A Discretely Formulated Option Pricing Model That, Absent Directness of Modeling of Volatility, Embeds the ‘Volatility Smile’

If a discretely formulated asset pricing model rivals efficacy of the Black and Scholes (1973) option pricing model, with canonical properties of option prices satisfied, rather counterfactually it spans a support space for call option prices that is continuous. Absent any directness of modeling of volatility, this study arrives at exactly such an option pricing model. In the model, maturity profiles for option contracts are independent of distributions for either of strike prices or option prices, and call option prices both increase with the volatility of the underlying asset price and are convex functions of strike prices. Novelly, yet consistent with the ideal, the equilibrium pricing function arises endogenously as a composition of dichotomous pricing functions that are formulated by each of option writers or buyers. In stated respect, option markets have economic viability `if and only if', the ex ante probability for emergence of `volatility smiles' strictly is positive. Economic viability of option markets is predicated on anticipation of volatility smiles, because call options expire in-the-money if and only if the ex ante probability for incidence of unanticipated positive jumps to asset prices, equivalently for emergence of volatility smiles, strictly is positive. Robustness of the option pricing model is evident in the finding, to wit, absent explicitness of adoption of a non-cooperative game theoretic structure, there is arrival at the Black and Scholes constant volatility process. Simulations of model predictions provide corroborating evidence that each of the volatility smile and risk neutral skew arise endogenously in context of adjustments to the option pricing function which, consequent on emergence of unanticipated positive jumps to the underlying asset price, are predicted to be necessary for maintenance of the rationality of each of option buyers and writers