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

A Bermudan swaption gives the holder the right to enter into a fixed versus floating swap at a set of future exercise dates. A standard Bermudan swaption (SBerm), sometimes called a ‘bullet’ Bermudan, is characterised by the fact that it is exercisable into a swap whose notional (and fixed rate) is the same for all coupons. In addition to the important and relatively liquid market for SBerms, there is considerable interest in accreting and amortising Bermudan swaptions (we will refer to these collectively as ABerms). Accreting Bermudans are exercisable into swaps whose notionals increase coupon by coupon, while amortising Bermudans are exercisable into swaps whose notionals decrease for each coupon. Economically, accreting Bermudans arise from the issuing of callable zero-coupon bonds, with the notional accreting at (or close to) the fixed rate of the bond (see, for example, Andersen & Piterbarg 2010, section 19.4.6). Amortising Bermudans can be used, for example, to hedge the prepayment risk on a loan, with the amortising notionals representing the loan repayment schedule.

In this article, we derive super-replication portfolios for both long and short ABerm positions in terms of SBerms. The motivation for seeking such portfolios is threefold. First, the super-replication provides an alternative, static risk management strategy for ABerms. Second, the values of the super-replication portfolios provide two-sided, model-independent bounds on the values of the ABerms. Such bounds are useful checks against model arbitrage, in a way that will be explained further below. Finally, as ABerms are more complex to model than SBerms, the bounds are useful to quantify valuation uncertainties for internal or regulatory prudent valuation adjustments. For similar applications of model-independent bounds in different contexts, see, for example, Laurence & Wang (2004) and Johnson & Nonas (2009). For an exact replication strategy of flexi-swaps using SBerms, see Evers & Jamshidian (2005).

In relation to model arbitrage, it is clear that, when using a globally calibrated model (eg, a Libor market model), there is no possibility of arbitrage between ABerm and SBerm prices. In practice, however, SBerms are often valued using low-dimensional models with local (ie, trade-dependent) calibration. For example, for performance reasons, practitioners typically use Gaussian or local volatility one-factor models, and price the Bermudan optionality using partial differential equation (PDE) solvers and backward induction. When such a low-dimensional model is used for SBerms, it is often considered desirable to use the same approach for ABerms. Given the limited calibration capabilities of the model, calibration strategies are then necessarily product dependent. For SBerms, the model is typically calibrated to the set of co-terminal European swaptions (at one or more strikes), and the mean-reversion parameter is adjusted to match any observable Bermudan premiums. For ABerms, the fundamental difference is that the corresponding calibration targets are the co-terminal amortising or accreting European swaptions (ASwaptions). These are generally no more liquid than the ABerm itself, so their prices must themselves be modelled. See Andersen & Piterbarg (2010) for an overview of ABerm calibration methodologies.

Whatever methodology is chosen to determine the calibration targets for ABerms, the resulting local calibrations can be significantly different from those used for SBerms. Such differences can translate into inconsistencies and even arbitrage between these closely related products. The bounds obtained in this article provide a useful safeguard against this kind of model arbitrage.

## Notation and preliminaries

We consider a schedule $\smash{\{T_{i}\}_{i=0}^{n}}$ constituting the fixed leg of a swap. Let $V^{\mathrm{a}}(t)$ be the time $t value of an amortising or accreting swap with notional schedule $\smash{\{N_{i}\}_{i=0}^{n-1}}$, and let $B^{\mathrm{a}}(t)$ be the value of the corresponding Bermudan swaption. Further, let $V(t;i,j)$ be the value of a unit notional vanilla swap spanning $T_{i}$ to $T_{j}$, with $B(t;i,j)$ being the value of the corresponding standard Bermudan swaption. For example, for a receiver swap,11A receiver swap receives the fixed leg (pays floating). A receiver swaption is an option to enter into a receiver swap. we have:22When explicitly writing out the floating leg, we will assume it has the same frequency and day count as the fixed leg. This is done solely to simplify the notation; all results are valid for general conventions.

