Adiabaticity conditions for volatility smile in Black-Scholes pricing model

Abstract

Our derivation of the distribution function for future returns is based on the risk neutral approach which gives a functional dependence for the European call (put) option price, C(K), given the strike price, K, and the distribution function of the returns. We derive this distribution function using for C(K) a Black-Scholes (BS) expression with volatility, σ, in the form of a volatility smile. We show that this approach based on a volatility smile leads to relative minima for the distribution function ("bad" probabilities) never observed in real data and, in the worst cases, negative probabilities. We show that these undesirable effects can be eliminated by requiring "adiabatic" conditions on the volatility smile.

Do your volatility smiles take care of extreme events?

“…We repeat the fitting procedure considering the volatility smile for different days, currencies and time to maturity T (Table I), then we analyze the relations between the fitting parameters. As already observed in [8], the following relation between n, T, g holds:…”

“…We started from the pricing equation of the Black-Scholes model for an European call and we considered the effect of the VS correction on the implied PDF. Our approach comes from statistical physics and it is related to the adiabatic interpretation in [8]. We showed that similar fits of a VS could imply strong differences on the implied returns PDF with obvious consequences on the risk estimation.…”

“…More generally, if one considers the dependence, σ = σ(K), in Eq. ( 3), it is possible to get the analytical expression of the implied distribution of financial returns [8]:…”

“…Let us now discuss the case χ = 1: in the light of the adiabatic interpretation presented in [8], we expect that on increasing χ, the PDF will present, soon or later a minimum. This means that the PDF should be flatter than before, so that µ should decrease.…”

“…Simplicity and a sort of reluctance to changes explain, in our opinion, the reason why, after about 37 years from its publication, BS model is still used by practitioners and justify the importance of our study to get a correct calibration procedure for the volatility smile (VS) effect also from a theoretical point of view. In a recent paper [8] it is shown a new calibration procedure that can be obtained using an adiabatic approach to avoid arbitrage opportunities. The term "adiabatic" comes from comes from statistical physics and is related to the slowness of the variation of a parameter λ that specifies the properties of a system or an external field.…”