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A Generalized Fourier Transform Approach to Risk Measures

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  • G. Bormetti
  • V. Cazzola
  • G. Livan
  • G. Montagna
  • O. Nicrosini

Abstract

We introduce the formalism of generalized Fourier transforms in the context of risk management. We develop a general framework to efficiently compute the most popular risk measures, Value-at-Risk and Expected Shortfall (also known as Conditional Value-at-Risk). The only ingredient required by our approach is the knowledge of the characteristic function describing the financial data in use. This allows to extend risk analysis to those non-Gaussian models defined in the Fourier space, such as Levy noise driven processes and stochastic volatility models. We test our analytical results on data sets coming from various financial indexes, finding that our predictions outperform those provided by the standard Log-Normal dynamics and are in remarkable agreement with those of the benchmark historical approach.

Suggested Citation

  • G. Bormetti & V. Cazzola & G. Livan & G. Montagna & O. Nicrosini, 2009. "A Generalized Fourier Transform Approach to Risk Measures," Papers 0909.3978, arXiv.org, revised May 2012.
    • Handle: RePEc:arx:papers:0909.3978
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    Cited by:

    1. Giacomo Bormetti & Sofia Cazzaniga, 2011. "Multiplicative noise, fast convolution, and pricing," Papers 1107.1451, arXiv.org.
    2. Dobrislav Dobrev∗ & Travis D. Nesmith & Dong Hwan Oh, 2017. "Accurate Evaluation of Expected Shortfall for Linear Portfolios with Elliptically Distributed Risk Factors," JRFM, MDPI, vol. 10(1), pages 1-14, February.
    3. Alessandro Ramponi, 2016. "On a Transform Method for the Efficient Computation of Conditional V@R (and V@R) with Application to Loss Models with Jumps and Stochastic Volatility," Methodology and Computing in Applied Probability, Springer, vol. 18(2), pages 575-596, June.
    4. Giacomo Bormetti & Sofia Cazzaniga, 2014. "Multiplicative noise, fast convolution and pricing," Quantitative Finance, Taylor & Francis Journals, vol. 14(3), pages 481-494, March.

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