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 volatility follows an Ornstein-Uhlenbeck process Itô's lemma then shows that \nu_t, 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: If the parameters obey the following condition (known as the Feller condition) then the process \nu_t is strictly positive

Risk-neutral measure

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: 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 Rm, the space of possible drifts. Consider the set of equivalent martingale measures to be isomorphic to a manifold M embedded in Rm; initially, consider the situation where we have no assets and M 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 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 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

Calibration

The calibration of the Heston model is often formulated as a least squares problem, with the objective function minimizing the squared difference between the prices observed in the market and those calculated from the model. The prices are typically those of vanilla options. Sometimes the model is also calibrated to the variance swap term-structure as in Guillaume and Schoutens. Yet another approach is to include forward start options, or barrier options as well, in order to capture the forward smile. Under the Heston model, the price of vanilla options is given analytically, but requires a numerical method to compute the integral. Le Floc'h summarized the various quadratures applied and proposed an efficient adaptive Filon quadrature. Calibration usually requires the gradient of the objective function with respect to the model parameters. This was usually computed with a finite difference approximation although it is less accurate, less efficient and less elegant than an analytical gradient because an insightful expression of the latter became available only when a new representation of the characteristic function was introduced by Cui et al. in 2017. Another possibility is to resort to automatic differentiation. For example, the tangent mode of algorithmic differentiation may be applied using dual numbers in a straightforward manner.