# AM CFAI Mock - Q51 Effect of shape curve change on put option value

The yield curve move from flat to upward sloping and they say the value of the put option will increase. Why is that?

If the yield curve move from flat to upward sloping isn’t the overall level of yields increase? If so shouldn’t the put option worth less?

No, when the yield curve moves up, yields rise and bond prices fall. Bond yields and prices have inverse relationships. If prices fall, that means that put options increase.

Example: Think of a bond as a series of fixed cash flows. Let’s say the bond is discounted at 2%

100/(1+.02) + 100/(1+.02)^2 + 100/(1+.02)^2 = Bond Price

Now, replace .02 with 0.03 to simulate an increase in yields. What happens to the bond price? You’ll see that the bond price will decrease.

I cant seem to get my head around a FI question (Q15, R44) where they set the coupon in arrears and floor/cap it at a certain level. I can figure out the coupon payment for the nod at the top and the bottom but i can not understand the process for the node which lie in the middle.

To give you a little more insight, the question is of valuing a bond with embeded options with a life of 3 yrs. They have given IRs for yr1-yr3 (1 IR for yr 1, 2 for yr 2 and 3 for yr3) … I discounted the CFs of the last year by taking all the relevant cash flows and have the DR for each as well. For 1 year on i am left with 3 CFs but i’ve 2 discount rates. That is ok when i am discounting the cash flows but i dont know what to do with the coupon payments? Should i take an average?

I hope i was able to explain you the problem … if not then please look at the EOC question and help me … needless to say i’ll be grateful for the help!!!

Actually in the question they say that the price of Bond A (option free) doesn’t move, so that means the yields overall level doesn’t change.

It’s only the shape change who makes the put option worth more. They say “as the yield curve move from flat to upward sloping the value of the put option increase”.

Why is that?!?

This is the way I saw it, and it yielded me to the right answer (somehow).

The price of the putable bond is equal to… Putable Bond = Straight Bond + Put Option

An increase in the yield curve, makes the bond more risky to a price decline. Therefore, in a riskier bond the price of a put option is higher. (kind of like for a more volatile stock, the price of its options increase).

Since the straight bond didn’t change in price, then at least the option must reflect the higher risk. Therefore, the option price increased.

Good luck and I hope it helps.

I used that as well… they said the price of the straight bond is the same, so we can look at the effect of the change on the put option. Remember, the put option effectively adds a floor to the value of the putable bond; when interest rates increase, the straight-bond price should decline, but the put option allows you to put the bond back to the issuer in this case. So, as interest rates appear to be increasing, the value of our put on the bond become more valuable (since we will probably need to use it). Using the equation cebrach mentioned, we can see the putable bond has increased because the option has become more valuable, and the straight bond has not changed.

double post

This would explain why the price decrease is less with a putable bond in comparison to other, nonputable bonds. The question states that the value of the straight bond is unchanged; we need only to look at the effect on the put’s value.

The last question also tells us that The effective convexity of a putable bond can be less than that of an otherwise identical option-free bond.

I’m not sure how though.

I would think that rising interest rates would show this example. If interest rates rise, the putable bond is less sensitive to the change than an identical option-free bond because the put has the effective floor value.

So in that sense, a straight identical bond would always have the greatest convexity compared to callable/putable because they have the greatest durations?

Hmm, makes sense. Thanks.