# Volatility Surface in the Heston Model

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The plot shows the volatility surface generated by the Heston stochastic volatility model (Heston 1993). This is implied volatility based on the Heston price, which depends on the time to expiration and on moneyness. Recall that for a call option, moneyness is the ratio of spot price to strike price. The Heston model is described by the following stochastic differential equations (SDE):

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Contributed by: Slava Solganik (March 2011)

Open content licensed under CC BY-NC-SA

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The Heston stochastic volatility (SV) model originates from work by Heston (1993). One of the benefits of this model compared to other SV models is that prices of vanilla options can be expressed as a single integral. Thus given the volatility surface, the Heston model can be calibrated to fit it. There are some technical difficulties in computing option prices. In particular, the integral to be computed oscillates quickly at infinity. Another subtlety is that the function under the integral contains complex logarithms and requires careful counting of winding numbers (see Kahl and Jackel 2005). Because of this a number of approaches for pricing vanilla instruments in this framework were developed. The approach here follows Lewis (2000), who used the generalized Fourier transform.

This Demonstration builds the volatility surface based on the parameters of the model and enhances an intuitive understanding of the Heston model.

Here are a few references to original papers and books relevant to this Demonstration. The list is incomplete and includes only the author's favorite selection from an enormous amount of literature related to Heston and SV models.

References

S. Heston, "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options", *Review of Financial Studies*, 6, 1993 pp. 327–343.

J. Gatheral, *The Volatility Surface: A Practitioner's Guide*, Hoboken: John Wiley & Sons, Inc., 2006.

P. Jackel and C. Kahl, "Not-So-Complex Logarithms in the Heston Model", *Wilmott Magazine*, 19, 2005 pp. 94–103.

A. L. Lewis, *Option Valuation under Stochastic Volatility: With Mathematica Code*, Newport Beach: Finance Press, 2000.

Web Reference

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