i i i i Cutting edge: Option pricing
Bounding Bermudans
Thomas Roos derives model-independent bounds for amortising and accreting Bermudan swaptions in terms of portfolios of standard Bermudan swaptions. In addition to the well-known upper bounds, the author derives new lower bounds for both amortisers and accreters. Numerical results show the bounds to be quite tight in many situations. Applications include model arbitrage checks, limiting valuation uncertainty, and super-replication of long and short positions in the non-standard Bermudan
Bermudan swaption gives the holder the right to enter into a fixed no more liquid than the ABerm itself, so their prices must themselves be versus floating swap at a set of future exercise dates. A standard modelled. See Andersen & Piterbarg (2010) for an overview of ABerm A Bermudan swaption (SBerm), sometimes called a ‘bullet’Bermu- calibration methodologies. dan, is characterised by the fact that it is exercisable into a swap whose Whatever methodology is chosen to determine the calibration targets for notional (and fixed rate) is the same for all coupons. In addition to the ABerms, the resulting local calibrationsMedia can be significantly different from important and relatively liquid market for SBerms, there is considerable those used for SBerms. Such differences can translate into inconsistencies interest in accreting and amortising Bermudan swaptions (we will refer and even arbitrage between these closely related products. The bounds to these collectively as ABerms). Accreting Bermudans are exercisable obtained in this article provide a useful safeguard against this kind of into swaps whose notionals increase coupon by coupon, while amortis- model arbitrage. ing Bermudans are exercisable into swaps whose notionals decrease for each coupon. Economically, accreting Bermudans arise from the issuing Notation and preliminaries n of callable zero-coupon bonds, with the notional accreting at (or close to) We consider a schedule fTi giD0 constituting the fixed leg of a swap. Let a the fixed rate of the bond (see, for example, Andersen & Piterbarg 2010, V .t/ be the time t ation adjustments. For similar applications of model-independent bounds where P.t;T/ is the discount bond at time t with maturity T , and i in different contexts, see, for example, Laurence & Wang (2004) and John- and Li .t/ are the day count fraction and forward Libor rate for period son & Nonas (2009). For an exact replication strategy of flexi-swaps using (Ti , TiC1), respectively. For future use, we note that the ASwap can be SBerms, see Evers & Jamshidian (2005). decomposed into a portfolio of co-initial standard swaps: In relation to model arbitrage, it is clear that, when using a globally cali- Xn a brated model (eg, a Libor market model), there is no possibility of arbitrage V .t/ D ıNkV.tI 0; k/ (1) betweenABerm and SBerm prices. In practice, however, SBerms are often kD1 valued using low-dimensional models with local (ie, trade-dependent) cali- as well as into a portfolio of co-terminal standard swaps: bration. For example, for performance reasons, practitioners typically use nX 1 a Gaussian or local volatility one-factor models, and price the Bermudan V .t/ D ıNkV.tI k; n/ (2) optionality using partial differential equation (PDE) solvers and back- kD0 ward induction. When such a low-dimensional model is used for SBerms, where we have defined ıNk Nk Nk 1, with N 1 D Nn 0. We will it is often considered desirable to use the same approach for ABerms. refer to (1) and (2) as the ‘co-initial’ and ‘co-terminal’ decompositions, Given the limited calibration capabilities of the model, calibration strate- respectively. gies are then necessarily product dependent. For SBerms, the model is typically calibrated to the set of co-terminal European swaptions (at one 1 A receiver swap receives the fixed leg (pays floating). A receiver swaption or more strikes), and the mean-reversion parameter is adjusted to match any is an option to enter into a receiver swap. observable BermudanCopyright premiums. ForABerms, the fundamental difference 2 When explicitly writing out the floating leg, we will assume it has the same is that the corresponding calibration targets are the co-terminal amortis- frequency