Purpose The purpose of this paper is to present the method for efficient computation of risk measures using Fourier transform technique. Another objective is to demonstrate that this technique enables an efficient computation of risk measures beyond value-at-risk and expected shortfall. Finally, this paper highlights the importance of validating assumptions behind the risk model and describes its application in the affine model framework. Design/methodology/approach The method proposed is based on Fourier transform methods for computing risk measures. The authors obtain the loss distribution by fitting a cubic spline through the points where Fourier inversion of the characteristic function is applied. From the loss distribution, the authors calculate value-at-risk and expected shortfall. As for the calculation of the entropic value-at-risk, it involves the moment generating function which is closely related to the characteristic function. The expectile risk measure is calculated based on call and put option prices which are available in a semi-closed form by Fourier inversion of the characteristic function. We also consider mean loss, standard deviation and semivariance which are calculated in a similar manner. Findings The study offers practical insights into the efficient computation of risk measures as well as validation of the risk models. It also provides a detailed description of algorithms to compute each of the risk measures considered. While the main focus of the paper is on portfolio-level risk metrics, all algorithms are also applicable to single instruments. Practical implications The algorithms presented in this paper require little computational effort which makes them very suitable for real-world applications. In addition, the mathematical setup adopted in this paper provides a natural framework for risk model validation which makes the approach presented in this paper particularly appealing in practice. Originality/value This is the first study to consider the computation of entropic value-at-risk, semivariance as well as expectile risk measure using Fourier transform method.
This paper examines the empirical performance of four stochastic volatility option pricing models: Heston, Heston[Formula: see text], Bates and Heston–Hull–White. To compare these models, we use individual stock options data from January 1996 to August 2014. The comparison is made with respect to pricing and hedging performance, implied volatility surface and risk-neutral return distribution characteristics, as well as performance across industries and time. We find that the Heston model outperforms the other models in terms of in-sample pricing, whereas Heston[Formula: see text] model outperforms the other models in terms of out-of-sample hedging. This suggests that taking jumps or stochastic interest rates into account does not improve the model performance after accounting for stochastic volatility. We also find that the model performance deteriorates during the crises as well as when the implied volatility surface is steep in the maturity or strike dimension.
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