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 **

**Reference***: An excellent reference is Jim Gatheral’s work The Volatility Surface*

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