and day count as the fixed leg. This is done solely to simplify ing or accreting European swaptions (ASwaptions). These are generally the notation; all results are valid for general conventions. risk.net 1 i i i i i i i i Cutting edge: Option pricing Model-independent bounds on the right-hand side of (7) into Bermudan swaptions. However, the neg- Upper bounds: amortisers. From (1), we see that amortising swaps ative notionals present a problem in that we cannot be short options in (where Nk 1 > Nk) can be decomposed into a sum of standard co-initial our super-replicating portfolio, as we need to have control over when the swaps with positive notionals. Explicitly, we have: options are exercised to match the ABerm. We could make the notionals positive by simply viewing a negative notional payer swap as a positive n 1 X notional receiver swap (and vice versa). This leads to an upper bound in V amo.t/ D jıN jV.tI 0; k/ C N V.tI 0; n/ (3) k n 1 terms of a mixed portfolio of payers and receivers, which is not useful in kD1 practice (and always inferior to the upper bound in (4)). If, instead, we We now consider a Bermudan swaption to enter into this amortising swap move the negative notional swaps to the left-hand side: (ABerm), and the portfolio of standard Bermudans (SBerms) to enter nX 1 into the standard swaps on the right-hand side of (3). It is evident this amo V .t/ C jıNkjV.tI k; n/ D N0V.tI 0; n/ portfolio of SBerms enables us to replicate all possible payouts of the kD1 ABerm: the strategy is simply to exercise all SBerms at the same time the and apply our previous reasoning, we conclude that if we owned a port- ABerm is exercised. Since the SBerm portfolio must be worth at least as folio of Bermudan swaptions that could be exercised into the swaps on much as the ABerm it can replicate, this gives us an upper bound for the the left-hand side, then this portfolio would constitute a super-replicating value of the amortising Bermudan in terms of the portfolio of standard Media strategy for the Bermudan to enter into the standard swap on the right-hand Bermudans: side. This gives us an upper bound on the value of this standard Bermu- nX 1 dan, which we write as a lower bound on the value of the amortising amo B .t/ 6 jıNkjB.tI 0; k/ C Nn 1B.tI 0; n/ (4) Bermudan: kD1 nX 1 amo B .t/ > N0B.tI 0I n/ jıN jB.tI k; n/ (8) Upper bounds: accreters. The reasoning behind accreting Bermudan k kD1 swaptions (where Nk > Nk 1) is similar, except that we need to start from Lower bounds: accreters. Decomposing the accreter using the co- (2) to get a decomposition of the accreting swap into a sum of standard initial decomposition in (1), we find: swaps with positive notionals. These swaps are co-terminal, starting at fT gn 1 and ending at T . Explicitly, we have: nX 1 i iD0 n acc V .t/ D Nn 1V.tI 0I n/ jıNkjV.tI 0; k/ (9) nX 1 kD1 acc V .t/ D N0V.tI 0I n/ C jıNkjV.tI k; n/ (5) Again, only the longest tenor swap has positive notional in this repre- kD1 sentation. Following the approach of the previous section, we move the As in the case of the amortiser, it is clear that if we own options that allow negative notional swaps to the left-hand side and deduce that the corre- us, at all possible exercise times of the ABerm, to enter into the standard sponding portfolio of Bermudan swaptions is a super-replication of the Incisivestandard Bermudan swaption corresponding to the right-hand side. This swaps on the right-hand side of (5), then we can replicate all possible payouts of the ABerm. The same reasoning as before gives us an upper gives us an upper bound for this standard Bermudan, which we write as a bound on the value of the accreting Bermudan in terms of a portfolio of lower bound for the accreter: standard Bermudans: nX 1 acc B .t/ > Nn 1B.tI 0I n/ jıNkjB.tI 0; k/ (10) n 1 X kD1 acc B .t/ 6 N0B.tI 0I n/ C jıNkjB.tI k; n/ (6) Discussion. In the previous sections, we obtained two-sided bounds kD1 for amortising and accreting Bermudan swaptions using super-replication One difference to the case in the previous section, where all SBerms start at arguments. A more formal proof can be found in appendix A. The upper the same time as the ABerm, is that, here, some of the co-terminal SBerms bounds in (4) and (6), and their usefulness in providing arbitrage checks in our portfolio may not have started at the time the ABerm is exercised. for ABerm prices, are well known to many practitioners (see, for example, However, this presents no difficulty for the replication strategy, as we can Andersen & Piterbarg 2010, section 19.4.5). From a risk management point simply wait for the Bermudan to start and exercise it then. of view, these bounds provide a static super-replication (ie, overhedge) for Lower bounds: amortisers. Previously, we used the co-initial a short position in the ABerm: whenever the short ABerm is exercised by decomposition in (1) to write the amortising swap as a sum of standard the counterparty, the portfolio of SBerms is able to reproduce the ASwap swaps with positive notionals. If we instead look at the co-terminal decom- payout. While the reasoning above for deriving upper and lower bounds position in (2), we find that only the notional of the longest tenor swap is is similar, the lower bounds in (8) and (10) do not seem to be known in positive, with all others being negative. Explicitly, we have: the literature or by many practitioners. In addition to providing arbitrage checks from the ‘other side’, they also represent overhedge strategies for nX 1 amo long positions in the ABerm. For example, (10) tells us that if we can buy V .t/ D N0V.tI 0I n/ jıNkjV.tI k; n/ (7) the accreting Bermudan at the lower bound value of: CopyrightkD1 nX 1 We would like to follow our previous reasoning and obtain a super- Nn 1B.tI 0I n/ jıNkjB.tI 0; k/ replicating portfolio for the amortiser by turning all the standard swaps kD1 2 risk.net June 2017 i i i i i i i i Cutting edge: Option pricing then we can sell the longest tenor standard Bermudan and purchase the tradable bids and offers that allow static hedging of ABerms via super- set of shorter tenor standard Bermudans, and so create a statically hedged replication. In this section, we provide numerical examples to explore this portfolio that requires no rebalancing over time. question. The modelling setup is as follows. Both SBerms and ABerms If the long tenor Bermudan is exercised against us, we simply exer- are priced in a one-factor Gaussian (Hull-White) model, using standard cise the ABerm and all remaining SBerms, cancelling the resulting swap PDE backward propagation. For each SBerm, the model is calibrated to position. In this context, it is important to note that, while our derivations the corresponding co-terminal standard European swaptions at the strike. assume the entire super-replicating portfolio is exercised at once, this is For each ABerm, the model is calibrated to the corresponding co-terminal not what one would do when running the strategy in practice. Instead, one amortising or accreting European swaptions, priced using the basket model would only exercise those Berms for which it was optimal (if any), and in appendix B. We use US dollar yield curve and swaption volatility data sell the rest back to the market, producing a windfall. The difference in from November 2016. The mean reversion is set to zero throughout to value at inception between the ABerm and its super-replicating portfolio reflect the low mean reversions typically required to match US dollar is therefore the maximum possible cost of running the strategy. The actual Bermudan premiums in practice. cost will usually be less. At first sight, the use of a one-factor model to price the Bermudans may We will explore the tightness of the bounds numerically in the next seem overly simplistic. We note, however, that the bounds only depend on section, but we can deduce some general patterns from our derivation in SBerm prices, and for these the useMedia of locally calibrated, low-dimensional terms of super-replication. In particular, we expect the tightness to depend models is both common practice and well established as being sufficient on how suboptimal it is (in expectation) to exercise all Bermudans in the (see, for example, Andersen & Andreasen 2001). For ABerms, especially portfolio at the optimal exercise time of the replicated Bermudan. The more accreters, the adequacy of one-factor models is less clear. However, we optionality we ‘give away’ when exercising the