Good-Deal Bounds for Option Prices under Value-at-Risk and Expected Shortfall Constraints

Sascha Desmettre, Christian Laudagé and Jörn Sass
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Sascha Desmettre: Institute of Financial Mathematics and Applied Number Theory, Johannes Kepler University Linz, Altenbergerstraße 69, 4040 Linz, Austria
Christian Laudagé: Department of Financial Mathematics, Fraunhofer Institute for Industrial Mathematics ITWM, Fraunhofer-Platz 1, 67663 Kaiserslautern, Germany
Jörn Sass: Department of Mathematics, University of Kaiserslautern, Erwin-Schrödinger-Straße, 67663 Kaiserslautern, Germany

Risks, 2020, vol. 8, issue 4, 1-22

Abstract: In this paper, we deal with the pricing of European options in an incomplete market. We use the common risk measures Value-at-Risk and Expected Shortfall to define good-deals on a financial market with log-normally distributed rate of returns. We show that the pricing bounds obtained from the Value-at-Risk admit a non-smooth behavior under parameter changes. Additionally, we find situations in which the seller’s bound for a call option is smaller than the buyer’s bound. We identify the missing convexity of the Value-at-Risk as main reason for this behavior. Due to the strong connection between good-deal bounds and the theory of risk measures, we further obtain new insights in the finiteness and the continuity of risk measures based on multiple eligible assets in our setting.

Keywords: good-deal bounds; risk measures; multiple eligible assets; Value-at-Risk; Expected Shortfall (search for similar items in EconPapers)
JEL-codes: C G0 G1 G2 G3 K2 M2 M4 (search for similar items in EconPapers)
Date: 2020
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