 $\displaystyle V^{\mathrm{a}}(t)$ $\displaystyle=\sum_{k=0}^{n-1}{N_{k}\tau_{k}}(K-L_{k}(t))P(t,T_{k+1})$ and $\displaystyle V(t;i,j)$ $\displaystyle=\sum_{k=i}^{j-1}{\tau_{k}}(K-L_{k}(t))P(t,T_{k+1})$

where $P(t,T)$ is the discount bond at time $t$ with maturity $T$, and $\tau_{i}$ and $L_{i}(t)$ are the day count fraction and forward Libor rate for period ($T_{i}$, $T_{i+1}$), respectively. For future use, we note that the ASwap can be decomposed into a portfolio of co-initial standard swaps:

 $V^{\mathrm{a}}(t)=\sum_{k=1}^{n}-\delta N_{k}V(t;0,k)$ (1)

as well as into a portfolio of co-terminal standard swaps:

 $V^{\mathrm{a}}(t)=\sum_{k=0}^{n-1}\delta N_{k}V(t;k,n)$ (2)

where we have defined $\delta N_{k}\equiv N_{k}-N_{k-1}$, with $N_{-1}=N_{n}\equiv 0$. We will refer to (1) and (2) as the ‘co-initial’ and ‘co-terminal’ decompositions, respectively.

## Model-independent bounds

### Upper bounds: amortisers.

From (1), we see that amortising swaps (where $N_{k-1}\geq N_{k}$) can be decomposed into a sum of standard co-initial swaps with positive notionals. Explicitly, we have:

 $V^{\mathrm{amo}}(t)=\sum_{k=1}^{n-1}|\delta N_{k}|V(t;0,k)+N_{n-1}V(t;0,n)$ (3)

We now consider a Bermudan swaption to enter into this amortising swap (ABerm), and the portfolio of standard Bermudans (SBerms) to enter into the standard swaps on the right-hand side of (3). It is evident this portfolio of SBerms enables us to replicate all possible payouts of the ABerm: the strategy is simply to exercise all SBerms at the same time the ABerm is exercised. Since the SBerm portfolio must be worth at least as much as the ABerm it can replicate, this gives us an upper bound for the value of the amortising Bermudan in terms of the portfolio of standard Bermudans:

 $B^{\mathrm{amo}}(t)\leq\sum_{k=1}^{n-1}|\delta N_{k}|B(t;0,k)+N_{n-1}B(t;0,n)$ (4)

### Upper bounds: accreters.

The reasoning behind accreting Bermudan swaptions (where $N_{k}\geq N_{k-1}$) is similar, except that we need to start from (2) to get a decomposition of the accreting swap into a sum of standard swaps with positive notionals. These swaps are co-terminal, starting at $\{T_{i}\}_{i=0}^{n-1}$ and ending at $T_{n}$. Explicitly, we have:

 $V^{\mathrm{acc}}(t)=N_{0}V(t;0;n)+\sum_{k=1}^{n-1}|\delta N_{k}|V(t;k,n)$ (5)

As in the case of the amortiser, it is clear that if we own options that allow us, at all possible exercise times of the ABerm, to enter into the standard 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 bound on the value of the accreting Bermudan in terms of a portfolio of standard Bermudans:

 $B^{\mathrm{acc}}(t)\leq N_{0}B(t;0;n)+\sum_{k=1}^{n-1}|\delta N_{k}|B(t;k,n)$ (6)

One difference to the case in the previous section, where all SBerms start at the same time as the ABerm, is that, here, some of the co-terminal SBerms in our portfolio may not have started at the time the ABerm is exercised. However, this presents no difficulty for the replication strategy, as we can simply wait for the Bermudan to start and exercise it then.

### Lower bounds: amortisers.

Previously, we used the co-initial decomposition in (1) to write the amortising swap as a sum of standard swaps with positive notionals. If we instead look at the co-terminal decomposition in (2), we find that only the notional of the longest tenor swap is positive, with all others being negative. Explicitly, we have:

 $V^{\mathrm{amo}}(t)=N_{0}V(t;0;n)-\sum_{k=1}^{n-1}|\delta N_{k}|V(t;k,n)$ (7)

We would like to follow our previous reasoning and obtain a super-replicating portfolio for the amortiser by turning all the standard swaps on the right-hand side of (7) into Bermudan swaptions. However, the negative notionals present a problem in that we cannot be short options in our super-replicating portfolio, as we need to have control over when the options are exercised to match the ABerm. We could make the notionals positive by simply viewing a negative notional payer swap as a positive notional receiver swap (and vice versa). This leads to an upper bound in terms of a mixed portfolio of payers and receivers, which is not useful in practice (and always inferior to the upper bound in (4)). If, instead, we move the negative notional swaps to the left-hand side:

