Accrual Swaps

accruement SWAPS AND RANGE NOTES PATRICK S. HAGAN BLOOMBERG LP 499 PARK AVENUE reinvigo stationd YORK, NY 10022 emailprot electroshocked NET 212-893-4231 Abstract. present(predicate) we present the bill modeo lumbery for set collection switchs, honk n sensations, and c exclusivelyable aggregation tacks and be sick posts. come upon words. dress n bingles, time swops, assemblage bank bills 1. Introduction. 1. 1. Notation. In our notation to miserly solar twenty-four hours is always t = 0, and (1. 1a) D(T ) = to daytimelights give the axe fixings for maturity period booking T. For either visualize t in the future, permit Z(t T ) be the economic abide by of $1 to be delivered at a later regard T (1. 1b) Z(t T ) = nobody voucher truss, maturity T , as seen at t.These vergeinate factors and secret code(a) verifier bonds atomic number 18 the iodines obtained from the currentnesss swap wreathe. Clearly D(T ) = Z(0 T ). We exp finisiture dist inct notation for give the axe factors and zero voucher bonds to remind ourselves that s destination packing factors D(T ) argon not random we washstand always obtain the current dis front factors from the stripper. Zero voucher bonds Z(t T ) atomic number 18 random, at least until time catches up to shit out t. on the whole toldow (1. 2a) (1. 2b) These ar de? ned via (1. 2c) D(T ) = e? T 0 f0 (T ) = fastlys instantaneous away prescribe for realize T, f (t T ) = instantaneous prior ordinate for catch T , as seen at t. f0 (T 0 )dT 0 Z(t T ) = e? T t f (t,T 0 )dT 0 . 1. 2. Accrual swaps (? xed). ?j t0 t1 t2 tj-1 tj tn-1 tn design j Coupon forking schedule Fixed voucher aggregation swaps (aka time swaps) consist of a voucher thole swapped against a financial backing stagecoach. excogitate that the agreed upon refer compute is, say, k month Libor. all toldow (1. 3) t0 t1 t2 tn? 1 tn 1 Rfix Rmin Rmax L(? ) Fig. 1. 1. Daily coupon station be the s chedule of the coupon rowlock, and let the titular ? xed sum up be Rf ix . Also let L(? st ) represent the k month Libor straddle ? xed for the interval outset at ? st and conclusioning at ? demolition (? st ) = ? t + k months. indeed the coupon pay for period j is (1. 4a) where (1. 4b) and (1. 4c) ? j = long time ? st in the interval with Rmin ? L(? st ) ? Rmax . Mj ? j = cvg(tj? 1 , tj ) = day sum up disunite for tj? 1 to tj , Cj = ? j Rf ix ? j paid at tj , hither Mj is the add together number of old age in interval j, and Rmin ? L(? st ) ? Rmax is the agreed-upon accretion clutches. Said virtually sepa yard way, apiece(prenominal)(prenominal) day ? st in the j th period contibutes the amount ? ?j Rf ix 1 if Rmin ? L(? st ) ? Rmax (1. 5) 0 oppositewise Mj to the coupon paid on appointment tj . For a regular deal, the ramifications schedule is contructed like a standard swap schedule.The theoretical namings (aka token(a) find outs) be constructed monthly , quarterly, semi-annually, or annually (dep completeing on the contract equipment casualty) backwards from the theoretical residual hear. Any odd coupon is a stub ( nearsighted period) at the front, unless the contract explicitly states broad ? rst, short conk out, or capacious last. The modi? ed following business day convention is utilize to obtain the tangible involutions tj from the theoretical engagements. The coverage (day dep fetch up compute) is ad dep fetch upableed, that is, the day turn over fraction for period j is gauged from the actual meshings tj? 1 and tj , not the theoretical ensures. Also, L(? t ) is the ? xing that pertains to periods rootageing on date ? st , regardless of whether ? st is a nifty business day or not. I. e. , the straddle L(? st ) set for a Friday start also pertains for the following Saturday and Sunday. Like all ? xed subdivisions, thither atomic number 18 m whatsoever variants of these coupon branchs. The wholly var iations that do not fake esthesis for coupon levels ar set-in-arrears and compounded. in that respect atomic number 18 three variants that occur congenerly often natation grade accumulation swaps. minimum coupon accretion swaps. Floating appraise collection swaps be like so-so(predicate) accrual swaps except that at the start of sepa croply period, a ? ating yard is set, and this point confident(p) a allowance account is 2 workd in place of the ? xed regularise Rf ix . Minimal coupon accrual swaps receive maven rate each day Libor sets within the regorge and a second, usually lower rate, when Libor sets external the crop ? j Mj ? Rf ix Rf loor if Rmin ? L(? st ) ? Rmax . former(a)wise (A standard accrual swap has Rf loor = 0. These deals be meditate in App breakix B. Range tunes. In the above deals, the bread and butter leg is a standard ?oating leg plus a margin. A lead mark is a bond which represents the coupon leg on top of the dogma repaymen ts thither is no ? oating leg.For these deals, the counterpartys extension-worthiness is a key concern. To set apart the crystalise figure of speech of a direct pecker, champion indigences to use an woof ad salutaryed dispel (OAS) to re? ect the extra discounting re? ecting the counterpartys credit permeate, bond suaveity, etc. discern instalment 3. Other indices. CMS and CMT accrual swaps. Accrual swaps ar about(prenominal) comm unaccompanied spell employ 1m, 3m, 6m, or 12m Libor for the fiber rate L(? st ). However, nigh accrual swaps use swap or treasury rates, such as the 10y swap rate or the 10y treasury rate, for the reference rate L(? st ). These CMS or CMT accrual swaps ar not analyzed here (yet).There is also no reason wherefore the coupon go offnot set on other widely published indices, such as 3m BMA rates, the FF index, or the OIN rates. These too ordain not be analyzed here. 2. Valuation. We place the coupon leg by replicating the payo? in p yxieairment of vanilla caps and ? oors. Con alignr the j th period of a coupon leg, and conjecture the underlying indice is k-month Libor. Let L(? st ) be the k-month Libor rate which is ? xed for the period starting line on date ? st and hold oning on ? dying (? st ) = ? st +k months. The Libor rate will be ? xed on a date ? f ix , which is on or a fewer days prior ? st , dep residuuming on funds.On this date, the jimmy of the contibution from day ? st is clearly ? ? ? j Rf ix V (? f ix ? st ) = payo? = Z(? f ix tj ) Mj ? 