The skew Brownian motion should not be used as a risk-neutral returns process: a well-posed skew-normal alternative

Abstract

Return models for risk-neutral financial valuation based on skew Brownian motions (SBMs) have been introduced about twenty years ago, and have recently enjoying growing popularity. Unfortunately, the story behind their development is one of mistakes and erroneous interpretations, beginning from the foundational misrepresentations that the prevalent financial model is based on the Itô-McKean SBM -- which, in fact, it is not. Besides, and more seriously, the clarification of \cite{rossello2012arbitrage} that price models with a local time in their returns, such as the SBM, are arbitrageable has been, by and large, ignored. In this paper, we try to clear the field from the confusions and misconceptions lurking in the standing option pricing literature on SBM, by exposing all the errors we could trace in the treatment so far. Recognizing however the potential of the SBM skew-normal marginals for risk-neutral valuation, as a positive contribution, we reformulate the putative SBM call pricing formula and show that, even if the SBM return model admits arbitrage, its option pricing formula does not. The correct Markovian SDE with skew-normal marginals is then identified, its strong well-posedness shown, and by exploiting the availability of closed formulae, an asymptotic analysis of the implied volatility surface is offered. En route to our conclusions, we obtain a novel normal/skew-normal stochastic dominance property of independent interest.