call option value positively or negatively related to Interest rates?

CFA Level II Fixed Income
#1

Hi

In derivatives we learn that the call option value is positively related to interest rates.

However in fixed income, under the callable bond section, we are told that the decline in interest rates cause the call option value to increase.

can someone explain this please.

Thanks

#2

It depends on the underlying!

For a call option on a bond, when interest rates are lower, the bond value is higher, therefore your call option is more valuable

#3

When we say that call options increase in value as interest rates rise, we’re assuming that the price of the underlying stays the same.

For bonds, that’s not the case.

Thus, when interest rates rise, there are two effects on a call option on the bond:

  • The general interest rate effect, which increases the value of the call option
  • The decrease in the value of the underlying bond, which decreases the value of the call option

Usually, the latter has a bigger impact than the former.

#4

Hi S,

I am still a little unclear on your explanation above.

Why would the general interest rate rise increase the value of a call option? As per bullet point 1. Surely as rates rise, prices fall. Therefore the value of the call becomes less.

Further, in relation to the OP, why is the call and interest rate positively related with black scholes? By the same principle as the statement above should the Call not be negatively related to interest rates. Hence when rates fall and prices rise call becomes more valuable. When rates rise the call becomes less valuable as the price is now lower so no need to excercise?

#5

Put Call Parity!

Throw some numbers in the formula and solve for the value of the call.

C0 + (X / 1+ r) = S0 + P0

C0 + (20 / 1.04) = 25 + 3

Solve for C0 = 8.77

Now increase the interest rate from 4% to 5% …

C0 + (20 / 1.05) = 25 + 3

Solve for C0 = 8.95

The value of the call went up? Why? The higher rate discounted the bond at a higher rate resulting in a lower PV. That lower PV is subtracted from the right side of the equation, thus a higher left side of the equation (our call in this example).

Edit - Using this you can also see how things can change if the value of S0 (25 in this simple example) falls. Let’s say we are dealing with long term bond with high sensitivity to interest rate changes, and the 1% increase in rates causes the value of the underlying bond to fall by 10%. S0 is now 22.5, and C0 falls to 6.45.

#6

Awesome Jaywill - thanks. But this concept is not the same for Fixed Income right? i presume in FI you can use put call parity as well so wouldn’t dont the two answers clash?

#7

Do not use put-call parity for options on bonds. It’ll only confuse you.

#8

Guys I’m sorry this is still bothering me. I an really imagine a question like this coming up in the exam ‘analyst one says this analyst two says this’ style.

Can I just clarify this:

In FIXED INCOME. When interest rates rise. The price of a bond falls. This makes the call option less valuable. Hence, there is an inverse relationship.

in DERIVATIVES. When interest rates rise, what is happening to the call value? Why is it increasing?

I just want to make sure I have a clear handle on this. Sorry!

#9

For fixed income, think about the value of the underlying when interest rates change; that’s what drives the value of call options and put options.

For derivatives, the underlying is assumed to be a stock or commodity or something else whose price isn’t directly influenced by interest rates; in that case, look at put-call parity assuming that the price of the underlying is constant.

#10

S2000 you have just cleared it up!

Derivatives BSM- RHO, Call option increases with Interest.

Fixed income Embedded options- interest rate falls, BOND price increases- call at the Money- Call Value increases.

#11

So for derivatives interest rates go up calls and puts go up interest rates go down calls and puts go down?

#12

Oh if we use the greeks like Max has suggested then interest rates go up calls go up, puts go down. interest rates go down calls go down puts go up?

#13

Yup.

#14

For anyone still struggling with this, I think a summary is:

  • For fixed income securities (e.g. callable bonds) the impact of rate changes on the price of the bond - and the subsequent impact on the likelihood of the option being exercised - is greater than the ‘pure’ impact on the embedded option
  • So if the interest rate increases:
    • The bond price decreases, which makes the call option less likely to be called, which reduces the value of the call option (and therefore increases the value of the bond)
    • The option value all else equal increases (as described in the Derivatives portion of the curriculum) _ but _ that increase in value is smaller than the decrease in value driven by the above (i.e. it is not ‘all else equal’ because the price of the underlying has dropped)

Please correct me if I’m wrong

#15

I also understand this way. This was a good explanation, thanks all of you