Simulation Methods for Infinite-Activity Lévy Processes
Summary
The document surveys ways to simulate Lévy processes with infinitely many jumps, including CGMY and Meixner models. One approach uses the characteristic function to recover a distribution for inverse-transform sampling. Other methods exploit representations as time-changed Brownian motion or mixtures involving gamma subordinators and normal draws; cited constructions can make Monte Carlo sampling more direct.
The discussion highlights an approximation in which small jumps are replaced by their expected value, introducing simulation error. Series representations based on independent uniform variables instead leave truncation as the remaining error source. It also points to a change of measure for exact simulation of CGMY increments and describes sampling paths for a variance-gamma process with stochastic arrival. These are method options rather than a benchmark: implementation detail is deferred to cited papers, and the document does not compare computational cost or accuracy across approaches.
Key ideas
- Characteristic-function inversion can support inverse-transform simulation from a uniform draw.
- Subordination and time-changed Brownian representations can simplify simulation of some Lévy models.
- Replacing the contribution of small jumps by its expectation introduces approximation error.
- Infinite-series constructions can reduce simulation error to the effects of truncation.
- A change of measure can enable exact simulation of CGMY increments.
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# How to simulate Levy processes
# How to simulate Levy processes
Hey how to simulate Levy processes? I have no problem with Wiener process and compound Poisson process, I also know how to simulate Variance Gamma process but I have no idea how to simulate for example Meixner process, CGMY process and other Levy processes with infinite activity.
## Answer by Kevin (score 11, accepted)
https://quant.stackexchange.com/a/65969
You have many different options. Firstly, you know the characteristic function for the log stock price and, using inversion, you can recover the (inverse) distribution and density function and simulate from these using a uniform draw. That's the brute force approach.
The variance gamma process is typically represented as a difference of gamma processes or a subordinated (time changed) Brownian motion. This makes Monte Carlo simulation easy. Madan and Yor (2008, JCF) show how to do this analysis for CGMY and Meixner processes. In more detail, they show how to simulate the CGMY process via the usual $$ X=\theta Y_t+W_{Y_t},$$ where $Y_t$ is the subordinator.
Alternatively, they set \begin{align*} X=\frac{G-M}{2}H_t+\sqrt{H_t}Z, \end{align*} where $Z\sim N(0,1)$ and \begin{align*} H_t = \delta t + \sum_{j=1}^\infty y_j\mathbf{1}_{\{\Gamma_j<t\}}\mathbf{1}_{\{h_y>u_{3j}\}}, \end{align*} where $\delta$ is a constant, $\Gamma_j$ a sum of jump times and $u_{3j}$ a sequence of uniform random variables. [All details are in the paper, Section 4.1.] A similar expression exists for the Meixner process where \begin{align} X=\frac{a}{b}\tau+\sqrt{\tau}Z. \end{align}
A problem is that the above analysis replaces small jumps with their expectation. This introduces some error in the simulations. Rosinski (2007, SPA) finds an expressions using an infinite sum of independent uniform distributions. The only remaining simulation error is then due to truncation.
Poirot and Tankov (2006) find a change of measure that reduces the CGMY process to a simpler process that can be simulated exactly. This way, they can simulate the increments of a CGMY process.
Hirsa (2013, Section 6.7.6) shows how to simulate sample paths for the variance gamma with stochastic arrival (VGSA) process, a stochastic volatility Lévy process from Carr et al. (2003, MF), by drawing from the gamma and normal distribution.
Finally, Ballotta and Kyriakou (2014, JFM) also look at the simulation of sample paths from the CGMY model. They essentially suggest the brute force approach mentioned above, see their section 4.3.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.