An Asymptotic Expansion with Push-Down of Malliavin Weights

Takahashi, Akihiko; Yamada, Toshihiro

doi:10.2139/ssrn.1514017

Cited by 27 publications

(51 citation statements)

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“…Remark 4.3 (Approximation of the square root). It is shown in [25,Section 4.1] that the impact of the approximation of the square root in the context of small noise expansions is asymptotically negligible and hence we will mostly omit mentioning it.…”

Section: Vol-of-vol Expansion: the Regular Casementioning

confidence: 99%

“…The so-called vol-of-vol expansion is particularly popular, as the leading term agrees with the Black & Scholes price and higher order terms are usually given in closed form. An early version of this expansion for one dimensional stochastic volatility models is given in [19] and a generalization to finite-dimensional stochastic volatility models in [25]. A further formal generalization came with the popular Bergomi-Guyon expansion [7] (henceforth BG expansion) which is formulated in terms of potentially infinite dimensional forward variance models and was incremental in the introduction of the rough Bergomi model (cf.…”

Section: Introductionmentioning

confidence: 99%

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Vol-of-Vol Expansion for (Rough) Stochastic Volatility Models

Akdogan

2019

SSRN Journal

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We introduce an asymptotic small noise expansion, a so called vol-of-vol expansion, for potentially infinite dimensional and rough stochastic volatility models. Thereby we extend the scope of existing results for finite dimensional models and validate claims for infinite dimensional models. Furthermore we provide new, explicit (in the sense of non-recursive) representations of the so-called push-down Malliavin weights that utilizes a precise understanding of the terms of this expansion.Date: December 6, 2019.

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Section: Vol-of-vol Expansion: the Regular Casementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Vol-of-Vol Expansion for (Rough) Stochastic Volatility Models

Akdogan

2019

SSRN Journal

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“…There is a substantial body of research on the approximation of Black's IV. Gatheral, Hsu, Laurence, Ouyang, and Wang (2012), Takahashi and Yamada (2012), and Lorig, Pagliarani, and Pascucci (2017) derived an approximation method of IVs under LV, SV, and LSV models, respectively. For example, Forde and Jacquier used the method to study the shortterm behavior of IVs (Forde and Jacquier, 2009).…”

Section: Introductionmentioning

confidence: 99%

“…Hagan, Kumar, Lesniewski, and Woodward (2002) used the asymptotic expansion in stochastic alpha beta rho (SABR) LSV model. Gatheral, Hsu, Laurence, Ouyang, and Wang (2012), Takahashi and Yamada (2012), and Lorig, Pagliarani, and Pascucci (2017) derived an approximation method of IVs under LV, SV, and LSV models, respectively. Homescu (2011) provided an extensive survey of the works on the IV surface.…”

Section: Introductionmentioning

confidence: 99%

An approximation formula for normal implied volatility under general local stochastic volatility models

Karami

Shiraya

2018

Journal of Futures Markets

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We approximate normal implied volatilities by means of an asymptotic expansion method. The contribution of this paper is twofold: to our knowledge, this paper is the first to provide a unified approximation method for the normal implied volatility under general local stochastic volatility models. Second, we applied our framework to polynomial local stochastic volatility models with various degrees and could replicate the swaptions market data accurately. In addition we examined the accuracy of the results by comparison with the Monte‐Carlo simulations.

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“…For (shifted) log-normal local volatility cases in (jump-diffusion) stochastic volatility models, the same technique is applied. (For instance, see [15,29,31,44,48].) This paper introduces a change of variable technique in order to obtain the flexibility for setting a benchmark distribution.…”

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confidence: 99%

Note on an Extension of an Asymptotic Expansion Scheme

Takahashi

Toda

2013

Int. J. Theor. Appl. Finan.

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This paper presents an extension of a general computational scheme for asymptotic expansions proposed in earlier works by the authors and coworkers. In the earlier works, a new method was developed for the computation of an arbitrary-order expansion with a normal benchmark distribution in a multidimensional diffusion setting. In particular, a new algorithm was proposed for calculating coefficients in an expansion by solving a system of ordinary differential equations. In the present note, by a change of variable technique, and by various ways of setting the perturbation parameters in the expansion, we provide the flexibility of setting the benchmark distribution around which the expansion is made and an automatic way for computation up to any order in the expansion. For instance, we introduce new expansions, called the lognormal expansion and the CEV expansion. We also show some concrete examples with numerical experiments, which imply that a high-order CEV expansion will produce more a precise and stable approximation for option pricing under the SABR model than other approximation methods such as the log-normal expansion and the well-known normal expansion. In the application of the asymptotic expansion based on Watanabe theory, we need to calculate certain conditional expectations which appear in the expansions and play a key role in computation. In the first place [52,53] have developed the formulas necessary for the second-order expansion. Subsequently, [33,34,[38][39][40] have derived new formulas up to the third order. (Also, multi-dimensional formulas were provided in [33,34].) In many applications, these formulas give sufficiently accurate approximation, but in some cases such as in the cases with long maturities or/and with highly volatile underlying variables, the approximation up to the third order may not provide satisfactory accuracies. Thus, the formulas for the higher order computation are desirable.Recently, [41,42,48] have proposed two alternative computational schemes for any order expansions in an automatic manner. In fact, one of their new methods does not rely on direct evaluation of the conditional expectations, but on solving a certain system of ordinal differential equations with grading structure. Independently, [19] has developed a new computational method for the conditional expectations necessary for high order expansions. As a consequence, their approximations generally showed sufficient accuracy with computation of high order expansions, which was confirmed by numerical experiments.Furthermore, in terms of approximation it is important to set the limiting or benchmark distribution around which an expansion is made. Typically a normal distribution is chosen, which enables us to compute higher order correction terms due to the nice and well-known Gaussian properties. For (shifted) log-normal local volatility cases in (jump-diffusion) stochastic volatility models, the same technique is applied. (For instance, see [15,29,31,44,48].) This paper introduces a change of variable technique i...

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An Asymptotic Expansion with Push-Down of Malliavin Weights

Cited by 27 publications

References 0 publications

Vol-of-Vol Expansion for (Rough) Stochastic Volatility Models

Vol-of-Vol Expansion for (Rough) Stochastic Volatility Models

An approximation formula for normal implied volatility under general local stochastic volatility models

Note on an Extension of an Asymptotic Expansion Scheme

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