Asymptotic Formulas with Error Estimates for Call Pricing Functions and the Implied Volatility at Extreme Strikes

Gulisashvili, Archil

doi:10.1137/090762713

Cited by 61 publications

(76 citation statements)

References 32 publications

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“…For instance, the lim sup can be removed by a simple limit, under the additional assumptions that S T has a regularly varying density, see [11]. More recently, Gulisashvili [12] and his co-authors have proved refined expansions of the form where the reminder is locally uniform in the log-moneneyss m = x 0 − k.…”

Section: An Overview Of Approximation Resultsmentioning

confidence: 99%

Asymptotic and non asymptotic approximations for option valuation

Bompis

Gobet

2012

Recent Developments in Computational Finance

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We give a broad overview of approximation methods to derive analytical formulas for accurate and quick evaluation of option prices. We compare different approaches, from the theoretical point of view regarding the tools they require, and also from the numerical point of view regarding their performances. In the case of local volatility models with general time-dependency, we derive new formulas using the local volatility function at the mid-point between strike and spot: in general, our approximations outperform previous ones by Hagan and HenryLabordère. We also provide approximations of the option delta.

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Section: An Overview Of Approximation Resultsmentioning

confidence: 99%

Asymptotic and non asymptotic approximations for option valuation

Bompis

Gobet

2012

Recent Developments in Computational Finance

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“…PDE methods for continuous-time diffusions [9,32,51], large deviations [15,18], saddlepoint methods [20], Malliavin calculus [7,41] and differential geometry [26,33] are among the main methods used to tackle the small-maturity case. Extreme strike asymptotics arose with the seminal paper by Roger Lee [43] and have been further extended by Benaim and Friz [4,5] and in [30,31,23,15]. Comparatively, large-maturity asymptotics have only been studied in [55,19,36,35,21] using large deviations and saddlepoint methods.…”

Section: Xt T≥0mentioning

confidence: 99%

Asymptotics of Forward Implied Volatility

Roome

2015

SIAM J. Finan. Math.

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Abstract. We prove here a general closed-form expansion formula for forward-start options and the forward implied volatility smile in a large class of models, including the Heston stochastic volatility and time-changed exponential Lévy models. This expansion applies to both small and large maturities and is based solely on the knowledge of the forward characteristic function of the underlying process. The method is based on sharp large deviations techniques, and allows us to recover (in particular) many results for the spot implied volatility smile.In passing we show (i) that the small-maturity exploding behaviour of forward smiles depends on whether the quadratic variation of the underlying is bounded or not, and (ii) that the forward-start date also has to be rescaled in order to obtain non-trivial small-maturity asymptotics.

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“…The natural tool they used is the regular variation theory. Other works concerning the option pricing with extreme strikes include, e.g., Gulisashvili [26], where error estimates for the implied volatility were obtained. In this paper, we take a different perspective.…”

Section: Introductionmentioning

confidence: 99%