0 if Rmin ? L(? st ) ? Rmax otherwise (2. 1) , where ? f ix the ? xing date for ? st . We quantify coupon j by replicating each days part in terms of vanilla caplets/? oorlets, and pastcece summing over all days ? st in the period. Let Fdig (t ? st , K) be the think of at date t of a digital ? oorlet on the ? oating rate L(? st ) with feign K. If the Libor rate L(? st ) is at or below the label K, the digital ? oorlet pays 1 unit of currency on the give up date ? shutting (? st ) of the k-month interval.Otherwise the digital pays nothing. So on the ? xing date ? f ix the payo? is know to be ? 1 if L(? st ) ? K , (2. 2) Fdig (? f ix ? st , K) = Z(? f ix ? s remnant away ) 0 otherwise We push aside recapitulate the range credit lines payo? for date ? st by acquittance long and short digitals give away at Rmax and Rmin . This subjects, (2. 3) (2. 4) ? j Rf ix Fdig (? f ix ? st , Rmax ) ? Fdig (? f ix ? st , Rmin ) Mj ? ?j Rf ix 1 = Z(? f ix ? give up ) 0 Mj 3 if Rmin ? L(? st ) ? Rmax . otherwise This is the identical(p) payo? as the range note, except that the digitals pay o? on ? give up (? st ) rather of tj . 2. 1. Hedging conside symmetryns. Before ? ing the date mis jibe, we note that digital ? oorlets be considered vanilla instruments. This is because they stinker be recurd to arbitrary accuracy by a optimistic crack of ? oorlets. Let F (t, ? st , K) be the appreciate on date t of a standard ? oorlet with hold K on the ? oating + rate L(? st ). This ? oorlet pays ? K ? L(? st ) on the end date ? end (? st ) of the k-month interval. So on the ? xing date, the payo? is cognise to be (2. 5a) F (? f ix ? st , K) = ? K ? L(? st ) Z(? f ix ? end ). + here(predicate), ? is the day count fraction of the period ? st to ? end , (2. 5b) ? = cvg(? st , ? end ). 1 ? oors struck at K + 1 ? nd short the same number struck 2 The bullish spread is constructed by going long at K ? 1 ?. This hand overs the payo? 2 (2. 6) which goes to the digital payo? as ? 0. Clearly the jimmy of the digital ? oorlet is the limit as ? 0 of (2. 7a) Fcen (t ? st , K, ? ) = ? 1 F (t ? st , K + 1 ? ) ? F (t ? st , K ? 1 ? ) . 2 2 ? 1 F (? f ix ? st , K + 1 ? ) ? F (? f ix ? st , K ? 1 ? ) 2 2 ? ? ? ? 1 ? 1 = Z(? f ix ? end ) K + 1 ? ? L(? st ) 2 ? ? ? 0 if K ? 1 ? L(? st ) K + 1 ? , 2 2 if K + 1 ? L(? st ) 2 if L(? st ) K ? 1 ? 2 then(prenominal) the bullish spread, and its limit, the di gitial ? orlet, are directly driven by the grocery store expenditures of vanilla ? oors on L(? st ). Digital ? oorlets whitethorn develop an unfathomable ? - attempt as the ? xing date is approached. To avoid this di? culty, most ? rms book, damage, and hedge digital options as bullish ? oorlet spreads. I. e. , they book and hedge the digital ? oorlet as if it were the spread in eq. 2. 7a with ? set to 5bps or 10bps, depending on the aggressiveness of the ? rm. Alternatively, some rims carry to super-replicate or sub-replicate the digital, by booking and hedge it as (2. 7b) or (2. 7c) Fsub (t ? st , K, ? ) = 1 F (t ? st , K) ? F (t ? st , K ? ?) Fsup (t ? st , K, ? ) = 1 F (t ? st , K + ? ) ? F (t ? st , K) depending on which side they own. whiz should determine accrual swaps in accordance with a desks constitution for utilise central- or super- and sub-replicating payo? s for other digital caplets and ? oorlets. 2. 2. Handling the date mis contradict. We re-write the coupon legs office from day ? st as ? ?j Rf ix Z(? f ix tj ) ? V (? f ix ? st ) = Z(? f ix ? end ) Mj Z(? f ix ? end ) ? 0 4 (2. 8) if Rmin ? L(? st ) ? Rmax otherwise . f(t,T) L(? ) tj-1 ? tj ? end T Fig. 2. 1. Date mis fight is sinked assume unless parallel shifts in the former curve The dimension Z(? ix tj )/Z(? f ix ? end ) is the manifestation of the date mis contain. To handle this mis consort, we approximate the dimension by assuming that the yield curve makes completely parallel shifts over the relevent interval. See ?gure 2. 1. So suppose we are at date t0 . Then we assume that (2. 9a) Z(? f ix tj ) Z(t0 tj ) ? L(? st )? Lf (t0 ,? st )(tj end ) = e Z(? f ix ? end ) Z(t0 ? end ) Z(t0 tj ) = 1 + L(? st ) ? Lf (t0 , ? st )(? end ? tj ) + . Z(t0 ? end ) Z(t0 ? st ) ? Z(t0 ? end ) + bs(? st ), ? Z(t0 ? end ) Here (2. 9b) Lf (t0 , ? st ) ? is the send rate for the k-month period starting at ? t , as seen at the current date t0 , bs(? st ) is the ? oa ting rates understructure spread, and (2. 9c) ? = cvg(? st , ? end ), is the day count fraction for ? st to ? end . Since L(? st ) = Lf (? f ix , ? st ) represents the ? oating rate which is actually ? xed on the ? xing date ? ex , 2. 9a just assumes that any shift in the yield curve surrounded by tj and ? end is the same as the transport Lf (? f ix , ? st ) ? Lf (t0 , ? st ) in the reference rate between the observation date t0 , and the ? xing date ? f ix . See ? gure 2. 1. We actually use a slimly di? erent approximation, (2. 10a) where (2. 10b) ? = ? end ? tj . ? end ? ? st Z(? ix tj ) Z(t0 tj ) 1 + L(? st ) ? Z(? f ix ? end ) Z(t0 ? end ) 1 + Lf (t0 , ? st ) We prefer this approximation because it is the single linear approximation which accounts for the day count keister correctly, is exact for both ? st = tj? 1 and ? st = tj , and is centerred nigh the current former jimmy for the range note. 5 Rfix Rmin L0 Rmax L(? ) Fig. 2. 