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Heston Model: The Variance Swap Calibration

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Abstract

This paper features a market implied methodology to infer adequate starting values for the spot and long-run variances and for the mean reversion rate of a calibration exercise under the Heston model. More particularly, these initial parameters are obtained by matching the term structure of the future expected total variance, inferred from the volatility surface, with the model term structure. In the numerical study, we compare the goodness of fit and the parameter stability of the Heston model calibrated by using either plausible random or market implied starting values for a one-year sample period including the recent credit crunch. In particular, we show that the proposed methodology avoids getting stuck in one “bad” local minimum and stabilizes the calibrated parameters through time.

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Notes

  1. An out-of-the-money call option is characterized by K i >F 0 and an out-of-the-money put option by K i <F 0.

  2. Arbitrage opportunities can only be detected from the term structure of the cumulated variance v(T) and not from the term structure of the annualized variance, i.e., \(\text{VIX}^{2}(T) := \frac{1}{T} \mbox{v}(T)\), due to the scaling by the time horizon T in the annualized variance. This explains why we have opted for fitting the total variance curve instead of the annualized variance one.

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Authors and Affiliations

  1. Department of Mathematics, K.U. Leuven, Celestijnenlaan 200 B, 3001, Leuven, Belgium

    Florence Guillaume & Wim Schoutens

Authors
  1. Florence Guillaume
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  2. Wim Schoutens
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Correspondence to Florence Guillaume.

Additional information

Florence Guillaume is a postdoctoral fellow of the Fund for Scientific Research–Flanders (Belgium) (F.W.O.).

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Guillaume, F., Schoutens, W. Heston Model: The Variance Swap Calibration. J Optim Theory Appl 161, 76–89 (2014). https://doi.org/10.1007/s10957-013-0331-7

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  • DOI: https://doi.org/10.1007/s10957-013-0331-7

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