# Vasicek Model (estimation of parameters)

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

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.

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 $$\theta^\mathbb Q$$.