Interest Rate and Bonds with embedded options

Hi all

I’m reading the chapter about interest rates and bonds with embedded options, I find it quite confusing so I was hoping someone could help me clarify. In specific:

  1. Why is OAS call decrease when interest rates vol. increases? I understand call (and put) option values increase when volatility increase, so the callable bond should have less value since straight bond’s value is unchanged. But then how is that related to OAS? In the notes it says it’s because the callable bond price will be closer to the market price. I don’t find it very logical.

  2. An upward sloping yield curve will have the same effect on bonds with options as an interest rate increase?

  3. How do changing interest rate influence the price of callable bonds and putable bonds? I kindda got a gist of it but wouldn’t say I fully understood it…

Thanks in advance!

I will tackle question 3 first :slight_smile: The two formulae are self explaining.

  1. V(callable)=V(normal)-V(call option) As interest rates increase , V(normal) decreases. Thus, V(callable) falls even faster because of the [-V(call option)] component. As interest rates _ decrease _, V(normal) increases. V(callable) increases too but at a _ slower pace _ because it is pulled down by the [-V(call option)] component. 2) V(putable)=V(normal)+V(put option)

As interest rates increase , V(normal) decreases. However, V(putable) does not decrease as fast because of the [+V(put option)] component. As interest rates _ decrease _, V(normal) increases. V(putable) _ increases even faster _ because of the additional [+V(put option)] component.

You can find the answer to 1) in the article I wrote on OAS:

Two things are puzzling for me.

  1. If OAS=Option Adjusted Spread why do we still call it OAS for a straight bond? Why not call it The Spread instead since there are no options involved?

  2. If the market price is already known, why bother calculate the value of the bond using the binomial tree in the first place? I mean bonds are not like stocks and there can’t be two different interpretations about its value by two different valuers. Is that exercise targeted just to deduce the OAS?

  1. Call it OAS when embedded options are present, otherwise it can be just named spread, where there are many: G-spread, I-spread, Z-spread etc.

  2. The market price already known is about the benchmark bond (treasury securities for example) which is what we try to arrive at using a constant spread: OAS.

Hope this helps.

The idea is that you can compare the OAS of a variety of securities and choose to invest in the one with the highest OAS. Calling it OAS for all securities simplifies the language and the thinking.

We use the binomial tree to determine the OAS: the value we get from the tree has to equal the market price.

The difference is that the OAS is added to the forward rates in a binomial tree, whereas the G-spread and I-spread are added to par rates, and the Z-spread is added to spot rates.

Well explained. One last remark before turning to Miss’ original post.

Isn’t the assumption that OAS is a constant a fallacy? It is a dynamic function of forward rates right as opposed to an intrinsic bond characteristic?

I don’t know that anyone assumes that OAS is a constant.

A constant spread instead?


Hi guys,

Thank you very much for your help as always! Just a few last comments,


Really appreciate this comprehensive article! It might have been mentioned in the notes, but is OAS always positive? i.e. is the market price always lower than calculated price using binomial tree without adding OAS? If yes, then it makes a lot more sense.


Are you assuming V(call option) and V(put option) always being positive? Because that is the confusing part for me. In the notes it saids interest rates decrease -> V(straight bond) or V(normal) increase -> -V(call option) limit bond’s upside, so I assume V(call option) decreases.

I believe BS model indicates: V(call option) is positively related to interest rate level and V(put option) inversely related. But in the notes it says Call option value is inversely related to the level of interest rate…under LOS45.e.I dont know if “call option value” here means the callable bond.

If the underlying yield curve is risk-free (e.g., a Treasury curve), then the OAS always should be 0 or positive.

Options prices can’t be negative if you think about it…

But I remember doing a Harvard case study during my Masters (no I didn’t study at Harvard) which featured an anomaly in the US Treasury callable bonds market implying negative option prices. The case is extremely interesting and engaging (especially after covering CFA fixed income) so have a look if you are interested.

The OAS being negative is not the same as option prices being negative; it depends on the nature of the par curve underlying the binomial tree.

I understand. I was referring to V(call option) and V(put option) parameters.

Thank you guys. I will have a look at the case study. Really appreciate it.

great explanation. Simplistic approach to the concept.