R. Sircar's work partially supported by NSF grant DMS-0807440, and S. Sturm's work partially supported by NSF grant DMS-0739195. S. Sturm completed this work while Postdoctoral Research Associate at Princeton University.
FROM SMILE ASYMPTOTICS TO MARKET RISK MEASURES
Article first published online: 2 NOV 2012
© 2012 Wiley Periodicals, Inc.
Volume 25, Issue 2, pages 400–425, April 2015
How to Cite
Sircar, R. and Sturm, S. (2015), FROM SMILE ASYMPTOTICS TO MARKET RISK MEASURES. Mathematical Finance, 25: 400–425. doi: 10.1111/mafi.12015
- Issue published online: 2 MAR 2015
- Article first published online: 2 NOV 2012
- Manuscript Accepted: JUL 2012
- Manuscript Received: JUL 2011
- dynamic convex risk measures;
- volatility skew;
- stochastic volatility models;
- indifference pricing;
- backward stochastic differential equations
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, 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 partial differential equation and provide a small time-to-maturity expansion and numerical solutions. This procedure allows to choose convex risk measures in a conveniently parameterized 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.