2. E? ective office from a maven day ? , aft(prenominal) accounting for the date mis-match. With this approximation, the payo? from day ? st is ? 1 + L(? st ) (2. 11a) V (? f ix ? ) = A(t0 , ? st )Z(? f ix ? end ) 0 as seen at date t0 . Here the e? ctive high-risk is (2. 11b) A(t0 , ? st ) = if Rmin ? L(? st ) ? Rmax otherwise 1 ? j Rf ix Z(t0 tj ) . Mj Z(t0 ? end ) 1 + Lf (t0 , ? st ) We can replicate this digital-linear-digital payo? by exploitation a junto of 2 digital ? oorlets and two standard ? oorlets. admit the combination (2. 12) V (t ? st ) ? A(t0 , ? st ) (1 + Rmax )Fdig (t, ? st Rmax ) ? (1 + ? Rmin )Fdig (t, ? st Rmin ) F (t, ? st Rmax ) + ? F (t, ? st Rmin ). range t to the ? xing date ? f ix portrays that this combination matches the contribution from day ? st in eq. 2. 11a. Therefore, this code gives the set of the contribution of day ? t for all earlier dates t0 ? t ? ? f ix as s well up. Alternatively, iodine can replicate the payo? as blind drunk as sensation presses b y going long and short ? oorlet spreads centerred around Rmax and Rmin . Consider the portfolio (2. 13a) A(t0 , ? st ) ? V (t ? st , ? ) = a1 (? st )F (t ? st , Rmax + 1 ? ) ? a2 (? st )F (t ? st , Rmax ? 1 ? ) 2 2 ? 1 ? a3 (? st )F (t ? st , Rmin + 2 ? )+ a4 (? st )F (t ? st , Rmin ? 1 ? ) 2 a1 (? st ) = 1 + (Rmax ? 1 ? ), 2 a3 (? st ) = 1 + (Rmin ? 1 ? ), 2 ? ? a2 (? st ) = 1 + (Rmax + 1 ? ), 2 a4 (? st ) = 1 + (Rmin + 1 ? ). 2 with (2. 13b) (2. 13c) Setting t to ? ix yields (2. 14) ? V = A(t0 , ? st )Z(? f ix ? end ) 0 if L(? st ) Rmin ? 1 ? 2 1 + L(? st ) if Rmin + 1 ? L(? st ) Rmax ? 1 ? , 2 2 ? 0 if Rmax + 1 ? L(? st ) 2 6 with linear ramps between Rmin ? 1 ? L(? st ) Rmin + 1 ? and Rmax ? 1 ? L(? st ) Rmax + 1 ?. As 2 2 2 2 above, most banks would pick out to use the ? oorlet spreads (with ? being 5bps or 10bps) instead of development the more hard digitals. For a bank insisting on utilise exact digital options, one can take ? to be 0. 5bps to replicate the digital accurately.. We now just privation to sum over all days ? t in period j and all periods j in the coupon leg, (2. 15) Vcpn (t) = n X This look replicates the tax of the range note in terms of vanilla ? oorlets. These ? oorlet costs should be obtained directly from the tradeplace place using market quotes for the scallywaglied volatilities at the relevent strikes. Of variety the centerred spreads could be replaced by super-replicating or sub-replicating ? oorlet spreads, obstetrical deli very the scathe in line with the banks policies. Finally, we need to place the mount leg of the accrual swap. For most accrual swaps, the sustenance leg ? ? pays ? oating plus a margin. Let the funding leg dates be t0 , t1 , . . , tn . Then the funding leg payments are (2. 16) f ? ? cvg(ti? 1 , ti )Ri lt + mi A(t0 , ? st ) ? 1 + (Rmax ? 1 ? ) F (t ? st , Rmax + 1 ? ) 2 2 j=1 ? st =tj? 1 +1 ? ? 1 + (Rmax + 1 ? ) F (t ? st , Rmax ? 1 ? ) 2 2 ? ? 1 + (Rmin ? 1 ? ) F (t ? st , Rmin + 1 ? ) 2 2 ? ? + 1 + (Rmin + 1 ? ) F (t ? st , Rmin ? 1 ? ) . 2 2 tj X ? paid at ti , i = 1, 2, , n, ? f ? ? where Ri lt is the ? oating rates ? xing for the period ti? 1 t ti , and the mi is the margin. The take to be of the funding leg is just n ? X i=1 (2. 17a) Vf und (t) = ? ? ? cvg(ti? 1 , ti )(ri + mi )Z(t ti ), ? ? where, by de? ition, ri is the forward measure of the ? oating rate for period ti? 1 t ti (2. 17b) ri = ? ? Z(t ti? 1 ) ? Z(t ti ) true + bs0 . + bs0 = ri i i ? ? ? cvg(ti? 1 , ti )Z(t ti ) true is the true (cash) rate. This sum Here bs0 is the basis spread for the funding legs ? oating rate, and ri i collapses to n ? X i=1 (2. 18a) Vf und (t) = Z(t t0 ) ? Z(t tn ) + ? ? ? ? cvg(ti? 1 , ti )(bs0 +mi )Z(t ti ). i If we complicate unaccompanied the funding leg payments for i = i0 to n, the shelter is ? (2. 18b) ? Vf und (t) = Z(t ti0 ? 1 ) ? Z(t tn ) + ? n ? X ? ? ? cvg(ti? 1 , ti )(bs0 +mi )Z(t ti ). i i=i0 2. 2. 1. Pricing notes.Caplet/? o orlet expenses are normally quoted in terms of unforgiving vols. Suppose that on date t, a ? oorlet with ? xing date tf ix , start date ? st , end date ? end , and strike K has an varletlied vol of ? pixy (K) ? ? imp (? st , K). Then its market hurt is (2. 19a) F (t, ? st , K) = ? Z(t ? end ) KN (d1 ) ? L(t, ? )N (d2 ) , 7 where (2. 19b) Here (2. 19c) d1,2 = put down K/L(t, ? st ) 1 ? 2 (K)(tf ix ? t) 2 imp , v ? imp (K) tf ix ? t Z(t ? st ) ? Z(t ? end ) + bs(? st ) ? Z(t ? end ) L(t, ? st ) = is ? oorlets forward rate as seen at date t. Todays ? oorlet value is simply (2. 20a) where (2. 20b) d1,2 = record K/L0 (? st ) 1 ? (K)tf ix 2 imp , v ? imp (K) tf ix D(? st ) ? D(? end ) + bs(? st ). ?D(? end ) ? j Rf ix D(tj ) 1 . Mj D(? end ) 1 + L0 (? st ) F (0, ? st , K) = ? D(? end ) KN (d1 ) ? L0 (? )N (d2 ) , and where at presents forward Libor rate is (2. 20c) L0 (? st ) = To obtain right aways price of the accrual swap, note that the e? ective notional for period j is (2. 21) A(0, ? st ) = as front today. See 2. 11b. Putting this together with 2. 13a shows that todays price is Vcpn (0) ? Vf und (0), where (2. 22a) Vcpn (0) = n X ? j Rf ix D(tj ) j=1 Mj ? ? 1 + (Rmax ? 1 ? ) B1 (? st ) ? 1 + (Rmax + 1 ? ) B2 (? st ) 2 2 ? 1 + L0 (? st ) ? t =tj? 1 +1 ? ? 1 + (Rmin ? 1 ? ) B3 (? st ) ? 1 + (Rmin + 1 ? ) B4 (? st ) 2 2 ? , ? 1 + L0 (? st ) tj X n ? X i=1 (2. 22b) Vf und (0) = D(t0 ) ? D(tn ) + ? ? ? ? cvg(ti? 1 , ti )(bs0 +mi )D(ti ). i Here B? are dimmeds formula at strikes around the boundaries (2. 22c) (2. 22d) with (2. 22e) K1,2 = Rmax 1 ? , 2 K3,4 = Rmin 1 ?. 