t j-1 t j t n-1 t n period j

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1 ACCRUAL SWAPS AND RANGE NOTES PATRICK S. HAGAN BLOOMBERG LP 499 PARK AVENUE NEW YORK, NY Abstract. Here we present the standard methodology for pricing accrual swaps, range notes, and callable accrual swaps and range notes. Key words. range notes, time swaps, accrual notes 1. Introduction Notation. In our notation today is always t =0,and (1.1a) D(T )=today s discount factor for maturity T. For any date t in the future, let Z(t; T ) be the value of $1 to be delivered at a later date T : (1.1b) Z(t; T ) =zero coupon bond, maturity T, as seen at t. These discount factors and zero coupon bonds are the ones obtained from the currency s swap curve. Clearly D(T ) =Z(0; T ). We use distinct notation for discount factors and zero coupon bonds to remind ourselves that discount factors D(T ) are not random; we can always obtain the current discount factors from the stripper. Zero coupon bonds Z(t; T ) are random, at least until time catches up to date t. Let (1.2a) (1.2b) f 0 (T )=today s instantaneous forward rate for date T, f(t; T )=instantaneous forward rate for date T,asseenatt. These are defined via (1.2c) D(T )=e T 0 f 0(T 0 )dt 0, Z(t; T )=e T t f(t,t 0 )dt Accrual swaps (fixed). α j t 0 t 1 t 2 t j-1 t j t n-1 t n period j Coupon leg schedule Fixedcouponaccrualswaps(aa time swaps) consist of a coupon leg swapped against a funding leg. Suppose that the agreed upon reference rate is, say, month Libor. Let (1.3) t 0 <t 1 <t 2 <t n 1 <t n 1

2 R fix R min R max L(τ) Fig Daily coupon rate be the schedule of the coupon leg, and let the nominal fixed rate be R fix. Also let L(τ st ) represent the month Libor rate fixed for the interval starting at τ st and ending at τ end (τ st )=τ st + months. Then the coupon paid for period j is (1.4a) C j = α j R fix θ j paid at t j, where (1.4b) α j =cvg(t j 1,t j )= day count fraction for t j 1 to t j, and (1.4c) θ j = #days τ st in the interval with R min L(τ st ) R max M j. Here M j is the total number of days in interval j, andr min L(τ st ) R max is the agreed-upon accrual range. Said another way, each day τ st in the j th period contibutes the amount (1.5) α j R fix M j ½ 1 if Rmin L(τ st ) R max 0 otherwise to the coupon paid on date t j. For a standard deal, the leg s schedule is contructed lie a standard swap schedule. The theoretical dates (aa nominal dates) are constructed monthly, quarterly, semi-annually, or annually (depending on the contract terms) bacwards from the theoretical enddate. Anyoddcouponisastub(shortperiod) atthe front, unless the contract explicitly states long first, short last, or long last. The modified following business dayconventionisusedtoobtaintheactual dates t j from the theoretical dates. The coverage (day count fraction) is adjusted, that is, the day count fraction for period j is calculated from the actual dates t j 1 and t j, not the theoretical dates. Also, L(τ st ) is the fixing that pertains to periods starting on date τ st, regardless of whether τ st is a good business day or not. I.e., the rate L(τ st ) set for a Friday start also pertains for the following Saturday and Sunday. Lie all fixed legs, there are many variants of these coupon legs. The only variations that do not mae sense for coupon legs are set-in-arrears and compounded. There are three variants that occur relatively frequently: Floating rate accrual swaps. Minimal coupon accrual swaps. Floating rate accrual swaps are lie ordinary accrual swaps except that at the start of each period, a floating rate is set, and this rate plus a margin is 2

3 used in place of the fixed rate R fix. Minimal coupon accrual swaps receive one rate each day Libor sets within the range and a second, usually lower rate, when Libor sets outside the range α j ½ Rfix M j R floor if R min L(τ st ) R max otherwise. (A standard accrual swap has R floor =0. These deals are analyzed in Appendix B. Range notes. In the above deals, the funding leg is a standard floating leg plus a margin. A range note is a bond which pays the coupon leg on top of the principle repayments; there is no floating leg. For these deals, the counterparty s credit-worthiness is a ey concern. To determine the correct value of a range note, one needs to use an option adjusted spread (OAS) to reflect the extra discounting reflecting the counterparty s credit spread, bond liquidity, etc. See section 3. Other indices. CMS and CMT accrual swaps. Accrual swaps are most commonly written using 1m, 3m, 6m, or 12m Libor for the reference rate L(τ st ). However, some accrual swaps use swap or treasury rates, such as the 10y swap rate or the 10y treasury rate, for the reference rate L(τ st ).TheseCMSorCMT accrual swaps are not analyzed here (yet). There is also no reason why the coupon cannot set on other widely published indices, such as 3m BMA rates, the FF index, or the OIN rates. These too will not be analyzed here. 2. Valuation. We value the coupon leg by replicating the payoff in terms of vanilla caps and floors. Consider the j th period of a coupon leg, and suppose the underlying indice is -month Libor. Let L(τ st ) be the -month Libor rate which is fixedfortheperiodstartingondateτ st and ending on τ end (τ st )=τ st + months. The Libor rate will be fixed on a date τ fix, which is on or a few days before τ st, depending on currency. On this date, the value of the contibution from day τ st is clearly α j R fix if R min L(τ st ) R max (2.1) V (τ fix ; τ st )=payoff = Z(τ fix ; t j ) M j 0 otherwise where τ fix the fixing date for τ st. We value coupon j by replicating each day s contribution in terms of vanilla caplets/floorlets, and then summing over all days τ st in the period. Let F dig (t; τ st,k) be the value at date t of a digital floorlet on the floating rate L(τ st ) with strie K. If the Libor rate L(τ st ) is at or below the strie K, the digital floorlet pays 1 unit of currency on the end date τ end (τ st ) of the -month interval. Otherwise the digital pays nothing. So on the fixing date τ fix the payoff is nown to be, ½ 1 if L(τ (2.2) F dig (τ fix ; τ st,k)=z(τ fix ; τ end ) st ) K 0 otherwise, We can replicate the range note s payoff for date τ st by going long and short digitals struc at R max and R min. This yields, (2.3) (2.4) α j R fix [F dig (τ fix ; τ st,r max ) F dig (τ fix ; τ st,r min )] M j = Z(τ fix ; τ end ) α ½ jr fix 1 if Rmin L(τ st ) R max M j 0 otherwise. This is the same payoff as the range note, except that the digitals pay off on τ end (τ st ) instead of t j. 3

