Volatility Surface in the Heston Model

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):

((d S)/S)=μ d t+(Sqrt[(Subscript[V, t])]) d (Subscript[W, t]),

d (Subscript[V, t])=κ((Subscript[V, ∞])-(Subscript[V, t]))d t+σ(Sqrt[(Subscript[V, t])])d (Subscript[Z, t]),

where d (Subscript[W, t]) and d (Subscript[Z, t]) are correlated Brownian motions with <d (Subscript[W, t]),d (Subscript[Z, t])>=ρ.

The spot price follows the process with drift μ and variance Vt, which is itself a stochastic process defined by the second equation. The second SDE is mean-reverting (the Cox–Ingersoll–Ross model, similar to the Ornstein–Uhlenbeck process). Here the long-term variance is (Subscript[V, ∞]), the mean reversion (or "speed of reversion") is κ, and the volatility of variance is σ. And finally there is another parameter that does not appear in the SDE, the initial condition for variance evolution.

In this Demonstration, the "original volatility" is used, that is, the square root of initial variance; volatility is used instead of variance as it is a more frequently quoted quantity. The plot generation is computationally intensive, so allow some time for it to regenerate.

THINGS TO TRY

MouseOver ▪ MouseOver

DETAILS

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 References

Heston Model at Quantlet

Stochastic Volatility at Wikipedia

Contributed by: Slava Solganik

© The Wolfram Demonstrations Project & Contributors   |   Terms of Use