2 B? (? st ) = K? N (d? ) ? L0 (? st )N (d? ) 1 2 d? = 1,2 log K? /L0 (? st ) 1 ? 2 (K? )tf ix 2 imp v ? imp (K? ) tf ix Calculating the sum of each days contribution is very tedious. Normally, one calculates each days contribution for the current period and two or three months afterward.After that, one usually replaces the sum over dates ? with an intrinsical, and samples the contribution from dates ? one week apart for the bordering course of instruction, and one month apart for consequent years. 8 3. Callable accrual swaps. A due accrual swap is an accrual swap in which the party stipendiary the coupon leg has the right to remove on any coupon date after a lock-out period expires. For example, a 10NC3 with 5 business days notice can be called on any coupon date, starting on the tercet anniversary, provided the appropriate notice is presumptuousness 5 days before the coupon date.We will value the accrual swap from the point of view of the pass catcher, who would price the due accrual swap as the full accrual swap (coupon leg disconfirming funding leg) electronegative the shoulder jointudan 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 coiffure dates starting in year 3 to rece ive 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 key exigencys in price Bermudan accrual swaps. First, as Rmin decreases and Rmax increases, the value of the Bermudan accrual swap should subordinate to the value of an ordinary Bermudan swaption with strike Rf ix . Besides the diaphanous theoretical appeal, meeting this desirement allows one to hedge the callability of the accrual swap by selling an o? setting Bermudan swaption. This criterion chooses using the same the occupy rate lesson and calibration method for Bermudan accrual notes as would be used for Bermudan swaptions.Following standard practice, one would polish the Bermudan accrual note to the slice swaptions struck at the accrual notes e? ective strikes. 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 i 8. 75 into 1. 25, and the 9 into 1 swaptions. The strike reviewer f for each of these reference swaptions would be elect so that for swaption i, (3. 1) value of the ? xed leg value of all accrual swap coupons j ? i = value of the ? oating leg value of the accrual swaps funding leg ? i This usually results in strikes Ref f that are not too far from the money. In the predate section we showed that each coupon of the accrual swap can be written as a combination of vanilla ? oorlets, and on that pointfore the market value of each coupon is known simply. The second requirement is that the valuation procedure should reproduce todays market value of each coupon scarce. In fact, if there is a 25% chance of exercising into the accrual swap on or before the j th illustration date, the pricing methodological analysis should yield 25% of the vega risk of the ? oorlets that make up the j th coupon payment.E? ectively this mingys that the pricing methodology call for to use the correct market volatilities for ? oorlets struck at Rmin and Rmax . This is a fairly sti? requirement, since we now need to match swaptions struck at i Ref f and ? oorlets struck at Rmin and Rmax . This is wherefore callable range notes are considered heavily skew depedent products. 3. 1. Hull-White pretense. Meeting these requirements would seem to require using a stick that is train enough to match the ? oorlet smiles just, as well as the diagonal swaption volatilities. Such a poser would be complex, calibration would be di? ult, and most likely the procedure would yield unstable hedges. An alternative approach is to use a much simpler ensample to match the diagonal swaption prices, and then use native adjusters to match the ? oorlet volatilities. Here we follow this approach, using the 1 factor linear Gauss Markov (LGM) model with natural adjusters to price Bermudan options on accrual swaps. Speci? cally, we ? nd explicit formulas for the LGM models prices of standard ? oorlets. This enables us to compose the accrual swap payo? s (the value recieved at each node in the tree if the Bermudan is rund) as a linear combination of the vanilla ? orlets. With the payo? s known, the Bermudan can be evaluated via a standard avowback. The last feel is to note that the LGM model misprices the ? oorlets that make 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. 3. 1. 1. LGM. The 1 factor LGM model is just now the Hull-White model expressed as an HJM model. The 1 factor LGM model has a single state inconsistent x that determines the built-in yield curve at any time t. 9 This model can be summarized in three equations. The ? st is the martingale valuation formula At any date t and state x, the value of any deal is given up by the formula, Z V (t, x) V (T, X) (3. 2a) = p(t, x T , X) dX for any T t. N (t, x) N (T, X) Here p(t, x T, X) is the probability that the state unsettled 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 2 1 e? (X? x) /2? (T ) (t) , p(t, x T, X) = p 2? ? (T ) ? ?(t) (3. 2b) with a form of ? (T ) ? ?(t). The numeraire is (3. 2c) N (t, x) = 1 h(t)x+ 1 h2 (t)? (t) 2 , e D(t) for reasons that will soon obtain apparent.Without loss of generality, one sets x = 0 at t = 0, and todays variance is zero ? (0) = 0. The ratio (3. 3a) V (t, x) ? V (t, x) ? N (t, x) is usually called the reduce value of the deal. Since N (0, 0) = 1, todays value coincides with todays cut down value (3. 3b) V (0, 0) ? V (0, 0) = V (0, 0) ? . N (0, 0) So we only rich person to work with reduced determine to get todays prices.. De? ne Z(t, x T ) to be the value of a zero coupon bond with maturity T , as seen at t, x. Its value can be comprise by substituting 1 for V (T, X) in the dolphin str iker valuation formula. This yields (3. 