How to choose model parameters?

Question

I'm studying math and attend this semester a course about interest rates. Now, some questions show up how exactly things are working in the real world.

My examples will be about interest rates model, but I guess there is no need to restrict ourselves to this case.

When you want to price for example bond prices, you will do the following:

1. modelling
2. pricing
3. calibration

Let's assume there is an equivalent martingale measure $Q$ such that all the bond prices are martingales. The density $\frac{dQ}{dP}$ is of a Girsanov type. Concerning the above list, I would start by choosing a model for my interest rate, i.e. I would directly write down the $Q$ dynamics of $r$. For simplicity assume $r$ has the following dynamics:

$$dr(t)=(b+\beta r(t))dt + \sigma dW^*(t)$$

where $b,\beta,\sigma$ are parameters and $W^*$ is one dimensional $Q$-Brownian motion.

For the second point I would start pricing using the usual formula

$$ \pi(t,T) = E_Q[\exp{(-\int_t^Tr(u) du)}|\mathcal{F}_t]$$

where $\pi(t,T)$ denotes the bond prices at $t$ with maturity $T$. Since in this case $r(t)$ is normal distributed I can perfectly calculate the prices dependent on the parameters $b,\beta,\sigma$.

Now the questions show up in the last point, i.e. in 3. Here I want to choose my parameters in such a way, that the computed prices match with the market prices. So far I was just working with the risk-neutral measure $Q$. Do I really compare the, under $Q$, estimated prices with the market prices? Or how exactly does this calibration work in reality?

Thanks in advance for your help.

Answer 1

Yes, you do really use market prices to calibrate models derived under the risk-neutral measure. That is the whole reason why risk-neutral measures are utilized, to a) ease the calculations but mostly b) because under no arbitrage and one price for each security assumptions (among couple other other assumptions) the price derived under the risk neutral measure must equal the market price (otherwise you could arbitrage which you just assumed does not not exist). Thus, you can use market prices to calibrate risk-neutral models.