. Accordingly, any polynomial which satisﬁes those criteria is admissible for price discovery and or market timing. Therefore, at least for empirical purposes, elaborate equations 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, our canonical representation of call option price is (i) convex in time to maturity, and (ii) depends on an algebraic number for the underlying – with coefﬁcients based on observables in a subﬁeld. Thus, paving the way for Wold decomposition of option prices, and providing theoretical contrast with Duan (1995) heuristic 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 premium, and (2) permits estimation of risk factor exposure for Greeks by standard [polynomial] regression procedures. Again, providing a theoretical (a) basis for option pricing of Greeks, and (b) solving Scott’s dual call option problem a fortiori by virtue of Riesz representation theory. Moreover, we show how our regenerative decomposition extends to (Wang and Caﬂish, 2010, pp. 4-5) Snell‘s envelope representation of a dual [American] call option premium. 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. This approach is distinguished from Chernov (2003) two stage procedure.
Keywords: algebraic number theory, price discovery, derivatives pricing, asset pricing, market timing, Wold decomposition, empirical pricing kernel, option Greeks, dual option pricing
JEL Classification: C02, D81, D84, G11, G12, G13, G17
Suggested Citation: Suggested Citation