The basic Heston model assumes that St, the price of the asset, is determined by a stochastic process,[1][2] where the volatility
is strictly positive [3] A fundamental concept in derivatives pricing is the risk-neutral measure; this is explained in further depth in the above article.
The set of equivalent measures is isomorphic to Rm, the space of possible drifts.
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
Hence we can see that in the general situation described above, the dimension of the set of equivalent martingale measures is
[citation needed] 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.
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).
Sometimes the model is also calibrated to the variance swap term-structure as in Guillaume and Schoutens.
Under the Heston model, the price of vanilla options is given analytically, but requires a numerical method to compute the integral.
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 [10].
For example, the tangent mode of algorithmic differentiation may be applied using dual numbers in a straightforward manner.