portfolio, the looser the expect our basket calibration coupled with different correlation scenarios bound. Since it is always suboptimal to make an exercise decision before to produce a representative range of ABerm prices; ultimately, the key a Bermudan’s start date, we expect the co-initial bounds (amortiser upper point is the ABerm price must lie within the bounds, independent of the and accreter lower) to be tighter than the co-terminal bounds (amortiser model used. lower and accreter upper). In the following examples, amortiser notionals decrease to zero in equal Following similar reasoning, we expect the bounds to be tighter for annual steps over the life of the trade, while accreter notionals increase payers in a rising curve environment, and vice versa. This is because annually by the fixed rate (strike) of the swaption. The initial notional is for increasing forward rates, the payer coupons become more positive set to one in both cases. as maturity increases, meaning that longer-dated Bermudans are more In the tables that follow, ‘Opt’ is the option type (payer or receiver), K in-the-money than shorter ones. This means that when it is optimal to is the strike, and ‘UB’ and ‘LB’ are the upper bound and the lower bound, exercise the longest tenor Bermudan, we also expect it to be close to opti- respectively. All Bermudans are exercisable annually. For the ABerm, we mal to exercise all shorter tenor co-initial and co-terminal Bermudans. run two correlation scenarios: ij D 100% for all fi;jg, and ij D 90% This is because one will wait to exercise the long tenor Bermudan if the for all fi ¤ j g (see appendix B). In each scenario, Ba is the value of shorter dated co-initial Bermudans are not optimal, while the shorterIncisive tenor the amortising or accreting Bermudan swaption, and a is its vega for a 1 co-terminal Bermudans will be deeper in-the-money than the long tenor basis point (10 4) parallel shift in (normal) European swaption volatilities. Bermudan, and therefore also close to optimal exercise. Finally, ı U and ı L are the changes in ABerm volatility (in bp) required Finally, we note that so far we have assumed the notional change dates to hit the upper bound and the lower bound, respectively (defined as ı U a a a a match the exercise schedule of the Bermudan. In general, exercise oppor- .UB B /= and ı L .B LB/= ). This is a measure of the tightness tunities may be more frequent (eg, notionals step up/down every five years, of the bound in terms of normal volatility, which can easily be compared while the trade is callable annually) or less frequent (eg, the trade is callable with volatility bid/offers, for example. Note that, given the definitions of every five years while notionals step up/down annually). The first case the ı , a negative value indicates a violation of the corresponding bound. is already covered by our formulas, with ıNi D 0 on exercise dates for We begin by looking at a ‘10-year no-call one-year’(10-nc-1) Bermudan which the notional does not change. The second case is more interesting. It swaption.3 Tables A and B show the results for the amortiser and accreter, requires one option for each date on which the notional changes (regardless respectively. of whether this is an ABerm exercise date or not), with the exercise sched- As expected from the discussion in the previous section, the bounds are ule of each such option matching the ABerm exercise schedule for sub- tighter for the payers than the receivers for the currently upward-sloping sequent dates. In particular, this means that for European amortising and US dollar yield curve. The payer bounds are very tight indeed, with the accreting swaptions, the formulas for the bounds are the same as derived upper and lower bounds separated by a few bp in value, corresponding to above, except with all SBerms replaced by standard European swaptions. anywhere from a fraction of a vega to a few vegas difference over a range of Since the EuropeanASwaptions must themselves be modelled, the bounds strikes. For the accreter, the D 90% Bermudan value is actually through can also be useful in this context as arbitrage checks and super-replication the upper bound for K > 2%, indicating (mild) arbitrage for this model