Consider a classical Black Scholes model ,

$$\frac{dS}{S} = \mu dt + \sigma dW$$ , where $dW$ is a Brownian motion, that $W(t_1) - W(t_0) \sim N(0, t_1 - t_0)$.

The back-testing strategy is straight-forward: Once $\mu$ and $\sigma$ is recognized from the samples, $ dS / S_t \sim N(\mu \cdot dt, \sigma^2 dt)$. So back-testing of the model becomes hypothesis testing of a normal distribution's mean and standard deviation.

However, I'm coming out with such a modified Vasicek model:

$$ dr_t = a(b-r_t)dt + (c + d \cdot r_t) dW $$

This modifies the original Vasicek model: $dr_t = a(b-r_t)dt + \sigma dW$ as I notice the samples shows time-variant $\sigma(t)$ which has a strong linear correlation with $r_t$.

Now, how could I design the back-testing to validate my model?

I thought to back-test the $a$ and $b$ first, at least, $$E[dr_t + a \cdot r_t dt] = a \cdot b \cdot dt$$ , this is a constant.

But $dr_t + a \cdot dt$ 's standard deviation is not constant, I'm lost how to set the Hypothesis testing's criterion! Leaving alone how to back-test the $c$ and $d$ part?


1 Answer 1


Your SDE has no closed-form solution, so you'll have to apply the Euler method to obtain an approximate terminal distribution. Once you have the terminal distributions, any time series you want to validate has a highly multivariate probability density (due to the fact that each day's data comes from a slightly different distribution).

You can transform this into normal space by forming z-scores of each data point. Hypothesis testing now becomes a trivial exercise on those z-scores arising from the standard gaussian.

  • $\begingroup$ sorry Brian, is it that my question is not put clearly or I misunderstood your reply? I take that you are recommending a way to fit parameters, which I appreciate. However, my question is how to validate the model, e.g. to back-test the model with 2012 data after the parameters are fit based on 2009-2011 data. $\endgroup$
    – athos
    Commented Aug 7, 2013 at 1:42
  • $\begingroup$ Just a bit more on numerical method to solve the SDE - I was planning to use Monte Carlo method, say, to estimate the future year, I'm to run 100,000 samples, and take the estimation of each day in the coming year as the average number of the 100,000 data points in that day. Is this OK? $\endgroup$
    – athos
    Commented Aug 7, 2013 at 1:45
  • $\begingroup$ I understand the question better and I think I fixed the answer $\endgroup$
    – Brian B
    Commented Aug 7, 2013 at 13:45

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