Accrual Swaps

ACCRUAL SWAPS AND RANGE NOTES PATRICK S. HAGAN BLOOMBERG LP 499 PARK AVENUE NEW YORK, NY 10022 [email  protected] NET 212-893-4231 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. 1. 1. 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) These are de? ned via (1. 2c) D(T ) = e? T 0 f0 (T ) = today’s instantaneous forward rate for date T, f (t; T ) = instantaneous forward rate for date 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 period j Coupon leg schedule Fixed coupon accrual swaps (aka time swaps) consist of a coupon leg swapped against a funding leg. Suppose that the agreed upon reference rate is, say, k month Libor. Let (1. 3) t0 < t1 < t2  ·  ·  · < tn? 1 < tn 1 Rfix Rmin Rmax L( ? ) Fig. 1. 1. Daily coupon rate be the schedule of the coupon leg, and let the nominal ? xed rate be Rf ix . Also let L(? st ) represent the k month Libor rate ? xed for the interval starting at ? st and ending at ? end (? st ) = ? t + k months. Then the coupon paid for period j is (1. 4a) where (1. 4b) and (1. 4c) ? j = #days ? st in the interval with Rmin ? L(? st ) ? Rmax . Mj ? j = cvg(tj? 1 , tj ) = day count fraction for tj? 1 to tj , Cj = ? j Rf ix ? j paid at tj , Here Mj is the total number of days in interval j, and Rmin ? L(? st ) ? Rmax is the agreed-upon accrual range. Said another way, each day ? st in the j th period contibutes the amount ? ?j Rf ix 1 if Rmin ? L(? st ) ? Rmax (1. 5) 0 otherwise Mj to the coupon paid on date tj . For a standard deal, the leg’s schedule is contructed like a standard swap schedule. The theoretical dates (aka nominal dates) are constructed monthly, quarterly, semi-annually, or annually (depending on the contract terms) backwards from the â€Å"theoretical end date. † Any odd coupon is a stub (short period) at the front, unless the contract explicitly states long ? rst, short last, or long last. The modi? ed following business day convention is used to obtain the actual dates tj 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 tj? 1 and tj , not the theoretical dates. Also, L(? t ) is the ? xing that pertains to periods starting on date ? st , regardless of whether ? st is a good business day or not. I. e. , the rate L(? st ) set for a Friday start also pertains for the following Saturday and Sunday. Like all ? xed legs, there are many variants of these coupon legs. The only variations that do not make sense for coupon legs are â€Å"set-in-arrearsâ €  and â€Å"compounded. † There are three variants that occur relatively frequently: Floating rate accrual swaps. Minimal coupon accrual swaps. Floating rate accrual swaps are like ordinary accrual swaps except that at the start of each period, a ? ating rate is set, and this rate plus a margin is 2 used in place of the ? xed rate Rf ix . 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 ? j Mj ? Rf ix Rf loor if Rmin ? L(? st ) ? Rmax . otherwise (A standard accrual swap has Rf loor = 0. These deals are analyzed in Appendix B. Range notes. In the above deals, the funding leg is a standard ?oating leg plus a margin. A range note is a bond which pays the coupon leg on top of the principle repayments; there is no ? oating leg. For these deals, the counterparty’s credit-worthiness is a key concern. To determine the correct value of a range note, one needs to use an option adjusted spread (OAS) to re? ect the extra discounting re? ecting 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(? 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(? st ). These CMS or CMT 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 payo? in terms of vanilla caps and ? oors. Consider the j th period of a coupon leg, and suppose the underlying indice is k-month Libor. Let L(? st ) be the k-month Libor rate which is ? xed for the period starting on date ? st and ending on ? end (? st ) = ? st +k months. The Libor rate will be ? xed on a date ? f ix , which is on or a few days before ? st , depending on currency. On this date, the value 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 value coupon j by replicating each day’s contribution in terms of vanilla caplets/? oorlets, and then summing over all days ? st in the period. Let Fdig (t; ? st , K) be the value at date t of a digital ? oorlet on the ? oating rate L(? st ) with strike K. If the Libor rate L(? st ) is at or below the strike K, the digital ? oorlet pays 1 unit of currency on the end date ? end (? st ) of the k-month interval. Otherwise the digital pays nothing. So on the ? xing date ? f ix the payo? is known to be ? 1 if L(? st ) ? K , (2. 2) Fdig (? f ix ; ? st , K) = Z(? f ix ; ? end ) 0 otherwise We can replicate the range note’s payo? for date ? st by going long and short digitals struck at Rmax and Rmin . This yields, (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 ; ? end ) 0 Mj 3 if Rmin ? L(? st ) ? Rmax . otherwise This is the same payo? as the range note, except that the digitals pay o? on ? end (? st ) instead of tj . 2. 1. Hedging considerations. Before ? ing the date mismatch, we note that digital ? oorlets are considered vanilla instruments. This is because they can be replicated to arbitrary accuracy by a bullish spread of ? oorlets. Let F (t, ? st , K) be the value on date t of a standard ? oorlet with strike K on the ? oating + rate L(? st ). This ? oorlet pays ? [K ? L(? st )] on the end date ? end (? st ) of the k-m onth interval. So on the ? xing date, the payo? is known to be (2. 