Measures of model uncertainty and calibrated option bounds

Abstract

Recently, Cont introduced a quantitative framework for measuring model uncertainty in the context of derivative pricing [Model uncertainty and its impact on the pricing of derivative instruments, Math. Finance, 16(3) (2006), pp. 519–547]. Two measures of model uncertainty were proposed: one measure based on a coherent risk measure compatible with market prices of derivatives and another measure based on convex risk measures. We show in a discrete time, finite state probability setting, that the two measures introduced by Cont are closely related to calibrated option bounds studied recently by King et al. [Calibrated option bounds, Inf. J. Ther. Appl. Financ., 8(2) (2005), pp. 141–159]. The precise relationship is established through convex programming duality. As a result, the model uncertainty measures can be computed efficiently by solving convex programming or linear programming problems after a suitable discretization. Numerical results using S&P 500 options are given.

Keywords:

• model uncertainty
• option pricing
• incomplete markets
• coherent risk measures
• convex risk measures
• calibrated option bounds
• duality

AMS Subject Classifications:

• 91B28
• 90C90

Acknowledgements

The programming assistance of Ahmet Camcı is gratefully acknowledged. The comments of two anonymous referees were useful in improving this article. This research is partially supported by TUBITAK Grant 107K250, and a scholarship from the Fulbright Commission.