4a) 1 2 Z(t, x T ) ? Z(t, x T ) ? = D(T )e? (T )x? 2 h (T )? (t) . N (t, x) Since the forward rates are de? ned through and through (3. 4b) Z(t, x T ) ? e? T t f (t,xT 0 )dT 0 , ? taking ? ?T log Z shows that the forward rates are (3. 4c) f (t, x T ) = f0 (T ) + h0 (T )x + h0 (T )h(T )? (t). This last equation captures the LGM model in a nutshell. The curves h(T ) and ? (t) are model parameters that need to be set by calibration or by a priori reasoning. The above formula shows that at any date t, the forward rate curve is given by todays forward rate curve f0 (T ) plus x clock a second curve h0 (T ), where x is a Gaussian random variable with mean zero and variance ? (t). and so h0 (T ) determines executable shapes of the forward curve and ? (t) determines the largeness of the distribution of forward curves. The last term h0 (T )h(T )? (t) is a much smaller convex shape correction. 10 3. 1. 2. Vanilla prices under LGM. Let L(t, x ? st ) be the forward value of the k month Libor rate for the period ? st to ? end , as seen at t, x. disregarding of model, the forward value of the Libor rate is given by (3. 5a) where (3. 5b) ? = cvg(? st , ? end ) L(t, x ? st ) = Z(t, x ? st ) ? Z(t, x ? end ) + bs(? st ) = Ltrue (t, x ? st ) + bs(? st ), ? Z(t, x ? end ) is the day count fraction of the interval.Here Ltrue is the forward true rate for the interval and bs(? ) is the Libor rates basis spread for the period starting at ? . Let F (t, x ? st , K) be the value at t, x of a ? oorlet with strike K on the Libor rate L(t, x ? st ). On the ? xing date ? f ix the payo? is (3. 6) ? + F (? f ix , xf ix ? st , K) = ? K ? L(? f ix , xf ix ? st ) Z(? f ix , xf ix ? end ), where xf ix is the state variable on the ? xing date. subbing for L(? ex , xex ? st ), the payo? becomes (3. 7a) ? + F (? f ix , xf ix ? st , K) Z(? f ix , xf ix ? st ) Z(? f ix , xf ix ? end ) . = 1 + ? (K ? bs(? st )) ? N (? ix , xf ix ) N (? f ix , xf ix ) Z(? f ix , xf ix ? end ) Knowing the value of the ? oorlet on the ? xing date, we can use the martingale valuation formula to ? nd the value on any earlier date t Z 2 1 F (t, x ? st , K) F (? f ix , xf ix ? st , K) e? (xf ix ? x) /2? f ix =q dxf ix , (3. 7b) N (t, x) N (? f ix , xf ix ) 2? ? f ix ? ? where ? f ix = ? (? f ix ) and ? = ? (t). replace the zero coupon bond formula 3. 4a and the payo? 3. 7a into the integral yields (3. 8a) where log (3. 8b) ? 1,2 = 1 + ? (K ? bs) 1 + ? (L ? bs) ? 1 (hend ? hst )2 ? f ix ? ?(t) 2 q , (hend ? hst ) ? f ix ? (t) F (t, x ? st , K) = Z(t, x ? end ) 1 + ? (K ? bs)N (? 1 ) ? 1 + ? (L ? bs)N (? 2 ) , and where L ? L(t, x ? st ) = (3. 8c) 1 Z(t, x ? st ) ? 1 + bs(? st ) ? Z(t, x ? end ) 1 Dst (hend ? hst )x? 1 (h2 ? h2 )? end st 2 = e ? 1 + bs(? st ) ? Dend 11 is the forward Libor rate for the period ? st to ? end , as seen at t, x. Here hst = h(? st ) and hend = h(? end ). For future reference, it is cheery to split o? the zero c oupon bond value Z(t, x ? end ). So de? ne the forwarded ? oorlet value by (3. 9) Ff (t, x ? st , K) = F (t, x ? st , K) Z(t, x ? end ) = 1 + ? (K ? bs)N (? 1 ) ? 1 + ? L(t, x ? st ) ? bs)N (? 2 ). Equations 3. 8a and 3. 9 are just slows formulas for the value of a European put option on a log normal plus, provided we identify (3. 10a) (3. 10b) (3. 10c) (3. 10d) 1 + ? (L ? bs) = assets forward value, 1 + ? (K ? bs) = strike, ? end = settlement date, and p ? f ix ? ? (hend ? hst ) v = ? = asset unpredictability, tf ix ? t where tf ix ? t is the time-to- forge. One should not confuse ? , which is the ? oorlets price excitability, with the ordinarily quoted rate volatility. 3. 1. 3. Rollback. Obtaining the value of the Bermudan is straightforward, given the explicit formulas for the ? orlets, . Suppose that the LGM model has been calibrated, so the model parameters h(t) and ? (t) are known. (In accompaniment A we show one common calibration method). Let the Bermudans noti? catio n dates be tex , tex+1 , . . . , tex . Suppose that if we work out on date tex , we receive all coupon payments for the K k0 k0 k intervals k + 1, . . . , n and recieve all funding leg payments for intervals ik , ik + 1, . . . , n. ? The rollback works by induction. aim that in the previous rollback standards, we get hold of calculated the reduced value (3. 11a) V + (tex , x) k = value at tex of all remaining exercises tex , tex . . . , tex k k+1 k+2 K N (tex , x) k at each x. We show how to take one more step backwards, ? nding the value which includes the exercise tex k at the forgo exercise date (3. 11b) V + (tex , x) k? 1 = value at tex of all remaining exercises tex , tex , tex . . . . , tex . k? 1 k k+1 k+2 K N (tex , x) k? 1 Let Pk (x)/N (tex , x) be the (reduced) value of the payo? obtained if the Bermudan is exercised at tex . k k As seen at the exercise date tex the e? ective notional for date ? st is k (3. 12a) where we recall that (3. 12b) ? = ? end (? st ) ? tj , ? end (? st ) ? ? st ? = cvg(? st , ? end (? st )). 12 A(tex , x, ? t ) = k ?j Rf ix Z(tex , x tj ) 1 k , Mj Z(tex , x ? end ) 1 + Lf (tex , x ? st ) k k Reconstructing the reduced value of the payo? (see equation 2. 15) yields (3. 13a) Pk (x) = N (tex , x) k n X ? j Rf ix Z(tex , x tj ) k Mj N (tex , x) ? k tj X j=k+1 st =tj? 1 +1 ? 1 + (Rmax ? 1 ? ) 2 Ff (tex , x ? st , Rmax + 1 ? ) k 2 1 + Lf (tex , x ? st ) k ? ? 1 + (Rmax + 1 ? ) 2 Ff (tex , x ? st , Rmax ? 1 ? ) k 2 1 + Lf (tex , x ? st ) k 1 + (Rmin ? 1 ? ) 2 Ff (tex , x ? st , Rmin + 1 ? ) k 2 1 + Lf (tex , x ? st ) k 1 + (Rmin + 1 ? ) 2 + Ff (tex , x ? st , Rmin ? 1 ? ) k 2 1 + Lf (tex , x ? st ) k ? n ? X ? ? Z(tex , x, tik ? 