5a) F (? f ix ; ? st , K) = ? [K ? L(? st )] Z(? f ix ; ? end ). + Here, ? 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 yields the payo? 2 (2. 6) which goes to the digital payo? as ? > 0. Clearly the value 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 Thus the bullish spread, and its limit, the digitial ? orlet, are directly determined by the market prices of vanilla ? oors on L(? st ). Digital ? oorlets may develop an unbounded ? - risk as the ? xing date is approached. To avoid this di? culty, most ? rms book, price, 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 banks choose to super-replicate or sub-replicate the digital, by booking and hedging 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. One should price accrual swaps in accordance with a desk’s policy for using central- or super- and sub-replicating payo? s for other digital caplets and ? oorlets. 2. 2. Handling the date mismatch. We re-write the coupon leg’s contribution 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 mismatch is corrected assuming only parallel shifts in the forward curve The ratio Z(? ix ; tj )/Z(? f ix ; ? end ) is the manifestation of the date mismatch. To handle this mismatch, we approximate the ratio by assuming that the yield curve makes only 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 en d ) = 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 forward rate for the k-month period starting at ? t , as seen at the current date t0 , bs(? st ) is the ? oating rate’s basis 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 change in the yield curve between tj and ? end is the same as the change 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 slightly 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 only linear approximation which accounts for the day count basis correctly, is exact for both ? st = tj? 1 and ? st = tj , and is centerred around the current forward value for the range note. 5 Rfix Rmin L0 Rmax L(? ) Fig. 2. 2. E? ective contribution from a single day ? , after 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 notional 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 using a combination of two digital ? oorlets and two standard ? oorlets. Consider 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 ; Rmi n ). Setting t to the ? xing date ? f ix shows that this combination matches the contribution from day ? st in eq. 2. 11a. Therefore, this formula gives the value of the contribution of day ? t for all earlier dates t0 ? t ? ? f ix as well. Alternatively, one can replicate the payo? as close as one wishes by 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 choose to use the ? oorlet spreads (with ? being 5bps or 10bps) instead of using the more troublesome digitals. For a bank insisting on using exact digital options, one can take ? to be 0. 5bps to replicate the digital accurately.. We now just need to sum over all days ? t in period j and all periods j in the coupon leg, (2. 15) Vcpn (t) = n X This formula replicates the value of the range note in terms of vanilla ? oorlets. These ? oorlet prices should be obtained directly from the marketplace using market quotes for the implied volatilities at the relevent strikes. Of course the centerred spreads could be replaced by super-replicating or sub-replicating ? oorlet spreads, bringing the pricing in line with the bank’s policies. Finally, we need to value the funding leg of the accrual swap. For most accrual swaps, the funding leg ? ? pays ? oating plus a margin. Let th e 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 rate’s ? xing for the period ti? 1 < t < ti , and the mi is the margin. The value 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 value 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 leg’s ? oating rate, and ri i collapses t o 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 include only the funding leg payments for i = i0 to n, the value 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/? oorlet prices are normally quoted in terms of Black vols. Suppose that on date t, a ? oorlet with ? xing date tf ix , start date ? st , end date ? end , and strike K has an implied vol of ? imp (K) ? ? imp (? st , K). Then its market price 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 = log 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 ? oorlet’s forward rate as seen at date t. Today’s ? oorlet value is simply (2. 20a) where (2. 20b) d1,2 = log 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 today’s forward Libor rate is (2. 20c) L0 (? st ) = To obtain today’s price of the accrual swap, note that the e? ective notional for period j is (2. 21) A(0, ? st ) = as seem today. See 2. 11b. Putting this together with 2. 13a shows that today’s 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 Black’s 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 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 ? with an integral, and samples the contribution from dates ? one week apart for the next year, and one month apart for subsequent years. 