# 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

• Hi, this is a nice question. What do you think about changing the title to "return distribution with ..." ? And could you also make a reference to your desired application, e.g. derivatives valuation, asset allocation, ... ? Depending on the application, there may (not) exist a solution at all... Dec 1 '20 at 11:54
• Nevertheless, you could find some ideas in here: arxiv.org/pdf/0906.4092.pdf Dec 1 '20 at 11:57

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?

• Thanks. Wanted to understand a few things (This exercise is to understand the "range" of the FX spot price over a time horizon [t,T] )... 1. If I already have a term-structure of fwd volatility , say \sigma(t), \sigma(t+1)....\ sigma(T) of the returns (fitted using a garch(p,q) model with student-t distribution assumption), how can I use those to come up with a spot price "range" over that period? Dec 1 '20 at 13:19
• aren't the moments of the t-distribution calculable for a certain range of degrees of freedom $\nu$ Dec 1 '20 at 13:20
• You could come up with quantiles (say from 5% to 95%) of the return distribution and transform those into a price quantiles - and thereby obtain a range. would that help? Dec 1 '20 at 13:23
• @develarist The moments, yes. But The MGF itself does not exist, hence the expectation $\mathbb{E}\left(e^{X}\right)$ does not exist. Put differently: The (non-)existence of a MGF does not imply the (non-)existence of moments. Dec 1 '20 at 13:24
• @Kermittfrog If I fit a parametric distribution, such as the t-distribution, to the histogram of an empirical dataset's samples (using MLE of the parameters), why doesn't the parametric distribution's moments ($\mu, \sigma$) not match the empirical data's moments? If I can't even get the moments to match, doesn't this defeat the whole purpose of modeling real data parametrically Dec 1 '20 at 13:28