Monte Carlo · Return processes

Multivariate Student-t Process

Comparison

student_t

A fat-tailed parametric alternative to the Gaussian, with covariance matched exactly to history and a single shared mixing variable per day so the whole cross-section fattens together. It is a comparison process, not a planning model, and the reasons are specific rather than decorative.

Construction

The multivariate Student-t is built as a scale mixture of normals. Each day draws

This has covariance , so to hit the historical covariance exactly the engine takes as the Cholesky factor of rather than of itself. The default is degrees of freedom, which requires for the covariance to exist at all.

The critical detail is that is drawn once per (path, day), not once per asset. A single mixing variable scales the entire cross-section together, so on a day the mixture makes extreme, every holding is extreme simultaneously. That produces genuine tail dependence, which a Gaussian copula with the same covariance cannot generate. It is the main thing this process adds over the Gaussian process.

Why it is labeled a comparison process

It emits unbounded simple returns.Unlike the log-domain processes, a Student-t draw on simple returns can produce a gross return at or below zero, which would mean an asset losing more than its entire value in one day. The engine's disclosed lower gross-return clip prevents that, but with the clip is not a rare formality. It is a real and load-bearing part of the model, and the result reports how often it bound so you can judge for yourself.

The tail thickness is assumed, not fitted.The degrees-of-freedom parameter sets how fat the tails are, and it is a fixed default rather than an estimate from your portfolio's data. The result is a distribution with the right covariance and an assumed tail index, which is a different thing from a distribution calibrated to your holdings.

Symmetric tails. The Student-t is symmetric, so it gives extreme gains exactly the same probability as extreme losses. Equity returns are negatively skewed: crashes are sharper and faster than rallies. The bootstrap families reproduce that skew exactly because they resample it; this process cannot represent it.

No volatility clustering. Days are IID. The mixing variable creates isolated fat days rather than the sustained turbulent regimes that actually damage a withdrawal plan.

When to use it

Use it to isolate one question: how much of my tail risk comes from fat tails, as opposed to from the shape of the historical sample? Run Gaussian, Student-t and block bootstrap under identical settings. The Gaussian-to-Student-t gap prices the fat tails on their own; the Student-t-to-bootstrap gap is everything the parametric family still misses, which is mainly skew and clustering.

For actual planning, use the block bootstrap, which gets fat tails, negative skew and cross-asset tail dependence from the data without assuming any of them, or the FHS processes, which add conditional volatility on top of empirical residuals.

References

Kotz, S., & Nadarajah, S. (2004). Multivariate t Distributions and Their Applications. Cambridge University Press.
McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management, 2nd edition, Chapter 6. Princeton University Press.
Cont, R. (2001). Empirical Properties of Asset Returns: Stylized Facts and Statistical Issues. Quantitative Finance, 1(2), 223–236.
Embrechts, P., McNeil, A. J., & Straumann, D. (2002). Correlation and Dependence in Risk Management: Properties and Pitfalls. In Risk Management: Value at Risk and Beyond, 176–223.