Heston model

In finance, the Heston model, named after Steven Heston, is a mathematical model describing the evolution of the volatility of an underlying asset.^[1] 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:^[2]

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

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

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

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

The parameters in the above equations represent the following:

μ is the rate of return of the asset.
θ is the long variance, or long run average price variance; 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 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 $\nu_t$ is strictly positive ^[3]

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

Extensions

In order to take into account all the features from the volatility surface, the Heston model may be a too rigid framework. It may be necessary to add degrees of freedom to the original model. A first straightforward extension is to allow the parameters to be time-dependent. The model dynamics are then written as:

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

Here $\nu_t$ , the instantaneous variance, is a time-dependent CIR process:

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

and ${\scriptstyle dW_{t}^{S},dW_{t}^{{\nu }}}$ are Wiener processes (i.e., random walks) with correlation ρ. In order to retain model tractability, one may require parameters to be piecewise-constant.

Another approach is to add a second process of variance, independent of the first one.

dS_{t}=\mu S_{t}\,dt+{\sqrt {\nu _{t}^{1}}}S_{t}\,dW_{t}^{{S,1}}+{\sqrt {\nu _{t}^{2}}}S_{t}\,dW_{t}^{{S,2}}\,

d\nu _{t}^{1}=\kappa ^{1}(\theta ^{1}-\nu _{t}^{1})\,dt+\xi ^{1}{\sqrt {\nu _{t}^{1}}}\,dW_{t}^{{\nu ^{1}}}\,

d\nu _{t}^{2}=\kappa ^{2}(\theta ^{2}-\nu _{t}^{2})\,dt+\xi ^{2}{\sqrt {\nu _{t}^{2}}}\,dW_{t}^{{\nu ^{2}}}\,

A significant extension of Heston model to make both volatility and mean stochastic is given by Lin Chen (1996). In the Chen model the dynamics of the instantaneous interest rate are specified by

dr_{t}=(\theta _{t}-\alpha _{t})\,dt+{\sqrt {r_{t}}}\,\sigma _{t}\,dW_{t},

d\alpha _{t}=(\zeta _{t}-\alpha _{t})\,dt+{\sqrt {\alpha _{t}}}\,\sigma _{t}\,dW_{t},

d\sigma _{t}=(\beta _{t}-\sigma _{t})\,dt+{\sqrt {\sigma _{t}}}\,\eta _{t}\,dW_{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 price of each of the underlying assets is a martingale. See Girsanov's theorem.
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 $e^{{-\rho t}}S_{t}$ 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.

Implementation

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]

There exist few known parametrisation of the volatility surface based on the Heston model (Schonbusher, SVI and gSVI) as well as their de-arbitraging methodologies.^[10]

Calibration

The calibration of the Heston model is often formulated as a least squares problem, with the objective function minimizing the difference between the price observed in the market and that calculated from the Heston model. The price refers to that of vanilla options. Under the Heston model, this price is given analytically but the expression is overly complicated. This inhibits the derivation of the gradient of the objective function with respect to the Heston parameters.

The analytical form of the gradient has not been available for a long time and hence the calibration has been a challenging problem at trading desks. Practitioners often treated it using a numerical gradient, the Excel built-in solver, or stochastic optimizers. These methods are too slow or unstable, as many authors reported that their results varied largely with the initial guess. Also, practitioners applied many heuristic or asymptotic rules to deal with the 5 parameters.

This has been an obstacle until Cui et al.^[11] proposed a full and fast Heston calibrator based on an analytical gradient. The latter was obtained by introducing an equivalent but tractable form of the Heston characteristic function. This calibrator works just in seconds on a normal PC. They also report no local minima. The previous observation of local minima could have been caused by an objective function shaped as a narrow valley and a loose tolerance.

References

↑ Heston, Steven L. (1993). "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options". The Review of Financial Studies. 6 (2): 327–343. doi:10.1093/rfs/6.2.327. JSTOR 2962057.
↑ Wilmott, P. (2006), Paul Wilmott on quantitative finance (2nd ed.), p. 861
↑ Albrecher, H.; Mayer, P.; Schoutens, W.; Tistaert, J. (January 2007), "The Little Heston Trap", Wilmott Magazine: 83–92, CiteSeerX 10.1.1.170.9335
↑ Kahl, C.; Jäckel, P. (2005). "Not-so-complex logarithms in the Heston model" (PDF). Wilmott Magazine. pp. 74–103.
↑ Carr, P.; Madan, D. (1999). "Option valuation using the fast Fourier transform" (PDF). Journal of Computational Finance. pp. 61–73.
↑ Grzelak, L.A.; Oosterlee, C.W. (2011). "On the Heston Model with Stochastic Interest Rates". SIAM J. Fin. Math. pp. 255–286.
↑ Benhamou, E.; Gobet, E.; Miri, M. (2009). "Time Dependent Heston Model". doi:10.2139/ssrn.1367955. SSRN 1367955.
↑ Christoffersen, P.; Heston, S.; Jacobs, K. (2009). "The Shape and Term Structure of the Index Option Smirk: Why Multifactor Stochastic Volatility Models Work so Well". SSRN 1447362.
↑ Gauthier, P.; Possamai, D. (2009), Efficient Simulation of the Double Heston Model, SSRN 1434853
↑ Babak Mahdavi Damghani (2013). "De-arbitraging with a weak smile". Wilmott. http://www.readcube.com/articles/10.1002/wilm.10201?locale=en
↑ Yiran Cui; Sebastian del Baño Rollin; Guido Germano (26 May 2016). "Full and fast calibration of the Heston stochastic volatility model". arXiv:1511.08718.
↑ Mahdavi Damghani, Babak (2013). "De-arbitraging With a Weak Smile: Application to Skew Risk". Wilmott. 2013 (1): 40–49. doi:10.1002/wilm.10201.

Stochastic processes

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