Large Deviations for the Extended Heston Model: The Large-Time Case

Large-time asymptotics are established for the SABR model with β = 1, ρ ≤ 0 and β < 1, ρ = 0. We also compute large-time asymptotics for the constant elasticity of variance (CEV) model in the large-time, fixed-strike regime and a new large-time, large-strike regime, and for the uncorrelated CEV-Heston model. Finally, we translate these results into a large-time estimates for implied volatility using the recent work of Gao and Lee (2011) and Tehranchi (2009).

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Large deviations for metastable states of Markov processes with absorbing states with applications to population models in stable or randomly switching environment

Journal of Statistical Mechanics Theory and Experiment ◽

10.1088/1742-5468/ac4519 ◽

2022 ◽

Vol 2022 (1) ◽

pp. 013206

Author(s):

Cécile Monthus

Keyword(s):

Large Deviations ◽

Markov Processes ◽

Large Time ◽

Diffusion Processes ◽

Large Deviation ◽

Time Window ◽

Extinction Rate ◽

Empirical Density ◽

Full Optimization ◽

Absorbing States

Abstract The large deviations at level 2.5 are applied to Markov processes with absorbing states in order to obtain the explicit extinction rate of metastable quasi-stationary states in terms of their empirical time-averaged density and of their time-averaged empirical flows over a large time-window T. The standard spectral problem for the slowest relaxation mode can be recovered from the full optimization of the extinction rate over all these empirical observables and the equivalence can be understood via the Doob generator of the process conditioned to survive up to time T. The large deviation properties of any time-additive observable of the Markov trajectory before extinction can be derived from the level 2.5 via the decomposition of the time-additive observable in terms of the empirical density and the empirical flows. This general formalism is described for continuous-time Markov chains, with applications to population birth–death model in a stable or in a switching environment, and for diffusion processes in dimension d.

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Large-Time Asymptotics of the Heston Model: From IG to NIG

SSRN Electronic Journal ◽

10.2139/ssrn.1647290 ◽

2010 ◽

Author(s):

Shu Tong Tse

Keyword(s):

Large Time ◽

Heston Model ◽

Large Time Asymptotics ◽

Time Asymptotics

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Small-time moderate deviations for the randomised Heston model

Journal of Applied Probability ◽

10.1017/jpr.2019.73 ◽

2020 ◽

Vol 57 (1) ◽

pp. 19-28

Author(s):

Antoine Jacquier ◽

Fangwei Shi

Keyword(s):

Large Deviations ◽

Small Time ◽

Moderate Deviations ◽

Heston Model ◽

Sharp Large Deviations

AbstractWe extend previous large deviations results for the randomised Heston model to the case of moderate deviations. The proofs involve the Gärtner–Ellis theorem and sharp large deviations tools.

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