strategies. specification (there are similar small violations for the amortising payer Numerical examplesCopyright For the bounds derived above to be useful, they must be tight enough to 3 That is, a Bermudan swaption starting in one year’s time on a swap act as meaningful checks against model arbitrage, or, ideally, to provide ending in 10 years’ time. risk.net 3 i i i i i i i i Cutting edge: Option pricing A. 10-nc-1 amortiser C. 30-nc-1 amortiser D 100% D 90% D 100% D 90% a a a a a a a a Opt K UB LB B ı U ı L B ı U ı L Opt K UB LB B ı U ı L B ı U ı L Pay 100 396 382 396 2.5 0.1 5.5 388 2.4 3.1 2.5 Pay 100 1591 1502 1583 9.7 0.8 8.3 1549 9.1 4.6 5.2 Pay 200 180 172 180 2.7 0.1 2.9 172 2.5 3.3 0.0 Pay 200 918 831 906 12 1.0 6.4 866 11 4.6 3.1 Pay 300 81 75 80 2.0 0.7 2.6 74 1.8 4.1 0.4 Pay 300 503 436 491 11 1.1 4.9 455 10 4.7 1.9 Pay 400 45 37 42 1.3 1.8 4.1 38 1.2 5.5 0.9 Pay 400 288 243 278 8.6 1.2 4.1 249 8.0 4.9 0.8 Pay 500 24 19 22 0.8 1.5 3.8 20 0.8 5.0 0.3 Pay 500 177 149 170 6.6 1.1 3.2 147 5.9 5.1 0.4 Rec 100 96 77 91 2.3 2.0 6.2 84 2.2 5.2 3.4 Rec 100 331 241 308 8.6 2.7 7.9 280 8.0 6.4 4.9 Rec 200 298 233 286 2.4 5.3 22 278 2.4 8.5 19 Rec 200 718 540 673 10 4.6 13 638 9.5 8.5 10 Rec 300 670 593 665 1.0 5.5 72 661 0.8 11 86 Rec 300 1439 1184 1401 6.7 5.7 32 1377 5.7 11 34 Rec 400 1120 1068 1119 0.2 6.5 256 1118 0.2 10 253 Rec 400 2502 2266 2490 1.9 6.3 118 2482 1.5 13 144 Rec 500 1586 1555 1586 0.0 4.2 637 1586 0.0 15 1127 Rec 500 3699 3534 3696 0.6 4.3 271 3695 0.3 14 536 All values in bp. Violations of the bounds are in bold. Source: Thompson Reuters All values in bp. Violations of the bounds are in bold. Source: Thompson Reuters B. 10-nc-1 accreter D. 30-nc-1 accreter D 100% D 90% D 100% D 90% a a a a Opt K UB LB B ı U ı L B ı U ı L a aMediaa a Opt K UB LB B ı U ı L B ı U ı L Pay 100 912 911 911 5.3 0.2 0.0 912 5.3 0.0 0.2 Pay 100 3570 3541 3548 25 0.9 0.3 3566 25 0.1 1.0 Pay 200 504 502 502 6.4 0.2 0.0 505 6.4 0.1 0.4 Pay 200 2541 2473 2497 36 1.2 0.7 2544 36 0.1 2.0 Pay 300 265 263 264 5.7 0.2 0.1 266 5.7 0.2 0.5 Pay 300 1748 1657 1698 41 1.2 1.0 1771 42 0.5 2.7 Pay 400 156 152 154 4.4 0.3 0.4 156 4.4 0.2 0.9 Pay 400 1296 1201 1250 41 1.1 1.2 1342 43 1.1 3.3 Pay 500 96 94 95 3.2 0.3 0.4 97 3.3 0.3 1.0 Pay 500 1040 957 1006 40 0.9 1.2 1115 43 1.7 3.7 Rec 100 201 199 200 5.0 0.2 0.2 201 5.0 0.0 0.3 Rec 100 752 722 738 22 0.7 0.7 750 22 0.1 1.3 Rec 200 529 516 522 6.1 1.1 1.0 524 6.1 0.7 1.3 Rec 200 1770 1638 1706 30 2.2 2.3 1748 30 0.7 3.6 Rec 300 1192 1169 1175 3.6 4.8 1.5 1177 3.9 3.7 2.1 Rec 300 3825 3513 3646 31 5.8 4.4 3728 31 3.1 6.9 Rec 400 2125 2104 2106 1.2 16 1.9 2108 1.3 14 3.1 Rec 400 7433 6991 7105 18 18 6.3 7205 22 10 9.6 Rec 500 3186 3169 3170 0.3 53 2.3 3171 0.4 38 3.5 Rec 500 12610 12158 12217 9.2 43 6.3 12303 13 23 11 All values in bp. Violations of the bounds are in bold. Source: Thompson Reuters All values in bp. Violations of the bounds are in bold. Source: Thompson Reuters at K D 1% (upper bound) and K D 3% (lower bound)). The receiver dynamic hedge for the ABerm, which requires expensive vega rebalancing bounds are somewhat wider, but they are still only a few vegas for most for duration-extending or contracting curve moves. strikes for the accreter. For the amortiser, the upper bound is significantly More generally, we found the bounds are tight enough to provide mean- better than the lower; this is expected for the co-initial versus co-terminal ingful tests of model arbitrage for locally calibrated ABerm models. For portfolio. Incisiveexample, our Hull-White pricer with basket calibration violates the amor- Tables C and D show the results for 30-nc-1 amortisers and accreters, tiser lower and accreter upper bounds already at D 90%, with these respectively. While the bounds are wider in absolute terms for these long- violations growing larger for lower correlations. In this context, it is dated trades, the difference is not significant when measured in terms important to remember that the tightness of the bounds depends both of vega (ı and ı ). As before, the payer bounds are tighter than the U L on the shape of the volatility surface and, sensitively, on the shape of receiver bounds due to the upward-sloping curve. the yield curve. Since any local ABerm calibration