# Estimating Parameters - Vasicek

The Vasicek model for the short rate $r_t$ is given by the SDE $$dr_t = \alpha(\beta - r_t)dt + \sigma dW_t,$$ where $W_t$ is a Brownian motion under the physical measure.

I'd like to compute bond prices under this model, so I need to estimate the three parameters $\alpha$, $\beta$ and $\sigma$. Of course, $r_t$ isn't observable, but the yields $R(t,T)$ are, which are computed from actual bond prices and so we should be using the risk-neutral measure. The Vasicek model after this change of measure is $$dr_t = (\alpha(\beta - r_t) - \lambda\sigma)dt + \sigma d\tilde{W}_t,$$ where $\tilde{W}_t$ is a Brownian motion under the risk-neutral measure and $\lambda$ is the market price of risk. However, we may write this as $$dr_t = a(b - r_t)dt + \sigma dW_t,$$ where $$a = \alpha, \qquad b = \frac{\alpha\beta - \lambda\sigma}{\alpha}.$$ Thus we still have only three parameters to estimate, $a,b$ and $\sigma$, while the market price of risk $\lambda$ is simply implicit.

Finally, recall the bond price $P(t,T)$ may be written in two ways as $$P(t,T) = e^{-R(t,T)(T-t)} = e^{A_t(a,b,\sigma) - B_t(a,b,\sigma) r_t(a,b,\sigma)},$$ where $A_t$ and $B_t$ are deterministic functions of the risk-neutral parameters $a$, $b$ and $\sigma$ (is this right?). Thus given a bond yield $R(t,T)$, the short rate may be recovered through the affine function $$r_t(a,b,\sigma) = \frac{R(t,T)(T-t) + A_t(a,b,\sigma)}{B_t(a,b,\sigma)}.$$ We may then use some fancy estimation procedure to estimate the parameters $a,b$ and $\sigma$ through the observed yields.

My question is, is it indeed the risk-neutral parameters we would be estimating? That is, would we be estimating the $a,b,$ and $\sigma$, or would we need to include $\lambda$ and estimate $\alpha, \beta, \sigma$ and $\lambda$?

• The book Interest Rate Models by Brigo & Mercurio has a nice discussion about this. You may want to check it. Dec 2, 2015 at 8:32
• Sorry if I misunderstand this, but isn't it usually assumed that the model (in this case the Vasicek model) describes the interest rates in the risk neutral measure? Then you go directly from the closed-form solution for the prices in the model to the quotes prices and just fit \alpha$,$\beta \sigma and \lambda\$. Which of course will not work to well as neither yield curve shapes nor vola surfaces will be fitted. Dec 22, 2018 at 19:09

## 3 Answers

**first answer, please give advice/edit if you see fit!

Assuming that the bond has affine structure, when lambda is not 0 which is your case, $$\frac{dB}{dt}$$ equals to something like $$\frac{dB}{dt} = \mu + 0.5 * \gamma B^{2} -1$$ (where here gamma is similar to your lambda term), so we will be estimating for lambda as well.

• Welcome @JojoTang on Quantitative Finance Stack Exchange! As you are asking for advice, i would like to recommend you to read the markdown help. QF Stack Exchange uses MathJax (see here), so there is no need for using CodeCogs or other third part code formatting software for LaTex here. Oct 4, 2018 at 8:47
• @skoestlmeier thanks for the tip! the format looks much better now! Oct 4, 2018 at 16:46

If you are estimating the parameters from a time series of observations of a rolling maturity yield, the problem is not identified.

Time series observations of the bond yield (continuously compounded) at fixed intervals (daily, monthly, whatever) follow an AR(1) process, which has three parameters. There are infinitely many combinations of the four parameters $$\alpha$$, $$\beta$$, $$b$$, and $$\sigma$$ that give you the same AR(1) process.

So really, you can't estimate either the risk-neutral or the true parameters, as long as all you've got is a time series of a fixed maturity yield.

Yes, you are estimating the risk neutral parameters; and need to estimate lambda as well

The change in numeraire to the risk neutral world is a change in drift as per adding the market-price-of-risk.

Given you convert the drift term and adjusted brownian motion, then your parameters are under the risk neutral world.

However, change in drift for stohastic short-rates require a volatility term as well, which I do not think you have included.

Kind regards