8 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 lock-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 key requirements in pricing Bermudan accrual swaps. First, as Rmin decreases and Rmax increases, the value of the Bermudan accrual swap should reduce to the value of an ordinary Bermudan swaption with strike Rf ix . Besides the obvious theoretical appeal, meeting this requirement allows one to hedge the callability of the accrual swap by selling an o? setting 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 â€Å"diagonal swaptions† struck at the accrual note’s â€Å"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 Ref f for each of these â€Å"reference swaptions† would be chosen 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 swap’s funding leg ? i This usually results in strikes Ref f that are not too far from the money. In the preceding section we showed that each coupon of the accrual swap can be written as a combination of vanilla ? oorlets, and therefore the market value of each coupon is known exactly. The second requirement is that the valuation procedure should reproduce today’s m arket 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 risk of the ? oorlets that make up the j th coupon payment. E? ectively this means that the pricing methodology needs 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 why callable range notes are considered heavily skew depedent products. 3. 1. Hull-White model. Meeting these requirements would seem to require using a model that is sophisticated enough to match the ? oorlet smiles exactly, as well as the diagonal swaption volatilities. Such a model 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 model to match the diagonal swaption prices, and then use â€Å"internal adjusters† to match the ? oorlet volatilities. Here we follow this approach, using the 1 factor linear Gauss Markov (LGM) model with internal adjusters to price Bermudan options on accrual swaps. Speci ? cally, we ? nd explicit formulas for the LGM model’s prices of standard ? oorlets. This enables us to compose the accrual swap â€Å"payo? s† (the value recieved at each node in the tree if the Bermudan is exercised) as a linear combination of the vanilla ? orlets. With the payo? s known, the Bermudan can be evaluated via a standard rollback. The last step 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 exactly the Hull-White model expressed as an HJM model. The 1 factor LGM model has a single state variable x that determines the entire 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 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 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 2 1 e? (X? x) /2[? (T ) (t)] , p(t, x; T, X) = p 2? [? (T ) ? ?(t)] (3. 2b) with a variance 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 become apparent. Without loss of generality, one sets x = 0 at t = 0, and today’s variance is zero: ? (0) = 0. The ratio (3. 3a) V (t, x) ? V (t, x) ? N (t, x) is usually called the reduced value of the deal. Since N (0, 0) = 1, today’s value coincides with today’s reduced value: (3. 3b) V (0, 0) ? V (0, 0) = V (0, 0) ? . N (0, 0) So we only have to work with reduced values to get today’s prices.. De? ne Z(t, x; T ) to be the value of a zero coupon bond with maturity T , as seen at t, x. It’s value can be found by substituting 1 for V (T, X) in the Martingale 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 (3. 4b) Z(t, x; T ) ? e? T t f (t,x;T 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 today’s forward rate curve f0 (T ) plus x times a second curve h0 (T ), where x is a Gaussian random variable with mean zero and variance ? (t). Thus h0 (T ) determines possible shapes of the forward curve and ? (t) determines the width of the distribution of forward curves. The last term h0 (T )h(T )? (t) is a much smaller convexity 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. Regardless 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 â€Å"true rate† for the interval and bs(? ) is the Libor rate’s 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. Substituting 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). Substituting the zero coupon bond formula 3. 4a and the payo? 3. 7a into the integral yields (3. 8a) where log (3. 8b) ? 1,2 =  µ 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)  µ  ¶ 1 Z(t, x; ? st ) ? 1 + bs(? st ) ? Z(t, x; ? end )  ¶  µ 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 convenient to split o? the zero coupon 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 Black’s formul as 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 + ? (L ? bs) = asset’s forward value, 1 + ? (K ? bs) = strike, ? end = settlement date, and p ? f ix ? ? (hend ? hst ) v = ? = asset volatility, tf ix ? t where tf ix ? t is the time-to-exercise. One should not confuse ? , which is the ? oorlet’s â€Å"price volatility,† with the commonly quoted â€Å"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 â€Å"model parameters† h(t) and ? (t) are known. (In Appendix A we show one popular calibration method). Let the Bermudan’s noti? cation dates be tex , tex+1 , . . . , tex . Suppose that if we exercise on date tex , we receive all coupon payments for the K k0 k0 k intervals k + 1, . . . , n and recieve all funding leg payments f or intervals ik , ik + 1, . . . , n. ? The rollback works by induction. Assume that in the previous rollback steps, we have 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 preceding 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 previously derived formula 3. 