4 2.1. Hedging considerations. Before fixing the date mismatch, we note that digital floorlets are considered vanilla instruments. This is because they can be replicated to arbitrary accuracy by a bullish spread of floorlets. Let F (t, τ st,k) be the value on date t of a standard floorlet with strie K on the floating rate L(τ st ).Thisfloorlet pays β [K L(τ st )] + on the end date τ end (τ st ) of the -month interval. So on the fixing date, the payoff is nown to be (2.5a) F (τ fix ; τ st,k)=β [K L(τ st )] + Z(τ fix ; τ end ). Here, β is the day count fraction of the period τ st to τ end, (2.5b) β =cvg(τ st,τ end ). The bullish spread is constructed by going long 1 εβ floors struc at K + 1 2ε and short the same number struc at K 1 2ε. This yields the payoff (2.6) 1 F (τ fix ; τ st,k+ 1 εβ 2 ε) F (τ fix; τ st,k 1 2 ε)ª = Z(τ fix ; τ end ) 1 ε 1 if L(τ st ) <K 1 2 ε K ε L(τ st) if K 1 2 ε<l(τ st) <K+ 1 2 ε 0 if K ε<l(τ st), which goes to the digital payoff as ε 0. Clearly the value of the digital floorlet is the limit as ε 0 of (2.7a) F cen (t; τ st,k,ε)= 1 F (t; τ st,k+ 1 εβ 2 ε) F (t; τ st,k 1 2 ε)ª. Thus the bullish spread, and its limit, the digitial floorlet, are directly determined by the maret prices of vanilla floors on L(τ st ). Digital floorletsmaydevelopanunbounded -ris as the fixing date is approached. To avoid this difficulty, most firms boo, price, and hedge digital options as bullish floorlet spreads. I.e., they boo and hedge the digital floorlet as if it were the spread in eq. 2.7a with ε set to 5bps or 10bps, depending on the aggressiveness of the firm. Alternatively, some bans choose to super-replicate or sub-replicate the digital, by booing and hedging it as (2.7b) F sup (t; τ st,k,ε)= 1 εβ {F (t; τ st,k+ ε) F (t; τ st,k)} or (2.7c) F sub (t; τ st,k,ε)= 1 εβ {F (t; τ st,k) F (t; τ st,k ε)} depending on which side they own. One should price accrual swaps in accordance with a des s policy for using central- or super- and sub-replicating payoffs for other digital caplets and floorlets Handling the date mismatch. We re-write the coupon leg s contribution from day τ st as (2.8) V (τ fix ; τ st )=Z(τ fix ; τ end ) Z(τ fix; t j ) Z(τ fix ; τ end ) 4 α j R fix M j 0 otherwise if R min L(τ st ) R max.

5 f(t,t) L(τ) t j-1 τ t j τ end T Fig Date mismatch is corrected assuming only parallel shifts in the forward curve The ratio Z(τ fix ; t j )/Z(τ fix ; τ end ) is the manifestation of the date mismatch. To handle this mismatch, we approximate the ratio by assuming that the yield curve maes only parallel shifts over the relevent interval. See figure 2.1. So suppose we are at date t 0. Then we assume that (2.9a) Here Z(τ fix ; t j ) Z(τ fix ; τ end ) = Z(t 0; t j ) Z(t 0 ; τ end ) e [L(τ st) L f (t 0,τ st )](t j τ end ) = Z(t 0; t j ) Z(t 0 ; τ end ) {1+[L(τ st) L f (t 0,τ st )](τ end t j )+ }. (2.9b) L f (t 0,τ st ) Z(t 0; τ st ) Z(t 0 ; τ end ) +bs(τ st ), βz(t 0 ; τ end ) is the forward rate for the -month period starting at τ st, as seen at the current date t 0, bs(τ st ) is the floating rate s basis spread, and (2.9c) β =cvg(τ st,τ end ), is the day count fraction for τ st to τ end. Since L(τ st )=L f (τ fix,τ st ) represents the floating rate which is actually fixedonthefixing date τ ex, 2.9a just assumes that any change in the yield curve between t j and τ end is the same as the change L f (τ fix,τ st ) L f (t 0,τ st ) in the reference rate between the observation date t 0,andthefixing date τ fix.seefigure 2.1. We actually use a slightly different approximation, (2.10a) where Z(τ fix ; t j ) Z(τ fix ; τ end ) Z(t 0; t j ) Z(t 0 ; τ end ) 1+ηβL(τ st ) 1+ηβL f (t 0,τ st ) (2.10b) η = τ end t j. τ end τ st We prefer this approximation because it is the only linear approximation which accounts for the day count basis correctly, is exact for τ st = t j, and is centerred around the current forward value for the range note. 5

6 R fix R min L 0 R max L(τ) Fig Effective contribution from a single day τ, after accounting for the date mis-match. With this approximation, the payoff from day τ st is ½ 1+ηβL(τ (2.11a) V (τ fix ; τ) =A(t 0,τ st )Z(τ fix ; τ end ) st ) if R min L(τ st ) R max 0 otherwise as seen at date t 0.Heretheeffective notional is (2.11b) A(t 0,τ st )= α jr fix Z(t 0 ; t j ) M j Z(t 0 ; τ end ) 1 1+ηβL f (t 0,τ st ). We can replicate this digital-linear-digital payoff by using a combination of two digital floorlets and two standard floorlets. Consider the combination (2.12) V (t; τ st ) A(t 0,τ st ) {(1 + ηβr max )F dig (t, τ st ; R max ) (1 + ηβr min )F dig (t, τ st ; R min ) ηf(t, τ st ; R max )+ηf(t, τ st ; R min ). Setting t to the fixing date τ fix shows that this combination matches the contribution from day τ st in eq. 2.11a. Therefore, this formula gives the value of the contribution of day τ st for all earlier dates t 0 t τ fix as well. Alternatively, one can replicate the payoff as close as one wishes by going long and short floorlet spreads centerred around R max and R min. Consider the portfolio (2.13a) Ṽ (t; τ st,ε)= A(t 0,τ st ) εβ a1 (τ st )F (t; τ st,r max ε) a 2(τ st )F (t; τ st,r max 1 2 ε) a 3 (τ st )F (t; τ st,r min ε)+ a 4(τ st )F (t; τ st,r min 1 2 ε)ª with (2.13b) a 1 (τ st )=1+ηβ(R max 1 2 ε), a 2(τ st )=1+ηβ(R max ε), (2.13c) a 3 (τ st) =1+ηβ(R min 1 2 ε), a 4(τ st )=1+ηβ(R min ε). Setting t to τ fix yields (2.14) 0 if L(τ st ) <R min 1 2 ε Ṽ = A(t 0,τ st )Z(τ fix ; τ end ) 1+ηβL(τ st ) if R min ε<l(τ st) <R max 1 2 ε 0 if R max ε<l(τ st) 6,

