How to design back-testing (validation) for such modified Vasicek model?

Question

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?

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.