Stochastic volatility

When variance is a random variable

In statistics, stochastic volatility models are those in which the variance of a stochastic process is itself randomly distributed. 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.

The SABR model (Stochastic Alpha, Beta, Rho), introduced by Hagan et al. describes a single forward (related to any asset e. g. an index, interest rate, bond, currency or equity) under stochastic volatility :

The initial values and are the current forward price and volatility, whereas and are two correlated Wiener processes (i. e. Brownian motions) with correlation coefficient The constant parameters are such that
The main feature of the SABR model is to be able to reproduce the smile effect of the volatility smile.

The Generalized Autoregressive Conditional Heteroskedasticity (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 continuous variance differential:

The GARCH model has been extended via numerous variants, including the NGARCH, TGARCH, IGARCH, LGARCH, EGARCH, GJR GARCH, Power GARCH, Component GARCH, etc. Strictly, however, the conditional volatilities from GARCH models are not stochastic since at time t the volatility is completely pre determined (deterministic) given previous values.

The 3/2 model is similar to the Heston model, but assumes that the randomness of the variance process varies with The form of the variance differential is:

However the meaning of the parameters is different from Heston model. In this model, both mean reverting and volatility of variance parameters are stochastic quantities given by and respectively.

Using estimation of volatility from high frequency data, smoothness of the volatility process has been questioned. It has been found that log volatility behaves as a fractional Brownian motion with Hurst exponent of order , at any reasonable timescale. This led to adopting a fractional stochastic volatility (FSV) model, leading to an overall Rough FSV (RFSV) where "rough" is to highlight that The RFSV model is consistent with time series data, allowing for improved forecasts of realized volatility.