7 with linear ramps between R min 1 2 ε<l(τ st) <R min ε and R max 1 2 ε<l(τ st) <R max + 1 2ε. As above, most bans would choose to use the floorlet spreads (with ε being 5bps or 10bps) instead of using the more troublesome digitals. For a ban insisting on using exact digital options, one can tae ε to be 0.5bps to replicate the digital accurately.. We now just need to sum over all days τ st in period j and all periods j in the coupon leg, (2.15) V cpn (t) = t j X j=1 τ st =t j 1 +1 A(t 0,τ st ) 1+ηβ(Rmax 1 εβ 2 ε) F (t; τ st,r max ε) 1+ηβ(R max ε) F (t; τ st,r max 1 2 ε) 1+ηβ(R min 1 2 ε) F (t; τ st,r min ε) + 1+ηβ(R min ε) F (t; τ st,r min 1 2 ε)ª. This formula replicates the value of the range note in terms of vanilla floorlets. These floorlet prices should be obtained directly from the maretplace using maret quotes for the implied volatilities at the relevent stries. Of course the centerred spreads could be replaced by super-replicating or sub-replicating floorlet spreads, bringing the pricing in line with the ban s policies. Finally, we need to value the funding leg of the accrual swap. For most accrual swaps, the funding leg pays floating plus a margin. Let the funding leg dates be t 0, t 1,..., tñ. Then the funding leg payments are (2.16) cvg( t i 1, t i )[R flt i + m i ] paid at t i, i =1, 2,..., ñ, where R flt i is the floating rate s fixing for the period t i 1 <t< t i,andthem i is the margin. The value of the funding leg is just (2.17a) V fund (t) = ñx cvg( t i 1, t i )(r i + m i )Z(t; t i ), i=1 where, by definition, r i is the forward value of the floating rate for period t i 1 <t< t i : (2.17b) r i = Z(t; t i 1 ) Z(t; t i ) cvg( t i 1, t i )Z(t; t i ) +bs0 i = rtrue i +bs 0 i. Here bs 0 i is the basis spread for the funding leg s floating rate, and ri true collapses to is the true (cash) rate. This sum ñx (2.18a) V fund (t) =Z(t; t 0 ) Z(t; tñ)+ cvg( t i 1, t i )(bs 0 i +m i )Z(t; t i ). If we include only the funding leg payments for i = i 0 to ñ, thevalueis i=1 (2.18b) V fund (t) =Z(t; t i0 1) Z(t; tñ)+ ñx i=i 0 cvg( t i 1, t i )(bs 0 i +m i )Z(t; t i ) Pricing notes. Caplet/floorlet prices are normally quoted in terms of Blac vols. Suppose that on date t, afloorlet with fixing date t fix, start date τ st,enddateτ end,andstriek has an implied vol of σ imp (K) σ imp (τ st,k). Then its maret price is (2.19a) F (t, τ st,k)=βz(t; τ end ) {KN (d 1 ) L(t, τ)n (d 2 )}, 7

8 where (2.19b) d 1,2 = log K/L(t, τ st) ± 1 2 σ2 imp (K)(t fix t) σ imp (K), t fix t Here (2.19c) L(t, τ st )= Z(t; τ st) Z(t; τ end ) +bs(τ st ) βz(t; τ end ) is floorlet s forward rate as seen at date t. Today sfloorlet value is simply (2.20a) F (0,τ st,k)=βd(τ end ) {KN (d 1 ) L 0 (τ)n (d 2 )}, where (2.20b) d 1,2 = log K/L 0(τ st ) ± 1 2 σ2 imp (K)t fix σ imp (K), t fix and where today s forward Libor rate is (2.20c) L 0 (τ st )= D(τ st) D(τ end ) +bs(τ st ). βd(τ end ) To obtain today s price of the accrual swap, note that the effective notional for period j is (2.21) A(0,τ st )= α jr fix D(t j ) 1 M j D(τ end ) 1+ηβL 0 (τ st ). as seem today. See 2.11b. Putting this together with 2.13a shows that today s price is V cpn (0) V fund (0), where (2.22a) V cpn (0) = (2.22b) j=1 α j R fix D(t j ) M j t j X τ st =t j ηβ(Rmax 1 2 ε) B 1 (τ st ) 1+ηβ(R max ε) B 2 (τ st ) ε [1 + ηβl 0 (τ st )] 1+ηβ(Rmin 1 2 ε) B 3 (τ st ) 1+ηβ(R min ε) B 4 (τ st ), ε [1 + ηβl 0 (τ st )] V fund (0) = D(t 0 ) D(tñ)+ ñx cvg( t i 1, t i )(bs 0 i +m i )D( t i ). Here B ω are Blac s formula at stries around the boundaries: (2.22c) B ω (τ st )=K ω N (d ω 1 ) L 0 (τ st )N (d ω 2 ) (2.22d) i=1 d ω 1,2 = log K ω/l 0 (τ st ) ± 1 2 σ2 imp (K ω)t fix σ imp (K ω ) t fix with (2.22e) K 1,2 = R max ± 1 2 ε, K 3,4 = R min ± 1 2 ε. Calculating the sum of each day s contribution is very tedious. Normally, one calculates each day s contribution for the current period and two or three months afterward. After that, one usually replaces the sum over dates τ with an integral, and samples the contribution from dates τ one wee apart for the next year, and one month apart for subsequent years. 8

