Short-Time Implied Volatility in Exponential Lévy Models

Ekström, Erik; Lu, Bing

doi:10.1142/s0219024915500259

Cited by 3 publications

(3 citation statements)

References 13 publications

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“…We derive in this section the small-time asymptotics of the premium of a call option and its associated implied volatility by applying the small-time asymptotics for the probability density obtained in Section 3 when H ≤ 1 2 . It is documented, for example, in Ekström and Lu (2015), that if the underlying asset is governed by an exponential Lévy model, the induced implied volatilities of non-ATM options may explode if jumps exist and the underlying process jumps towards the strike. As we shall see in the following, when H < 1 2 , the small time approximation of implied volatility also explodes; creating a jumplike behavior in the underlying process.…”

Section: Small Time Approximation Of Option Price and Implied Volatilitymentioning

confidence: 99%

Probability Density of Lognormal Fractional SABR Model

Akahori

Song

Wang

2022

Risks

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Instantaneous volatility of logarithmic return in the lognormal fractional SABR model is driven by the exponentiation of a correlated fractional Brownian motion. Due to the mixed nature of driving Brownian and fractional Brownian motions, probability density for such a model is less studied in the literature. We show in this paper a bridge representation for the joint density of the lognormal fractional SABR model in a Fourier space. Evaluating the bridge representation along a properly chosen deterministic path yields a small time asymptotic expansion to the leading order for the probability density of the fractional SABR model. A direct generalization of the representation of joint density often leads to a heuristic derivation of the large deviations principle for joint density in a small time. Approximation of implied volatility is readily obtained by applying the Laplace asymptotic formula to the call or put prices and comparing coefficients.

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Section: Small Time Approximation Of Option Price and Implied Volatilitymentioning

confidence: 99%

Probability Density of Lognormal Fractional SABR Model

Akahori

Song

Wang

2022

Risks

View full text Add to dashboard Cite

show abstract

“…We derive in this section the small time asymptotics of the premium of a call option and its associated implied volatility by applying the small time asymptotics for the probability density obtained in Section 3 when H ≤ 1 2 . It is documented, for exmaple, in Ekström and Lu [6], that if the underlying asset is governed by an exponential Lévy model, the induced implied volatilities of non ATM options may explode if jumps exist and the underlying process jumps towards the strike. As we shall see in the following, when H < 1 2 , the small time approximation of implied volatility also explodes; creating a jump like behavior in the underlying process.…”

Section: Small Time Approximation Of Option Price and Implied Volatilitymentioning

confidence: 99%

Probability density of lognormal fractional SABR model

Akahori¹,

Song²,

Wang³

2017

Preprint

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Instantaneous volatility of logarithmic return in the lognormal fractional SABR model is driven by the exponentiation of a correlated fractional Brownian motion. Due to the mixed nature of driving Brownian and fractional Brownian motions, probability density for such a model is less studied in the literature. We show in this paper a bridge representation for the joint density of the lognormal fractional SABR model in a Fourier space. Evaluating the bridge representation along a properly chosen deterministic path yields a small time asymptotic expansion to the leading order for the probability density of the fractional SABR model. A direct generalization of the representation to joint density at multiple times leads to a heuristic derivation of the large deviations principle for the joint density in small time. Approximation of implied volatility is readily obtained by applying the Laplace asymptotic formula to the call or put prices and comparing coefficients.

show abstract

“…However, these are not necessarily precise mathematical results, but should rather be viewed as rules of thumb based on numerical evidence. In fact, it is shown in [9] that spectrally negative jump models exhibit the opposite monotonicity (i.e. increasing implied volatility) for short-time implied volatility.…”

Section: Introductionmentioning

confidence: 99%