title: A Kramers-Moyal approach to the analysis of third-order noise with applications in option valuation.
abstract: We propose the use of the Kramers-Moyal expansion in the analysis of third-order noise. In particular, we show how the approach can be applied in the theoretical study of option valuation. Despite Pawula's theorem, which states that a truncated model may exhibit poor statistical properties, we show that for a third-order Kramers-Moyal truncation model of an option's and its underlier's price, important properties emerge: (i) the option price can be written in a closed analytical form that involves the Airy function, (ii) the price is a positive function for positive skewness in the distribution, (iii) for negative skewness, the price becomes negative only for price values that are close to zero. Moreover, using third-order noise in option valuation reveals additional properties: (iv) the inconsistencies between two popular option pricing approaches (using a "delta-hedged" portfolio and using an option replicating portfolio) that are otherwise equivalent up to the second moment, (v) the ability to develop a measure R of how accurately an option can be replicated by a mixture of the underlying stocks and cash, (vi) further limitations of second-order models revealed by introducing third-order noise.
authors.affiliation: Department of Physics, University of Richmond, Richmond, Virginia, United States of America.
authors.forename: Dan M
authors.initials: DM
authors.lastname: Popescu
authors.affiliation: Department of Physics, University of Richmond, Richmond, Virginia, United States of America.
authors.forename: Ovidiu
authors.initials: O
authors.lastname: Lipan
mesh_terms: D000465:Algorithms; D003365:Costs and Cost Analysis; D003627:Data Interpretation, Statistical; D018803:Models, Economic; D011336:Probability; D013269:Stochastic Processes