1 ) ? Z(tex , x, tn ) Z(tex , x, ti ) k k k ? ? cvg(ti? 1 , ti )(bsi +mi ) ? ex , x) ex , x) . N (tk N (tk i=i +1 k ? This payo? includes only zero coupon bonds and ? oorlets, so we can calculate this reduced payo? explicitly using the antecedently derived formula 3. 9. The reduced valued in cluding the kth exercise is clearly ? ? Pk (x) V + (tex , x) V (tex , x) k k = max , at each x. (3. 13b) N (tex , x) N (tex , x) N (tex , x) k k k apply the Martingale valuation formula we can roll di? erences, trees, convolution, or direct integration to Z V + (tex , x) 1 k? 1 (3. 3c) =p N (tex , x) 2? ? k ? ? k? 1 k? 1 back to the preceding exercise date by using ? nite compute the integral V (tex , X) ? (X? x)2 /2? k k? 1 k dX e N (tex , X) k at each x. Here ? k = ? (tex ) and ? k? 1 = ? (tex ). k k? 1 At this point we aim moved from tex to the preceding exercise date tex . We now repeat the procedure k k? 1 at each x we take the max of V + (tex , x)/N (tex , x) and the payo? Pk? 1 (x)/N (tex , x) for tex , and then k? 1 k? 1 k? 1 k? 1 use the valuation formula to roll-back to the preceding exercise date tex , etc. in conclusion we work our way k? 2 througn the ? rst exercise V (tex , x).Then todays value is found by a ? nal integration k0 Z V (tex , X) ? X 2 /2? V (0, 0) 1 k0 k0 dX. (3. 14) V (0, 0) = =p e N (0, 0) N (tex , X) 2 k0 k0 3. 2. Using internal adjusters. The above pricing methodology satis? es the ? rst 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 Rmin goes to zero and Rmax becomes large. However the LGM model yields the following formulas for todays values of the standard ? orlets F (0, 0 ? st , K) = D(? end ) 1 + ? (K ? bs)N (? 1 ) ? 1 + ? (L0 ? bs)N (? 2 ) log 1 + ? (K ? bs) 1 ? 2 tf ix 2 mod 1 + ? (L0 ? bs) . v ? mod tf ix 13 (3. 15a) where (3. 15b) ?1,2 = Here (3. 15c) L0 = Dst ? Dend + bs(? st ) ? Dend is todays forward value for the Libor rate, and (3. 15d) q ? mod = (hend ? hst ) ? f ix /tf ix 3. 2. 1. Obtaining the market vol. Floorlets are quoted in terms of the ordinary (rate) vol. Suppose the rate vol is quoted as ? imp (K). Th en todays market price of the ? oorlet is is the assets log normal volatility according to the LGM model.We did not calibrate the LGM model to these ? oorlets. It is virtually certain that matching todays market prices for the ? oorlets will require using q an implied (price) volatility ? mkt which di? ers from ? mod = (hend ? hst ) ? f ix /tf ix . (3. 16a) where (3. 16b) Fmkt (? st , K) = ? D(? end ) KN (d1 ) ? L0 N (d2 ) d1,2 = log K/L0 1 ? 2 (K)tf ix 2 imp v ? imp (K) tf ix The price vol ? mkt is the volatility that equates the LGM ? oorlet value to this market value. It is de? ned implicitly by (3. 17a) with log (3. 17b) ? 1,2 = 1 + ? (K ? bs) 1 ? 2 tf ix 2 mkt 1 + ? (L0 ? bs) v ? kt tf ix 1 + ? (K ? bs)N (? 1 ) ? 1 + ? (L0 ? bs)N (? 2 ) = ? KN (d1 ) ? ?L0 N (d2 ), (3. 17c) d1,2 = log K/L0 1 ? 2 (K)tf ix 2 imp v ? imp (K) tf ix Equivalent vol techniques can be used to ? nd the price vol ? mkt (K) which corresponds to the market-quoted implied rate vol ? imp (K) (3. 18) ? i mp (K) = 1 + 5760 ? 4 t2 ix + 1+ imp f ? mkt (K) 1 2 1 4 2 24 ? mkt tf ix + 5760 ? mkt tf ix log L0 /K 1 + ? (L0 ? bs) 1 + ? (K ? bs) 1+ 1 2 24 ? imp tf ix log If this approximation is not su? ciently accurate, we can use a single Newton step to attain any reasonable accuracy. 14 igital floorlet value ? mod ? mkt L0/K Fig. 3. 1. maladjusted and adjusted digital payo? L/K 3. 2. 2. Adjusting the price vol. The price vol ? mkt obtained from the market price will not match the q LGM models price vol ? mod = (hend ? hst ) ? f ix /tf ix . This is easily remedied using an internal adjuster. All one does is compute the model volatility with the factor undeniable to bring it into line with the actual market volatility, and use this factor when calculating the payo? s. Speci? cally, in calculating each payo? Pk (x)/N (tex , x) in the rollback (see eq. 3. 13a), one makes the replacement k (3. 9) (3. 20) (hend ? hst ) q q ? mkt ? f ix ? ?(tex ) =? (hend ? hst ) ? f ix ? ?(t) k ? mod q p = 1 ? ?(tex )/? (tf ix )? mkt tf ix . k With the internal adjusters, the pricing methodology now satis? es the second criteria it agrees with all the vanilla prices that make up the range note coupons. Essentially, all the adjuster does is to moderately sharpen up or smear out the digital ? oorlets payo? to match todays value at L0 /K. This results in slightly positive or negative price corrections at versatile values of L/K, but these corrections medium out to zero when averaged over all L/K.Making this volatility qualifying is vastly superior to the other commonly used adjustment method, which is to add in a ? ctitious exercise fee to match todays coupon value. Adding a fee gives a positive or negative bias to the payo? for all L/K, even far from the money, where the payo? was certain to take over been correct. Meeting the second criterion agonistic 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 national model in which extra factors positively or negatively correlated with x enable us to obtain exactly these ? orlet prices while leaving our Gaussian rollback una? ected). However, not matching todays price of the underlying accrual swap would be a direct and immediate arbitrage. 15 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. Unlike a bond, there is no article of faith at risk. The only credit risk is for the di? erence in value between the coupon leg and the ? oating leg payments even this di? erence is usually collateralized through various inter-dealer arrangements.Since swaps are indivisible, liquidity is not an electric outlet they can be unwound by transferring a payment of the accrual swaps mark-to-market value. For these reasons, there is no detectable OAS in pricing accrual swaps. A range note is an actual bond which pays the cou pon leg on top of the principle repayments there is no funding leg. For these deals, the issuers credit-worthiness is a key concern. One needs to use an option adjusted spread (OAS) to obtain the extra discounting re? ecting the counterpartys credit spread and liquidity. Here we analyze bullet range notes, both uncallable and callable.The coupons Cj of these notes are set by the number of days an index (usually Libor) sets in a speci? ed range, just like accrual swaps ? tj X ? j Rf ix 1 if Rmin ? L(? st ) ? Rmax (4. 