9 3. Callable accrual swaps. A callable accrual swap is an accrual swap in which the party paying the coupon leg has the right to cancel on any coupon date after a loc-out period expires. For example, a 10NC3 with 5 business days notice can be called on any coupon date, starting on the third anniversary, provided the appropriate notice is given 5 days before the coupon date. We will value the accrual swap from the viewpoint of the receiver, who would price the callable accrual swap as the full accrual swap (coupon leg minus funding leg) minus the Bermudan option to enter into the receiver accrual swap. So a 10NC3 cancellable quarterly accrual swap would be priced as the 10 year bullet quarterly receiver accrual swap minus the Bermudan option with quarterly exercise dates starting in year 3 to receive the remainder of the coupon leg and pay the remainder of the funding leg. Accordingly, here we price Bermudan options into receiver accrual swaps. Bermudan options on payer accrual swaps can be priced similarly. There are two ey requirements in pricing Bermudan accrual swaps. First, as R min decreases and R max increases, the value of the Bermudan accrual swap should reduce to the value of an ordinary Bermudan swaption with strie R fix. Besides the obvious theoretical appeal, meeting this requirement allows one to hedge the callability of the accrual swap by selling an offsetting Bermudan swaption. This criterion requires using the same the interest rate model and calibration method for Bermudan accrual notes as would be used for Bermudan swaptions. Following standard practice, one would calibrate the Bermudan accrual note to the diagonal swaptions struc at the accrual note s effective stries. For example, a 10NC3 accrual swap which is callable quarterly starting in year 3 would be calibrated to the 3 into 7, the 3.25 into 6.75,..., the 8.75 into 1.25, and the 9 into 1 swaptions. The strie Reff i for each of these reference swaptions would be chosen so that for swaption i, (3.1) value of the fixed leg value of the floating leg = value of all accrual swap coupons j i value of the accrual swap s funding leg i This usually results in stries Reff i that are not too far from the money. Intheprecedingsectionweshowedthateachcouponoftheaccrualswapcanbewrittenasacombination of vanilla floorlets, and therefore the maret value of each coupon is nown exactly. The second requirement is that the valuation procedure should reproduce today s maret value of each coupon exactly. In fact, if there is a 25% chance of exercising into the accrual swap on or before the j th exercise date, the pricing methodology should yield 25% of the vega ris of the floorlets that mae up the j th coupon payment. Effectively this means that the pricing methodology needs to use the correct maret volatilities for floorlets struc at R min and R max. This is a fairly stiff requirement, since we now need to match swaptions struc at Reff i and floorlets struc at R min and R max. This is why callable range notes are considered heavily sew depedent products Hull-White model. Meeting these requirements would seem to require using a model that is sophisticated enough to match the floorlet smiles exactly, as well as the diagonal swaption volatilities. Such a model would be complex, calibration would be difficult, and most liely the procedure would yield unstable hedges. An alternative approach is to use a much simpler model to match the diagonal swaption prices, and then use internal adjusters to match the floorlet volatilities. Here we follow this approach, using the 1 factor linear Gauss Marov (LGM) model with internal adjusters to price Bermudan options on accrual swaps. Specifically, we find explicit formulas for the LGM model s prices of standard floorlets. This enables us to compose the accrual swap payoffs (the value recieved at each node in the tree if the Bermudan is exercised) as a linear combination of the vanilla floorlets. With the payoffs nown, the Bermudan can be evaluated via a standard rollbac. The last step is to note that the LGM model misprices the floorlets that mae up the accrual swap coupons, and use internal adjusters to correct this mis-pricing. Internal adjusters can be used with other models, but the mathematics is more complex LGM. The 1 factor LGM model is exactly thehull-whitemodelexpressedasanhjmmodel. The 1 factor LGM model has a single state variable x thatdeterminestheentireyieldcurveatanytimet. 9

10 This model can be summarized in three equations. The first is the Martingale valuation formula: At any date t and state x, the value of any deal is given by the formula, Z V (t, x) (3.2a) N(t, x) = p(t, x; T,X) V (T,X) N(T,X) dx for any T>t. Here p(t, x; T,X) is the probability that the state variable is in state X at date T, given that it is in state x at date t. For the LGM model, the transition density is Gaussian (3.2b) p(t, x; T, X) = 1 p 2π[ζ(T ) ζ(t)] e (X x)2 /2[ζ(T ) ζ(t)], with a variance of ζ(t ) ζ(t). Thenumeraireis (3.2c) N(t, x) = 1 D(t) eh(t)x+ 1 2 h2 (t)ζ(t), for reasons that will soon become apparent. Without loss of generality, one sets x =0at t =0,andtoday s variance is zero: ζ(0) = 0. The ratio (3.3a) Ṽ (t, x) is usually called the reduced value of the deal. reduced value: (3.3b) V (0, 0) = V (t, x) N(t, x) Since N(0, 0) = 1, today s value coincides with today s Ṽ (0, 0) V (0, 0) N(0, 0). So we only have to wor with reduced values to get today s prices. Define Z(t, x; T ) to be the value of a zero coupon bond with maturity T,asseenatt, x. It s value can be found by substituting 1 for V (T,X) in the Martingale valuation formula. This yields (3.4a) Z(t, x; T ) Z(t, x; T ) N(t, x) Since the forward rates are defined through = D(T )e h(t )x 1 2 h2 (T )ζ(t). (3.4b) Z(t, x; T ) e T t f(t,x;t 0 )dt 0, taing T log Z shows that the forward rates are (3.4c) f(t, x; T )=f 0 (T )+h 0 (T )x + h 0 (T )h(t )ζ(t). This last equation captures the LGM model in a nutshell. The curves h(t ) and ζ(t) are model parameters that need to be set by calibration or by apriorireasoning. The above formula shows that at any date t, the forward rate curve is given by today s forward rate curve f 0 (T ) plus x times a second curve h 0 (T ), where x is a Gaussian random variable with mean zero and variance ζ(t). Thush 0 (T ) determines possible shapes of the forward curve and ζ (t) determines the width of the distribution of forward curves. The last term h 0 (T )h(t )ζ(t) is a much smaller convexity correction. 10