 $V^{\mathrm{amo}}(t)+\sum_{k=1}^{n-1}|\delta N_{k}|V(t;k,n)=N_{0}V(t;0,n)$

and apply our previous reasoning, we conclude that if we owned a portfolio of Bermudan swaptions that could be exercised into the swaps on the left-hand side, then this portfolio would constitute a super-replicating strategy for the Bermudan to enter into the standard swap on the right-hand side. This gives us an upper bound on the value of this standard Bermudan, which we write as a lower bound on the value of the amortising Bermudan:

 $B^{\mathrm{amo}}(t)\geq N_{0}B(t;0;n)-\sum_{k=1}^{n-1}|\delta N_{k}|B(t;k,n)$ (8)

### Lower bounds: accreters.

Decomposing the accreter using the co-initial decomposition in (1), we find:

 $V^{\mathrm{acc}}(t)=N_{n-1}V(t;0;n)-\sum_{k=1}^{n-1}|\delta N_{k}|V(t;0,k)$ (9)

Again, only the longest tenor swap has positive notional in this representation. Following the approach of the previous section, we move the negative notional swaps to the left-hand side and deduce that the corresponding portfolio of Bermudan swaptions is a super-replication of the standard Bermudan swaption corresponding to the right-hand side. This gives us an upper bound for this standard Bermudan, which we write as a lower bound for the accreter:

 $B^{\mathrm{acc}}(t)\geq N_{n-1}B(t;0;n)-\sum_{k=1}^{n-1}|\delta N_{k}|B(t;0,k)$ (10)

### Discussion.

In the previous sections, we obtained two-sided bounds for amortising and accreting Bermudan swaptions using super-replication arguments. A more formal proof can be found in appendix A. The upper bounds in (4) and (6), and their usefulness in providing arbitrage checks for ABerm prices, are well known to many practitioners (see, for example, Andersen & Piterbarg 2010, section 19.4.5). From a risk management point of view, these bounds provide a static super-replication (ie, overhedge) for a short position in the ABerm: whenever the short ABerm is exercised by the counterparty, the portfolio of SBerms is able to reproduce the ASwap payout. While the reasoning above for deriving upper and lower bounds is similar, the lower bounds in (8) and (10) do not seem to be known in the literature or by many practitioners. In addition to providing arbitrage checks from the ‘other side’, they also represent overhedge strategies for long positions in the ABerm. For example, (10) tells us that if we can buy the accreting Bermudan at the lower bound value of:

 $N_{n-1}B(t;0;n)-\sum_{k=1}^{n-1}|\delta N_{k}|B(t;0,k)$

then we can sell the longest tenor standard Bermudan and purchase the set of shorter tenor standard Bermudans, and so create a statically hedged portfolio that requires no rebalancing over time.

If the long tenor Bermudan is exercised against us, we simply exercise the ABerm and all remaining SBerms, cancelling the resulting swap position. In this context, it is important to note that, while our derivations assume the entire super-replicating portfolio is exercised at once, this is not what one would do when running the strategy in practice. Instead, one would only exercise those Berms for which it was optimal (if any), and sell the rest back to the market, producing a windfall. The difference in value at inception between the ABerm and its super-replicating portfolio is therefore the maximum possible cost of running the strategy. The actual cost will usually be less.

We will explore the tightness of the bounds numerically in the next section, but we can deduce some general patterns from our derivation in terms of super-replication. In particular, we expect the tightness to depend on how suboptimal it is (in expectation) to exercise all Bermudans in the portfolio at the optimal exercise time of the replicated Bermudan. The more optionality we ‘give away’ when exercising the portfolio, the looser the bound. Since it is always suboptimal to make an exercise decision before a Bermudan’s start date, we expect the co-initial bounds (amortiser upper and accreter lower) to be tighter than the co-terminal bounds (amortiser lower and accreter upper).

Following similar reasoning, we expect the bounds to be tighter for payers in a rising curve environment, and vice versa. This is because for increasing forward rates, the payer coupons become more positive as maturity increases, meaning that longer-dated Bermudans are more in-the-money than shorter ones. This means that when it is optimal to exercise the longest tenor Bermudan, we also expect it to be close to optimal to exercise all shorter tenor co-initial and co-terminal Bermudans. This is because one will wait to exercise the long tenor Bermudan if the shorter dated co-initial Bermudans are not optimal, while the shorter tenor co-terminal Bermudans will be deeper in-the-money than the long tenor Bermudan, and therefore also close to optimal exercise.