9. The reduced valued including 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 Using the Martingale valuation formula we can â€Å"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 have moved from tex to the preceding exercise date tex . We now repeat the procedure: k k? 1 at each x we t ake 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. Eventually we work our way k? 2 througn the ? rst exercise V (tex , x). Then today’s 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 today’s values of the standard ? orlets: F (0, 0; ? st , K) = D(? end ) {[1 + ? (K ? bs)]N (? 1 ) ? [1 + ? (L0 ? bs)]N (? 2 )} log  µ  ¶ 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 today’s 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). Then today’s market price of the ? oorlet is is the asset’s log normal volatility according to the LGM model. We did not calibrate the LGM model to these ? oorlets. It is virtually certain that matching today’s 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 =  µ  ¶ 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) ? imp (K) = 1 + 5760 ? 4 t2 ix +  ·  ·  · 1+ imp f ? mkt (K) 1 2 1 4 2 24 ? mkt tf ix + 5760 ? mkt tf ix  µ 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. Unadjusted 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 model’s price vol ? mod = (hend ? hst ) ? f ix /tf ix . 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 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 slightly â€Å"sharpen up† or â€Å"smear out† the digital ? oorlet’s payo? to match today’s value at L0 /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. Making this volatility adjustment is vastly superior to the other commonly used adjustment method, which is to add in a ? ctitious â€Å"exercise fee† to match today’s 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 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 ? orlet prices while leaving our Gaussian rollback una? ected). However, not matching today’s 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. U nlike a bond, there is no principle 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 issue: they can be unwound by transferring a payment of the accrual swap’s 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 coupon leg on top of the principle repayments; there is no funding leg. For these deals, the issuer’s credit-worthiness is a key concern. One needs to use an option adjusted spread (OAS) to obtain the extra discounting re? ecting the counterparty’s 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 dates be tex for k = k0 , k0 + 1, . . . , K ? 1, K with K < n. k Assume 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 choice of the discount curve DA (? ) depends on what we wish the OAS to measure. If one wishes to ? nd the range note’s value relative to the issuer’s other bonds, then one should use the issuer’s discount curve for DA (? ); the OAS then measures the note’s richness or cheapness compared to the other bonds of issuer A. If one wishes to ? nd the note’s value relative to its credit risk, then the OAS calculation should use the issuer’s â€Å"risky discount curve† or for the issuer’s credit rating’s risky discount curve for DA (? ). If one wishes to ? nd the absolute OAS, then one should use the swap market’s 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 market’s idiosyncratic preference or adversion of the bond. When valuing a callable range note, we are ma king 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 today’s bond. 16 4. 2. Non-callable range notes. Range note coupons are ? xed 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(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 Black’s 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 today’s forward rate: (4. 4e) Finally, (4. 4f) ? = ? end ? tj . ? en d ? ? 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 adjust 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 today’s value of the non-callable range note in 4. 4a, and VA (0) is today’s 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 =  µ  ¶ 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 = ?  µ and the forward Libor value is (4. 6e) (4. 6f) L? L (tex , x; ? st ) k  µ Dst Dend 1 = ? 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 strikes 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 ? 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 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 ? , and then calibrate on the diagonal European swaptions making up the Bermudan. In the LGM model, a â€Å"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 tenor 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 fraction s for interval i using the ? xed leg and ? oating leg day count bases. (For simplicity, we are cheating slightly by applying the ? oating leg’s 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 Under the LGM model, today’s 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 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 iteration. T o price a Bermudan swaption, one typcially calibrates on the component 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 swaption’s exercise date. One generally uses piecewise linear interpolation to obtain ? (t) at dates between the exercise dates. The remaining problem 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 Black’s 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 ?. 2 This 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 exotics, 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 deal’s duration as well as its moneyness. Appendix B. Floating rate accrual notes. 21