Stochastic volatility

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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.

1 Basic Model
2 Calibration
3 References

[edit] Basic Model

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

$dS_t = \mu S_t\,dt + \sigma S_t\,dW_t \,$

where $\mu \,$ is the constant drift (i.e. expected return) of the security price $S_t \,$ , $\sigma \,$ is the constant volatility, and $dW_t \,$ 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-Rubinstein.

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

$dS_t = \mu S_t\,dt + \sqrt{\nu_t} S_t\,dW_t \,$

$d\nu_t = \alpha_{S,t}\,dt + \beta_{S,t}\,dB_t \,$

where $\alpha_{S,t} \,$ and $\beta_{S,t} \,$ are some functions of $\nu \,$ .

[edit] Heston Model

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:

$d\nu_t = \theta(\omega - \nu_t)dt + \xi \sqrt{\nu_t}\,dB_t \,$

where $\omega \,$ is the mean long-term volatility, $\theta \,$ is the rate at which the volatility reverts toward its long-term mean, $\xi \,$ is the volatility of the volatility process, and $dB_t \,$ is, like $dW_t \,$ , a gaussian with zero mean and unit standard deviation. However, $dW_t \,$ and $dB_t \,$ are correlated with the constant correlation value $\rho \,$ .

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

exhibits a tendency to revert towards a long-term mean volatility $\omega \,$ at a rate $\theta \,$ ,
exhibits its own (constant) volatility, $\xi \,$ ,
and whose source of randomness is correlated (with correlation $\rho \,$ ) with the randomness of the underlier's price processes.

[edit] GARCH Model

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:

$d\nu_t = \theta(\omega - \nu_t)dt + \xi \nu_t\,dB_t \,$

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

[edit] 3/2 Model

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

$d\nu_t = \theta(\omega - \nu_t)dt + \xi \nu_t^\frac{3}{2}\,dB_t \,$

[edit] 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 $\Psi_0 = \{\omega, \theta, \xi, \rho\} \,$ can be estimated applying an MLE algorithm such as the Powell Directed Set method [1] to observations of historic underlying security prices.

In this case, you start with an estimate for $\Psi_0 \,$ , compute the residual errors when applying the historic price data to the resulting model, and then adjust $\Psi \,$ to try to minimize these errors. Once the calibration has been performed, it is standard practice to re-calibrate the model over time.

[edit] 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).

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