11 Vanilla prices under LGM. Let L(t, x; τ st ) be the forward value of the month Libor rate for the period τ st to τ end, as seen at t, x. Regardless of model, the forward value of the Libor rate is given by (3.5a) L(t, x; τ st )= Z(t, x; τ st) Z(t, x; τ end ) βz(t, x; τ end ) where (3.5b) β =cvg(τ st,τ end ) +bs(τ st )=L true (t, x; τ st )+bs(τ st ), is the day count fraction of the interval. Here L true is the forward true rate for the interval and bs(τ) is the Libor rate s basis spread for the period starting at τ. Let F (t, x; τ st,k) be the value at t, x of a floorlet with strie K on the Libor rate L(t, x; τ st ). On the fixing date τ fix the payoff is (3.6) F (τ fix,x fix ; τ st,k)=β K L(τ fix,x fix ; τ st ) + Z(τ fix,x fix ; τ end ), where x fix (3.7a) is the state variable on the fixing date. Substituting for L(τ ex,x ex ; τ st ),thepayoff becomes F (τ fix,x fix ; τ st,k) N(τ fix,x fix ) = Z(τ fix,x fix ; τ end ) N(τ fix,x fix ) 1+β(K bs(τ st )) Z(τ + fix,x fix ; τ st ). Z(τ fix,x fix ; τ end ) Knowing the value of the floorlet on the fixing date, we can use the Martingale valuation formula to find the value on any earlier date t: (3.7b) F (t, x; τ st,k) N(t, x) Z 1 = q 2π[ζ fix ζ] e (x fix x) 2 /2[ζ fix ζ] F (τ fix,x fix ; τ st,k) dx fix, N(τ fix,x fix ) where ζ fix = ζ(τ fix ) and ζ = ζ(t). Substituting the zero coupon bond formula 3.4a and the payoff 3.7a into the integral yields (3.8a) F (t, x; τ st,k)=z(t, x; τ end ) {[1 + β(k bs)]n (λ 1 ) [1 + β(l bs)]n (λ 2 )}, where µ 1+β(K bs) log ± 1 1+β(L bs) 2 (h end h st ) 2 ζ fix ζ(t) (3.8b) λ 1,2 = q, (h end h st ) ζ fix ζ(t) and where (3.8c) L L(t, x; τ st )= 1 µ Z(t, x; τ st ) β Z(t, x; τ end ) 1 +bs(τ st ) = 1 µ Dst e (h end h st )x+ 1 2 (h2 end h2 st )ζ 1 +bs(τ st ) β D end is the forward Libor rate for the period τ st to τ end, as seen at t, x. Hereh st = h(τ st ) and h end = h(τ end ). 11

12 For future reference, it is convenient to split off the zero coupon bond value Z(t, x; τ end ).Sodefine the forwarded floorlet value by (3.9) F f (t, x; τ st,k)= F (t, x; τ st,k) Z(t, x; τ end ) =[1+β(K bs)]n (λ 1 ) [1 + β(l(t, x; τ st ) bs)]n (λ 2 ). Equations 3.8a and 3.9 are just Blac s formulas for the value of a European put option on a log normal asset, provided we identify (3.10a) (3.10b) (3.10c) (3.10d) 1+β(L bs) = asset s forward value, 1+β(K bs) = strie, τ end = settlement date, and p ζfix ζ (h end h st ) = σ = asset volatility, tfix t where t fix t is the time-to-exercise. One should not confuse σ, whichisthefloorlet s price volatility, with the commonly quoted rate volatility Rollbac. Obtaining the value of the Bermudan is straightforward, given the explicit formulas for the floorlets,. Suppose that the LGM model has been calibrated, so the model parameters h(t) and ζ(t) are nown. (In Appendix A we show one popular calibration method). Let the Bermudan s notification dates be t ex 0,t ex 0 +1,...,tex K. Suppose that if we exercise on date tex, we receive all coupon payments for the intervals +1,...,n and recieve all funding leg payments for intervals i,i +1,...,ñ. The rollbac wors by induction. Assume that in the previous rollbac steps, we have calculated the reduced value (3.11a) V + (t ex,x) = value at tex,x) of all remaining exercises t ex +1,t ex +2...,t ex K at each x. We show how to tae one more step bacwards, finding the value which includes the exercise t ex at the preceding exercise date: (3.11b) V + (t ex 1,x) 1 = value at tex 1 of all remaining exercises t ex,x),t ex +1,t ex +2...,t ex K. Let P (x)/n (t ex,x) be the (reduced) value of the payoff obtained if the Bermudan is exercised at tex Asseenattheexercisedatet ex the effective notional for date τ st is. (3.12a) A(t ex,x,τ st )= α jr fix Z(t ex,x; t j ) 1 M j Z(t ex,x; τ end) 1+ηβL f (t ex,x; τ st), where we recall that (3.12b) η = τ end(τ st ) t j τ end (τ st ) τ st, β =cvg(τ st,τ end (τ st )). 12

13 Reconstructing the reduced value of the payoff (see equation 2.15) yields (3.13a) P (x),x) = α j R fix Z(t ex,x; t t j) X j ½ 1+ηβ(Rmax 1 2 ε) εβm j,x) 1+ηβL f (t ex,x; τ st) F f(t ex,x; τ st,r max ε) j=+1 τ st=t j ηβ(R max ε) 1+ηβL f (t ex,x; τ st) F f(t ex,x; τ st,r max 1 2 ε) 1+ηβ(R min 1 2 ε) 1+ηβL f (t ex,x; τ st) F f(t ex,x; τ st,r min ε) + 1+ηβ(R min ε) ¾ 1+ηβL f (t ex,x; τ st) F f(t ex,x; τ st,r min 1 2 ε) Z(tex,x, t i 1) Z(t ex,x, tñ) ñx,x) cvg( t i 1, t i )(bs i +m i ) Z(tex,x, t i ) i=i +1,x). This payoff includes only zero coupon bonds and floorlets, so we can calculate this reduced payoff explicitly using the previously derived formula 3.9. The reduced valued including the th exercise is clearly V (t ex (3.13b),x) ½ ¾ + P (x) V (tex =max,x),x),x),,x) at each x. Using the Martingale valuation formula we can roll bac to the preceding exercise date by using finite differences, trees, convolution, or direct integration to compute the integral V + (t ex 1 (3.13c),x) Z 1,x) = 1 V (t ex p,x) 2 /2[ζ ζ ] 1 2π[ζ ζ 1 ] dx,x)e (X x) at each x. Hereζ = ζ(t ex ) and ζ 1 = ζ(t ex 1 ). At this point we have moved from t ex to the preceding exercise date tex 1. Wenowrepeattheprocedure: at each x we tae the max of V + (t ex 1,x)/N (tex 1,x) and the payoff P 1(x)/N (t ex 1,x) for tex 1,andthen use the valuation formula to roll-bac to the preceding exercise date t ex 2,etc.Eventuallyweworourway througn the first exercise V (t ex 0,x). Then today s value is found by a final integration: (3.14) V (0, 0) = V (0, 0) N(0, 0) = 1 Z V (t ex p 0,X) 2πζ0 2 /2ζ 0 dx. 0,X) e X 3.2. Using internal adjusters. The above pricing methodology satisfies the first criterion: Provided we use LGM (Hull-White) to price our Bermudan swaptions, and provided we use the same calibration method for accrual swaps as for Bermudan swaptions, the above procedure will yield prices that reduce to the Bermudan prices as R min goes to zero and R max becomes large. However the LGM model yields the following formulas for today s values of the standard floorlets: (3.15a) F (0, 0; τ st,k)=d(τ end ) {[1 + β(k bs)]n (λ 1 ) [1 + β(l 0 bs)]n (λ 2 )} where µ 1+β(K bs) log ± 1 1+β(L 0 bs) 2 σ2 mod t fix (3.15b) λ 1,2 =. σ mod tfix 13

