The previous chapter laid out the basic theory of pricing plain-vanilla interest rate swaps. As we saw, swap valuation boils down to extracting discount factors and using this discount factor curve to project and discount the cash flows. In this chapter, we consider the details of swap markets in practice using the USD market as our prime example.

For the USD swap market, the benchmark floating index is the 3-month London Inter-Bank Offered Rate (LIBOR), which is the prevailing 3-month interest rate for dollar-deposits between London banks. The default settlement date, also known as the Spot Date, is 2 London business days after trade date, rolled forward if necessary to be a good New York and London business day.

Every day, at 11:00 a.m. London time, the British Bankers Association (BBA) polls various dealers for cash deposit rates for various terms: Overnight (O/N), Tomorrow Next (T/N) (although no longer quoted), 1 week, 2 week, 1M, 2M, 3M, . . . , 12M, and publishes the resulting averages for various currencies. When USD swaps are traded during the day, these Libor fixings (rather than their live quoted values) are used to construct the front end of the Libor curve.

An n × m forward rate agreement (pronounced n-by-m FRA ′frä) is an agreement based on the economic value of a forward deposit starting n months from today, and maturing m months from today, with the payoff based on the difference between the actual ″m - n″-month rate versus the agreed-upon deposit rate K. So a 2 × 5 FRA is based on the 3m (forward) rate, starting in 2m. At inception, the rate K is chosen so that the FRA has zero value, that is, K is equated to the forward rate. A seasoned FRA has positive or negative value, depending on how the fixed rate K compares to the current market forward rate. The payoff of n × m FRA occurs at rate fixing date—n-months from now—and uses the actual rate for fixing and discounting (called ″FRA discounting″). For example, if one owns a 2 × 5 FRA struck at K, the payment in 2 months is: per unit of notional, where α ≈ 1/4 is the length (Act/360) of the [2m,5m] period.

Euro-dollar contracts are exchange-traded derivative contracts traded in the International Money Market (IMM) pit of the Chicago Mercantile Exchange (CME, or ″Merc″). Each contract settles on 2 London business days prior to the third Wednesday of the contract month based on that day′s BBA Libor fixing.

ED futures trade based on price, and have an implied futures rate (100-Price). They are forward contracts and have 0 value at trade time. Other than opening a futures (margin) account and posting initial margin, and paying brokerage (about $1 per contract), no money is exchanged when trading them. So if ED1 is trading at 95.00 (implied futures rate 5%), and you buy 100 contracts, you pay no money (except $100 for brokerage). However, unlike true forward contracts (FRAs) which require no cash exchange until expiration, ED contracts are cash-settled daily.

As stated, ED futures are not exactly forward rate contracts, as they have daily mark-to-market settlement, while true FRAs do not. For FRAs the P&L of the contract (the difference between the contracted value and the realized value) is only paid at the final settlement date, and hence its value today is the properly discounted value of this future settlement value. In contrast, the P&L of the future contract, while based on the same forward interest rate, is broken out into a series of daily P&Ls and paid daily without discounting. Since interest rates are highly correlated (overnight rates versus forward rates in our case), this daily settlement confers an advantage to a short position in the ED future contract, as he earns money and gets to reinvest it daily at overnight rates when interest rates are high, and loses money and hence has to borrow overnight money when interest rates are low. This systematic advantage for a short position does not go unnoticed by the market. The short is charged for this free lunch in the form of being required to sell the contract at a cheaper level than would have otherwise been if it was a true FRA. Therefore, the implied rate in the future contract (100 - price) is higher than the true forward rate, and this difference is known as the future-forward convexity adjustment. The size of convexity adjustment depends on the expected extra P&L of the short position over the life of the contract, which in turn depends on the volatility of the rates, and requires a model to calculate it. In general, it can be shown that the implied future rate is the expected rate at future′s settlement (see Appendix B). A commonly used formula based on the Ho-Lee model is as follows: for a future contract that settles in t-years, and has Normalized volatility of σ_N. As can be seen, the effect of the convexity adjustment gets larger (t² order) for later settlements, and this is one of the reasons that later contracts (3-years and out) are less liquid, as their value depends on the proper modeling of the convexity adjustment. Armed with a convexity adjustment model, one can construct a Libor curve purely based on a strip of ED futures, and calculate par-swap rate from this curve. The resulting par-swap rate is called the strip rate, and can be compared to quoted par-swap rates to potentially take advantage of differences.

