# Modelling callable bonds in a risk model (historical simulation)

What is a best-practice example on how to model callable bonds in a risk model - I focus on historical simulation (HS).

For plain-vanilla bonds the input factors for historical simulation could be

• the zero curve of the market (the currency)

Then HS would model changes in interst rates of the currency (as systematic risk) and spreads either in issuer level (idiosyncratic) or rating level (rather systematic risk). Then we could reprice the bond in these scenarios.

Looking at callable bonds on the other hand we have to simulate/estimate the chance that the bond is called and when. To do this we could use an interest rate model which we would have to calibrate on future interest rate uncertainty. Then we can simulate the future and price the bond in these scenarios.

Market data that reflect this that I know are swaptions and captions. But these are instruments for the money market/capital market of a currency. However, the decisin of the issuer to call the bonds will depend on the interest rate level of the currency and the issuers spread.

How can we find a risk model that can be calibrated to readily available market data and that models the systematic as well as the idiosyncratic part of the call risk? How do industry solutions look like?

Callable bonds are exposed to interest rates, spreads, and your interest rate model. You could link your spreads to interest rates, but then you will need a systematic spread model. In most pricing models that I have seen, spreads are not evolving through time (which is incorrect). The problem is one doesn't have any market instrument to calibrate the evolution of spreads in a risk neutral manner, and pricing is done in risk neutral world. For rates, rate evolution is calibrated on other instruments like swaptions or futures, which are market instruments, thus rate evolution models can be made risk neutral.

• What would you say dominates for the decision to call: rates or spreads? Commented Feb 25, 2016 at 10:26

It says I have to have 50 reputations points to comment, and I don't, and actually don't even know what that means so here is my "Answer." I think you're getting into apples and oranges. There is no risk neutral anything here. For historical simulation, you already have all of the rates to apply to your prospective cash flows, and in your simulation you have the amounts and dates of those cash flows from the bond description. Fine. Now the only addition is to add a decision function that terminates the cash flows to principal plus call spread if the bond qualifies under certain conditions: price justifies call and call date is far enough in the future to justify being triggered, etc. You are making this way more difficult than it should be.

• There are two things here. HS requires risk factor simulation, then you need to price the callable bond at each scenario of HS. Pricing is risk neutral