A Kramers-Moyal Approach to the Analysis of Third-Order Noise with Applications in Option Valuation

Dan M Popescu and Ovidiu Lipan

PLOS ONE, 2015, vol. 10, issue 1, 1-12

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.

Date: 2015
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Persistent link: https://EconPapers.repec.org/RePEc:plo:pone00:0116752

DOI: 10.1371/journal.pone.0116752

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