The first ED future′s settlement is rarely 1m or 3m away. One usually needs to obtain the discount factor for the ED1′s settlement, and some brokers quote a stub rate, which is for a deposit starting on Spot Date, and maturing on the ED1′s settlement date (2 London business days prior to the third Wed of the contract month).

Par-swap rates for various maturities are quoted and are used to construct a Libor curve. In USD, the following maturities are quoted: 2y-10y, 12y, 15y, 20y, 25y, 30y. Longer par-swap rates (40y, 50y, 60y) are quoted as spreads to 30y rates. Also, USD par-swap rates are typically quoted as (swap) spreads to current treasuries with similarly quoted maturity (CT2, CT3, CT5, CT10, CT30), or interpolated treasury yields for other maturities, though 12y spread is usually quoted as a spread to CT10. In most other currencies, the swap rates are directly quoted, and not as spreads to government yields. For 1y maturity, the swap rate is derived and calculated from the money market futures (ED for USD). As the 1y point is the cross over point between money market and capital markets, it is often quoted with a different frequency and/or day-count basis than other rates, so care needs to be taken when trading 1y swaps.

USD par-swap rates are quoted as spreads to benchmark current (on-the-run) treasuries, and a typical broker swap screen is shown in Table 3.1, and the graph of the swap rates or spread versus maturity is known as the Swap and Spread curves, Figure 3.1. For example, a 2y spread market of 55-55.5 means that the dealer is willing to pay the yield of CT2 treasury, y₂, plus 55 bp, and receive y₂ plus 55.5 bp.

TABLE 3.1 A Typical USD Swap Broker Screen

FIGURE 3.1 USD On-the-Run Treasury, Par-Swap, and Swap-Spread Curves

When one is trading rates, one is taking duration risk. For example, receiving in $100M 2y-swap has similar risk of buying $100M 2y treasuries. In this case, the inquiry should specify that swap rates are needed: ″My client wants to receive in $100M 2y swaps. Rates! Where will you pay?″ Dealer response: ″Spreads are 55-55.5. With 2y treasury trading at 100-09/100- 092 (3.84235%-3.84657%), I will pay 4.39235% (=3.84235% + 55 bp).″ The reason that the dealer is using the offered side of the treasury (100-092 price, 3.84235% yield) is that he is paying in the swap, so he has short duration risk. U.S. swap traders are not in the business of taking duration risk (that belongs to the cash/treasury desk), only spread risk. As soon as the trade is done, the trader will cover his short and buy treasuries, that is, lift the offer of 100-092. This will leave him with spread risk, where he is paying the bid side (55 bp) of the swap-spread market. With enough 2-way flows, he will try to pay the bid side of the swap-spread market, and receive the offered side, and make a living out of the bid-offer spread (of the swap-spread market).

On the other hand, one can directly enter into a spread trade by entering into a swap and simultaneously providing the treasury hedge.

For swap maturities where an equivalent-maturity Current Treasury does not exist, one uses an interpolated treasury yield, and quotes swap spread relative to this interpolated yield. For example, a quoted 7y swap-spread market of 67-67.5 bp is relative to 3/5 y₅ + 2/5 y₁₀ where y₅, y₁₀ are the yields of the current 5y, 10y treasuries (CT5, CT10).

Similar to the treasury curve, one can express views or hedge exposure to the slope (longer rate - shorter rate) of the swap curve via curve trades. For 2 given swap maturities, one can put on a steepener by buying the curve, that is, receiving in the shorter-maturity swap and paying in the longer-maturity, with the notional of each swap chosen so that each swap has the same PV01. Similarly, one can put on a flattener by selling the curve: paying in the shorter-maturity and receiving in PV01-equivalent amount of the longer swap. For example, one can buy the 2′s/10′s swap curve by receiving in $100M 2y swap and paying in $24.54M (= $100M × 7.62/1.87) 10y swaps, where PV01 (2y) = 1.87, PV01 ( 10y) = 7.62 cents. In this trade, one is immune to parallel moves in the swap curve, but is carrying $18,700-per-01 2′s/10′s curve risk. For each 1 bp steepening/flattening of the 2′s/10′s curve, one makes/loses $18,700.