14 Here (3.15c) L 0 = D st D end βd end +bs(τ st ) is today s forward value for the Libor rate, and (3.15d) σ mod =(h end h st ) q ζ fix /t fix is the asset s log normal volatility according to the LGM model. We did not calibrate the LGM model to these floorlets. It is virtually certain that matching today s maret pricesq for the floorlets will require using an implied (price) volatility σ mt which differs from σ mod =(h end h st ) ζ fix /t fix Obtaining the maret vol. Floorlets are quoted in terms of the ordinary (rate) vol. Suppose the rate vol is quoted as σ imp (K). Then today s maret price of the floorlet is (3.16a) F mt (τ st,k)=βd(τ end ) {KN (d 1 ) L 0 N (d 2 )} where (3.16b) d 1,2 = log K/L 0 ± 1 2 σ2 imp (K)t fix σ imp (K) t fix The price vol σ mt is the volatility that equates the LGM floorlet value to this maret value. It is defined implicitly by (3.17a) [1 + β(k bs)]n (λ 1 ) [1 + β(l 0 bs)]n (λ 2 )=βkn (d 1 ) βl 0 N (d 2 ), with µ 1+β(K bs) log ± 1 1+β(L 0 bs) 2 σ2 mt t fix (3.17b) λ 1,2 = σ mt tfix (3.17c) d 1,2 = log K/L 0 ± 1 2 σ2 imp (K)t fix σ imp (K) t fix Equivalent vol techniques can be used to find the price vol σ mt (K) which corresponds to the maret-quoted implied rate vol σ imp (K) : (3.18) σ imp (K) σ2 imp t fix σ4 imp t2 fix + = σ mt (K) log L 0 /K σ2 mt t µ fix σ4 mt t2 1+β(L0 bs) fix log 1+β(K bs) If this approximation is not sufficiently accurate, we can use a single Newton step to attain any reasonable accuracy. 14

15 digital floorlet value σ mt σ mod L 0 /K L/K Fig Unadjusted and adjusted digital payoff Adjusting the price vol. The price q vol σ mt obtained from the maret price will not match the LGM model s price vol σ mod =(h end h st ) ζ fix /t fix. This is easily remedied using an internal adjuster. All one does is multiply the model volatility with the factor needed to bring it into line with the actual maret volatility, and use this factor when calculating the payoffs. Specifically, in calculating each payoff P (x)/n (t ex,x) in the rollbac (see eq. 3.13a), one maes the replacement (3.19) (3.20) q q (h end h st ) ζ fix ζ(t ex )= (h end h st ) ζ fix ζ(t) σ mt σ q mod = 1 ζ(t ex )/ζ(t p fix)σ mt tfix. With the internal adjusters, the pricing methodology now satisfies the second criteria: it agrees with all the vanilla prices that mae up the range note coupons. Essentially, all the adjuster does is to slightly sharpen up or smear out the digital floorlet s payoff to match today s value at L 0 /K. This results in slightly positive or negative price corrections at various values of L/K, but these corrections average out to zero when averaged over all L/K. Maing this volatility adjustment is vastly superior to the other commonly used adjustment method, which is to add in a fictitious exercise fee to match today s coupon value. Adding a fee gives a positive or negative bias to the payoff for all L/K, even far from the money, where the payoff was certain to have been correct. Meeting the second criterion forced us to go outside the model. It is possible that there is a subtle arbitrage to our pricing methodology. (There may or may not be an arbitrage free model in which extra factors positively or negatively correlated with x enable us to obtain exactly these floorlet prices while leaving our Gaussian rollbac unaffected). However, not matching today s price of the underlying accrual swap would be a direct and immediate arbitrage. 15

16 4. Range notes and callable range notes. In an accrual swap, the coupon leg is exchanged for a funding leg, which is normally a standard Libor leg plus a margin. Unlie a bond, there is no principle at ris. The only credit ris is for the difference in value between the coupon leg and the floating leg payments; even this difference is usually collateralized through various inter-dealer arrangements. Since swaps are indivisible, liquidity is not an issue: they can be unwound by transferring a payment of the accrual swap s mar-to-maret value. For these reasons, there is no detectable OAS in pricing accrual swaps. A range note is an actual bond which pays the coupon leg on top of the principle repayments; there is no funding leg. For these deals, the issuer s credit-worthiness is a ey concern. One needs to use an option adjusted spread (OAS) to obtain the extra discounting reflecting the counterparty s credit spread and liquidity. Here we analyze bullet range notes, both uncallable and callable. The coupons C j of these notes are set by the number of days an index (usually Libor) sets in a specified range, just lie accrual swaps: (4.1a) C j = t j X τ st =t j 1 +1 α j R fix M j ½ 1 if Rmin L(τ st ) R max 0 otherwise where L(τ st ) is month Libor for the interval τ st to τ end (τ st ),andwhereα j and M j are the day count fraction and the total number of days in the j th coupon interval t j 1 to t j. In addition, these range notes repay the principle on the final pay date, so the (bullet) range note payments are: (4.1b) C j paid on t j, j =1, 2,...n 1, (4.1c) 1+C n paid on t n. For callable range notes, let the notification on dates be t ex for = 0, 0 +1,...,K 1,K with K<n. Assume that if the range note is called on t ex, then the strie price K is paid on coupon date t and the payments C j are cancelled for j = +1,...,n Modeling option adjusted spreads. Suppose a range note is issued by issuer A. Define Z A (t, x; T ) to be the value of a dollar paid by the note on date T,asseenatt, x. We assume that (4.2) Z A (t, x; T )=Z(t, x; T ) Λ(T ) Λ(t), where Z(t, x; T ) is the value according to the Libor curve, and (4.3) Λ(τ) = D A(τ) D(τ) e γτ. Here γ is the OAS of the range note. The choice of the discount curve D A (τ) depends on what we wish the OAS to measure. If one wishes to find the range note s value relative to the issuer s other bonds, then one should use the issuer s discount curve for D A (τ); the OAS then measures the note s richness or cheapness compared to the other bonds of issuer A. Ifonewishestofind the note s value relative to its credit ris, then the OAS calculation should use the issuer s risy discount curve or for the issuer s credit rating s risy discount curve for D A (τ). If one wishes to find the absolute OAS, then one should use the swap maret s discount curve D(τ), sothatλ(τ) is just e γτ. When valuing a non-callable range note, we are just determining which OAS γ is needed to match the current price. I.e., the OAS needed to match the maret s idiosyncratic preference or adversion of the bond. When valuing a callable range note, we are maing a much more powerful assumption. By assuming that the same γ can be used in evaluating the calls, we are assuming that (1) the issuer would re-issue the bonds if it could do so more cheaply, and (2) on each exercise date in the future, the issuer could issue debt at the same OAS that prevails on today s bond. 16,

