Student-t Returns and the Limits of Lognormal Spot Modeling

Summary

This note considers how to model future spot when log returns are assumed to follow a Student-t distribution rather than a Gaussian one. It highlights a key theoretical problem: the Student-t distribution lacks a moment-generating function, so the exponential return does not have a finite expected value under the stated model. Without that expectation, the usual martingale representation for the future asset value cannot be established directly.

For simulation, the response suggests drawing Student-t variates, exponentiating them, and empirically rescaling the resulting values by their sample mean. This can be a practical adjustment for some applications, such as pricing or allocation, but it does not resolve the underlying theoretical issue. Results depend on the simulation and scaling choices, and the note does not provide a closed-form alternative or establish that the rescaled process is a valid pricing model.

Key ideas

• Student-t log returns do not have a moment-generating function under the stated model.
• The exponential of a Student-t return therefore lacks a finite expected value in the theoretical setup described.
• Without a finite expected future spot, the usual martingale representation cannot be obtained directly.
• Simulation can rescale exponentiated draws empirically, but this does not solve the theoretical limitation.

Tags

# FX spot distribution with student-t returns


# FX spot distribution with student-t returns












If I am modelling my returns as $\sim N(0, \sigma^2)$, then I can evolve my spot distribution as: $$S_{t} = S_{0}e^{(\mu - \frac{1}{2}\sigma^{2})t + \sigma dW_{t}}$$ where $S_{0}$ is the spot, $\mu$ is the mean , and $\sigma$ is the returns volatility and $dW_{t}$ is the gaussian noise.

How should I amend my Spot (lognormal) distribution if I assuming my returns follow a student-t distribution $\sim t-dist(\nu)$

Thanks

## Answer by Kermittfrog (score 4, accepted)

https://quant.stackexchange.com/a/59683

1. Theory

The Student $t$ distribution does not exhibit a moment generating function

$$ M_X(t)=\mathbb{E}\left(e^{tX} \right) $$ Hence, there exist no closed form solution for $M_X(t=1)=\mathbb{E}\left(e^X\right)$, i.e. the expected future spot price. Thus, at least theoretically, we are not able to pinpoint the expectation of the future asset value, thereby preventing us from finding a martingal representation.

2. Practical simulation Depending on your application (derivatives pricing, asset allocation), you can of course resort to some arcane methods. I understand that you know how to simulate from a $t$ distribution. Then, in order to properly centre your random variates $x\sim e^{t(\nu)}$, you could empirically re-scale your draws as $\tilde{x}=\frac{x}{\bar{e^x}}$. Again, that will not help you with theory-building, though...

HTH?

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