In the USD swap market, swap traders are primarily trading swap spreads, and hedge most of their duration risk, either with other swaps or with U.S. treasuries. As such, they are mostly focused on swap spreads, and manage that risk. To understand swap spreads, one has to remember that they primarily represent the average credit spread of the Libor Panel—consisting of the banks polled by BBA to determine its estimate of Libor—generally thought to be equivalent to AA-, versus U.S. government credit. While many explanations have been offered as to what drives swap spreads, they all come back to relative supply/demand of bank versus government credit. For example, in the early 1990s government-sponsored agencies like Fannie-Mae, Freddie-Mac, would hedge the duration mismatch in their mortgage portfolios with U.S. treasuries, resulting in tightening swap spreads. When later in the decade, they switched to use swaps, swap spreads in general widened. Similarly, in a steep curve environment, corporates will swap their fixed-rate debt (either existing or new issue) by receiving in swaps to take advantage of lower short-term funding costs. This increased demand for receiving in swaps results in tightening swap spreads. Another supply/demand driver occurs when the government is running a deficit and issuing more debt via U.S. treasuries: this results in the tightening of swap spreads. On the flip side, during periods of economic turmoil and flight-to-safety, there is high demand for U.S. treasuries, resulting in widening swap spreads. As noted, all of these drivers boil down to relative supply/demand.

A similar measure of swap spreads is the Matched-Maturity (sometimes called Yield-Yield) swap spreads, which measure the yield of a given treasury security to the par-swap rate of a swap maturing on the same date as the given treasury. When one buys/sells a matched-maturity spread, one buys/sell a given treasury and pays/receives in PV01-equivalent amount of a swap with the same maturity date. Therefore it is 2 trades, done as a package. When dealing with short-term treasuries, say treasury bills or treasuries less than 2 years remaining maturity, instead of swaps, one can buy/sell a strip of ED futures versus selling/buying the treasury. This yield spread is referred to as the Treasury-ED (TED) spread, expressed usually as the semi-annual yield of both (Treasury, ED) components.

Another way of trading swap spreads is via asset-swaps where the swap fixed leg′s payment and dates are required to match exactly those of a given bond, and an asset-swap spread is added to the Libor leg of the swap, with either the notional of the floating leg matching the principal amount of the bond, par-par asset swap, or the initial dirty price of the bond, Market Value asset swap. In either case, the asset-swap spread is primarily the difference between the funding level for the treasury, that is, its repo rate, versus Libor until the maturity of swap/bond.

The purest expression of swap spreads is the swap rate for a zero-coupon swap versus the yield of a similar-maturity zero-coupon treasury bond, as each instrument has a single cash flow, and hence their yield spread is just a measure of credit quality for the maturity point. A Zero-Coupon Swap consists of a fixed and a floating leg, with both legs having a single (net) payment at maturity. The floating leg is usually based on the benchmark swap index, Libor-3m for USD, and the final payment is based on the compounded interest of current and future Libor settings: where T is the maturity of the swap, and N₀ is the initial notional of the swap. Similarly, the fixed leg′s single payment is based on the compounded interest based on the quoted zero-coupon rate. For example, the fixed payment of an N-year swap with semi-annual zero-coupon rate of Z is

In practice, one has to specify the initial notional (N₀) or final notional (N_Final) of the swap, related by

The fixed leg′s single payment is then N_Final—N₀ and is called the fixed payment, paid at swap maturity (T).

In this case, since the par zero-Coupon rate is quoted so that the value of the swap at inception is zero, it is simply a way of expressing the Libor discount factor for T:

Finally, another measure of swap spreads is the zero rate spread, most often called z-spread. The idea is to apply a parallel shift to the Libor discount curve so that the market (dirty) price of the bond is recovered if all the remaining cash flows are discounted using this shifted curve. The size of the required parallel shift applied to the zero rates is the z-spread. Note that to be precise, one has to pin down the quote convention of the zero rates (compounding frequency, if any, and treatment of fractional years), so there is no universal z-spread. Moreover, while popular with quants, z-spreads are not traded: they are merely a (and yet another) measure of swap spreads, and often provide pretty much the same information or cheap/rich signals as other (traded) spreads (matched-maturity, asset swap).

