Exact simulation of gamma-driven Ornstein–Uhlenbeck processes with finite and infinite activity jumps

Abstract We develop a distributional decomposition approach for exactly simulating two types of Gamma-driven Ornstein–Uhlenbeck (OU) processes with time-varying marginal distributions: the Gamma-OU process and the OU-Gamma process. The former has finite-activity jumps, and its marginal distribution is asymptotically Gamma; the latter has infinite-activity jumps that are driven by a Gamma process. We prove that the transition distributions of the two processes at any given time can be exactly decomposed into simple elements: at any given time, the former is equal in distribution to the sum of one deterministic trend and one compound Poisson random variable (r.v.); the latter is equal in distribution to the sum of one deterministic trend, one compound Poisson r.v., and one Gamma r.v. The results immediately lead to very efficient algorithms for their exact simulations without numerical inversion. Extensive numerical experiments are reported to demonstrate the accuracy and efficiency of our algorithms.

[1]  Nonparametric Inference , 2021, Medical Statistics.

[2]  Angelos Dassios,et al.  Efficient simulation of Lévy-driven point processes , 2019, Advances in Applied Probability.

[3]  Michele Leonardo Bianchi,et al.  Forward-looking portfolio selection with multivariate non-Gaussian models , 2018, Quantitative Finance.

[4]  Yan Qu,et al.  Exact Simulation for a Class of Tempered Stable and Related Distributions , 2018, ACM Trans. Model. Comput. Simul..

[5]  Angelos Dassios,et al.  Efficient Simulation of Clustering Jumps with CIR Intensity , 2017, Oper. Res..

[6]  Svetlozar T. Rachev,et al.  Tempered stable Ornstein– Uhlenbeck processes: A practical view , 2017, Commun. Stat. Simul. Comput..

[7]  Almut E. D. Veraart,et al.  Modeling the Variance Risk Premium of Equity Indices: The Role of Dependence and Contagion , 2016, SIAM J. Financial Math..

[8]  Marcos Lopez de Prado,et al.  Algorithmic and High Frequency Trading , 2016 .

[9]  Frank J. Fabozzi,et al.  Investigating the Performance of Non-Gaussian Stochastic Intensity Models in the Calibration of Credit Default Swap Spreads , 2015 .

[10]  Sebastian Jaimungal,et al.  A Closed-Form Execution Strategy to Target Volume Weighted Average Price , 2015, SIAM J. Financial Math..

[11]  Hilmar Mai,et al.  Efficient maximum likelihood estimation for Lévy-driven Ornstein–Uhlenbeck processes , 2014, 1403.2954.

[12]  Nan Chen,et al.  Localization and Exact Simulation of Brownian Motion-Driven Stochastic Differential Equations , 2013, Math. Oper. Res..

[13]  Chayawat Ornthanalai,et al.  Lévy Jump Risk: Evidence from Options and Returns , 2013 .

[14]  A. Dassios,et al.  Exact Simulation of Hawkes Process with Exponentially Decaying Intensity , 2013 .

[15]  M. Yor,et al.  A Simple Stochastic Rate Model for Rate Equity Hybrid Products , 2013 .

[16]  Reiichiro Kawai,et al.  Exact discrete sampling of finite variation tempered stable Ornstein–Uhlenbeck processes , 2011, Monte Carlo Methods Appl..

[17]  A. Dassios,et al.  A dynamic contagion process , 2011, Advances in Applied Probability.

[18]  Jean Jacod,et al.  Testing whether jumps have finite or infinite activity , 2011, 1211.5219.

[19]  Shibin Zhang,et al.  Exact simulation of tempered stable Ornstein–Uhlenbeck processes , 2011 .

[20]  Hiroki Masuda,et al.  On simulation of tempered stable random variates , 2010, J. Comput. Appl. Math..

[21]  Reiichiro Kawai,et al.  Infinite Variation Tempered Stable Ornstein–Uhlenbeck Processes with Discrete Observations , 2010, Commun. Stat. Simul. Comput..

[22]  Xinsheng Zhang,et al.  On the Transition Law of Tempered Stable Ornstein–Uhlenbeck Processes , 2009, Journal of Applied Probability.

[23]  W. Schoutens,et al.  Levy Processes in Credit Risk , 2009 .

[24]  Jean Jacod,et al.  Estimating the degree of activity of jumps in high frequency data , 2009, 0908.3095.

[25]  S. Frühwirth-Schnatter,et al.  Bayesian estimation of stochastic volatility models based on OU processes with marginal Gamma law , 2009 .

[26]  Xinsheng Zhang,et al.  Exact Simulation of IG-OU Processes , 2008 .

[27]  P. Devolder,et al.  Mortality modelling with Lévy processes , 2008 .

