Vasicek Model (estimation of parameters)

Question

I have a question concerning the "choice" of parameters for the Vasicek model (formula below).

Consider me as a moron with below average level in maths haha. What I've done is basically run the Vasicek on Excel and compare it with the term structure of bonds. Then I used the solver to give me the parameters r, sigma, theta and kappa to minimize the squared error between my Vasicek and the market data.

Obviously it's a kitchen sink result, I have a very good fit but the underlying economic meaning is lost. So I thought about constraining sigma and theta in my solver, but how should I choose my constraint levels?

I was thinking about checking the average or moving average of the short term rate and its volatility over a few years and set them as a boundary.

I'm not looking for something perfect or complicated at all, I just want my parameters to make a little more sense, am I in the right direction?

Thanks

Answer 1

A simple way to proceed is by matching empirical and theoretical moments. For starters, if you apply the Euler-Maruyama approximation, you get \begin{align} r_{t+1} &\simeq r_t + \kappa \theta - \kappa r_t + \sigma \epsilon_t, \; \; \epsilon_t \sim N(0,1) \\ r_{t+1} &\simeq \kappa \theta + (1 - \kappa) r_t + \sigma \epsilon_t. \end{align} This is an AR(1). You have three parameters, hence you need at least three moments. If you restrict yourself to $|1 - \kappa| < 1$, then this is covariance stationary and you could use (1) the expectation, (2) the variance and (3) the first order autocovariance. Stack them up in a vector as a function of your parameter and match them with their empirical counterpart. Minimize a square norm and you got yourself the GMM estimator.

It's not exactly right because you discretization is technically justified as the step size shrinks, but it's good enough.

Answer 2

Your SDE may be discretized as $$\Delta r_t = \alpha + \beta r_t + \eta$$ where $\alpha=\kappa\theta\Delta t, \beta = -\kappa\Delta t$ and $\eta\sim\mathrm N\left(0,\varsigma^2\right)$, with $\varsigma^2 = \sigma^2\Delta t$. In principle, you could estimate these parameters from the short-term interest rate movements using OLS.

If you follow this path, you may add these terms to your objective function: force the estimates obtained from matching the prices to be not that far from the historical values. Moreover, each error term might be multiplied by the statistical error of $\hat\alpha,\hat\beta,\hat{\sigma^2}$.

However, keep in mind that, while the parameters used to price the bonds correspond to the risk-neutral world, the parameters involved in the historical dynamics correspond to the physical measure. In other words, you might need to consider a risk-premium $\lambda$ which might lead, for example, to have two different parameters $\theta^\mathbb P$ and $\theta^\mathbb Q$.