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A PDE approach to risk measures of derivatives

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  • Tak Kuen Siu
  • Hailiang Yang

Abstract

This paper proposes a partial differential equation (PDE) approach to calculate coherent risk measures for portfolios of derivatives under the Black-Scholes economy. It enables us to define the risk measures in a dynamic way and to deal with American options in a relatively effective way. Our risk measure is based on the representation form of coherent risk measures. Through the use of some earlier results the PDE satisfied by the risk measures are derived. The PDE resembles the standard Black-Scholes type PDE which can be solved using standard techniques from the mathematical finance literature. Indeed, these results reveal that the PDE approach can provide practitioners with a more applicable and flexible way to implement coherent risk measures for derivatives in the context of the Black-Scholes model.

Suggested Citation

  • Tak Kuen Siu & Hailiang Yang, 2000. "A PDE approach to risk measures of derivatives," Applied Mathematical Finance, Taylor & Francis Journals, vol. 7(3), pages 211-228.
    • Handle: RePEc:taf:apmtfi:v:7:y:2000:i:3:p:211-228
    DOI: 10.1080/13504860110045741
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    References listed on IDEAS

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    1. Ioannis Karatzas & Jaksa Cvitanic, 1999. "On dynamic measures of risk," Finance and Stochastics, Springer, vol. 3(4), pages 451-482.
    2. Leland, Hayne E, 1985. "Option Pricing and Replication with Transactions Costs," Journal of Finance, American Finance Association, vol. 40(5), pages 1283-1301, December.
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    4. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    5. Robert Jarrow, 2017. "Derivatives," World Scientific Book Chapters, in: THE ECONOMIC FOUNDATIONS OF RISK MANAGEMENT Theory, Practice, and Applications, chapter 3, pages 19-28, World Scientific Publishing Co. Pte. Ltd..
    6. Margrabe, William, 1978. "The Value of an Option to Exchange One Asset for Another," Journal of Finance, American Finance Association, vol. 33(1), pages 177-186, March.
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    Cited by:

    1. Fei Lung Yuen & Hailiang Yang, 2012. "Optimal Asset Allocation: A Worst Scenario Expectation Approach," Journal of Optimization Theory and Applications, Springer, vol. 153(3), pages 794-811, June.

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