In this chapter, we begin a detailed discussion of pricing and risk management of interest rate derivatives'.'. W e start with interest rate swaps'. The simplest version of an interest-rate swap is the exchange of fixed-rate interest payments for floating-rate interest payments. For example, corporation ABC might receive the fixed rate
E
and pay a floating rate. Here is a picture of a plain-vanilla swap3.Fixed vs. Floating Swap
ABC Corporation
r E l Floating Wall Street Broker- Dealer Swap Desk
Acknowledgements to Traders, Brokers, Sales: I thank brokers at Eurobrokers and traders at FCMC for breaking me into real-world aspects of interest-rate derivatives. I also thank many traders at Citibank, Smith Barney, Salomon Smith Barney, and Citigroup through the various mergers, for helpful interactions. I thank sales people for informative conversations. Much of the presentation of the practical real-world Risk Lab part of the book comes from what I learned from these people.
1
History: I got direct exposure to practical fixed-income derivatives at Eurobrokers, and as the Middle-Office risk manager for FCMC. This work was based on pricing and risk- management software that I designed and wrote.
Swap Picture: Diagrams are used to clarify deals. With complicated transactions involving various counterparties and many swap legs, diagrams are essential.
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The most common floating rate is Libor. Other rates are Prime and CMT, which we discuss Ch. 10 on interest-rate caps.
Why is this risk management for ABC? ABC Corporation may have issued fixed rate debt but prefers, for its own reasons (for example asset-liability matching), to pay floating rate debt. Therefore, ABC exchanges the interest rate payments. Alternatively, in the case that ABC issued floating rate debt but prefers to pay fixed, the swap would go the other way. From the point of view of the broker-dealer BD, the swap in the picture is a “pay-fixed” swap. The BD swap desk will hedge the swap in a manner that we will consider in some detail.
Of course, ABC may not really be performing risk management. Maybe ABC thinks that rates will decrease, due to its own analysis. Now usually the forward curve is upward sloping, implying that the markets “expectation” is that rates will increase. Thus, ABC is “betting against the forward curve”. Sometimes this has worked in the past for corporations, and other times it has failed.
We next show a plain-vanilla USD fixed-float swap in a spreadsheet-like format, maybe similar to a system you might encounter, along with comments4.
We begin with the deal definition. Again, these quantities give the deal
“kinematics” that would be stored by the books and records of the firm.
Deal Definition 1 I D
2 Fixed Rate 3 Floating Rate 4 Deal date 5 Start date 6 End Date 7 Notional 8 Currency
9 Notional Schedule?
10 Floating Rate Spread 1 1 Floating rate type 12 Cancelable?
13 Payments in Arrears?
ABC Corp. 0001 6.35%
3 months 31 1 9/96 3/27/96 312710 1
$100 MM USD No 0
USD Libor No Yes Comments: Deal Definition
Deal identification that will be put into the database.
Conventions for rate units will be specified (301360, actl360 etc.) Time between successive specifications (“fixings”) of the tloating rate Date the deal is being done. This is an old deal.
First date specified in the contract for some sort of action.
1 2 3 4 5
Systems Appearance: These range from actual spreadsheets to GUIs of whatever eclectic esthetics the systems designer chooses (naturally with feedback from the traders who have to use it). Probably the appearance would be much more attractive than what I use here for illustration. The comments would not appear; they are for the reader.
Inrerest-Rare Swaps
6 Last date specified in the contract. This deal has expired.
7 Normalization for calculating cash flows.
8 Payments in different currencies lead to “cross currency swaps“.
9 “No” means that the normalization is the same for each date.
10 Means no additional interest paid above the floating rate.
1 1 Other interest rates are possible but less common.
I 2 If cancelable, options called “swaptions” are involved.
13 Payments are made 3 months after floating rates are determined.
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Next, we give the input parameters for pricing the swap. These parameters are the forward-rate curve, where the forwards correspond to the particular floating rate on which the swap is based (3-month Libor is the most common).
Normally this curve would be produced in another part of the system and read into the module used for pricing and hedging. Therefore, we have:
Parameters
1 Forward Rate Curve Comments: Parameters
See previous chapter for discussion 1
The graph shows the forward rate curve used for the swap in this particular case along with the break-even (BE) rate of the swap.
Forward Rates and Break-Even Rate I--cFwdi?ates +BERate
1
:::I
I I I I I I I ~I ~ I I ~ I I I I ,4.0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Swaplet number
The BE rate is the average of all 20 rates with discounting included (see the section at the end for the math). The “swaplets” are the components of the swap and are discussed below.
We now give the representative pricing and overview of hedging results for this swap5.
Pricing and Overview of Hedging
1 Swap value $000 $40.5
2 BESwapRate 6.359%
3 Delta total 1714.5
4 Gamma total -87.2
Comments: Pricing and Overview of Hedging
This swap was practically “at the money“ with very little value.
Equivalent fixed rate that would have led to zero swap value.
The initial hedge would be to buy 17 14 ED futures contracts ($43IUbp).