17 4.2. Non-callable range notes. Range note coupons are fixed by Libor settings and other issuerindependent criteria. Thus the value of a range note is obtained by leaving the coupon calculations alone, and replacing the coupon s discount factors D(t j ) with the bond-appropriate D A (t j )e γt j : (4.4a) V A (0) = α j R fix D A (t j )e γtj M j=1 j t j X τ st =t j ηβ(Rmax 1 2 ε) B 1 (τ st ) 1+ηβ(R max ε) B 2 (τ st ) ε [1 + ηβl 0 (τ st )] 1+ηβ(Rmin 1 2 ε) B 3 (τ st ) 1+ηβ(R min ε) B 4 (τ st ) ε [1 + ηβl 0 (τ st )] +D A (t n )e γtn. Here the last term D A (t n )e γt n is the value of the notional repaid at maturity. As before, the B ω are Blac s formulas, (4.4b) B ω (τ st )=K j N (d ω 1 ) L 0 (τ st )N (d ω 2 ) (4.4c) d ω 1,2 = log K ω/l 0 (τ st ) ± 1 2 σ2 imp (K ω)t fix σ imp (K ω ) t fix (4.4d) K 1,2 = R max ± 1 2 ε, K 3,4 = R min ± 1 2 ε, and L 0 (τ) is today s forward rate: (4.4e) L 0 (τ st )= D(τ st) D(τ end ) βd(τ end ) Finally, (4.4f) η = τ end t j τ end τ st Callable range notes. We price the callable range notes via the same Hull-White model as used to price the cancelable accrual swap. We just need to adjust the coupon discounting in the payoff function. Clearly the value of the callable range note is the value of the non-callable range note minus the value of the call: (4.5) V callable A (0) = VA bullet (0) VA Berm (0). Here VA bullet (0) is the today s value of the non-callable range note in 4.4a, and VA Berm (0) is today s value of the Bermudan option. This Bermudan option is valued using exactly the same rollbac procedure as before, 17

18 except that now the payoff is (4.6a) (4.6b) j=+1 α j R fix Z A (t ex,x; t j) εβm j,x) P (x),x) = t j X τ st =t j 1 +1 ½ 1+ηβ(Rmax 1 2 ε) 1+ηβL f (t ex,x; τ st) F f(t ex,x; τ st,r max ε) 1+ηβ(R max ε) 1+ηβL f (t ex,x; τ st) F f (t ex,x; τ st,r max 1 2 ε) 1+ηβ(R min 1 2 ε) 1+ηβL f (t ex,x; τ st) F f (t ex,x; τ st,r min ε) + 1+ηβ(R min ε) ¾ 1+ηβL f (t ex,x; τ st) F f(t ex,x; τ st,r min 1 2 ε) + Z A(t ex,x,t n) Z A,x) K (tex,x,t ),x) Here the bond specific reduced zero coupon bond value is Z A (t ex (4.6c),x,T),x) = D(tex ) D A (t ex )D A(T )e the (adjusted) forwarded floorlet value is ex γ(t t ) e h(t )x 1 2 h2 (T )ζ ex, F f (t ex,x; τ st,k)=[1+β(k bs)]n (λ 1 ) [1 + β(l(t ex,x; τ st ) bs)]n (λ 2 ) µ 1+β(K bs) log 1+β(L bs) (4.6d) λ 1,2 = and the forward Libor value is L L (t ex,x; τ st) = 1 µ Z(t ex (4.6e) β (4.6f) = 1 β ± 1 2 [1 ζ(tex )/ζ(t fix)]σ 2 mt t fix p 1 ζ(t ex )/ζ(t fix )σ mt tfix,,x; τ st ) Z(t ex,x; τ end) 1 +bs(τ st ) µ Dst D end e (h end h st )x 1 2 (h2 end h2 st )ζex 1 +bs(τ st ) The only remaining issue is calibration. For range notes, we should use constant mean reversion and calibrate along the diagonal, exactly as we did for the cancelable accrual swaps. We only need to specify the stries of the reference swaptions. A good method is to transfer the basis spreads and margin to the coupon leg, and then match the ratio of the coupon leg to the floating leg. For exercise on date t, this ratio yields (4.7a) j=+1 α j R fix D A (t j )e γ(t j t ) M j K D A (t ) λ = t j X τ st=t j 1+1 ( 1+ηβ(Rmax 1 2 ε) B 1 (τ st ) 1+ηβ(R max ε) B 1 (τ st ) ε [1 + ηβl 0 (τ st )] 1+ηβ(Rmin 1 2 ε) B 3 (τ st ) 1+ηβ(R min ε) ) B 3 (τ st ) 1+ηβL f (t ex,x; τ st) + D A(t n )e γ(t n t ) K D A (t ) 18

