0
$\begingroup$

I would like a general framework for the calibration of the unknown parameters in an arbitrary stochastic differential equation. I have a proposed method that seems reasonable in theory, but is problematic in practice. Can you help?

The method

Suppose we have the SDE $dX=\mu(t,X)dt+\sigma(t,X)dW$, where $W(t)$ is a Wiener process. And suppose we know a time series realization ("historical data") $X(t_n)=X_n$ for $n=0,\ldots,N$. And further suppose that the functions $\mu$ and $\sigma$ contain unknown real parameters.

The idea is to convert the SDE into a finite difference equation (which holds approximately): $$X_{n+1}-X_{n} = \mu(t, X_n) (t_{n+1}-t_n)+ \sigma(t, X_n)\sqrt{t_{n+1}-t_n} Z_n.$$

Here: $n=0,\ldots,N-1$ and $Z_n$ are independent random variables, each with a standard normal distribution. Here, I used the heuristic fact that $dW$ is normal with mean 0 and variance $dt$.

Solving for $Z_n$ gives $Z_n$ as a function of the unknown parameters: $$Z_n =\frac{X_{n+1} - X_{n} - \mu(t, X_n)(t_{n+1}-t_n)}{\sigma(t, X_n) \sqrt{t_{n+1} - t_n}}.$$

The liklihood of the time series is given by $$\prod_{n=0}^{N-1} \phi(Z_n) = \left( \frac{1}{\sqrt{2 \pi}} \right)^N \exp \left( -\frac{1}{2} \sum Z_{n}^{2} \right).$$ Here, $\phi$ is the probability density of the standard normal. Maximizing the liklihood requires minimization of $\sum Z_n^2$. And this gives the maximum liklihood estimators.

The problem

Consider the SDE $dX = \sigma dW$, with unknown $\sigma$. Given historical data, and following the above method, we get $$\sum Z_n^2 = \frac{1}{\sigma} \sum \frac{X_{n+1} - X_{n}}{\sqrt{t_{n+1} - t_n}}.$$

However, this does not have a minimum wrt $\sigma$ (except for perhaps $\sigma = \infty$, which does not make sense).

$\endgroup$
  • 2
    $\begingroup$ I think you should use the backward pricing PDE of $V(t,X_t)$ where $V(t,X_t)$ is the price of a claim at time $t$ with current price $X_t$. $\endgroup$ – user16651 Jan 4 '17 at 15:46
  • $\begingroup$ Since you've transformed the problem into one of estimating parameters for a discrete model (as is the most common procedure), why don't you just use the Markov property and maximize $L(\Theta|X) = \prod^{T}_{t=1}p(x_t|x_{t-1}, \Theta)$? A complex reference on the subject could be arxiv.org/pdf/1408.2441v1.pdf. I might be totally off, but that is how I would do it $\endgroup$ – Tingiskhan Jan 4 '17 at 19:33
  • $\begingroup$ Or if you feel a bit more Bayesian, you could use a sequential algorithm like IBIS: academic.oup.com/biomet/article-abstract/89/3/539/251804/… $\endgroup$ – Tingiskhan Jan 4 '17 at 19:45
1
$\begingroup$

From the SDE $dX_t=\sigma dW_t$, \begin{align*} X_{n+1} - X_n =\sigma \sqrt{t_{n+1}-t_n} Z_n. \end{align*} That is, \begin{align*} X_{n+1}\mid X_n \sim N\left(X_n, \, \left(\sigma \sqrt{t_{n+1}-t_n}\,\right)^2 \right), \end{align*} and the conditional density function is given by \begin{align*} \phi(x\mid X_n) = \frac{1}{\sqrt{2\pi}\sigma \sqrt{t_{n+1}-t_n}}e^{-\frac{(x-X_n)^2}{2 \sigma^2 (t_{n+1}-t_n)}}. \end{align*} Then, the conditional log-likelihood function is given by \begin{align*} L(X_1, \ldots, X_{n+1}) &= \sum_{i=1}^n\ln \Big(\phi(X_{i+1}\mid X_i)\Big)\\ &=-\frac{n}{2}\ln(2\pi) -\frac{n}{2}\ln(t_{i+1}-t_i)-n\ln\sigma - \sum_{i=1}^n\frac{(X_{i+1}-X_i)^2}{2 \sigma^2 (t_{n+1}-t_n)}. \end{align*} Now, you can minimize the function to obtain the $\sigma$ estimation.

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.