Heston model

The basic Heston model assumes that S_t, the price of the asset, is determined by a stochastic process:[2]

${\displaystyle dS_{t}=\mu S_{t}\,dt+{\sqrt {\nu _{t}}}S_{t}\,dW_{t}^{S}\,}$

where ${\displaystyle \nu _{t}}$, the instantaneous variance, is a CIR process:

${\displaystyle d\nu _{t}=\kappa (\theta -\nu _{t})\,dt+\xi {\sqrt {\nu _{t}}}\,dW_{t}^{\nu }\,}$

and ${\displaystyle W_{t}^{S},W_{t}^{\nu }}$ are Wiener processes (i.e., continuous random walks) with correlation ρ, or equivalently, with covariance ρ dt.

The parameters in the above equations represent the following:

• ${\displaystyle \mu }$ is the rate of return of the asset.
• ${\displaystyle \theta }$ is the long variance, or long run average price variance; as t tends to infinity, the expected value of ν_t tends to θ.
• ${\displaystyle \kappa }$ is the rate at which ν_t reverts to θ.
• ${\displaystyle \xi }$ is the volatility of the volatility, or 'vol of vol', and determines the variance of ν_t.

If the parameters obey the following condition (known as the Feller condition) then the process ${\displaystyle \nu _{t}}$ is strictly positive [3]

${\displaystyle 2\kappa \theta >\xi ^{2}\,.}$

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:

1. 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.
2. 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 price of each of the underlying assets is a martingale. See Girsanov's theorem.
3. 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.
4. 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 ${\displaystyle n}$ underlying assets and a linearly independent set of ${\displaystyle m}$ Wiener processes. The set of equivalent measures is isomorphic to R^m, the space of possible drifts. Consider the set of equivalent martingale measures to be isomorphic to a manifold ${\displaystyle M}$ embedded in R^m; initially, consider the situation where we have no assets and ${\displaystyle M}$ is isomorphic to R^m.

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 ${\displaystyle M}$ by one dimension. Hence we can see that in the general situation described above, the dimension of the set of equivalent martingale measures is ${\displaystyle m-n}$.

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 ${\displaystyle e^{-\rho t}S_{t}}$ 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.

• A recent discussion of implementation of the Heston model is given in a paper by Kahl and Jäckel .[4]

• Information about how to use the Fourier transform to value options is given in a paper by Carr and Madan.[5]

• Extension of the Heston model with stochastic interest rates is given in the paper by Grzelak and Oosterlee.[6]

• Derivation of closed-form option prices for time-dependent Heston model is presented in the paper by Gobet et al.[7]

• Derivation of closed-form option prices for double Heston model are presented in papers by Christoffersen

[8] and Gauthier. [9]

• Explicit solution of the Heston price equation in terms of the volatility was developed in Kouritzin [10], which can be combined with known weak solutions for the volatility equation and Girsanov's theorem to produce explicit weak solutions to the Heston model. Such solutions are useful for efficient simulation.

• There exist few known parametrisation of the volatility surface based on the Heston model (Schonbusher, SVI and gSVI).
• Use of the model in a local stochastic volatility context is given in a paper by Van Der Weijst.[11]

The calibration of the Heston model is often formulated as a least squares problem, with the objective function minimizing the difference between the prices observed in the market and those calculated from the Heston 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.[12] 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[13] summarizes the various quadratures applied and proposes an efficient adaptive Filon quadrature.

The calibration problem involves the gradient of the objective function with respect to the Heston parameters. A finite difference approximation of the gradient has a tendency to create artificial numerical issues in the calibration. It is a much better idea to rely on automatic differentiation techniques. For example, the tangent mode of algorithmic differentiation may be applied using dual numbers in a straightforward manner. Alternatively, Cui et al.[14] give explicit formulas for the analytical gradient. The latter was obtained by introducing an equivalent but tractable form of the Heston characteristic function.

• Stochastic volatility
• Risk-neutral measure (another name for the equivalent martingale measure)
• Girsanov's theorem
• Martingale (probability theory)
• SABR Volatility Model
• MATLAB code for implementation by Kahl, Jäckel and Lord

• Damghani, Babak Mahdavi; Kos, Andrew (2013). "De-arbitraging With a Weak Smile: Application to Skew Risk". Wilmott. 2013 (1): 40–49. doi:10.1002/wilm.10201.