Time Dependent Heston Model

Benhamou, Eric; Gobet, Emmanuel; Miri, Mohammed

doi:10.2139/ssrn.1367955

Cited by 42 publications

(81 citation statements)

References 24 publications

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“…Notice that, when = 4; formula (6) coincides with (7). On the other hand, the accuracy of the approximation given in (7) becomes worse as tends to 2:…”

Section: Remarkmentioning

confidence: 89%

“…In all these techniques, the region of validity of the results is restricted to either short or long maturities.The obtained approximations for option prices allow for fast callibration and give a better understanding of the role of model parameters. More recently, another approach have been proposed by Benhamou, Gobet and Miri (2009a, 2009band 2009c, where the authors focus directly on the law of the log-stock price at maturity time, given its initial condition. They expand prices with respect o the volatility of the volatility, computing the correction terms using Malliavin calculus.…”

Section: Introductionmentioning

confidence: 99%

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A decomposition formula for option prices in the Heston model and applications to option pricing approximation

Alòs

2012

Finance Stoch

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By means of classical Itô's calculus we decompose option prices as the sum of the classical Black-Scholes formula with volatility parameter equal to the root-mean-square future average volatility plus a term due by correlation and a term due to the volatility of the volatility. This decomposition allows us to develop …rst and second-order approximation formulas for option prices and implied volatilities in the Heston volatility framework, as well as to study their accuracy. Numerical examples are given.

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“…Notice that, when = 4; formula (6) coincides with (7). On the other hand, the accuracy of the approximation given in (7) becomes worse as tends to 2:…”

Section: Remarkmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

A decomposition formula for option prices in the Heston model and applications to option pricing approximation

Alòs

2012

Finance Stoch

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“…In this paper, we extend the results of Benhamou et al (2010) in a simple way and, thus, develop the approximation formula under the general multifactor Heston model with time-dependent parameters. Our aim is to model the implied volatility surface more realistically, but, with reasonable computational intensities.…”

Section: Introductionmentioning

confidence: 87%

“…For example, Tanaka et al (2010) use the Gram-Charlier expansion to derive asymptotic approximation for interest rate derivatives, Papageorgiou and Sircar (2009) use singular perturbations to price single-name and multi-name credit derivatives under a stochastic volatility environment, and more recently, Bayraktar and Yang (2011) use similar techniques for equity-credit hybrid modeling. Benhamou et al (2010) employ Malliavin calculus to develop a fast and accurate approximation formula of option prices under the onefactor Heston model with time-dependent parameters. By the asymptotic expansion with respect to the volatility of volatility, they show that the European put option price can be approximated by the Black-Scholes formula, with a number of correction terms related to the Greeks of the option.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Expansion Formula of Option Price Under Multifactor Heston Model

Nagashima

Chung

Tánaka

2014

Asia-Pac Financ Markets

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The stochastic volatility model of Heston (Rev Financ Stud 6(2): [327][328][329][330][331][332][333][334][335][336][337][338][339][340][341][342][343] 1993) has found difficulty in describing some of the important features of implied volatility dynamics, leading to a quest for multifactor extensions as well as the incorporation of time-dependent model parameters. In this paper, an asymptotic expansion approach to the multifactor Heston model with time-dependent parameters is developed. The results of Benhamou et al. (SIAM J Financ Math 1(1):289-325, 2010) are extended and it is shown that the extension to the multifactor model involves an extra expansion term that captures the interaction between variance factors. The expansion formula under constant parameters can be explicitly computed and the incorporation of time-dependent parameters is straightforward under the framework. As illustration, a two-factor model is calibrated to data of index options and variance swaps and it is found that it is possible to distinguish a short-term and long-term variance factor from the implied volatility surface and variance swap rates. Moreover, the two-factor model is able to reproduce the shapes of the implied volatility surface during various market scenarios.

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“…For instance for ρ = 20% (resp., −20% and −50%), the mean absolute error for the price is 0.28 bps (resp., 0.33 bps and 0.84 bps). We refer to Table 5 for the prices when ρ = −50% (for other values of ρ, results can be found in the first version of this work [9]). We notice that the errors are smaller for a correlation close to zero and become larger when the absolute value of the correlation increases.…”

Section: T /Kmentioning

confidence: 99%

Time Dependent Heston Model

Benhamou¹,

Gobet²,

Miri³

2010

SIAM J. Finan. Math.

139

110

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Abstract. The use of the Heston model is still challenging because it has a closed formula only when the parameters are constant [S. Heston, Rev. Financ. Stud., 6 (1993) 1. Introduction. Stochastic volatility modeling emerged in the late nineties as a way to manage the smile. In this work, we focus on the Heston model, which is a lognormal model where the square of volatility follows a Cox-Ingersoll-Ross (CIR) 1 process. The call (and put) price has a closed formula in this model thanks to a Fourier inversion of the characteristic function (see Heston [22], Lewis [27], and Lipton [29]). When the parameters are piecewise constant, one can still derive a recursive closed formula using a PDE method (see Mikhailov and Nogel [31]) or a Markov argument in combination with affine models (see Elices [16]), but formula evaluation becomes increasingly time consuming. However, for general time dependent parameters there is no analytical formula and one usually has to perform Monte Carlo simulations. This explains the interest of recent works for designing more efficient Monte Carlo simulations: see Broadie and Kaya [13] for an exact simulation and bias-free scheme based on Fourier integral inversion; see Andersen [4] based on a Gaussian moment matching method and a user friendly algorithm; see Smith [40] relying on an almost exact scheme;

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Time Dependent Heston Model

Cited by 42 publications

References 24 publications

A decomposition formula for option prices in the Heston model and applications to option pricing approximation

A decomposition formula for option prices in the Heston model and applications to option pricing approximation

Asymptotic Expansion Formula of Option Price Under Multifactor Heston Model

Time Dependent Heston Model

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