Quantification of risk in classical models of finance

Alois Pichler and Ruben Schlotter

Quantitative Finance, 2022, vol. 22, issue 1, 31-45

Abstract: This paper treats optimal control problems and derivative pricing with regard to fixed levels of 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 for hedging risk. The results for European option pricing are extended to risk-averse American options, to study the impact of risk on the price and optimal time to exercise such options. We also extend Merton's optimal consumption problem to the risk-averse setting.

Date: 2022
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DOI: 10.1080/14697688.2021.1993613

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