In several papers it is argued that for many Itô diffusion processes, $$dX_t = a(t,X_t)dt+b(t,X_t)dB_t,$$ in mathematical finance the distribution of $X_T$ for fixed $T>0$ is unknown, which makes Monte Carlo simulations viable if not necessary even for the calculation of European options.

However, all models used in mathematical finance that come immediately to my mind are processes for which the distribution is known, such as geometric Brownian motion and the Cox-Ingersoll-Ross process.

What would be examples for diffusion processes, preferably widely used, of the above form for which the distribution of $X_T$ is unknown?

  • $\begingroup$ I understood your question, particular the meaning of "known" or "unknown" from the comments below. It would help clarify much if you would add that it is the closed or analytical form of the distribution that is known or unknown and that it is the closed or analytical forms that you are after. If a closed or analytical form is not available, Monte Carlo is not the only tool to obtain a numerical distribution. Plenty of numerical methods of PDE's are available. $\endgroup$
    – Hans
    Dec 23, 2013 at 2:25

2 Answers 2


Any of a wide variety of local vol models, where (from your equation) $b(\cdot,\cdot)$ is some fitted surface, are unlikely to have closed-form solutions for the terminal distribution. Indeed it's well-known that these models tend to have very unusual forward term structures of volatility.

As a specific example, take $b(\cdot,\cdot)$ to be an approximation derived from the first few terms of the 2-d Fourier representation of some high-resolution local vol fit $\tilde{b}(\cdot,\cdot)$.

Our SDE becomes $$ dX_t=\mu(t)dt+dB_t\sum_{\vec{k}} \omega_{\vec{k}} \exp\left( i\vec{c}_{\vec{k}}\cdot \binom{t}{X_t} \right) $$

I'm pretty sure there will be no closed form.


If you allow $X_t$ to be two dimensional then a model with a stock price $X_t^1$ and its variance process $X_t^2$ (stochastic volatility) would fit your definition.

In such cases to my knowledge we often don't have a closed form of the density of $X_T^1$ but in some cases we have a closed form of the Laplace transform.

An example is the Heston model.

  • $\begingroup$ Thank you; I should have been more specific in my question, as I was aware of multi-dimensional models. However, I am looking for one-dimensional models which are otherwise straightforward, but only difficult with respect to finding the distribution at a time instant $T$. $\endgroup$ Nov 29, 2013 at 17:33
  • $\begingroup$ I see - thanks for pointing out. I will think of one-dimensional exmaples. I guess you have already thought of most of the interest rate models? $\endgroup$
    – Richi Wa
    Dec 2, 2013 at 8:30
  • $\begingroup$ I did a quick check and I have to agree: one often finds: Gaussian distributions (Hull-White, Ho-Lee, Vasicek), Chi-squared (CIR) and Lognormal (Geom.BM, LIBOR market models). $\endgroup$
    – Richi Wa
    Dec 2, 2013 at 8:41
  • $\begingroup$ I find your question very interesting and important. Practitioners tend to simulated everything even if it is not necessary. On the other hand knowing the distribution one could probably simulated more efficiently (instead of simulating paths even if the path does not matter). $\endgroup$
    – Richi Wa
    Dec 2, 2013 at 12:07

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