Finally, we note that so far we have assumed the notional change dates match the exercise schedule of the Bermudan. In general, exercise opportunities may be more frequent (eg, notionals step up/down every five years, while the trade is callable annually) or less frequent (eg, the trade is callable every five years while notionals step up/down annually). The first case is already covered by our formulas, with $\delta N_{i}=0$ on exercise dates for which the notional does not change. The second case is more interesting. It requires one option for each date on which the notional changes (regardless of whether this is an ABerm exercise date or not), with the exercise schedule of each such option matching the ABerm exercise schedule for subsequent dates. In particular, this means that for European amortising and accreting swaptions, the formulas for the bounds are the same as derived above, except with all SBerms replaced by standard European swaptions. Since the European ASwaptions must themselves be modelled, the bounds can also be useful in this context as arbitrage checks and super-replication strategies.

## Numerical examples

For the bounds derived above to be useful, they must be tight enough to act as meaningful checks against model arbitrage, or, ideally, to provide tradable bids and offers that allow static hedging of ABerms via super-replication. In this section, we provide numerical examples to explore this question. The modelling setup is as follows. Both SBerms and ABerms are priced in a one-factor Gaussian (Hull-White) model, using standard PDE backward propagation. For each SBerm, the model is calibrated to the corresponding co-terminal standard European swaptions at the strike. For each ABerm, the model is calibrated to the corresponding co-terminal amortising or accreting European swaptions, priced using the basket model in appendix B. We use US dollar yield curve and swaption volatility data from November 2016. The mean reversion is set to zero throughout to reflect the low mean reversions typically required to match US dollar Bermudan premiums in practice.

At first sight, the use of a one-factor model to price the Bermudans may seem overly simplistic. We note, however, that the bounds only depend on SBerm prices, and for these the use of locally calibrated, low-dimensional models is both common practice and well established as being sufficient (see, for example, Andersen & Andreasen 2001). For ABerms, especially accreters, the adequacy of one-factor models is less clear. However, we expect our basket calibration coupled with different correlation scenarios to produce a representative range of ABerm prices; ultimately, the key point is the ABerm price must lie within the bounds, independent of the model used.

In the following examples, amortiser notionals decrease to zero in equal annual steps over the life of the trade, while accreter notionals increase annually by the fixed rate (strike) of the swaption. The initial notional is set to one in both cases.

In the tables that follow, ‘Opt’ is the option type (payer or receiver), $K$ is the strike, and ‘UB’ and ‘LB’ are the upper bound and the lower bound, respectively. All Bermudans are exercisable annually. For the ABerm, we run two correlation scenarios: $\rho_{ij}=100\%$ for all $\{i,j\}$, and $\rho_{ij}=90\%$ for all $\{i\neq j\}$ (see appendix B). In each scenario, $B^{\mathrm{a}}$ is the value of the amortising or accreting Bermudan swaption, and $\nu^{\mathrm{a}}$ is its vega for a 1 basis point ($10^{-4}$) parallel shift in (normal) European swaption volatilities. Finally, $\delta\sigma_{\mathrm{U}}$ and $\delta\sigma_{\mathrm{L}}$ are the changes in ABerm volatility (in bp) required to hit the upper bound and the lower bound, respectively (defined as $\delta\sigma_{\mathrm{U}}\equiv(\mathrm{UB}-B^{\mathrm{a}})/\nu^{\mathrm{a}}$ and $\delta\sigma_{\mathrm{L}}\equiv(B^{\mathrm{a}}-\mathrm{LB})/\nu^{\mathrm{a}}$). This is a measure of the tightness of the bound in terms of normal volatility, which can easily be compared with volatility bid/offers, for example. Note that, given the definitions of the $\delta\sigma$, a negative value indicates a violation of the corresponding bound.

We begin by looking at a ‘10-year no-call one-year’ (10-nc-1) Bermudan swaption.33That is, a Bermudan swaption starting in one year’s time on a swap ending in 10 years’ time. Tables A and B show the results for the amortiser and accreter, respectively.

