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Quantification of Risk in Classical Models of Finance

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  • Alois Pichler
  • Ruben Schlotter

Abstract

This paper enhances the pricing of derivatives as well as optimal control problems to a level comprising risk. We employ nested risk measures to quantify risk, investigate the limiting behavior of nested risk measures within the classical models in finance and characterize existence of the risk-averse limit. As a result we demonstrate that the nested limit is unique, irrespective of the initially chosen risk measure. Within the classical models risk aversion gives rise to a stream of risk premiums, comparable to dividend payments. In this context we connect coherent risk measures with the Sharpe ratio from modern portfolio theory and extract the Z-spread -- a widely accepted quantity in economics to hedge risk. The results for European option pricing are then extended to risk-averse American options, where we study the impact of risk on the price as well as the optimal time to exercise the option. We also extend Merton's optimal consumption problem to the risk-averse setting.

Suggested Citation

  • Alois Pichler & Ruben Schlotter, 2020. "Quantification of Risk in Classical Models of Finance," Papers 2004.04397, arXiv.org, revised Jul 2020.
  • Handle: RePEc:arx:papers:2004.04397
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    File URL: http://arxiv.org/pdf/2004.04397
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    References listed on IDEAS

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    1. Riedel, Frank, 2004. "Dynamic coherent risk measures," Stochastic Processes and their Applications, Elsevier, vol. 112(2), pages 185-200, August.
    2. Andrzej Ruszczyński & Alexander Shapiro, 2006. "Conditional Risk Mappings," Mathematics of Operations Research, INFORMS, vol. 31(3), pages 544-561, August.
    3. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.),Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    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. Andy Philpott & Vitor de Matos & Erlon Finardi, 2013. "On Solving Multistage Stochastic Programs with Coherent Risk Measures," Operations Research, INFORMS, vol. 61(4), pages 957-970, August.
    6. Garman, Mark B. & Kohlhagen, Steven W., 1983. "Foreign currency option values," Journal of International Money and Finance, Elsevier, vol. 2(3), pages 231-237, December.
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