Stochastic volatility

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 underlying, 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 are one approach to resolve a shortcoming of the Black-Scholes model. In particular, these models assume that the underlying volatility is constant over the life of the derivative, and unaffected by the changes in the price level of the underlying. However, these models cannot explain long-observed features of the implied volatility surface such as volatility smile and skew, which indicate that implied volatility does tend to vary with respect to strike price and expiration. By assuming that the volatility of the underlying price is a stochastic process rather than a constant, it becomes possible to model derivatives more accurately.

Basic Model

Starting from a constant volatility approach, assume that the derivative's underlying 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. The explicit solution of this stochastic differential equation is :$S\_t=\; S\_0\; e^\{(mu-\; frac\{1\}\{2\}\; sigma^2)\; t+\; sigma\; W\_t\}$.

The Maximum likelihood estimator to estimate the constant volatility $sigma\; ,$ for given stock prices $S\_t\; ,$ at different times $t\_i\; ,$ is :$hat\{sigma\}^2=\; left(frac\{1\}\{n\}\; sum\_\{i=1\}^n\; frac\{(ln\; S\_\{t\_i\}-\; ln\; S\_\{t\_\{i-1)^2\}\{t\_i-t\_\{i-1\; ight)\; -\; frac\{1\}\{n\}\; frac\{(ln\; S\_\{t\_n\}-\; ln\; S\_\{t\_0\})^2\}\{t\_n-t\_0\}$;its expectation value is $E\; left\; [\; hat\{sigma\}^2\; ight]\; =\; frac\{n-1\}\{n\}\; sigma^2$.

This basic model with constant volatility $sigma\; ,$ 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 $u\_t\; ,$, that models the variance of $S\_t\; ,$. This variance function is also modeled as brownian motion, and the form of $u\_t\; ,$ depends on the particular SV model under study. :$dS\_t\; =\; mu\; S\_t,dt\; +\; sqrt\{\; u\_t\}\; S\_t,dW\_t\; ,$

:

where $alpha\_\{S,t\}\; ,$ and are some functions of $u\; ,$ and $dB\_t\; ,$ is another standard gaussian that is correlated with $dW\_t\; ,$ with constant correlation factor $ho\; ,$.

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:

:$d\; u\_t\; =\; heta(omega\; -\; u\_t)dt\; +\; xi\; sqrt\{\; u\_t\},dB\_t\; ,$

where $omega\; ,$ is the mean long-term volatility, $heta\; ,$ 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 $ho\; ,$.

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 $heta\; ,$,
#exhibits its own (constant) volatility, $xi\; ,$,
#and whose source of randomness is correlated (with correlation $ho\; ,$) with the randomness of the underlying's price processes.

GARCH Model

The Generalized Autoregressive 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:

:$d\; u\_t\; =\; heta(omega\; -\; u\_t)dt\; +\; xi\; u\_t,dB\_t\; ,$

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

3/2 Model

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

:$d\; u\_t\; =\; heta(omega\; -\; u\_t)dt\; +\; xi\; u\_t^frac\{3\}\{2\},dB\_t\; ,$.

Chen Model

In interest rate modelings, Lin Chen in 1994 developed the first stochastic mean and stochastic volatility model, Chen model.Specifically, the dynamics of the instantaneous interest rate are given by following the stochastic differential equations:

:$dr\_t\; =\; (\; heta\_t-alpha\_t),dt\; +\; sqrt\{r\_t\},sigma\_t,\; dW\_t$,:$d\; alpha\_t\; =\; (zeta\_t-alpha\_t),dt\; +\; sqrt\{alpha\_t\},sigma\_t,\; dW\_t$, :.

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 $Psi\_0\; =\; \{omega,\; heta,\; xi,\; ho\}\; ,$ can be estimated applying an MLE algorithm such as the Powell Directed Set method [http://www.library.cornell.edu/nr/bookcpdf.html] 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.

ee also

*Volatility
*Local volatility
*Heston model
*SABR Volatility Model
*Risk-neutral measure
*Chen model

References

* [http://www.wilmott.com/detail.cfm?articleID=245 Stochastic Volatility and Mean-variance Analysis] , Hyungsok Ahn, Paul Wilmott, (2006).
* [http://www.javaquant.net/papers/Heston-original.pdf A closed-form solution for options with stochastic volatility] , SL Heston, (1993).
* [http://www.amazon.com/s?platform=gurupa&url=index%3Dblended&keywords=inside+volatility+arbitrage Inside Volatility Arbitrage] , Alireza Javaheri, (2005).
* [http://ssrn.com/abstract=982221 Accelerating the Calibration of Stochastic Volatility Models] , Kilin, Fiodar (2006).
*cite book | title = Stochastic Mean and Stochastic Volatility -- A Three-Factor Model of the Term Structure of Interest Rates and Its Application to the Pricing of Interest Rate Derivatives. Blackwell Publishers.
author = Lin Chen | publisher = Blackwell Publishers | year = 1996