Gamma is expressed as 100
*
change in number of contractshp.1 2 3 4
This swap is viewed from the point of view of the counterparty paying the fixed rate, at the deal date. Since the fixed rate (6.35 %) was slightly below the BE rate (6.359 %), the swap had a small positive value. A swap has a small convexity, i.e. second derivative with respect to rates producing gamma (y), due to the discount factors being nonlinear in the rates. However, gamma cannot be hedged with Eurodollar (ED) futures since ED futures have no convexity (a change in rates by 1 bp changes any ED future by $25 without any discounting correction). Therefore, gamma would need to be hedged with other instruments.
Alternative reporting specifications of A and y in $US are also used, multiplying by $25/contract/bp.
Looking Inside a Swap: Swaplets
The details of the swap involve the breakdown into “swaplets” as shown below.
Where Did this Swap Example Come From? This swap is the same as in my first 1996 CIFER tutorial. While rates have dropped dramatically since 1996, the principles are general and the results for current swaps will be similar. I wrote the swap pricer.
Interest-Rate Swaps 89 Swaplet Fwd Rate Start End $Value $000 Delta 100 Gamma
6 6.162 6/27/97 9/29/97 $ (45.00) 96 -4.7 7 6.277 9/29/97 12/29/97 $ (1 7.00) 91 -4.5 8
9 10 11
6.402 12/29/97 3/27/98 $ 11 .OO 86 -4.3 6.424 3/27/98 6/29/98 $ 17.00 91 -4.6
6.464 6/29/98 9/28/98 $25.00 86 -4.4
6.559 9/28/98 12/28/98 $44.00 85 -4.4 12
13 14 15 16
6.629 12/28/98 3/29/99 $ 58.00 83 -4.3
6.681 3/29/99 6/28/99 $68.00 81 -4.3
6.706 6/28/99 9/27/99 $72.00 80 -4.2
6.762 9/27/99 12/27/99 $82.00 78 -4.2 6.849 12/27/99 3/27/00 $98.00 77 -4.1 1 1 6.851 3/27/00( 6/27/00( $97.00
Swaplet Composition of a Swap
There are 20 swaplets in a 5-year swap. The swaplet hedges are individually given in equivalent numbers of ED futures contracts. These result from the sensitivities to forward rates for which a given swaplet depends. The biggest sensitivity is to the forward rate in the same line in the table as the swaplet, but there are other sensitivities due to the discount factors.
Here is a picture of one swaplet inside a swap.
761 -4.1
I Diagram for one swaplet I
181 6.8551 6/27/0d 9/27/001$96.00
Forward Rate
_____,
751 -4.1
Time forward from "now"
19 20
6.906 9/27/00 12/27/00 $ 103.00 72 -3.9 7.005 12/27/00 3/27/01 $ 11 8.00 70 -3.8
Simple Scenario Analysis for a Swap
Now we consider the change in the characteristics of a swap under a hypothetical simple scenario. The scenario is that all forward rates f(‘) regardless of maturity are raised by 10 bp keeping time fixed, i.e. dfJcLL,io = l o @ . Sometimes rates can change suddenly by this magnitude, e.g. in less than 1 day. We could also envision a time-dependent scenario, although the simplest and most common procedure is to separate out the time dependence by moving time forward while keeping rates fixed. The change in the forward rates (10 bp) is assumed equal for all forward rates-a “parallel shift”. Other scenarios, with the various forward rate changes taken as unequal, give the yield-curve shape risk.
The revalued or reval (“new”) results are shown below for the value of the swap and the Greeks along with the initial (“old”) results, and the changes.
Scenario Reval New Old Change
1. Swap value $000 $468.1 $40.5 $427.5
2. Delta Hedge (contracts) 1,705.8 1,714.5 (8.7) 3. Gamma (100
*
Change in # Contracts) -86.9 -87.2 0.34. Break-even Swap Rate 6.46% 6.36% 0.10%
Comments
1. Pay fixed swap makes money because the received floating rates increased 2. Fewer contracts needed to hedge the swap
3. Very little change in gamma
4. Break-even rate for the swap after rates increase also goes up 10 bp The “Math Calc” for Price Changes Using the Greeks
We now compare the approximate results using the Greeks to the exact results obtained through scenario revals. The delta contribution gives almost all the change in the value of a swap:
$25
.
df (*Obp)Scenario
bp
.
contract$ d s o e l t u = ‘contracts *
Inserting the old A = 1714.5 contracts gives $dSD,,,, = $428.6K. The convexity due to gamma is negative because the discount factors decrease with increasing rates. We have
$25
. ’[
df Scenario (IobP)]2
(8.2)SdSGumma = Ydcontracts I bp * bp 9 contract 2
Interest-Rate Swaps 91 Inserting y=-O.87 change in contracts per bp gives the small result
$dS, ,,,,,,,n = -$l.lK
.
The sum gives the “Math Calc” change $dSMo,,,~c ,The result for $dSM(,f,,,,c is equal to the scenario revaluation difference to this accuracy, $dSs,e,,,,i, = $427.5K. Actually, this result is “too accurate” because a real-world hedge would just involve an even multiple of 100, i.e. 1700 contracts.
Now we need to check the change in delta dA,,,,,, due to gamma. We should have to a good approximation
- df (‘ObP)
‘contracts - Ydcontrucis I bp ’ Scenurio
Plugging in gamma as y = -0.87 gives the change in the number of contracts as dAco,,fr~CfS = -8.7, in agreement with the reval result.