Stochastic Volatility for Lévy Processes

By Peter P. Carr, Helyette Geman, Dilip B Madan 5, Marc Yor. 2003

926 citations recorded by the source corpus.

Three processes reflecting persistence of volatility are initially formulated by evaluating three Lévy processes at a time change given by the integral of a mean‐reverting square root process. The model for the mean‐reverting time change is then generalized to include non‐Gaussian models that are solutions to Ornstein‐Uhlenbeck equations driven by one‐sided discontinuous Lévy processes permitting correlation with the stock. Positive stock price processes are obtained by exponentiating and mean correcting these processes, or alternatively by stochastically exponentiating these processes. The characteristic functions for the log price can be used to yield option prices via the fast Fourier transform. In general mean‐corrected exponentiation performs better than employing the stochastic exponential. It is observed that the mean‐corrected exponential model is not a martingale in the filtration in which it is originally defined. This leads us to formulate and investigate the important property of martingale marginals where we seek martingales in altered filtrations consistent with the one‐dimensional marginal distributions of the level of the process at each future date.

Subfields: Economics and Econometrics, Finance

View Paper: https://www.semanticscholar.org/paper/1f10b79c1ecb3174b919442de41da5ee71bc72e8

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Citations in the corpus

Option Pricing Under a Double Exponential Jump Diffusion Model (2004, methodology): “For the background of these alternative models, see, for example, Hull (2000) and Carr et al. (2003).”
Stochastic volatility with leverage: Fast and efficient likelihood inference (2007, result): “This is relevant empirically and also corresponds to the literature on timechange Lévy processes and Lévy-based SV models (Carr et al., 2003; Carr and Wu, 2004; Cont and Tankov, 2004).”
Meromorphic Lévy processes and their fluctuation identities. (2010, methodology): “It turns out that, using exactly the same method, we can also obtain similar results for Lévy processes with stochastic volatility, which are very popular models in Mathematical Finance; see, for example, [11].”
Tempering stable processes (2007, methodology): “They were also introduced in mathematical finance to model stochastic volatility (the CGMY model in [8, 9]), (the Ornstein-Uhlenbeck–based model in [3, 4]); option pricing based on such processes was considered in [6], just to mention a few.”
Electricity price modeling with stochastic time change (2017, neutral): “[10], Carr and Wu [11] or Kallsen and Muhle-Karbey [19]).”
EQUILIBRIUM ASSET AND OPTION PRICING UNDER JUMP DIFFUSION (2012, methodology): “The later development of option pricing is built on the analytical structure of the inverse Fourier transformation, see, for example, Heston (1993), Duffie, Pan, and Singleton’s (2000) affine jump diffusion, and Carr et al. (2003) and Carr and Wu’s (2004) time-changed Lévy process.”
Testing for pure-jump processes for high-frequency data (2015, methodology): “…the normal inverse Gaussian [Rydberg (1997); BarndorffNielsen (1997, 1998)], the variance gamma [Madan, Carr and Chang (1998)], the CGMY model of Carr et al. (2003b), the time-changed Lévy mdoels of Carr et al. (2003a), the non-Gaussian Ornstein–Uhlenbeck-based models of Barndorff-Nielsen and…”
Exponentially affine martingales, affine measure changes and exponential moments of affine processes (2010, background): “10, we consider the stochastic volatility model of [2].”
A class of Lévy process models with almost exact calibration to both barrier and vanilla FX options (2010, positive): “Carr et al. (2002, 2003) and Carr and Wu (2007) show how, in their ability to fit and explain implied volatility surfaces, finite activity Lévy processes are outperformed by infinite activity Lévy processes, which, in turn, are outperformed by time-changed Lévy processes.”
Valuing path dependent options in the variance-gamma model by Monte Carlo with a gamma bridge (2002, methodology): “Using the bridge together with stratified sampling we obtain considerable speed improvements over a plain Monte Carlo method when pricing path-dependent options.”
Fluctuations of Lévy Processes with Applications (2014, background): “…Good (1953) and Jørgensen (1982)), truncated stable (see Tweedie (1984), Hougaard (1986), Koponen (1995), Boyarchenko and Levendorskii (2002a) and Carr et al. (2003)), generalised hyperbolic (see Halgreen (1979)), Meixner (see Schoutens and Teugels (1998)), Pareto (see Steutel (1970) and Thorin…”
Static Hedging of Asian Options under Lévy Models (2005, neutral): “For more examples and applications of Levy processes in finance see [5, 13, 14, 21, 29, 36].”
Activity Signature Functions for High-Frequency Data Analysis (2010, methodology): “…1997; Barndorff-Nielsen, 1997, 1998), the variance gamma (Madan et al., 1998), the CGMY model of Carr et al. (2002), the timechanged Lévy models of Carr et al. (2003), the COGARCH model of Klüppelberg et al. (2004) for the financial prices as well as the non-Gaussian Ornstein–Uhlenbeck-based…”
Riding on the smiles (2011, neutral): “The market practice is to achieve a calibration per day and this is the path followed by Bakshi et al. (1997) and Carr et al. (2003).”