In finance, the Heston model, named after Steven L. Heston, is a mathematical model that describes the evolution of the volatility of an underlying asset. It is a stochastic volatility model: such a model assumes that the volatility of the asset is not constant, nor even deterministic, but follows a random process.
Basic Heston model
The basic Heston model assumes that St, the price of the asset, is determined by a stochastic process,
where , the instantaneous variance, is given by a Feller square-root or CIR process,
and are Wiener processes (i.e., continuous random walks) with correlation ρ.
The model has five parameters: , the initial variance. , the long variance, or long-run average variance of the price; as t tends to infinity, the expected value of νt tends to θ. , the correlation of the two Wiener processes. , the rate at which νt reverts to θ. , the volatility of the volatility, or 'vol of vol', which determines the variance of νt.
If the parameters obey the following condition (known as the Feller condition) then the process is strictly positive
Risk-neutral measure
See Risk-neutral measure for the complete article
A fundamental concept in derivatives pricing is the risk-neutral measure; this is explained in further depth in the above article. For our purposes, it is sufficient to note the following:
To price a derivative whose payoff is a function of one or more underlying assets, we evaluate the expected value of its discounted payoff under a risk-neutral measure. A risk-neutral measure, also known as an equivalent martingale measure, is one which is equivalent to the real-world measure, and which is arbitrage-free: under such a measure, the discounted price of each of the underlying assets is a martingale. See Girsanov's theorem. In the Black-Scholes and Heston frameworks (where filtrations are generated from a linearly independent set of Wiener processes alone), any equivalent measure can be described in a very loose sense by adding a drift to each of the Wiener processes. By selecting certain values for the drifts described above, we may obtain an equivalent measure which fulfills the arbitrage-free condition.
Consider a general situation where we have underlying assets and a linearly independent set of Wiener processes. The set of equivalent measures is isomorphic to Rm, the space of possible drifts. Consider the set of equivalent martingale measures to be isomorphic to a manifold embedded in Rm; initially, consider the situation where we have no assets and is isomorphic to Rm.
Now consider each of the underlying assets as providing a constraint on the set of equivalent measures, as its expected discount process must be equal to a constant (namely, its initial value). By adding one asset at a time, we may consider each additional constraint as reducing the dimension of by one dimension. Hence we can see that in the general situation described above, the dimension of the set of equivalent martingale measures is .
In the Black-Scholes model, we have one asset and one Wiener process. The dimension of the set of equivalent martingale measures is zero; hence it can be shown that there is a single value for the drift, and thus a single risk-neutral measure, under which the discounted asset will be a martingale.
In the Heston model, we still have one asset (volatility is not considered to be directly observable or tradeable in the market) but we now have two Wiener processes - the first in the Stochastic Differential Equation (SDE) for the stock price and the second in the SDE for the variance of the stock price. Here, the dimension of the set of equivalent martingale measures is one; there is no unique risk-free measure.
This is of course problematic; while any of the risk-free measures may theoretically be used to price a derivative, it is likely that each of them will give a different price. In theory, however, only one of these risk-free measures would be compatible with the market prices of volatility-dependent options (for example, European calls, or more explicitly, variance swaps). Hence we could add a volatility-dependent asset; by doing so, we add an additional constraint, and thus choose a single risk-free measure which is compatible with the market. This measure may be used for pricing.
Implementation
The use of the Fourier transform to value options was shown by Carr and Madan.
A discussion of the implementation of the Heston model was given by Kahl and Jäckel.
A derivation of closed-form option prices for the time-dependent Heston model was presented by Benhamou et al.