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.
Contents
- Basic model
- Heston model
- CEV model
- SABR volatility model
- GARCH model
- 32 model
- Chen model
- Calibration and Estimation
- References
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.
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
The Maximum likelihood estimator to estimate the constant volatility
its expectation value is
This basic model with constant volatility
For a stochastic volatility model, replace the constant volatility
where
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
In other words, the Heston SV model assumes that the variance is a random process that
- exhibits a tendency to revert towards a long-term mean
ω at a rateθ , - exhibits a volatility proportional to the square root of its level
- and whose source of randomness is correlated (with correlation
ρ ) with the randomness of the underlying's price processes.
There exist few known parametrisation of the volatility surface based on the heston model (Schonbusher, SVI and gSVI) as well as their de-arbitraging methodologies.
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
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) describes a single forward
The initial values
The main feature of the SABR model is to be able to reproduce the smile effect of the volatility smile.
GARCH model
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 variance differential:
The GARCH model has been extended via numerous variants, including the NGARCH, TGARCH, IGARCH, LGARCH, EGARCH, GJR-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.
3/2 model
The 3/2 model is similar to the Heston model, but assumes that the randomness of the variance process varies with
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
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:
Calibration and Estimation
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). For instance, in the Heston model, the set of model parameters
In this case, you start with an estimate for
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. The first three cater for GARCH-type models with deterministic volatilities; the fourth deals with stochastic volatility estimation.
There are also alternate statistical estimation libraries in other languages such as Python: