  We study the risk premium impact in the Perturbative Black Scholes model. The
Perturbative Black Scholes model, developed by Scotti, is a subjective
volatility model based on the classical Black Scholes one, where the volatility
used by the trader is an estimation of the market one and contains measurement
errors. In this article we analyze the correction to the pricing formulas due
to the presence of an underlying drift different from the risk free return. We
prove that, under some hypothesis on the parameters, if the asset price is a
sub-martingale under historical probability, then the implied volatility
presents a skewed structure, and the position of the minimum depends on the
risk premium $\lambda$.
