  The left tail of the implied volatility skew, coming from quotes on
out-of-the-money put options, can be thought to reflect the market's assessment
of the risk of a huge drop in stock prices. We analyze how this market
information can be integrated into the theoretical framework of convex monetary
measures of risk. In particular, we make use of indifference pricing by dynamic
convex risk measures, which are given as solutions of backward stochastic
differential equations (BSDEs), to establish a link between these two
approaches to risk measurement. We derive a characterization of the implied
volatility in terms of the solution of a nonlinear PDE and provide a small
time-to-maturity expansion and numerical solutions. This procedure allows to
choose convex risk measures in a conveniently parametrized class, distorted
entropic dynamic risk measures, which we introduce here, such that the
asymptotic volatility skew under indifference pricing can be matched with the
market skew. We demonstrate this in a calibration exercise to market implied
volatility data.
