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. which solves two open problems in option pricing theory. The first 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 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). 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, i.e., algebraic number, in an extended field, with coefficients in a subfield containing the underlying, is admissible for market timing. Thus, elaborate models of mathematical physics or otherwise are unnecessary for pricing derivatives because much simpler adaptive polynomials in admissible 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. Third, our representation 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 solving Scott’s dual call option problem, a fortiori, in tandem with Riesz representation theory.

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