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

Author

Listed:
  • 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)

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.

Suggested Citation

  • Sascha Desmettre & Christian Laudagé & Jörn Sass, 2020. "Good-Deal Bounds for Option Prices under Value-at-Risk and Expected Shortfall Constraints," Risks, MDPI, vol. 8(4), pages 1-22, October.
    • Handle: RePEc:gam:jrisks:v:8:y:2020:i:4:p:114-:d:437604
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    References listed on IDEAS

    as
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