[1] 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.
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.
In these models the underlying volatility does not feature any new randomness but it isn't a constant either.
Starting from a constant volatility approach, assume that the derivative's underlying asset price follows a standard model for geometric Brownian motion: where
is the constant drift (i.e. expected return) of the security price
is a standard Wiener process with zero mean and unit rate of variance.
This variance function is also modeled as Brownian motion, and the form of
In this case, the differential equation for variance takes the form: where
In other words, the Heston SV model assumes that the variance is a random process that Some parametrisation of the volatility surface, such as 'SVI',[2] are based on the Heston model.
In other markets, volatility tends to rise as prices fall, modelled with
The SABR model (Stochastic Alpha, Beta, Rho), introduced by Hagan et al.[3] describes a single forward
(related to any asset e.g. an index, interest rate, bond, currency or equity) under stochastic volatility
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
[6] It has been found that log-volatility behaves as a fractional Brownian motion with Hurst exponent of order
The RFSV model is consistent with time series data, allowing for improved forecasts of realized volatility.
[6][8] 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.
One popular technique is to use maximum likelihood estimation (MLE).
can be estimated applying an MLE algorithm such as the Powell Directed Set method [1] to observations of historic underlying security prices.
, compute the residual errors when applying the historic price data to the resulting model, and then adjust
Once the calibration has been performed, it is standard practice to re-calibrate the model periodically.
An alternative to calibration is statistical estimation, thereby accounting for parameter uncertainty.
Many frequentist and Bayesian methods have been proposed and implemented, typically for a subset of the abovementioned models.
The following list contains extension packages for the open source statistical software R that have been specifically designed for heteroskedasticity estimation.
Many numerical methods have been developed over time and have solved pricing financial assets such as options with stochastic volatility models.
A recent developed application is the local stochastic volatility model.
[13] This local stochastic volatility model gives better results in pricing new financial assets such as forex options.
There are also alternate statistical estimation libraries in other languages such as Python: