valuing an IRS

Discussion of Pricing, hedging and Trading Financial Instruments
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david
Posts: 22
Joined: Thu Apr 07, 2011 10:28 am

valuing an IRS

Postby david » Thu Aug 09, 2012 6:30 am

Hi Carol,

In the example of III.1.5 valuing an interest rate swap, in the solution you mentioned that for valuation purposes we can assume the realzied spot rates are the current forward rates. Just wondering why we can legitimately assume this when we value an IRS?

Thanks

David

coalexander
Posts: 815
Joined: Sun Sep 28, 2008 10:30 pm

Re: valuing an IRS

Postby coalexander » Thu Aug 09, 2012 11:30 am

Hi David,

The forward curve is derived using no-arbitrage based on market instruments i.e. spot rates available in the market now. For instance, long 5 year spot and short 4 year spot gives 1-year forward rate at 4 years. Thus, you get a no-arbitrage price for the swap, based on no-arbitrage forward rates. Of course, you don't HAVE to use these forward rates. If you think you have a model that is a better predictor of realized spot rates than the current forward rates, by all means use your model .... but if the rates you use lie outside the no-arbitrage range defined by the forward curve you will have introduced some extra risks.

You see, banks typically buy variable LIBOR - spread and pay fixed , with counter-parties that are corporate clients who have issued variable rate bonds for financing but want fixed rates on their books because corporations use cash accounting, not MtM. [If banks would underwrite fixed rate bonds for corporations in the first place, the IRS market would be much smaller] The banks aims to make money on the LIBOR spread [well, as well as artificially depressing LIBOR rates it seems, eh?]

Anyway, banks should be hedging the risks from the variable rates, to make money only from small spreads on high volume business. But if these rates are defined by a model, and not via no-arbitrage based on current market instruments, it is more difficult to hedge them.

Cheers, Carol

david
Posts: 22
Joined: Thu Apr 07, 2011 10:28 am

Re: valuing an IRS

Postby david » Thu Aug 09, 2012 3:58 pm

Haha...banks also aim to artificially depress Libor rates it seems....it really brought smiles to my face!!!

After digesting what you've written, it seems that i have come to understand that because a swap is essentially a forward strip and forward pricing is based on no-arb, it makes sense to price a swap using foward rates so that the swap price is also no-arb. Did i understand correctly?

2) Also, whenever forward curve is mentioned, do we typically mean that on this curve the set of forward rates share the same tenor but different terms???

3)in pricing a vanilla IRS with LIBOR as underlier, instead of deriving forward rates from the current set of Eurodollar time deposits, can we use directly the implied rates by the eurodollar futures prices? Or you would rather derive the forward curve from the LIBOR spot curve, for the reason that eurodollar futures prices are biased due to convexity effects? (Could you also explain what is meant by convexity here? i understand why the futures prices are biased upward relative to FRAs as you have explained in the book, just not quite sure what convexity really is here.)

4)what are the extra risks induced if one uses model-derived rates that lie outside the no-arb range??

5)in the real trading world, to hedge a vanilla IRS, traders use Eurodollar futures or FRAs as both seem to be quite liquid?

Much appreciation for your help!

David

coalexander
Posts: 815
Joined: Sun Sep 28, 2008 10:30 pm

Re: valuing an IRS

Postby coalexander » Fri Aug 10, 2012 8:16 am

Hi David

Yes, you understand correctly, and the answer to question 2) is also yes.

3) You can't use futures prices because they are biased by the margining. That's basically what we mean by the convexity. To explain it fully, consider this example:

Suppose you buy the futures at 96, so rate is 4%

After one month, the price can be 95 (rate is 5%) or 97 (so rate is 3%)

Rate moves randomly up or down during the month. If the rate ends up at 3% you will have been making money and funds would have been released from the margin account at a rate of about 3.5% on average, say -- but if it ends up at 5% then on average you have to finance extra margin at 4.5% (say)

These figures are rough and the example is toy. But the point to be illustrated should be clear: i.e. the rate at which you must finance extra margin will be greater than the rate at which you can realize money from margin account if rate goes down.

Now, the expected P&L from margin account is E[k \Delta F r] where k is the multiple which determines the margin, \Delta F is the change in futures price and r is the interest rate. Now, this is not zero, even though \Delta F = 0 (its a martingale) because there is a negative covariance term. To see this, our example showed that \Delta M (change in margin) = k \Delta F is correlated with r. That is, when \Delta M is positive (you lose money) r is higher then when \Delta M is negative (you make money).

Since E[XY] = E[X]E[Y] + COV[X,Y] and in this case the covariance term is negative, the expected P&L on the margin is non-zero. This margining effect biases futures prices relative to forwards (which have no margin account), so futures are a little bit cheaper than forwards.

Why call it convexity when actually its a covariance term? Practitioners tend to use 'convexity' as a general term referring to any second order effect, like a variance. In fact covariance is a first order effect of joint distribution, not a second order effect. However, the practitioners of old who coined the terminology would not have gone so far as to specify this distinction.

4) It is a hedging risk...you can hedge within the no-arbitrage range using instruments that are traded in the market today (spot rates) or any other instruments (eg other swaps, or bonds) with prices based on these same spot rates. That is because the no-arbitrage range is defined using these spot rates. But if you price the swap with another set of rates, not the forward rates derived from the spot rates, then what can you use to hedge them?

5) You can use anything to hedge the swap that is liquid and whose price is derived from same basic set of spot rates.

Cheers, Carol

david
Posts: 22
Joined: Thu Apr 07, 2011 10:28 am

Re: valuing an IRS

Postby david » Fri Aug 10, 2012 9:43 am

Perfect!! Got 'em all now!Thanks a million, Carol!!!

Blessings, Dave

coalexander
Posts: 815
Joined: Sun Sep 28, 2008 10:30 pm

Re: valuing an IRS

Postby coalexander » Fri Aug 10, 2012 6:56 pm

Jolly good, Cheers, Carol


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