Stochastic volatility models are used in the field of quantitative finance to evaluate derivative securities, such as options. The name derives from the models' treatment of the underlying security's volatility as a random process, governed by state variables such as the price level of the underlier, the tendency of volatility to revert to some long-run mean value, and the variance of the volatility process itself, among others.

Stochastic volatility models address many of the short-comings of popular option pricing models such as the Black-Scholes model and the Cox-Ross-Rubinstein model. In particular, these models assume that the underlier volatility is constant over the life of the derivative, and unaffected by the changes in the price level of the underlier. However, these models cannot explain long-observed anomalies such as volatility smile, which indicate that volatility does tend to vary over time. By assuming that the volatility of the underlying price is a stochastic process rather than a constant, it becomes possible to more accurately model derivatives.

=Basic Model=

Starting from a constant volatiltiy approach, assume that the derivative's underlier price follows a standard model for brownian motion:

$dS\_t\; =\; \backslash mu\; S\_t\backslash ,dt\; +\; \backslash sigma\; S\_t\backslash ,dW\_t\; \backslash ,$

where $\backslash mu\; \backslash ,$ is the constant drift (i.e. expected return) of the security price $S\_t\; \backslash ,$, $\backslash sigma\; \backslash ,$ is the constant volatility, and $dW\_t\; \backslash ,$ is a standard gaussian with zero mean and unit standard deviation. This is the starting point for non-stochastic volatility models such as Black-Scholes and Cox-Ross-Rubinstien.

For a stochastic volatility model, replace the constant volatility $\backslash sigma\; \backslash ,$ with a function $\backslash nu\_t\; \backslash ,$, that models the variance of $S\_t\; \backslash ,$. This variance function is also modeled as brownian motion, and the form of $\backslash nu\_t\; \backslash ,$ depends on the particular SV model under study.

$dS\_t\; =\; \backslash mu\; S\_t\backslash ,dt\; +\; \backslash nu\_t\; S\_t\backslash ,dW\_t\; \backslash ,$

$d\backslash nu\_t\; =\; \backslash alpha\_\{S,t\}\backslash ,dt\; +\; \backslash beta\_\{S,t\}\backslash ,dB\_t\; \backslash ,$

where $\backslash alpha\_\{S,t\}\; \backslash ,$ and $\backslash beta\_\{S,t\}\; \backslash ,$ are some functions of $\backslash nu\; \backslash ,$.

Heston Model

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The popular Heston model is a commonly used SV model, in which the randomness of the variance process varies as the square root of variance. In this case, the differential equation for variance takes the form:

$d\backslash nu\_t\; =\; (\backslash omega\; -\; \backslash theta\backslash ,\backslash nu\_t)dt\; +\; \backslash xi\; \backslash sqrt\{\backslash nu\_t\}\backslash ,dB\_t\; \backslash ,$

where $\backslash omega\; \backslash ,$ is the mean long-term volatility, $\backslash theta\; \backslash ,$ is the rate at which the volatility reverts towards it's long-term mean, $\backslash xi\; \backslash ,$ is the volatility of the volatility process, and $dB\_t\; \backslash ,$ is, like $dW\_t\; \backslash ,$, a guassian with zero mean and unit standard deviation. However, $dW\_t\; \backslash ,$ and $dB\_t\; \backslash ,$ are correlated with the constant correlation value $\backslash rho\; \backslash ,$.

In other words, the Heston SV model assumes that volatility is a random process that

1. exhibits a tendendecy to revert towards a long-term mean volatility $\backslash omega\; \backslash ,$ at a rate $\backslash theta\; \backslash ,$,
2. exhibits it's own (constant) volatility, $\backslash xi\; \backslash ,$,
3. and whose source of randomness is correlated (with correlation $\backslash rho\; \backslash ,$) with the randomness of the underlier's price process.

GARCH Model

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The Generalized Auto-Regression Conditional Heteroskedacity (GARCH) model is another popular model for estimating stochastic volatility. It assumes that the randomness of the variance process varies with the variance, as opposed to the square root of the variance as in the Heston model. The standard GARCH(1,1) model has the following form for the variance differential:

$d\backslash nu\_t\; =\; (\backslash omega\; -\; \backslash theta\backslash ,\backslash nu\_t)dt\; +\; \backslash xi\; \backslash nu\_t\backslash ,dB\_t\; \backslash ,$

The GARCH model has been extended via numerous variants, including the NGARCH, LGARCH, EGARCH, GJR-GARCH, etc.

3/2 Model

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The 3/2 model is similar to the Heston model, but assumes that the randomness of the variance process varies with $\backslash nu\_t^\backslash frac\{3\}\{2\}\; \backslash ,$. The form of the variance differential is:

$d\backslash nu\_t\; =\; (\backslash omega\; -\; \backslash theta\backslash ,\backslash nu\_t)dt\; +\; \backslash xi\; \backslash nu\_t^\backslash frac\{3\}\{2\}\backslash ,dB\_t\; \backslash ,$

=Calibration= Once a particular SV model is chosen, it must be calibrated against existing market data. Calibration is the process of identifying the set of model parameters that are most likely given the observed data. This process is called Maximum Likelihood Estimation (MLE). For instance, in the Heston model, the set of model parameters $\backslash Psi\_0\; =\; \backslash \{\backslash omega,\; \backslash theta,\; \backslash xi,\; \backslash rho\backslash \}\; \backslash ,$ can be estimated applying an MLE algorithm such as the Powell Directed Set method ^* to observations of historic underlying security prices.

In this case, you start with an estimate for $\backslash Psi\_0\; \backslash ,$, compute the residual errors when applying the historic price data to the resulting model, and then adjust $\backslash Psi\; \backslash ,$ to try to minimize these errors. Once the calibration has been performed, it is standard practice to re-calibrate the model over time.

=References=

• Stochastic Volatility and Mean-variance Analysis, Hyungsok Ahn, Paul Wilmott, (2006).
• A closed-form solution for options with stochastic volatility, SL Heston, (1993).
• Inside Volatility Arbitrage, Alireza Javaheri, (2005).

Mathematical finance | Options | Derivatives