Large deviations for the extended Heston model: the large-time case

Aleksandar Mijatovic; Antoine Jacquier

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Large deviations for the extended Heston model: the large-time case

Pricing of Securities · Submitted 22 Mar 2012

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The authors' main claim We study here the large-time behaviour of all continuous affine stochastic volatility models (in the sense of Keller-Ressel) and deduce a closed-form formula for the large-maturity implied volatility smile.

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We study here the large-time behaviour of all continuous affine stochastic volatility models (in the sense of Keller-Ressel) and deduce a closed-form formula for the large-maturity implied volatility smile. Based on refinements of the Gartner-Ellis theorem on the real line, our proof reveals pathological behaviours of the asymptotic smile. In particular, we show that the condition assumed in Gatheral and Jacquier under which the Heston implied volatility converges to the SVI parameterisation is necessary and sufficient.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We study here the large-time behaviour of all continuous affine stochastic volatility models (in the sense of Keller-Ressel) and deduce a closed-form formula for the large-maturity implied volatility smile.	contribution 0.98	0.95	0.60	asserted 1.00
s1	Based on refinements of the Gartner-Ellis theorem on the real line, our proof reveals pathological behaviours of the asymptotic smile.	method 0.49	0.91	0.12	asserted 1.00
s2	In particular, we show that the condition assumed in Gatheral and Jacquier under which the Heston implied volatility converges to the SVI parameterisation is necessary and sufficient.	result 0.99	0.90	0.30	asserted 1.00

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Original abstract

We study here the large-time behaviour of all continuous affine stochastic volatility models (in the sense of Keller-Ressel) and deduce a closed-form formula for the large-maturity implied volatility smile. Based on refinements of the Gartner-Ellis theorem on the real line, our proof reveals pathological behaviours of the asymptotic smile. In particular, we show that the condition assumed in Gatheral and Jacquier under which the Heston implied volatility converges to the SVI parameterisation is necessary and sufficient.

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2 arXiv papers cite this in 3 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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Cites it as background 1 paper

Comparatively, large-maturity asymptotics have only been studied in [ref] using large deviations and saddlepoint methods.

Asymptotics of forward implied volatility · Introduction · 0.84

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Builds on 1 paper

We show here that a large deviations principle still holds (as $t$ tends to infinity) for the process $(t^-1X_t)_t>0$ when the two conditions (i) and (ii) above fail, i.e. without the technical assumptions of [ref] .

Large deviations and stochastic volatility with jumps: asymptotic implied volatility for affine models · Introduction · 0.84

Uses the method or tool of 1 paper

The proof is analogous to the proof of [ref] , and we therefore omit it for brevity.

Large deviations and stochastic volatility with jumps: asymptotic implied volatility for affine models · Implied volatility asymptotics · 0.80

Compares with 1 paper

A similar issue arose in [ref] where the authors studied large deviations properties for the maximum likelihood estimator of an Ornstein-Uhlenbeck process.

Sharp large deviations for the non-stationary Ornstein-Uhlenbeck process · Proof of Theorem [ref] · 0.80

Cites as background 1 paper

The weak convergence of the process $(t^-1X_t)_t>0$ has been studied in [ref] for the Heston model and in [ref] for ASVM, via the Gärtner-Ellis theorem from large deviations theory.

Large deviations and stochastic volatility with jumps: asymptotic implied volatility for affine models · Introduction · 0.82

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