methodology must work in a wide range of market environments, such schemes should Conclusion be checked for arbitrage against a diverse set of historical and scenario The numerical results show that, in our model, the upper and lower bounds market data. are often within a few normal vegas of the ABerm price across a range of Finally, the tightness of the bounds makes them useful to set prudent strikes for both amortisers and accreters. From a market participant’s point valuation limits for ABerms in terms of more readily observable SBerms. of view, the most interesting cases are the amortising payer upper bound For example, the uncertainty related to the correlation parameters ij of and the accreting receiver lower bound. This is because the owner of a loan the ABerm model can define a significantly wider range of ABerm prices subject to prepayment risk is short an amortising payer, while the issuer than the upper and lower SBerm portfolios, which are not sensitive to this of a callable zero-coupon bond is long an accreting receiver. Moreover, correlation. as discussed in appendix B, both of these positions are short correlation, which we expect to be significantly bid/offered. Looking, therefore, at the D 100% model, we see that even for the long-dated (30Y) trades, ı U is Appendix A: proof of bounds around 1bp for the amortiser (table C), while ı L is 1–6bp for the accreter Here, we sketch a more formal derivation of the bounds that were obtained (table D). These values are of the same order as typical volatility bid/ using super-replication arguments. The proofs are straightforward, if offers for liquid SBerms. Bid/offers for ABerms are usually substantially somewhat tedious. wider. Assuming thatCopyright our model produces realistic SBerm prices, setting Co-initial decomposition. We start from the co-initial decomposition up a super-replicating portfolio for an ABerm is therefore not necessar- of the non-standard swap in (1), which will give us the upper bound for the ily impractical. Ultimately, this approach may be cheaper than running a amortising Bermudan and the lower bound for the accreting Bermudan. 4 risk.net June 2017 i i i i i i i i Cutting edge: Option pricing The proof proceeds by backward induction. For compactness, we define: Bermudan. At Tn 1, the value of the ABerm is given by (11). At Tn 2, we have: j;k C V V.Ti I j; k/ and X max.X; 0/ i amo acc n 2;n n 1;n B .Tn 2/ D max Nn 2Vn 2 jıNn 1jVn 2 ; At time Tn 1, the ABerm value is simply the payout of a European option ˇ n 1;n C ˇ Nn 1.Vn 1 / ˇ on the remaining swap, paid on the final notional: An 2E ˇ Tn 2 (15) An 1 amo acc n 1;n C B .Tn 1/ D Nn 1.Vn 1 / (11) Using: Nn 1 D Nn 2 jıNn 1j Let A.t/ be a numeraire, and define: inside the expectation, and splitting the max operator as before to match Ai A.Ti / terms with the same notional, we get the inequalities: amo acc At Tn 2, we have the choice between the ASwap at this time or continuing B .Tn 2/ ˇ without exercising: .V n 1;n/C ˇ > N max V n 2;n;EMedian 1 ˇ T 6 n 2 n 2 ˇ n 2 An 1 amo ˇ B acc .Tn 2/ n 1;n C ˇ n 1;n .Vn 1 / ˇ jıNn 1j max Vn 2 ; An 2E ˇ Tn 2 n 2;n 1 An 1 D max ˙jıNn 1jVn 2 ˇ (16) n 1;n C ˇ n 2;n Nn 1.Vn 1 / ˇ C Nn 1Vn 2 ; An 2E ˇ Tn 2 The second ‘max’ can be replaced with its second argument because: An 1 (12) ˇ .V n 1;n/C ˇ n 1;n 6 n 1;n n 1 ˇ Vn 2 Bn 2 D An 2E ˇ Tn 2 where the upper sign applies to the amortiser, and the lower sign applies An 1 to the accreter. Using: n 2;n Since the first ‘max’ is simply Bn 2 , we can write the inequalities as: max.a C b; c/ 6 max.a; 0/ C max.b; c/ amo acc > n 2;n n 1;n B .Tn 2/ 6 Nn 2Bn 2 jıNn 1jBn 2 (17) for the amortiser and: To complete the inductive proof, one shows these inequalities hold at Ti 1, assuming they hold at Ti . As above, this is done by repeatedly splitting max.a C b; c/ > max. a; 0/ C max.b; c/ Incisivethe max operator and collecting terms with the same notional. for the accreter, we obtain: Appendix B: basket model n 1;n ˇ amo V ˇ acc 6 n 2;n n 1 ˇ Here, we derive a simple model for amortising and accreting European B .Tn 2/ > Nn 1 max Vn 2 ; An 2E ˇ Tn 2 An 1 swaptions. Prices obtained from this model are used as calibration targets n 2;n 1 C ˙jıNn 1j.Vn 2 / (13) for the Hull-White based ABerm PDE pricer