Shot-noise cojumps: Exact simulation and option pricing

Abstract

We consider a parsimonious framework of jump-diffusion models for price dynamics with stochastic price volatilities and stochastic jump intensities in continuous time. They account for conditional heteroscedasticity and also incorporate key features appearing in financial time series of price volatilities and jump intensities, such as persistence of contemporaneous jumps (cojumps), mean reversion and feedback effects. More precisely, the stochastic variance and stochastic intensity are jointly modelled by a generalised bivariate shot-noise process sharing common jump arrivals with any non-negative jump-size distributions. This framework covers many classical and important models in the literature. The main contribution of this paper is that, we develop a very efficient scheme for its exact simulation based on perfect decomposition where neither numerical inversion nor acceptance/rejection scheme is required, which means that it is not only accurate but also the efficiency would not be sensitive to the parameter choice. Extensive numerical implementations and tests are reported to demonstrate the accuracy and effectiveness of this scheme. Our algorithm substantially outperforms the classical discretisation scheme. Moreover, we unbiasedly estimate the prices of discrete-barrier European options to show the applicability and flexibility of our algorithms.

Keywords:

• Exact simulation
• Monte Carlo simulation
• jump-diffusion models
• stochastic volatility models
• shot-noise process
• contemporaneous jumps
• cojumps
• shot-noise cojumps
• option pricing
• systemic risk

Mathematics Subject Classification (2010)::

• Primary: 91G60
• 91B25
• Secondary: 65C05
• 60H35
• 60G17
• 91G20

JEL Classification::

• C15
• G13
• C63

Acknowledgements

The authors would like to thank the reviewer for very helpful and constructive comments and suggestions.

Notes

1 Shot-noise processes have already been used as parsimonious stochastic intensity models (i.e. shot-noise stochastic intensity models) for event arrivals in finance and insurance, such as corporate defaults in Duffie and Singleton (1999, 2003) and catastrophes in Dassios and Jang (2003).

2 In fact, shot-noise process was used by Bookstaber and Pomerantz (1989) for modelling the volatility dynamics. We use it here for the variance dynamics, but we still name it as a shot-noise stochastic volatility (rather than shot-noise stochastic variance).

3 This approach has also been recently adopted by Dassios and Zhao (2013, 2017); Dassios et al. (2018); Qu et al. (2021a,b,c) to develop tailored algorithms for exactly sampling some important random variables and point processes, such as tempered stable distributions, Hawkes process, point processes with CIR-intensities, Lévy-driven OU processes and Lévy-driven point processes. The basic idea is simple: to exactly simulate a random variable or a stochastic process, we investigate whether it can be decomposed into simpler random variables exactly in law, each of which can be simulated exactly. However, it is only a generic principle and does not tell us anything in detail about how to simulate any targeted random variable or stochastic process, and we have to obtain their distributional properties case by case and identify a successful decomposition which is nontrivial.

4 In fact, the point process $N_t^{*}$ is a dynamic contagion process as introduced by Dassios and Zhao (2011).

5 V_t is the so-called gamma-OU process, since the asymptotic marginal distribution of volatility process is a simple gamma distribution.

6 Alterative approaches can be found in the literature. Petrella and Kou (2004) adopted the numerical inversion of Laplace transform. Broadie and Yamamoto (2005) developed a fast Gaussian transform for Gaussian models. Feng and Linetsky (2008) developed a Hilbert transform approach based on the convolution for general non-Gaussian Lévy models.

7 A positive tempered stable distribution can be constructed from a one-sided α-stable law by exponential tilting, see (Barndorff-Nielsen and Shephard, 2001b, p.3) and (Barndorff-Nielsen et al., 2002, p.14). It is a popular and basic tool in finance, see some recent applications in Carr et al. (2002), Bates (2012), Li and Linetsky (2013), Mendoza-Arriaga and Linetsky (2014), Todorov (2015) and Andersen et al. (2017). In particular, if $\alpha =\frac{1}{2},$ it reduces to a very important distribution, the inverse Gaussian (IG) distribution (which can be interpreted as the distribution of the first passage time of a Brownian motion to an absorbing barrier).

8 Copula functions are widely used for modelling dependency in finance, insurance and economics, see Embrechts et al. (2002) and Patton (2009) for more details.

Additional information

Funding

The corresponding author Hongbiao Zhao would like to acknowledge the financial support from the National Natural Science Foundation of China (#71401147) and the research fund provided by the Innovative Research Team of Shanghai University of Finance and Economics (#2020110930) and Shanghai Institute of International Finance and Economics.