Heston model

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In finance, the Heston model is a mathematical model describing 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.

[edit] Basic Heston model

The basic Heston model assumes that S_t, the price of the asset, is determined by a stochastic process:

$dS_t = \mu S_t\,dt + \sqrt{\nu_t} S_t\,dW^S_t \,$

where $ν t$ , the instantaneous variance, is a CIR process:

$d\nu_t = \kappa(\theta - \nu_t)dt + \xi \sqrt{\nu_t}\,dW^{\nu}_t \,$

and ${\scriptstyle dW^S_t, dW^{\nu}_t}$ are Wiener processes (i.e., random walks) with correlation ρ.

The parameters in the above equations represent the following:

μ is the rate of return of the asset.
θ is the long vol, or long run average price volatility; as t tends to infinity, the expected value of ν_t tends to θ.
κ is the rate at which ν_t reverts to θ.
ξ is the vol of vol, or volatility of the volatility; as the name suggests, this determines the variance of ν_t.

[edit] Risk-neutral measure

See Risk-neutral measure for the complete article

A fundamental concept in derivatives pricing is that of 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 process of each of the underlying assets is a martingale.
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 $n$ underlying assets and a linearly independent set of $m$ Wiener processes. The set of equivalent measures is isomorphic to R^m, the space of possible drifts. Let us consider the set of equivalent martingale measures to be isomorphic to a manifold $M$ embedded in R^m; initially, consider the situation where we have no assets and $M$ is isomorphic to R^m.

Now let us 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 $M$ by one dimension. Hence we can see that in the general situation described above, the dimension of the set of equivalent martingale measures is $m - n$ .

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 $e - ρ t S t$ 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 SDE for the asset and the second in the SDE for the stochastic volatility. 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.