discussed in the text. The payout at expiry T0 is: Recognising the term proportional to Nn 1 as the standard Bermudan: a C a V.T0/ D . .S .T0/ K// A .T0/ (18) n 2;n B.Tn 2I n 2;n/ Bn 2 where: n 2;n 1 P and the term proportional to ıNn 1 as the standard Bermudan Bn 2 , n 1 NkLk.t/ kP.t;TkC1/ we can write this as: Sa.t/ D kD0 Aa.t/ amo acc 6 n 2;n n 2;n 1 and: B .Tn 2/ > Nn 1Bn 2 ˙jıNn 1jBn 2 (14) nX 1 a To complete the proof, one assumes that the inequalities in (14) hold at Ti A .t/ D Nk kP.t;TkC1/ kD0 and shows they hold at Ti 1. This is done by repeatedly splitting the max operator, collecting terms proportional to the same notional and identifying are the amortising/accreting swap rate and fixed leg annuity, respectively, them as standard Bermudans, as in the previous step. and DC1 for a payer and 1 for a receiver (see footnote 2). Choosing Co-terminalCopyright decomposition. We now look at the co-terminal rep- Aa.t/ as the numeraire, the value today is: resentation of the non-standard swap in (2), which will give us the lower a a C bound for the amortising Bermudan and the upper bound for the accreting V .0/ D A .0/EAa Œ. .S .T0/ K// (19) risk.net 5 i i i i i i i i Cutting edge: Option pricing a a a We also know that EAa ŒS .T0/ D S .0/, since S .t/ is a martingale and N.z/ and n.z/ being the normal cumulative and probability distribu- in the measure induced by Aa.t/. In order to evaluate (19), it therefore tion functions, respectively. To complete the specification of the model, a remains to determine the volatility of S .T0/. To this end, we use (1) to it remains to specify the volatilities j and the correlations ij . We set write: j equal to the market-implied swaption volatility of rate S.T0I T0;Tj / Xn for strike K. The correlation specification is more difficult. Ideally, one Sa.T / D w S.T I T ;T / (20) 0 k 0 0 k would use market-implied correlations obtained, for example, from spread kD1 option prices. However, these trade only for a limited number of pairs; they where: are also often illiquid, and their correlations are strike dependent. Given a wk D ıNkA.T0I T0;Tk/=A .T0/ the absence of suitable hedge instruments and market-implied values, one can expect the correlation parameters to be marked conservatively for and S.T0I T0;Tk/ and A.T0I T0;Tk/ are the standard swap rate and annu- risk management and to incur substantial bid/offer spreads when quot- ity spanning .T0;Tk/, respectively. If we now freeze the relatively slow- ing. In this context, we note the correlation dependence of amortisers and moving weights wk at today’s values and assume that S.T0I T0;Tk/ is accreters is generally opposite: for amortisers, all weights wk are positive, a a normally distributed with volatility k, then the volatility of S .T0/ is: so the volatility in (21), and therefore the amortising swaption price, are v long correlation; for accreters, only the final weight w is positive (recall u n uXn Xn Media Nn 0), with all other weights being negative. The accreter is therefore a a D t w w (21) j k ij j k weighted spread option between a rate and a basket, and one can show that j D1 kD1 for reasonable volatility and correlation structures, the accreter swaption price is short correlation. where ij is the correlation between rates S.T0I T0;Ti / and S.T0I T0;Tj /. By (19), today’s value of the swaption is then given by Bachelier’s formula: p Thomas Roos is an independent consultant specialising in a a V .0/ D A .0/ T0ŒzN.z/ C n.z/ (22) derivatives, located in London. He was previously global head of interest rate quantitative analytics at Barclays. The author with: would like to thank Vladimir Piterbarg for stimulating discus- .Sa.0/ K/ sions. z D p a T0 Email: [email protected]. REFERENCES Andersen LBG and Andersen LBG and Evers I and F Jamshidian, Johnson S and B Nonas, 2009 J Andreasen, 2001 VV Piterbarg, 2010 Incisive2005 Arbitrage-free construction of Factor dependence of Interest Rate Modeling Replication of flexi-swaps the swaption cube Bermudan swaptions: fact or Atlantic Financial Press Risk March, pages 67–70 Working Paper, SSRN fiction? Journal of Financial Economics Laurence P and T-H Wang, 62(1), pages 3–37 2004 What’s a basket worth? 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