A graphical representation of any of the above spreads (headline, matched-maturity, asset swap, z, zero-coupon) versus maturity is referred to as the swap spread curve, or simply the spread curve. One can express views or hedge exposures to different points of the spread curve. When engaging in 2 simultaneous spread trades of different maturities, one is said to be trading spread of spreads or alternatively since each spread trade consists of two trades, one cash (treasury), and one swap, it is also referred to as a box trade. For example, when taking views on the slope of spread curve, say between 2y point and 5y point, buying a ″2′s-5′s spread of spreads″ means buying the 5y spreads (buying 5y cash, paying in 5y swaps), and selling the 2y spread (selling 2y cash, receiving in 2y swaps), thereby profiting from steepening of spread curve between the 2-year and 5-year maturities. The amount of treasuries (and hence swaps) on each leg is adjusted so that each will have the same spread PV01. In this way, the trade is immune to parallel moves in the spread curve.

A future headline swap spread can be locked via spread-lock trades, which come in 2 varieties: discrete-setting (also called European) spread-lock, or rolling spread-lock. In a discrete-setting spread-lock agreement for the N-year spread, the N-year headline swap spread is locked at a fixed level K for a future date, and at expiry, one enters into a N-year spread trade. If long a spread-lock, one pays in N-year swap with fixed rate set to expiry date′s par-swap rate S (as published by ISDAFIX 11:00 a.m. New York) and buying PV01-equivalent amount of US CT[N] at a yield of S—K , that is buying the headline spread at K. If one is short the spread-lock, then one receives in swaps at S and sells PV01-equivalent amount of CT[N] at S—K yield. Instead of physical settlement, one could instead opt for cash-settlement with cash value set to the difference between the N-year headline spread (as published by ISDAFIX 11:00 a.m. New York) versus the locked spread K , multiplied by PV01 of an N-year swap.

As opposed to bonds where we are dealing with a single yield-to-maturity, in swap-land we are dealing with a series of interest rates that are aggregated to construct a discount factor curve. This discount factor curve is then used to discount swap cash flows and calculate par-swap rates and forward rates. Therefore the value of a swap depends on all the input instruments used to construct the discount factor curve.

Swap and option traders are not only interested in the parallel/partial deltas, but also how their deltas change when the market moves, that is, convexity which in swap-land is also referred to as gamma. When the market moves, a completely hedged book can gain/lose duration, and needs to be rebalanced. Often when there is a large market movement (market gap), there is not enough time to recompute partial deltas for a large book. In order to be prepared for such movements, traders precompute parallel and partial deltas for a variety of scenarios (typically a series of parallel shifts) to come up with a gamma ladder. This allows them to quickly rebalance their books under fast market conditions.

Another risk that swap traders pay attention to is exposure of the floating leg to upcoming Libor fixings. Swap curves are usually constructed out of O/N, 1m, 2m, 3m, money-market futures, and par-swap quotes, and the bucketed risk to these rates are calculated. However, for the very front end, say the first 3 months, the difference between derived forward-3m rates versus their actual fixings, called reset risk, has to be monitored and hedged. The hedging is usually done via FRAs that start on the same date as the upcoming resets, and interdealer brokers routinely quote a strip of daily FRAs for, say, any day from today to 3 months from now.

TABLE 3.2 Parallel and Partial PV01s for a $100M 5.5y Swap

When trading swaps and their derivatives, they are generally quoted and traded based on terms, for example, a ″2-year″ swap, since the specification of all the relevant dates is cumbersome and time-consuming. Each swap market has a series of standard conventions used as default to generate most of the relevant dates (see Table 3.3). Therefore, only the most salient terms of a swap are communicated at trade time, and any remaining date ambiguity is generally resolved at the trade confirmation level, and if not captured at that stage, ultimately resolved (usually not amicably!) on payment dates. Note that each day of missed interest for a typical level of interest rates, say 5%, will cost 1.3889 cents, or $13,889.89 for a $100M swap, a typical size. A missed day here, and a missed day there, and pretty soon. . . .

Each swap leg is based on a series of contiguous calculation periods, book marked by the effective and the maturity dates. For each typical calculation period, there are 4 salient dates. Calculation start/end dates: the first/last day for the calculation period where interest accrues. These dates do not need to be, but often are, adjusted to be good business days; payment date: the date where the interest payment is made, which needs to be a good business day; index reset date: for floating payments, the date where the index, say 3m-Libor, is observed/reset.

