Heston model

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.

Basic Heston model

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

:$dS\_t\; =\; mu\; S\_t,dt\; +\; sqrt\{\; u\_t\}\; S\_t,dW^S\_t\; ,$

where $u\_t$, the instantaneous variance, is a CIR process:

:$d\; u\_t\; =\; kappa(\; heta\; -\; u\_t)dt\; +\; xi\; sqrt\{\; u\_t\},dW^\{\; u\}\_t\; ,$

and $\{scriptstyle\; dW^S\_t,\; dW^\{\; u\}\_t\}$ 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".

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 $e^\{-\; ho\; t\}S\_t$ 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 Stochastic Differential Equation (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.

See also

*Stochastic Volatility
*Risk-neutral measure (another name for the equivalent martingale measure)
*Martingale (probability theory)
*SABR Volatility Model

References

* A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options, by Steven L. Heston, The Review of Financial Studies 1993 Volume 6, number 2, pp- 327-343 [http://www.jstor.org/pss/2962057]