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Bounding Bermudans

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Bounding Bermudans 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<T0 value of an amortising or accreting swap with n1 a section 19.4.6).Amortising Bermudans can be used, for example, to hedge notional schedule fNi giD0 , and let B .t/ be the value of the corresponding the prepayment risk on a loan, with the amortising notionals representing Bermudan swaption. Further, let V.tI i;j/ be the value of a unit notional the loan repayment schedule. vanilla swap spanning Ti to Tj , with B.tI i;j/ being the value of the In this article, we derive super-replication portfolios for both long and corresponding standard Bermudan swaption. For example, for a receiver short ABerm positions in terms of SBerms. The motivation for seeking swap,1 we have:2 such portfolios is threefold. First, the super-replication provides an alter- nX1 a native, static risk management strategy for ABerms. Second, the values V .t/ D Nkk.K Lk.t//P.t; TkC1/ of the super-replication portfolios provide two-sided, model-independent kD0 bounds on the values of theABerms. Such bounds are useful checks againstIncisiveand model arbitrage, in a way that will be explained further below. Finally, as jX1 ABerms are more complex to model than SBerms, the bounds are useful V.tI i;j/ D k.K Lk.t//P.t; TkC1/ to quantify valuation uncertainties for internal or regulatory prudent valu- kDi 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 nX1 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 Á Nk Nk1, with N1 D Nn Á 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 Nk1 > 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 n1 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 n1 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 nX1 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- nX1 dan, which we write as a lower bound on the value of the amortising amo B .t/ 6 jıNkjB.tI 0; k/ C Nn1B.tI 0; n/ (4) Bermudan: kD1 nX1 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 > Nk1) 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 gn1 and ending at T . Explicitly, we have: nX1 i iD0 n acc V .t/ D Nn1V.tI 0I n/ jıNkjV.tI 0; k/ (9) nX1 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: nX1 acc B .t/ > Nn1B.tI 0I n/ jıNkjB.tI 0; k/ (10) n1 X kD1 acc B .t/ 6 N0B.tI 0I n/ C jıNkjB.tI k; n/ (6) Discussion.
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