A note on the derivative of the normal distribution's logarithm

Abstract.

In mathematical finance the model of Black & Scholes has become the standard model for pricing contingent claims and other derivative securities. Since the Black & Scholes model is based on the assumption that the price process of a risky security is driven by a geometric Brownian motion the problems in mathematical finance can often be reduced to problems concerning standard normal distributions. Inspired by such a problem the convexity of the derivative of the normal distribution's¶¶ logarithm $f(z)= \exp \biggl\{ - {z^2\over 2}\biggr\} \biggl(\int\limits_{t=-\infty}^z \exp \biggl\{-{t^2\over 2}\biggr\}dt\biggr)^{-1}$ is proved.