Hull Q 6.14 - BT solutions 6.14

[notjusttp]

Hi David,

I have a doubt in regards to way you have solved Q6.14 hull in your HULL section

In your solutions of the 3 Euro dollar quotes given you are saying that the 3rd quote is not required. However if you refer Page 140 of Hull the value which is been taken for Fi is the “forward rate for a 90 day period beginning in 400 days”

Eurodollar maturing in 300 days gives a libor equivalent as prevalent in the 300th day cycle.For calculating the Libor rate of 398th day the forward rate as of 398th day should be taken i.e we should be taking the Eurodollar quote for the contract maturing on 398 (viz 95.62) i.e 4.4167 in your solution .

My impression is the first Eurodollar quote ( 95.83) is not required and the rest 2 are required. Can you pls clarify my understanding.

Thanks & Rgds
Amit :roll:

 

[David Harper CFA FRM]

Amit,

Right, I struggle with this too, but i think answer is okay. For me, the confusion is that a ED contract maturing in 300 days refers to a forward rate that "starts" in 300 days and "ends" 90/91 days later. Put another way, the 300 day ED futures in an estimate of the 90 day LIBOR spot rate in 300 days (not in 210 days).

In regard to the Eurodollar contract maturing in 300 days quoted at 95.83, this refers to a forward for the 90-day period starting in 300 days
(i.e, to "cover" the period from 300 days to 390/391 days; the quirk in this problem is that it extends to 398 days).

So, the "no arbitrage" idea is:

1. I can invest in the spot rate today, call it zero_398days, and receive funds at 398 days, or
2. I can invest in spot for 300 days (zero_300days), then plan to "roll over" on day 300, for the subsequent 98 days, and received funds in 398 days. My best estimate of the return from 300 to 398 is not the 398 ED contract but rather the ED contract maturity in 300 days

so:

1. (zero_398_rate)*398 needs to =
2. (zero_300_rate)*300 then rollover at (ED forward_300 days) * 98 days
again, b/c the 300 day ED contract refers to a rate that *starts* in 300 days

so, under continuous

EXP(398 zero rate*398 days) = EXP(300 zero rate * 300 days) * EXP(Eurodollar 300 forward maturity in 300 days * 98 days)
then LN() of both sides, and solve for 398 days

....so i think this approach is fine: http://public.sheet.zoho.com/public/btzoho/hull-06-14-1

David

 

[notjusttp]

Hi David,

Thanks for the detailed explanation as usual.

However there is some area of confusion in my mind. Do you mean to say that

1) A 3 month Euro dollar which matures on 300th day and the one maturing in 398 day ( as i stand on the 300th day) would have the same quote OR

2) You mean to say that on 300 th day the contract which is maturing is actually stating a rate of ED depo kept for 3 months as on the date of maturity (300th day ) and for the quote for ED maturing on 398 days is actually a rate which market looks at today which would apply for 3 months from the 398th day (i.e 6 months from today).

The source of confusion is because if i want to fix a Libor rate today for 3 months i enter into a ED futures contract maturing in 3 months as per discussion in Hull. That means on the 210th day if i want to fix a 3 month rate then i enter into ED futures expiring on 300th day. Then on mat of the ED i have the realised rate which is different from the locked in rate. But that realised rate is for a ED 3 months deposit maturing on 300th day and not for the 1 which would start for 3 months on that day.

Sorry for extra push and thanks for your usual support.

rgds
Amit

 

[David Harper CFA FRM]

Hi Amit,

Not (1).
Re: "You mean to say that on 300th day the contract which is maturing is actually stating a rate of ED depo kept for 3 months as on the date of maturity (300th day )"
Yes, this is correct but the rest does not follow...

using Hull's example, if you want to lock in a 90-day interest rate as of June, you want a contract that *matures* on June because your gain/loss is based on the difference between the contract and the prevailing 90-day rate as of June.

so, in the above, if you want to "lock-in" a 90-day rate in 300 days, then you would buy a ED futures contract that matures in 300 days.
In the above, the price of that contract is 95.83 which is an impled forward rate of 4.17%;
i.e., the implied 90-day rate (4.17%) starting in 300 days.
The long position locks in the 4.17% rate...e.g., if we go forward and the actual 90-day spot rate in 300 days is only 3%, then the long profits on the future (his hedge works for him)...

so...i think for the no-arbitrage idea, the key is: today is T0. The price of a ED contract maturing in 300 days gives us the market's consensus estimate of a 90-day forward rate starting in 300 days; i..e, F(0,300,390).

David

 

[notjusttp]

thanks for your ever amazing clarifications..cheers..Amit