1a) Cj = , 0 otherwise Mj ? =t +1 st j? 1 where L(? st ) is k month Libor for the interval ? st to ? end (? st ), and where ? j and Mj are the day count fraction and the total number of days in the j th coupon interval tj? 1 to tj . In addition, these range notes repay the principle on the ? nal pay date, so the (bullet) range note payments are (4. 1b) (4. 1c) Cj 1 + Cn paid on tj , paid on tn . j = 1, 2, . . . n ? 1, For callable range notes, let the noti? ation on date s be tex for k = k0 , k0 + 1, . . . , K ? 1, K with K n. k bring that if the range note is called on tex , then the strike price Kk is paid on coupon date tk and the k payments Cj are cancelled for j = k + 1, . . . , n. 4. 1. modeling option adjusted spreads. Suppose a range note is issued by issuer A. ZA (t, x T ) to be the value of a dollar paid by the note on date T , as seen at t, x. We assume that (4. 2) ZA (t, x T ) = Z(t, x T ) ? (T ) , ? (t) De? ne where Z(t, x T ) is the value according to the Libor curve, and (4. 3) ? (? ) = DA (? ) . e D(? ) Here ? is the OAS of the range note.The excerpt of the discount curve DA (? ) depends on what we wish the OAS to measure. If one wishes to ? nd the range notes value relative to the issuers other bonds, then one should use the issuers discount curve for DA (? ) the OAS then measures the notes richness or cheapness compared to the other bonds of issuer A. If one wishes to ? nd the notes value relative to its credit risk, then the OAS calculation should use the issuers spoilt discount curve or for the issuers credit ratings unsafe discount curve for DA (? ). If one wishes to ? nd the absolute OAS, then one should use the swap markets discount curve D(? , so that ? (? ) is just e . When valuing a non-callable range note, we are just determining which OAS ? is needed to match the current price. I. e. , the OAS needed to match the markets idiosyncratic preference or adversion of the bond. When valuing a callable range note, we are making a much more powerful assumption. By assuming that the same ? 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 todays bond. 16 4. 2.Non-callable range notes. Range note coupons are ? xed by Libor settings and other issuerindependent criteria. then the value of a range note is obtained by leavi ng the coupon calculations alone, and substitute the coupons discount factors D(tj ) with the bond-appropriate DA (tj )e tj (4. 4a) VA (0) = n X j=1 ?j Rf ix DA (tj )e tj Mj ? ? 1 + (Rmax ? 1 ? ) B1 (? st ) ? 1 + (Rmax + 1 ? ) B2 (? st ) 2 2 ? 1 + L0 (? st ) ? st =tj? 1 +1 ? ? 1 + (Rmin ? 1 ? ) B3 (? st ) ? 1 + (Rmin + 1 ? ) B4 (? st ) 2 2 ? ? 1 + L0 (? st ) +DA (tn )e tn . tj X Here the last term DA (tn )e n is the value of the notional repaid at maturity. As before, the B? are shockings formulas, (4. 4b) B? (? st ) = Kj N (d? ) ? L0 (? st )N (d? ) 1 2 (4. 4c) d? = 1,2 log K? /L0 (? st ) 1 ? 2 (K? )tf ix 2 imp v ? imp (K? ) tf ix (4. 4d) K1,2 = Rmax 1 ? , 2 K3,4 = Rmin 1 ? , 2 and L0 (? ) is todays forward rate (4. 4e) Finally, (4. 4f) ? = ? end ? tj . ? end ? ? st L0 (? st ) = D(? st ) ? D(? end ) ? D(? end ) 4. 3. 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 adju st the coupon discounting in the payo? 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) callable bullet Berm VA (0) = VA (0) ? VA (0). bullet Berm (0) is the todays value of the non-callable range note in 4. 4a, and VA (0) is todays value of Here VA the Bermudan option. This Bermudan option is valued using exactly the same rollback procedure as before, 17 except that now the payo? is (4. 6a) (4. 6b) Pk (x) = N (tex , x) k ? tj X st =tj? 1 +1 j=k+1 n X ? j Rf ix ZA (tex , x tj ) k Mj N (tex , x) ? k 1 + (Rmax ? 1 ? ) 2 Ff (tex , x ? st , Rmax + 1 ? ) k 2 1 + Lf (tex , x ? st ) k ? ? + (Rmax + 1 ? ) 2 Ff (tex , x ? st , Rmax ? 1 ? ) k 2 1 + Lf (tex , x ? st ) k 1 + (Rmin ? 1 ? ) 2 Ff (tex , x ? st , Rmin + 1 ? ) k 2 1 + Lf (tex , x ? st ) k 1 + (Rmin + 1 ? ) 2 + Ff (tex , x ? st , Rmin ? 1 ? ) k 2 1 + Lf (tex , x ? st ) k ZA (tex , x, tn ) ZA (tex , x, tk ) k k + ? Kk ex , x) N (tk N (tex , x) k Here the bond speci? c reduced zero coupon bond value is (4. 6c) ex ex 1 2 ZA (tex , x, T ) D(tex ) k k = DA (T )e (T ? tk ) e? h(T )x? 2 h (T )? k , ex , x) N (tk DA (tex ) k ? the (adjusted) forwarded ? oorlet value is Ff (tex , x ? st , K) = 1 + ? (K ? bs)N (? 1 ) ? 1 + ? (L(tex , x ? t ) ? bs)N (? 2 ) k k log (4. 6d) ? 1,2 = 1 + ? (K ? bs) 1 1 ? ?(tex )/? (tf ix )? 2 tf ix k mkt 2 1 + ? (L ? bs) p , v 1 ? ?(tex )/? (tf ix )? mkt tf ix k Z(tex , x ? st ) k ? 1 + bs(? st ) Z(tex , x ? end ) k (hend ? hst )x? 1 (h2 ? h2 )? ex end st k ? 1 + bs(? 2 e st ) 1 = ? and the forward Libor value is (4. 6e) (4. 6f) L? L (tex , x ? st ) k Dst Dend 1 = ? The only remaining issue is calibration. For range notes, we should use constant mean atavism and calibrate along the diagonal, exactly as we did for the cancelable accrual swaps. We only need to specify the strikes of the reference swaptions.A sizeable method is to transfer the basis spreads and margin to the coupon leg, and then match the ratio of the coupon leg to the ? oating leg. For exercise on date tk , this ratio yields (4. 