TABLE 3.3 Typical Swap Conventions

The first step is to resolve the effective and maturity dates of the swap, as these control the total number of interest-accrual days for each leg. Unless otherwise specified, swaps are spot-starting, so the Effective date is the same as the spot date. The spot date, or the default settlement date is market-specific: for USD market, it is 2 London business days after the trade date, adjusted forward if necessary to be a good NY+LON business day. In our example, the trade date is Wed 27-Feb-2008, and the spot date and hence the effective date is Fri, 29-Feb-2008 (2008 is a leap year). Maturity date is ″1-year″ after the effective date, which gets us to Sat, 28-Feb-2009. Since USD swaps are adjusted MFOL, this means that the adjusted maturity date is Fri, 27-Feb-2009. So we have a swap from 29-Feb-2008 to 27-Feb-2009.

The fixed leg of USD swaps is paid semiannually, so we need to break the swap term into two calculation periods, [29-Feb-2008, X], [X, 27-Feb- 2009]. To determine the X, we have to first generate the date, and then adjust it according to the specified rule, MF—if ″unadjusted″ was specified, then we would leave it be even if it is a holiday.

The floating leg of USD swaps is based on the 3-month Libor index, and is reset, accrued Act/360, and paid quarterly, so we need to generate 3 new quarterly intermediate dates between the effective and maturity dates: [29-Feb- 2008, X₁], [X₁, X₂], [X₂,X₃],[X₃, 27-Feb-2009]. A new wrinkle that arises is whether these dates are generated recursively, or with respect to the effective/maturity dates. For example, assume that we are generating the dates backwards starting from the maturity date. Having generated the intermediate date 3 months prior to the maturity date, X₃, do we use X₃ to generate a date 3 months prior to it, X₂, or do we generate a date 6 months prior to the maturity date? The latter choice is the most common. A similar approach applies if generating dates forward from the effective date.

Having generated the calculation periods, the next step is to generate the payment dates corresponding to each calculation period Since calculation dates are usually adjusted to be good business days, payment dates equate the calculation end date for each period (as long as the payment/calcuation periodicity and holiday calendars match), however for unadjusted calculation periods, the calculation end date (if a holiday) needs to be adjusted to be a good business day so that payments can actually be made and settled.

Finally, for each calculation period of the floating leg, the rate reset date for the floating index needs to be generated. For USD swaps, the index is 3m-Libor and is observed/reset 2 London days prior to the beginning of each calculation period. On each calculation period′s index reset date, the market rate for a 3-month Libor deposit starting on the calculation start date, as published by BBA, is observed. These start/end dates of this underlying deposit are sometimes called rate effective/maturity dates or index start/end dates. While by construction, the rate effective date coincides with calculation start date, the rate end date does not necessarily coincide with the calculation end date, and these dates can be off by a few days. This is due to the way the swap calculation periods are generated (multiple months from the anchored unadjusted maturity date) and then rolled (MF, New York, and London) versus how the rate end date is generated (3 months after the calculation/rate start date) and rolled (MF, London only). Another way that this mismatch can arise is for IMM Swaps, where calculation periods are not 3 months apart, but cover inter-IMM intervals—sometimes called IMM gaps—that is, they run through 3rd Wednesdays of each quarter-end, and usually have 13 weeks (91 days), but occasionally 14 weeks (98 days). This date mismatch—while small, a few days for regular swaps, but as long as 6 days for IMM swaps—somewhat invalidates the perfect replication needed to establish the value of a floating leg as D(Effective Date) - D(Maturity Date), but is usually ignored in practice, or compensated for via a delay-of-payment convexity adjustment.

The actual cash flows of the swap for each calculation period consist of the interest rate—either fixed rate or the floating index—accrued for the length of the calculation period according to the specified day-count basis, and paid on the corresponding payment date. For a given calculation period, the day-count basis specifies the length of year for which the interest accrues. The most common type of bases are 30/360 (or any of its variants), Act/365, Act/360, and to a much lesser degree, Act/Act which is different and not to be confused with the Act/Act convention used to calculate accrued interest for bonds. The value of a swap is the PV of these cash flows, FV′ed to the settlement date—sometimes called the as-of date—which usually corresponds to the spot date.

The cash flows for a standard 1y USD swap with 28-th rolls traded on Wed 27-Oct-2008 are presented in Tables 3.4 and 3.5. Note that if the fixed leg′s basis was 30E/360 ISDA instead of 30/360, the first calculation period (29-Feb-08 to 28-Aug-08) would have 178 days rather than 179, and would result in a different par-swap rate or swap value.

TABLE 3.4 Fixed Leg for 1y USD Swap

TABLE 3.5 Floating Leg of a 1y USD Swap