What is it about?

This study develops, in discrete time, an option pricing model which generates option prices that are connected, as such that reside in a support space that is continuous. Consistent with the intuition in Cox and Ross (1976), the option pricing function has parameterization as a cumulative probability function, hence unequivocally is parameterized by each of the connectedness and continuity properties. Consistent with a demand that, hitherto, has been difficult to satisfy (see for example, Rubinstein 1994), with K_z and K_v as two exercise prices that satisfy, K_z < K_v, risk neutral probabilities are independent of specific location of the spot price, S. For concreteness, regardless of, K_z < S < K_v or K_z < K_v < S, risk neutral probabilities remain exactly the same. The rational mechanism that facilitates the difficult to achieve result? Suppose a call option writer writes N different exercise prices on the same asset. In presence of N feasible maturity dates corresponding to the N different exercise prices, the formal theory explicitly constrains the maturity of all of the call options to be exactly the same; and the option writer has available a scientific condition for the choice between the N feasible maturity dates. Whereas all prior realizations of option pricing models achieve risk neutrality via introduction of volatility parameters that feasibly benefit either of the writer or buyer of a call option, in the model that is developed in this study, risk neutrality specifically is parameterized as, 'whereas maturity profiles for options are, at the margin, determined by option buyers, simultaneously option buyers resign themselves to the necessity of a perturbation to the priors of the option writer for feasibility of an in-the-money expiration.

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Why is it important?

Combined, directness of modeling of volatility and assumption of jumps in the price processes for underlying assets facilitate robustness of option pricing models to generation of 'volatility smiles'. Consider, however, that modeling of volatility and assumption of jumps implies the resulting volatility smiles embed endogeneity problems, with outcome Derman (2003) concludes that while such models perform best, still they fall short of the ideal specification of an option pricing model. Whereas, typically endogeneity problems distort inferences in respect of phenomena that are studied, the endogeneity that was induced was not harmful, because models that embed each of stochastic volatility and jumps - the best, hitherto of all of the different approaches to the modeling of option prices - are not applied to the pricing of options, rather are applied towards extraction of risk neutral probabilities from option prices that already are formed by option writers. All such option pricing models are, as such 'academic' in their applications. Absent either of modeling of volatility or assumption of jumps, the option pricing model that is developed in this study endogenously conditions changes to the volatility of underlying assets and changes to the volatility of option prices on arrival of jump intensities that are not anticipated by option writers. Absent any assumptions to the effect, arrival of unanticipated jump intensities induce changes to each of the volatility of the asset price and the volatility of the option price. Stated outcome is facilitated via directness of modeling of risk neutral probabilities for each of the option writer and option buyer, and arrival at the equilibrium pricing function via conditions that facilitate composition of the two dichotomous series of risk neutral probabilities. Importantly, for any specific asset, and at some specific point in time, risk neutral probabilities that are formed by either of option writers or buyers are shown to be unique and embed two crossing properties that, simultaneously validate rationality of each of the option buyer and writer. Uniqueness of the risk neutral probabilities and presence, endogenously, of the two crossing properties are, combined, necessary conditions for option prices that both are well defined and parameterized by risk neutrality.


For transformation of the risk neutral probabilities into option prices, there is necessity of application of the Kolmogorov Backward Equation to the option pricing function. That exercise is, at the moment, deferred to a follow up study. With the option pricing function developed in context of modeling of native probabilities, the option pricing function has, explicitly characterization also as each of a model for pricing of contingent liabilities, or a model for pricing of any derivative asset. For transformation of the pricing function into dollar prices, practitioners need only arrive at their own proprietary transforms, of which the Kolmogorov Backward Equation seems, based on it's theory, to be the most appropriate.

Dr Oghenovo A Obrimah
Fisk University

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This page is a summary of: A Discretely Formulated Option Pricing Model That, Absent Directness of Modeling of Volatility, Embeds the ‘Volatility Smile’, SSRN Electronic Journal, January 2022, Elsevier,
DOI: 10.2139/ssrn.4058439.
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