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# FRA mapping

#### Liming

##### New Member
Dear David,

I would really appreciate it if you can help me out with the following topic: In your 2009 study notes on financial products, you mentioned that a FRA position can be decomposed into 1) Long 6 month bill 2) Short 12-month bill; I've been thing hard about this but still couldn't understand how the 12-month bill comes into play when the long FRA valuation formula should be L(M-K)(T2 - T1)exp(-rT2) and therefore should depend on M (prevailing market rate in 6 month) as well K(negotiated contract rate), nothing to do with 12-month?

Can I also just double check with you that:
1) to map a derivative position is to essentially valuate the position and find out the correct formula for calculating position value?
For example, long currency forward contract valuation (essentially) = (F-K)exp(-rt), so by replacing F with F=S*exp(r-rf)t, we can derive the forward valuation formula that is S*exp(-rf t) - K * exp(-rt) (where S and F are respectively the spot and forward price at the time of valuation)

2) Why N(d1) equals to delta? does it apply to Black-scholes option pricing as well?

Thank you very much!

Cheers
Liming
30/09/2009

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi Liming,

In reagard to the FRA, I started a fresh XLS to illustrate mapping the FRA (it's still rough; e.g., discrepancy due to compound conventions)...
...and because i think this is difficult...
note assumptions
notional of $100 3 month spot rate = 2% 6 month spot rate = 3% such that 3 month forward = 4%, so that's the FRA rate then i have three scenarios, where the 90-day LIBOR rate in 3 months is either 3%, 4% or 5% and compare 1. The FRA cash flow, to 2. the mapping so if you look at the last column, where actual LIBOR 5%, the FRA buyer, who locked in a 4% borrowing rate, is going to receive 250K (0.25 * 1% * 100) okay, so then the hard part is how the mapping is equivalent: below, you see this buyer instead is: long 3 month rate and short the 6 month rate so, he/she shorts the 6 month bond (which raises$100) and will have to be repaid in 6 months
takes the \$100 and invests at 3 month rate (2%)
...then in 3 months, he/she reinvest at the prevaling LIBOR
...if LIBOR is 5%, this reinvesment at higher LIBOR produces the gain

asja asked, "where is the preset fixed rate?" which vexed me until i did the XLS ... then you may be able to see that the fixed rate is essentially the embedded 3-month forward rate implied by the 6-month spot rate; i.e., shorting the 6 month rate is shorting the 3 month spot plus the 3 month forward @ 4% ...

1) in the above, we followed Jorion to *replicate* the value of an FRA as a function of two *primative* instruments (i.e., 3 month zero and 6 month zero). So, this is the essence of *analytical* mapping: to replicate (reproduce) the value with a combination of simpler instruments. It's looks ironic because our replication appears more complicated but we have:

one complex instrument (FRA) = two simple instruments (zeros)

but we would prefer to treat the latter as risk factors...

2) N(d1) is literally the first derivative of the option value with respect to a change in stock price; so, if you take the Black-Scholes; ie.., c = f[S, X, sigma, ....] and take first partial derivative:
dc/dS = N(d1)

Hope that helps, David

#### Liming

##### New Member
Dear David,

Thank you for your detailed answer concerning the FRA replication ! So for seller of FRA, is my following thought correct?

Conversely for seller of 3x6 FRA, he/she is borrowing 100 for 3 months and invests the proceeds for 6 months. The reason for this is that seller is effectively borrowing at 3 month spot rate for 3 month which will then need to be repaid in 3 months. The funds is used for investment for 6 months. At the expiration of the FRA (the end of the first 3 month period), the seller refinance the funding for another 3 months at the then-3 month spot rate. At the end of 6 months, the funding to repay is 100*exp[(r1+r2)*0.25] where r1 is 3 month spot rate at the beginning and r2 is 3 month spot rate at the end of first 3 month period. This contrasts with the investment return, which is 100*exp(R*0.5) where R is 6 month spot rate at the beginning. It can be seen that when r2 is small, the FRA seller gains.

Thanks.

Cheers
Liming
2/10/09

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi Liming,

Yes, I think you have described the seller leg perfectly! Thank you for sharing the thought process from the other side of the trade...
(and why have we gone to the trouble? By mapping the complex FRA to two primative spot rates, we can model the risk of the FRA by "simple" term structure [spot rate] changes...)

thanks, David

#### ajsa

##### New Member
Hi David,

It seems that the mapping can only work at the initiation of FRA when it is a zero-sum game. If after 1 month the market moves, we cannot use 2-mo long + 5 mo short to map anymore, since (R2 + original preset fixed forward rate) does not equal to (2* R5) anymore.. At the time, how should we map?

Just my 2 cents.. please let me know if you think otherwise.
Thanks.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
H asja - That's interesting...I agree we wouldn't re-map as time goes forward (to an initial value = 0, as you say)...
...but the original mapping is durable. The long and short are both zeros (rates are locked in); i.e., rows 17 and 19 in XLS above, copied here:
http://sheet.zoho.com/public/btzoho/fra-as2bonds-1
(note that nothing really changes the payoffs of the zeros b/c they are held to maturity effectively)

so as time goes forward the only revision, i think, it the 90-day libor when the shorter term position is reinvested (as Liming says, when "the seller refinance the funding for another 3 months at the then-3 month spot rate"). So, it looks to me like (in this case, the FRA), the original mapping is durable for the whole term of the FRA and only the forward 90-day LIBOR is being revised (which gives rise to variations in value)...David

#### ajsa

##### New Member
Hi David,

BTW, could you confirm which side is "long" for FRA? receving fixed or paying fixed? it seems Hull uses L(K-M)(T2 - T1)exp(-rT2), so receiving fixed is the long, so it should be long 12 mo and short 6 mo.. i know this is minor, but just to double check.

Thanks.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi ajsa - I am not aware that Hull introduces long/short (buyer/seller) re: FRA, do you have a page number?
...because it's the opposite: the long (i.e., buyer of FRA) is the one who hedges future borrowing; the *long* FRA is pay-fixed/receive-floating (Sources: Neftci & double checked with Steiner)....David

#### ajsa

##### New Member
Hi David,

Sorry for the confusion. I think you are correct. Hull did not say it was long, he just gave a formula on p87 (6th edition) and it seems it is actually for short side.

Thanks.

#### fullofquestions

##### New Member
I have reviewed the FRA screencast and various posts about FRAs. I believe I understand the product well but at the moment I am struggling with an FRA question from 2002. The wording simply isn't sitting well with me. Could someone please explain why the answer is b?

A long position in a FRA 2x5 is equivalent to the following positions in the spot market:
(a) Borrowing in 2 months to finance a 5-month investment.
(b) Borrowing in 5 months to finance a 2-month investment.
(c) Borrowing half a loan amount at 2 months and the remainder at 5 months.
(d) Borrowing in 2 months to finance a 3-month investment.