On the density of log-spot in the Heston volatility model

Rollin, Sebastian del Baño; Ferreiro-Castilla, Albert; Utzet, Frederic

doi:10.1016/j.spa.2010.06.003

Cited by 29 publications

(20 citation statements)

References 18 publications

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“…Proposition is consistent with proposition 2.5, page 34 of Andersen and Piterbarg (), also see remark 4.2, page 2052 of Del Baño Rollin, Ferreiro‐Castilla, and Utzet ()…”

supporting

confidence: 84%

On the Martingale Property in Stochastic Volatility Models Based on Time‐homogeneous Diffusions

Bernard

Cui

McLeish

2014

Mathematical Finance

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Lions and Musiela give sufficient conditions to verify when a stochastic exponential of a continuous local martingale is a martingale or a uniformly integrable martingale. Blei and Engelbert and Mijatović and Urusov give necessary and sufficient conditions in the case of perfect correlation (ρ = 1). For financial applications, such as checking the martingale property of the stock price process in correlated stochastic volatility models, we extend their work to the arbitrary correlation case (−1 ≤ ρ ≤ 1). We give a complete classification of the convergence properties of both perpetual and capped integral functionals of time-homogeneous diffusions and generalize results in Mijatović and Urusov with direct proofs avoiding the use of separating times (concept introduced by Cherny and Urusov and extensively used in the proofs of Mijatović and Urusov).KEY WORDS: Martingale property, local martingale, stochastic volatility, Engelbert Schmidt zero-one law.

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“…Proposition is consistent with proposition 2.5, page 34 of Andersen and Piterbarg (), also see remark 4.2, page 2052 of Del Baño Rollin, Ferreiro‐Castilla, and Utzet ()…”

supporting

confidence: 84%

On the Martingale Property in Stochastic Volatility Models Based on Time‐homogeneous Diffusions

Bernard

Cui

McLeish

2014

Mathematical Finance

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“…do 4 Solve the normal equations (33) for ∆θ k . 5 Compute θ k+1 = θ k + ∆θ k and r(θ k+1 ) . 6 Compute δ L = ∆θ k ⊺ (µ∆θ k + J k r(θ k )) and δ F = r(θ k ) − r(θ k+1 ) .…”

Section: Calibration Using the Levenberg-marquardt Methodsmentioning

confidence: 99%

“…where ε 1 , ε 2 and ε 3 are tolerance levels. The first condition (36a) indicates that the iteration is stopped by a desired value of the objective function (4)- (5). The second condition (36b) indicates that the iteration is stopped by a small gradient.…”

Section: Calibration Using the Levenberg-marquardt Methodsmentioning

confidence: 99%

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Full and fast calibration of the Heston stochastic volatility model

Cui

Rollin

Germano

2017

European Journal of Operational Research

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This paper presents an algorithm for a complete and efficient calibration of the Heston stochastic volatility model. We express the calibration as a nonlinear least squares problem. We exploit a suitable representation of the Heston characteristic function and modify it to avoid discontinuities caused by branch switchings of complex functions. Using this representation, we obtain the analytical gradient of the price of a vanilla option with respect to the model parameters, which is the key element of all variants of the objective function. The interdependency between the components of the gradient enables an efficient implementation which is around ten times faster than a numerical gradient. We choose the Levenberg-Marquardt method to calibrate the model and do not observe multiple local minima reported in previous research. Two-dimensional sections show that the objective function is shaped as a narrow valley with a flat bottom. Our method is the fastest calibration of the Heston model developed so far and meets the speed requirement of practical trading.

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“…In our numerical work we have used some of the formulas in Dufresne, Garrido, and Morales (2009), that allow for the very fast computation of European put and call prices. For those options there is no need to resort to the transition density function, as done in del Baño Rollin, Ferreiro-Castilla, and Utzet (2010) and Glasserman and Kim (2011), since Fourier inversion works and is much more efficient numerically.…”

Section: Resultsmentioning

confidence: 99%

Change of measure for the square-root process

Dufresne¹,

Vázquez-Abad²,

Chin³

2014

Proceedings of the Winter Simulation Conference 2014

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The square-root process is used to model interest rates and volatility in financial mathematics. The pricing of derivatives involving that process often requires simulating it, since there are often no explicit formulas for prices. We study how a change of measure (CM) may improve those simulations. We compare with Andersen's quadratic-exponential scheme (QE), which so far appears to be the most efficient technique to simulate the stochastic differential equation satisfied by the square-root process. An integer-dimension squared Bessel process, easy to simulate, is used to generate the law of the square-root process using a change of measure. The new method performs very well, and the two algorithms execute at similar speeds; however, CM is slower than QE if random number generation is taken into account, because CM requires more random numbers. The Radon-Nikodym derivative sometimes has a rather intriguing behavior, which is itself of interest. We propose an explanation.

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On the density of log-spot in the Heston volatility model

Cited by 29 publications

References 18 publications

On the Martingale Property in Stochastic Volatility Models Based on Time‐homogeneous Diffusions

On the Martingale Property in Stochastic Volatility Models Based on Time‐homogeneous Diffusions

Full and fast calibration of the Heston stochastic volatility model

Change of measure for the square-root process

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