Simulating Student’s t Returns with Lévy, GARCH, and Variance Mixtures
Summary
The document asks how to construct intraday returns whose aggregated daily returns follow a Student’s t distribution, contrasting this with the aggregation property of normal returns. One response points to Lévy processes as a framework that can include processes generating Student’s t variables. Another explains that a normal distribution mixed over a random variance drawn from an inverse-gamma distribution yields a Student’s t distribution; this suggests sampling a day’s variance first, then simulating returns conditional on it.
A separate alternative uses GARCH dynamics with normal innovations. Time-varying conditional variance can produce excess kurtosis and autocorrelated volatility in a sample, even though it does not make the sample exactly Student’s t distributed. These are distinct modeling goals: reproducing a specific marginal distribution differs from matching heavy tails and volatility clustering. The discussion offers conceptual routes rather than calibrated parameters or comparative validation, so a model still needs to be checked against the desired return properties and aggregation horizon.
Key ideas
- Lévy processes provide a framework for constructing returns with non-normal distributions.
- A Student’s t variable can be represented as a normal variable with an inverse-gamma mixture over variance.
- Conditional on a sampled daily variance, intraday returns can be simulated with a Brownian-motion approach.
- GARCH with normal innovations can create excess kurtosis and autocorrelated volatility.
- A GARCH-generated heavy-tailed sample is not necessarily exactly Student’s t distributed.
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Full text
# What stochastic process produces Student's t-distributed returns?
# What stochastic process produces Student's t-distributed returns?
If I think daily log returns have a normal distribution, I can simulate intraday log returns as normal, because the sum of normal variates is also normally distributed. What if I want to simulate intraday log returns consistent with daily log returns that follow a Student's t-distribution?
## Answer by Raskolnikov (score 2, accepted)
https://quant.stackexchange.com/a/45789
What you need is a Lévy process. Take a look at this primer. It contains a summary description of some Lévy processes and among them some that generate student t-distributed variables.
## Answer by Bob Jansen (score 2)
https://quant.stackexchange.com/a/45775
One way to generate excess kurtosis in your sample is the approach below. It doesn't give you a student-t distributed sample but from your comment I understand that might not be a hard requirement.
A simulation with GARCH where the intraday innovations are inputted into to the GARCH model would give you time varying volatility which appears as excess kurtosis if you calculate it from a sample where you assume constant volatility.
For example, you could simulate like this (using R):
```
library(moments)
N = 1000
# Assume some GARCH(1, 1) parameters
omega <- 0.000001
alpha <- 0.04
beta <- 0.95
# Simulate a return series with GARCH(1, 1)-based volatility
set.seed(1L)
normalInnovations <- rnorm(N)
returns <- c(normalInnovations[[1L]], rep(NA_real_, N - 1L))
variance <- c(0, rep(NA_real_, N - 1L))
for (i in 2:N) {
variance[[i]] <- omega +
alpha * normalInnovations[[i - 1L]] ^ 2 +
beta * variance[[i - 1L]] ^ 2
returns[[i]] <- normalInnovations[[i]] * sqrt(variance[[i]])
}
skewness(normalInnovations)
kurtosis(normalInnovations)
jarque.test(normalInnovations)
skewness(returns)
kurtosis(returns)
jarque.test(returns)
acf(returns^2, lag.max = 10L)
```
These returns will exhibit excess kurtosis and auto-correlated volatility.
## Answer by Fortranner (score 2)
https://quant.stackexchange.com/a/45780
Student's t distribution can be regarded as a Normal distribution with variance mixture Y, where Y follows the inverse gamma distribution (1). So to simulate intraday returns consistent with the daily return having a t distribution, you could first sample from the inverse gamma distribution to determine that day's variance and then simulate a Brownian motion with that variance.
(1) https://www.johndcook.com/t_normal_mixture.pdfShown 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.