[28]  Drew D. Creal.  Analysis of filtering and smoothing algorithms for Lévy-driven stochastic volatility models , 2008, Comput. Stat. Data Anal..

[29]  P. Brockwell,et al.  Estimation for Nonnegative Lévy-Driven Ornstein-Uhlenbeck Processes , 2007, Journal of Applied Probability.

[30]  M. Steel,et al.  Inference With Non-Gaussian Ornstein-Uhlenbeck Processes for Stochastic Volatility , 2006 .

[31]  A. Kyprianou.  Introductory Lectures on Fluctuations of Lévy Processes with Applications , 2006 .

[32]  Mark Broadie,et al.  Exact Simulation of Stochastic Volatility and Other Affine Jump Diffusion Processes , 2006, Oper. Res..

[33]  A. V. D. Vaart,et al.  Nonparametric inference for Lévy-driven Ornstein-Uhlenbeck processes , 2005 .

[34]  Nakahiro Yoshida,et al.  Asymptotic expansion for Barndorff-Nielsen and Shephard's stochastic volatility model , 2005 .

[35]  R. Norberg.  VASIČEK BEYOND THE NORMAL , 2004 .

[36]  G. Roberts,et al.  Bayesian inference for non‐Gaussian Ornstein–Uhlenbeck stochastic volatility processes , 2004 .

[37]  E. Nicolato,et al.  Option Pricing in Stochastic Volatility Models of the Ornstein‐Uhlenbeck type , 2003 .

[38]  Neil Shephard,et al.  Integrated OU Processes and Non‐Gaussian OU‐based Stochastic Volatility Models , 2003 .

[39]  W. Schoutens.  Lévy Processes in Finance: Pricing Financial Derivatives , 2003 .

[40]  N. Shephard,et al.  Realised power variation and stochastic volatility models , 2003 .

[41]  W. Schoutens.  Lévy Processes in Finance , 2003 .

[42]  N. Shephard,et al.  Econometric analysis of realized volatility and its use in estimating stochastic volatility models , 2002 .

[43]  O. Barndorff-Nielsen,et al.  Some stationary processes in discrete and continuous time , 1998, Advances in Applied Probability.

[44]  Ole E. Barndorff-Nielsen,et al.  Processes of normal inverse Gaussian type , 1997, Finance Stochastics.

[45]  O. Barndorff-Nielsen.  Normal Inverse Gaussian Distributions and Stochastic Volatility Modelling , 1997 .

[46]  Mark H. A. Davis.  Piecewise‐Deterministic Markov Processes: A General Class of Non‐Diffusion Stochastic Models , 1984 .

[47]  Ken-iti Sato,et al.  Operator-selfdecomposable distributions as limit distributions of processes of Ornstein-Uhlenbeck type , 1984 .

[48]  S. Wolfe.  On a continuous analogue of the stochastic difference equation Xn=ρXn-1+Bn , 1982 .

[49]  S. Wolfe.  On a continuous analogue of the stochastic difference equation Xn = ρ X n–1 + Bn , 1980, Advances in Applied Probability.

[50]  Oldrich A. Vasicek.  An equilibrium characterization of the term structure , 1977 .

[51]  G. Uhlenbeck,et al.  On the Theory of the Brownian Motion , 1930 .

[52]  R. Schilling.  Financial Modelling with Jump Processes , 2005 .

[53]  H. Masudaa,et al.  Asymptotic expansion for Barndorff-Nielsen and Shephard ’ s stochastic volatility model $ , 2005 .

[54]  Angelos Dassios,et al.  Pricing of catastrophe reinsurance and derivatives using the Cox process with shot noise intensity , 2003, Finance Stochastics.

[55]  N. Shephard.  Realized power variation and stochastic volatility models , 2003 .

[56]  M. Steel,et al.  Inference With Non-Gaussian Ornstein-Uhlenbeck Processes , 2003 .

[57]  N. Shephard,et al.  Modelling by Lévy Processess for Financial Econometrics , 2001 .

[58]  N. Shephard,et al.  Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics , 2001 .

[59]  N. Shephard,et al.  Modelling by L´ evy Processes for Financial Econometrics , 2000 .

[60]  R. Wolpert.  Lévy Processes , 2000 .

[61]  J. Bertoin.  Subordinators: Examples and Applications , 1999 .

[62]  P. Carr,et al.  The Variance Gamma Process and Option Pricing , 1998 .

[63]  E. Seneta,et al.  The Variance Gamma (V.G.) Model for Share Market Returns , 1990 .

[64]  Paul Embrechts,et al.  Martingales and insurance risk , 1989 .

[65]  R. Mazo.  On the theory of brownian motion , 1973 .