The Resurgence of Stochastic Volatility Models
Team Latte
August 10, 2008

Stochastic Volatility models have become popular amongst the global quant community of late. In the last ten years, and we reckon, especially since 2004 stochastic volatility models have moved from academia and quant research to mainstream front office derivatives pricing desks. What has brought about this change? What are stochastic volatility models and how do they differ from other volatility models? In this and the following two articles we will talk about these issues.

Even though Hull and White proposed the first stochastic volatility models in late eighties the first robust formulation of these class of models came about with Heston’s formulation of the problem in 19993. Heston (1993) stochastic volatility model and later on its variant, Heston-Nandi GARCH (2000) model still remains the dominant stochastic volatility model amongst the quant community. Of course, in the last couple of years SABR (stochastic alpha, beta and rho) model of Hagan, Kumar, Lesniewski and Woodward (which was put forth in 2002) has also gained in popularity, especially in the FX and interest rate options arena.

Stochastic volatility (SV) models, by saying that volatility is stochastic (random) and changing from one instant to the other, try to explain the varying (Black-Scholes) implied volatility across strikes and expirations in the options market and it does so in a self consistent way. Further, SV models assume volatility to be mean reverting. This is an important assumption and is in fact based on a very simple economic argument* (as explained brilliantly by Jim Gatheral in The Volatility Surface). If you take the distribution of volatility of Dollar-Yen over a period of 30 years (Gatheral takes a better example and talks about IBM vols over 100 years) and say that vols are not mean reverting then the probability of Dollar-yen vols staying between say, 1% and 20% over this period should be very low. However, it is highly unlikely, that Dollar-Yen vols will ever cross this range, in other words, we strongly believe that Dollar-Yen vols will lie in the range of 1% to 20% shows that volatility of Dollar-Yen conforms to some mean reverting phenomenon.

A stochastic volatility model assumes that volatility, like the underlying asset itself, follows a diffusion process and the Weiner process of this diffusion process is correlated with the Weiner process of the underlying asset. In short, volatility, or more appropriately the volatility process, is correlated with the underlying asset. In the particular case of Heston (1993) model, volatility is assumed to follow a square root process exactly like a Cox-Ingersoll-Ross (1985) process. Volatility by mimicking the CIR process, therefore follows a jump free affine jump diffusion (AJD) process. In a “square root” process the stochastic Weiner process (the random component of a Brownian motion) is anchored to the square root of the volatility.

In a SV model, volatility follows a mean reverting, square root stochastic process which is correlated with the asset price process. Therefore, we have a set of two stochastic differential equations which needs to be solved to get any closed form solution.

……………….To be continued

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