Heston model
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 "S", the price of the asset, is determined by a stochastic process, where formula_2, the instantaneous variance, is given by a Feller square-root or CIR process, and formula_4 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 formula_2 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 formula_12 underlying assets and a linearly independent set of formula_13 Wiener processes. The set of equivalent measures is isomorphic to R, the space of possible drifts. Consider the set of equivalent martingale measures  to be isomorphic to a manifold formula_14 embedded in R; initially, consider the situation where we have no assets and formula_14 is isomorphic to R. 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 formula_14 by one dimension. Hence we can see that in the general situation described above, the dimension of the set of equivalent martingale measures is  formula_17. 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 formula_18 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. 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.