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{{Short description|When variance is a random variable}}{{Use American English|date = January 2019}}{{Hatnote|See also Volatility (finance).}}In statistics, stochastic volatility models are those in which the variance of a stochastic process is itself randomly distributed.BOOK, Jim Gatheral, The Volatility Surface: A Practitioner's Guide,weblink 18 September 2006, Wiley, 978-0-470-06825-0, They are used in the field of mathematical 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 security, 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, models based on Black-Scholes 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 security. 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 expiry. 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.A middle ground between the bare Black-Scholes model and stochastic volatility models is covered by local volatility models. In these models the underlying volatility does not feature any new randomness but it isn't a constant either. In local volatility models the volatility is a non-trivial function of the underlying asset, without any extra randomness. According to this definition, models like constant elasticity of variance would be local volatility models, although they are sometimes classified as stochastic volatility models. The classification can be a little ambiguous in some cases. The early history of stochastic volatility has multiple roots (i.e. stochastic process, option pricing and econometrics), it is reviewed in Chapter 1 of Neil Shephard (2005) "Stochastic Volatility," Oxford University Press.

Basic model

Starting from a constant volatility approach, assume that the derivative's underlying asset price follows a standard model for geometric Brownian motion: 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 Wiener process with zero mean and unit rate of variance. The explicit solution of this stochastic differential equation is The maximum likelihood estimator to estimate the constant volatility sigma , for given stock prices S_t , at different times t_i , is begin{align}widehat{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}} right) - frac 1 n frac{(ln S_{t_n}- ln S_{t_0})^2}{t_n-t_0}& = frac 1 n sum_{i=1}^n (t_i-t_{i-1})left(frac{ln frac{S_{t_i}}{S_{t_{i-1}}}}{t_i-t_{i-1}} - frac{ln frac{S_{t_n}}{S_{t_0}}}{t_n-t_0}right)^2;end{align}its expected value is operatorname E left[ widehat{sigma}^2right]= 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 model and Cox–Ross–Rubinstein model.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. where alpha_{nu,t} and beta_{nu,t} are some functions of nu , and dB_t is another standard gaussian that is correlated with dW_t with constant correlation factor rho .

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: where omega is the mean long-term variance, theta is the rate at which the variance reverts toward its long-term mean, xi is the volatility of the variance process, and dB_t is, like dW_t, a gaussian with zero mean and dt variance. However, dW_t and dB_t are correlated with the constant correlation value rho.In other words, the Heston SV model assumes that the variance is a random process that

1. exhibits a tendency to revert towards a long-term mean omega at a rate theta,
2. exhibits a volatility proportional to the square root of its level
3. and whose source of randomness is correlated (with correlation rho) with the randomness of the underlying's price processes.

Some parametrisation of the volatility surface, such as 'SVI',JOURNAL, J Gatheral, A Jacquier, Arbitrage-free SVI volatility surfaces, Quantitative Finance, 2014, 10.1080/14697688.2013.819986, 14, 59–71, 1204.0646, 41434372, are based on the Heston model.

CEV model

The CEV model describes the relationship between volatility and price, introducing stochastic volatility: Conceptually, in some markets volatility rises when prices rise (e.g. commodities), so gamma > 1. In other markets, volatility tends to rise as prices fall, modelled with gamma < 1.Some argue that because the CEV model does not incorporate its own stochastic process for volatility, it is not truly a stochastic volatility model. Instead, they call it a local volatility model.

SABR volatility model

The SABR model (Stochastic Alpha, Beta, Rho), introduced by Hagan et al.PS Hagan, D Kumar, A Lesniewski, DE Woodward (2002) Managing smile risk, Wilmott, 84-108. describes a single forward F (related to any asset e.g. an index, interest rate, bond, currency or equity) under stochastic volatility sigma: The initial values F_0 and sigma_0 are the current forward price and volatility, whereas W_t and Z_t are two correlated Wiener processes (i.e. Brownian motions) with correlation coefficient -1