Fourier transform algorithms for pricing and hedging discretely sampled exotic variance products and volatility derivatives under additive processes

Zheng, Wendong; Kwok, Yue Kuen

doi:10.21314/jcf.2014.278

Cited by 9 publications

(2 citation statements)

References 27 publications

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“…The model also generalizes the bilateral gamma model that arises when the two Y parameters are zero. For inverse transform methods applied to discretely sampled variance products we reference Zheng and Kwok (2014).…”

Section: Restrictiveness Of Time Changed Brownian Motions Letmentioning

confidence: 99%

Quadratic variation, models, applications and lessons

Madan

Wang

2022

FMF

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<p style="text-indent:20px;">Time changes of Brownian motion impose restrictive jump structures in the motion of asset prices. Quadratic variations also depart from time changes. Quadratic variation options are observed to have a nonlinear exposure to risk neutral skewness. Joint Laplace Fourier transforms for quadratic variation and the stock are developed. They are used to study the multiple of the cap strike over the variance swap quote attaining a given percentage price reduction for the capped variance swap. Market prices for out-of-the-money options on variance are observed to be above those delivered by the calibrated models. Bootstrapped data and simulated paths spaces are used to study the multiple of the dynamic hedge return desired by a quadratic variation contract. It is observed that the optimized hedge multiple in the bootstrapped data is near unity. Furthermore, more generally, it is exposures to negative cubic variations in path spaces that raise variance swap prices, lower hedge multiples, increase residual risk charges and gaps to the log contract hedge. A case can be made for both, the physical process being closer to a continuous time change of Brownian motion, while simultaneously risk neutrally this may not be the case. It is recognized that in the context of discrete time there are no issues related to equivalence of probabilities.</p>

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Section: Restrictiveness Of Time Changed Brownian Motions Letmentioning

confidence: 99%

Quadratic variation, models, applications and lessons

Madan

Wang

2022

FMF

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“…These methods are based on Fourier Space Timestepping (FST) (Jackson et al, 2008), the CONV technique (Lord et al, 2008) or the COS algorithm (Fang and Oosterlee, 2008). Fourier methods have been applied to pricing of exotic variance products and volatility derivatives (Zheng and Kwok, 2014), guaranteed minimum withdrawal benefits (Ignatieva et al, 2018;Alonso-Garcia et al, 2018;Huang et al, 2017) and equity-indexed annuities (Deng et al, 2017) to name just a few.…”

Section: Introductionmentioning

confidence: 99%

ε-monotone Fourier methods for optimal stochastic control in finance

Labahn¹

2019

JCF

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Stochastic control problems in finance often involve complex controls at discrete times. As a result numerically solving such problems, for example using methods based on partial differential or integro-differential equations, inevitably give rise to low order accuracy, usually at most second order. In many cases one can make use of Fourier methods to efficiently advance solutions between control monitoring dates and then apply numerical optimization methods across decision times. However Fourier methods are not monotone and as a result give rise to possible violations of arbitrage inequalities. This is problematic in the context of control problems, where the control is determined by comparing value functions. In this paper we give a preprocessing step for Fourier methods which involves projecting the Green's function onto the set of linear basis functions. The resulting algorithm is guaranteed to be monotone (to within a tolerance), ℓ ∞ -stable and satisfies an ǫ-discrete comparison principle. In addition the algorithm has the same complexity per step as a standard Fourier method while at the same time having second order accuracy for smooth problems. Key Messages:• Current Fourier methods (FST/CONV/COS) are not necessarily monotone• We devise a pre-processing step for FST/CONV methods which are monotone to a user specified tolerance• The resulting methods can be used safely for optimal control problems in finance

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A superconvergent partial differential equation approach to price variance swaps under regime switching models

Dilloo

Tangman

2017

Journal of Computational and Applied Mathematics

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Fourier transform algorithms for pricing and hedging discretely sampled exotic variance products and volatility derivatives under additive processes

Cited by 9 publications

References 27 publications

Quadratic variation, models, applications and lessons

Quadratic variation, models, applications and lessons

ε-monotone Fourier methods for optimal stochastic control in finance

A superconvergent partial differential equation approach to price variance swaps under regime switching models

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