19 As before, the B j are dimensionless Blac formulas, (4.7b) B ω (τ st )=K ω N (d ω 1 ) L 0 (τ st )N (d ω 2 ) (4.7c) d ω 1,2 = log K ω/l 0 (τ st ) ± 1 2 σ2 imp (K ω)t fix σ imp (K ω ) t fix (4.7d) K 1,2 = R max ± 1 2 ε, K 3,4 = R min ± 1 2 ε, and L 0 (τ st ) is today s forward rate: Appendix A. Calibrating the LGM model. The are several methods of calibrating the LGM model for pricing a Bermudan swaption. The most popular method is to choose a constant mean reversion κ, and then calibrate on the diagonal European swaptions maing up the Bermudan. In the LGM model, a constant mean reversion κ means that the model function h(t) is given by (A.1) h(t) = 1 e κt. κ Usually the value of κ is selected from a table of values that are nown to yield the correct maret prices of liquid Bermudans; It is nown empirically that the needed mean reversion parameters are very, very stable, changing little from year to year. κ 1Y 2Y 3Y 4Y 5Y 7Y 10Y 1M -1.00% -0.50% -0.25% -0.25% -0.25% -0.25% -0.25% 3M -0.75% -0.25% 0.00% 0.00% 0.00% 0.00% 0.00% 6M -0.50% 0.00% 0.25% 0.25% 0.25% 0.25% 0.25% 1Y 0.00% 0.25% 0.50% 0.50% 0.50% 0.50% 0.50% 3Y 0.25% 0.50% 1.00% 1.00% 1.00% 1.00% 1.00% 5Y 0.50% 1.00% 1.25% 1.25% 1.25% 1.25% 1.25% 7Y 1.00% 1.25% 1.50% 1.50% 1.50% 1.50% 1.50% 10Y 1.50% 1.50% 1.75% 1.75% 1.75% 1.75% 1.75% Table A.1 Mean reverssion κ for Bermudan swaptions. Rows are time-to-first exercise; columns are tenor of the longest underlying swap obtained upon exercise. With h(t) nown, we only need determine ζ(t) by calibrating to European swaptions. Consider a European swaption with notification date t ex. Suppose that if one exercises the option, one recieves a fixed leg worth (A.2a) V fix (t, x) = and pays a floating leg worth R fix cvg(t i 1,t i, dcb fix )Z(t, x; t i ), i=1 (A.2b) V flt (t, x) =Z(t, x; t 0 ) Z(t, x; t n )+ cvg(t i 1,t i, dcb flt )bs i Z(t, x; t i ). i=1 19

20 Here cvg(t i 1,t i, dcb fix ) and cvg(t i 1,t i, dcb flt ) are the day count fractions for interval i using the fixed leg and floating leg day count bases. (For simplicity, we are cheating slightly by applying the floating leg s basis spread at the frequency of the fixed leg. Mea culpa). Adjusting the basis spread for the difference in the day count bases (A.3) bs new i = cvg(t i 1,t i, dcb flt ) cvg(t i 1,t i, dcb fix ) bs i allows us to write the value of the swap as (A.4) V swap (t, x) =V fix (t, x) V flt (t, x) = (R fix bs i )cvg(t i 1,t i, dcb fix )Z(t, x; t i )+Z(t, x; t n ) Z(t, x; t 0 ) i=1 Under the LGM model, today s value of the swaption is (A.5) V swptn (0, 0) = Z 1 p 2πζ(tex ) e x2 ex /2ζ(t ex) [V swap(t ex,x ex )] + dx ex,x ex ) Substituting the explicit formulas for the zero coupon bonds and woring out the integral yields (A.6a) V swptn (0, 0) = (R fix bs i )cvg(t i 1,t i, dcb fix )D(t i )N i=1 +D(t n )N Ã! y +[h(t i ) h(t 0 )] ζ p ex ζex Ã! Ã! y +[h(t n ) h(t 0 )] ζ p ex y D(t 0 )N p, ζex ζex where y is determined implicitly via (A.6b) (R fix bs i )cvg(t i 1,t i, dcb fix )e [h(t i) h(t 0 )]y 1 2 [h(t i) h(t 0 )] 2 ζ ex i=1 +D(t n )e [h(t n) h(t 0 )]y 1 2 [h(t n) h(t 0 )] 2 ζ ex = D(t0 ). The values of h(t) are nown for all t, so the only unnown parameter in this price is ζ(t ex ). One can show that the value of the swaption is an increasing function of ζ(t ex ), so there is exactly one ζ(t ex ) which matches the LGM value of the swaption to its maret price. This solution is easily found via a global Newton iteration. To price a Bermudan swaption, one typcially calibrates on the component Europeans. For, say, a 10NC3 Bermudan swaption struc at 8.2% and callable quarterly, one would calibrate to the 3 into 7 swaption struc at 8.2%, the 3.25 into 6.75 swaption struc at 8.2%,..., then 8.75 into 1.25 swaption struc at 8.25%, and finally the 9 into 1 swaption struc at 8.2%. Calibrating each swaption gives the value of ζ(t) on the swaption s exercise date. One generally uses piecewise linear interpolation to obtain ζ(t) at dates between the exercise dates. The remaining problem is to pic the strie of the reference swaptions. A good method is to transfer the basis spreads and margin to the coupon leg, and then match the ratio of the coupon leg to the funding leg to the equivalent ratio for a swaption. For the exercise on date t, this ratio is defined to be 20

21 (A.7a) λ = j=+1 α j D(t j ) M j D(t ) t j X τ st =t j D(t n n) D(t ) X cvg(t i 1,t i )(bs 0 i +m i ) D(t i) D(t ) i=1 where B ω are Blac s formula at stries around the boundaries: 1+ηβ(Rmax 1 2 ε) B 1 (τ st ) 1+ηβ(R max ε) B 2 (τ st ) ε [1 + ηβl 0 (τ st )] 1+ηβ(Rmin 1 2 ε) B 3 (τ st ) 1+ηβ(R min ε) B 4 (τ st ) ε [1 + ηβl 0 (τ st )] (A.7b) B ω (τ st )=βd(τ end ) {K ω N (d ω 1 ) L 0 (τ st )N (d ω 2 )} (A.7c) d ω 1,2 = log K ω/l 0 (τ st ) ± 1 2 σ2 imp (K ω)t fix σ imp (K ω ) t fix with (A.7d) K 1,2 = R max ± 1 2 ε, K 3,4 = R min ± 1 2 ε. This is to be matched to the swaption whose swap starts on t and ends on t n,withthestrier fix chosen so that the equivalent ratio matches the λ defined above: (A.7e) λ = (R fix bs i )cvg(t i 1,t i, dcb fix ) D(t i) D(t ) + D(t n) D(t ) i=+1 The above methodology wors well for deals that are similar to bullet swaptions. For some exotics, such as amortizing deals or zero coupon callables, one may wish to choose both the tenor of the and the strie of the reference swaptions. This allows one to match the exotic deal s duration as well as its moneyness. Appendix B. Floating rate accrual notes. 21