The Impact of Jump Distributions on the Implied Volatility of Variance

Nicolato, Elisa; Pisani, Camilla; Sloth, David

doi:10.1137/16m1059072

Cited by 9 publications

(22 citation statements)

References 31 publications

(38 reference statements)

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“…The proof of the right wing in Corollary 4.10, that is (32), is a direct consequence of (31) and Nicolato et al. (2017, Proposition 2.2‐(a)). The proof of the left wing is more technical and is postponed in Appendix.Remark Since the VIX is bounded from below by

\sqrt{B false(normalΔ false)}

, the price of a put option with strike below

\sqrt{B false(normalΔ false)}

is worth zero and then we have by convention that

{normalΣ}_{VIX} false(T, k false) = 0

for

k \leq \frac{1}{2} \log B false(normalΔ false)

.…”

Section: Asymptotic Behavior and Implied Volatilitymentioning

confidence: 59%

“…Then, it is not surprising that “the specific choice of jump distribution has a minor effect on the qualitative behavior of the skew and the term structure of the implied volatility surface” as noted in Nicolato et al. (2017) (see also Nicolato & Venardos, 2003), since the plasticity of the model is limited to the form of the auxiliary function

ϕ (t, ξ)

which is independent of the level of initial variance

V_{0}

in the cumulant generating function.…”

Section: Affine Characteristicsmentioning

confidence: 99%

“…For the stochastic volatility jump model in Nicolato et al. (2017), the examples share the same Riccati operator of the Heston model. As a consequence, the Laplace transform of the variance process has a certain form for the affine function.…”

Section: Affine Characteristicsmentioning

confidence: 99%

“…We now consider the variance and VIX options for which a large growing literature has been developed (see for instance (Goard & Mazur, 2013; Nicolato et al., 2017; Sepp, 2008)). As pointed by Kallsen et al.…”

Section: Asymptotic Behavior and Implied Volatilitymentioning

confidence: 99%

“…Nicolato et al. (2017) study the case where a jump term is added to the instantaneous variance process which depends on an increasing and driftless Lévy process, and they analyze the impact of jump diffusions on the realized variance smile and the implied volatility of VIX options. More generally, Duffie et al.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

The Alpha‐Heston stochastic volatility model

Jiao

Scotti

et al. 2021

Mathematical Finance

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We introduce an affine extension of the Heston model, called the α‐Heston model, where the instantaneous variance process contains a jump part driven by α‐stable processes with α∈(1,2]. In this framework, we examine the implied volatility and its asymptotic behavior for both asset and VIX options. Furthermore, we study the jump clustering phenomenon observed on the market. We provide a jump cluster decomposition for the variance process where each cluster is induced by a “mother jump” representing a triggering shock followed by “secondary jumps” characterizing the contagion impact.

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\sqrt{B false(normalΔ false)}

, the price of a put option with strike below

\sqrt{B false(normalΔ false)}

is worth zero and then we have by convention that

{normalΣ}_{VIX} false(T, k false) = 0

for

k \leq \frac{1}{2} \log B false(normalΔ false)

.…”

Section: Asymptotic Behavior and Implied Volatilitymentioning

confidence: 59%

ϕ (t, ξ)

which is independent of the level of initial variance

V_{0}

in the cumulant generating function.…”

Section: Affine Characteristicsmentioning

confidence: 99%

Section: Affine Characteristicsmentioning

confidence: 99%

Section: Asymptotic Behavior and Implied Volatilitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The Alpha‐Heston stochastic volatility model

Jiao

Scotti

et al. 2021

Mathematical Finance

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A Gamma Ornstein–Uhlenbeck model driven by a Hawkes process

Bernis¹,

Brignone

Scotti

et al. 2021

Math Finan Econ

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We propose an extension of the $$\Gamma $$ Γ -OU Barndorff-Nielsen and Shephard model taking into account jump clustering phenomena. We assume that the intensity process of the Hawkes driver coincides, up to a constant, with the variance process. By applying the theory of continuous-state branching processes with immigration, we prove existence and uniqueness of strong solutions of the SDE governing the asset price dynamics. We propose a measure change of self-exciting Esscher type in order to describe the relation between the risk-neutral and the historical dynamics, showing that the $$\Gamma $$ Γ -OU Hawkes framework is stable under this probability change. By exploiting the affine features of the model we provide an explicit form for the Laplace transform of the asset log-return, for its quadratic variation and for the ergodic distribution of the variance process. We show that the proposed model exhibits a larger flexibility in comparison with the $$\Gamma $$ Γ -OU model, in spite of the same number of parameters required. We calibrate the model on market vanilla option prices via characteristic function inversion techniques, we study the price sensitivities and propose an exact simulation scheme. The main financial achievement is that implied volatility of options written on VIX is upward shaped due to the self-exciting property of Hawkes processes, in contrast with the usual downward slope exhibited by the $$\Gamma $$ Γ -OU Barndorff-Nielsen and Shephard model.

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Orthogonal expansions for VIX options under affine jump diffusions

Barletta

Nicolato

2017

Quantitative Finance

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The Impact of Jump Distributions on the Implied Volatility of Variance

Cited by 9 publications

References 31 publications

The Alpha‐Heston stochastic volatility model

The Alpha‐Heston stochastic volatility model

A Gamma Ornstein–Uhlenbeck model driven by a Hawkes process

Orthogonal expansions for VIX options under affine jump diffusions

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