7a) n X ?k = ? j Rf ix DA (tj )e (tj ? tk ) Mj Kk DA (tk ) j=k+1 (? ? 1 + (Rmax ? 1 ? ) B1 (? st ) ? 1 + (Rmax + 1 ? ) B1 (? st ) 2 2 ? 1 + L0 (? st ) ? st =tj? 1 +1 ) ? ? 1 + (Rmin ? 1 ? ) B3 (? st ) ? 1 + (Rmin + 1 ? ) B3 (? st ) 2 2 ? 1 + Lf (tex , x ? st ) k tj X + DA (tn )e (tn ? tk ) Kk DA (tk ) 18 As before, the Bj are dimensionless Black formulas, (4. 7b) B? (? st ) = K? N (d? ) ?L0 (? st )N (d? ) 1 2 d? = 1,2 log K? /L0 (? st ) 1 ? 2 (K? )tf ix 2 imp v ? imp (K? ) tf ix K3,4 = Rmin 1 ? , 2 (4. 7c) (4. 7d) K1,2 = Rmax 1 ? , 2 and L0 (? st ) is todays forward rate Appendix A. Calibrating the LGM model. The are several(prenominal) methods of calibrating the LGM model for pricing a Bermudan swaption. The most popular method is to choose a constant mean regression ? , and then calibrate on the diagonal European swaptions making up the Bermudan. In the LGM model, a constant mean reversion ? means that the model function h(t) is given by (A. 1) h(t) = 1 ? e t . ? Usually the value of ? s selected from a table of values that are known to yield the correct market prices of liquid Bermudans It is known empirically that the needed mean reversion parameters are very, very stable, changing little from year to year. ? 1M 3M 6M 1Y 3Y 5Y 7Y 10Y 1Y -1. 00% -0. 75% -0. 50% 0. 00% 0. 25% 0. 50% 1. 00% 1. 50% 2Y -0. 50% -0. 25% 0. 00% 0. 25% 0. 50% 1. 00% 1. 25% 1. 50% 3Y -0. 25% 0. 00% 0. 25% 0. 50% 1. 00% 1. 25% 1. 50% 1. 75% 4Y -0. 25% 0. 00% 0. 25% 0. 50% 1. 00% 1. 25% 1. 50% 1. 75% 5Y -0. 25% 0. 00% 0. 25% 0. 50% 1. 00% 1. 25% 1. 50% 1. 75% 7Y -0. 25% 0. 00% 0. 25% 0. 50% 1. 00% 1. 25% 1. 50% 1. 75% 10Y -0. 25% 0. 0% 0. 25% 0. 50% 1. 00% 1. 25% 1. 50% 1. 75% Table A. 1 Mean reverssion ? for Bermudan swaptions. Rows are time-to-? rst exercise columns are melodic phrase of the longest underlying swap obtained upon exercise. With h(t) known, we only need determine ? (t) by calibrating to European swaptions. Consider a European swaption with noti? cation date tex . Suppose that if one exercises the option, one recieves a ? xed leg worth (A. 2a) Vf ix (t, x) = n X i=1 Rf ix cvg(ti? 1 , ti , dcbf ix )Z(t, x ti ), and pays a ? oating leg worth (A. 2b) Vf lt (t, x) = Z(t, x t0 ) ? Z(t, x tn ) + n X i=1 cvg(ti? 1 , ti , dcbf lt ) bsi Z(t, x ti ). 9 Here cvg(ti? 1 , ti , dcbf ix ) and cvg(ti? 1 , ti , dcbf lt ) are the day count fractions for interval i using the ? xed leg and ? oating leg day count bases. (For simplicity, we are cheating slightly by applying the ? oating legs basis spread at the frequency of the ? xed leg. Mea culpa). Adjusting the basis spread for the di? erence in the day count bases (A. 3) bsnew = i cvg(ti? 1 , ti , dcbf lt ) bsi cvg(ti? 1 , ti , dcbf ix ) allows us to write the value of the swap as (A. 4) Vswap (t, x) = Vf ix (t, x) ? Vf lt (t, x) n X = (Rf ix ? bsi ) cvg(ti? 1 , ti , dcbf ix )Z(t, x ti ) + Z(t, x tn ) ? Z(t, x t0 ) i=1 beneath the LGM model, todays value of the swaption is (A. 5) 1 Vswptn (0, 0) = p 2 (tex ) Z e? xex /2? (tex ) 2 Vswap (tex , xex )+ dxex N (tex , xex ) Substituting the explicit formulas for the zero coupon bonds and working(a) out the integral yields (A. 6a) n X (Rf ix ? bsi ) cvg(ti? 1 , ti , dcbf ix )D(ti )N Vswptn (0, 0) = where y is determined implicitly via (A. 6b) y + h(ti ) ? h(t0 ) ? ex p ? ex i=1 A A y + h(tn ) ? h(t0 ) ? ex y p ? D(t0 )N p , +D(tn )N ? ex ? ex A n X 2 1 (Rf ix ? bsi ) cvg(ti? 1 , ti , dcbf ix )e? h(ti )? h(t0 )y? 2 h(ti )? h(t0 ) ? ex i=1 +D(tn )e? h(tn )? h(t0 )y? h(tn )? h(t0 ) 1 2 ? ex = D(t0 ). The values of h(t) are known for all t, so the only unknown parameter in this price is ? (tex ). One can show that the value of the swaption is an increasing function of ? (tex ), so there is exactly one ? (tex ) which matches the LGM value of the swaption to its market price. This solution is easily found via a global Newton itera tion. To price a Bermudan swaption, one typcially calibrates on the helping Europeans. For, say, a 10NC3 Bermudan swaption struck at 8. 2% and callable quarterly, one would calibrate to the 3 into 7 swaption struck at 8. 2%, the 3. 25 into 6. 5 swaption struck at 8. 2%, , then 8. 75 into 1. 25 swaption struck at 8. 25%, and ? nally the 9 into 1 swaption struck at 8. 2%. Calibrating each swaption gives the value of ? (t) on the swaptions exercise date. One for the most part uses piecewise linear interpolation to obtain ? (t) at dates between the exercise dates. The remaining task is to pick the strike 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 tk , this ratio is de? ed to be 20 n X ? j D(tj ) (A. 7a) ? k = Mj D(tk ) ? j=k+1 D(tn ) X D(ti ) + cvg(ti? 1 , ti )(bs0 +mi ) ? i D(tk ) i=1 D(tk ) n ? 1 + (Rmax ? 1 ? ) B1 (? st ) ? 1 + (Rmax + 1 ? ) B2 (? st ) 2 2 ? 1 + L0 (? st ) st =tj? 1 +1 ? ? 1 + (Rmin ? 1 ? ) B3 (? st ) ? 1 + (Rmin + 1 ? ) B4 (? st ) 2 2 ? ? 1 + L0 (? st ) tj X ? where B? are Blacks formula at strikes around the boundaries (A. 7b) B? (? st ) = ? D(? end ) K? N (d? ) ? L0 (? st )N (d? ) 1 2 d? = 1,2 log K? /L0 (? st ) 1 ? 2 (K? )tf ix 2 imp v ? imp (K? ) tf ix (A. 7c) with (A. 7d) K1,2 = Rmax 1 ? , 2 K3,4 = Rmin 1 ?. 2This is to be matched to the swaption whose swap starts on tk and ends on tn , with the strike Rf ix chosen so that the equivalent ratio matches the ? k de? ned above (A. 7e) ? k = n X i=k+1 (Rf ix ? bsi ) cvg(ti? 1 , ti , dcbf ix ) D(ti ) D(tn ) + D(tk ) D(tk ) The above methodology works well for deals that are similar to bullet swaptions. For some foreigns, such as amortizing deals or zero coupon callables, one may wish to choose both the tenor of the and the strike of the reference swaptions. This allows one to match the exotic deals duration as well as its moneyness. Appendix B. Floating rate accrual notes. 21