As expected from the discussion in the previous section, the bounds are tighter for the payers than the receivers for the currently upward-sloping US dollar yield curve. The payer bounds are very tight indeed, with the upper and lower bounds separated by a few bp in value, corresponding to anywhere from a fraction of a vega to a few vegas difference over a range of strikes. For the accreter, the $\rho=90\%$ Bermudan value is actually through the upper bound for $K\geq 2\%$, indicating (mild) arbitrage for this model specification (there are similar small violations for the amortising payer at $K=1\%$ (upper bound) and $K=3\%$ (lower bound)). The receiver bounds are somewhat wider, but they are still only a few vegas for most strikes for the accreter. For the amortiser, the upper bound is significantly better than the lower; this is expected for the co-initial versus co-terminal portfolio.

Tables C and D show the results for 30-nc-1 amortisers and accreters, respectively. While the bounds are wider in absolute terms for these long-dated trades, the difference is not significant when measured in terms of vega ($\delta\sigma_{\mathrm{U}}$ and $\delta\sigma_{\mathrm{L}}$). As before, the payer bounds are tighter than the receiver bounds due to the upward-sloping curve.

## Conclusion

The numerical results show that, in our model, the upper and lower bounds are often within a few normal vegas of the ABerm price across a range of strikes for both amortisers and accreters. From a market participant’s point of view, the most interesting cases are the amortising payer upper bound and the accreting receiver lower bound. This is because the owner of a loan subject to prepayment risk is short an amortising payer, while the issuer of a callable zero-coupon bond is long an accreting receiver. Moreover, as discussed in appendix B, both of these positions are short correlation, which we expect to be significantly bid/offered. Looking, therefore, at the $\rho=100\%$ model, we see that even for the long-dated (30Y) trades, $\delta\sigma_{\mathrm{U}}$ is around 1bp for the amortiser (table C), while $\delta\sigma_{\mathrm{L}}$ is 1–6bp for the accreter (table D). These values are of the same order as typical volatility bid/offers for liquid SBerms. Bid/offers for ABerms are usually substantially wider. Assuming that our model produces realistic SBerm prices, setting up a super-replicating portfolio for an ABerm is therefore not necessarily impractical. Ultimately, this approach may be cheaper than running a dynamic hedge for the ABerm, which requires expensive vega rebalancing for duration-extending or contracting curve moves.

More generally, we found the bounds are tight enough to provide meaningful tests of model arbitrage for locally calibrated ABerm models. For example, our Hull-White pricer with basket calibration violates the amortiser lower and accreter upper bounds already at $\rho=90\%$, with these violations growing larger for lower correlations. In this context, it is important to remember that the tightness of the bounds depends both on the shape of the volatility surface and, sensitively, on the shape of the yield curve. Since any local ABerm calibration methodology must work in a wide range of market environments, such schemes should be checked for arbitrage against a diverse set of historical and scenario market data.

Finally, the tightness of the bounds makes them useful to set prudent valuation limits for ABerms in terms of more readily observable SBerms. For example, the uncertainty related to the correlation parameters $\rho_{ij}$ of the ABerm model can define a significantly wider range of ABerm prices than the upper and lower SBerm portfolios, which are not sensitive to this correlation.

## Appendix A: proof of bounds

Here, we sketch a more formal derivation of the bounds that were obtained using super-replication arguments. The proofs are straightforward, if somewhat tedious.

### Co-initial decomposition.

We start from the co-initial decomposition of the non-standard swap in (1), which will give us the upper bound for the amortising Bermudan and the lower bound for the accreting Bermudan.

The proof proceeds by backward induction. For compactness, we define:

 $V_{i}^{j,k}\equiv V(T_{i};j,k)\quad\text{and}\quad X^{+}\equiv\max(X,0)$

At time $T_{n-1}$, the ABerm value is simply the payout of a European option on the remaining swap, paid on the final notional:

 $B^{\substack{\mathrm{amo}\\ \mathrm{acc}}}(T_{n-1})=N_{n-1}(V_{n-1}^{n-1,n})^{+}$ (11)

Let $\mathcal{A}(t)$ be a numeraire, and define:

 $\mathcal{A}_{i}\equiv\mathcal{A}(T_{i})$

At $T_{n-2}$, we have the choice between the ASwap at this time or continuing without exercising:

 $\displaystyle B^{\substack{\mathrm{amo}\\ \mathrm{acc}}}(T_{n-2})$ $\displaystyle=\max\bigg(\pm|\delta N_{n-1}|V_{n-2}^{n-2,n-1}+N_{n-1}V_{n-2}^{n% -2,n},\mathcal{A}_{n-2}E\bigg[\frac{N_{n-1}(V_{n-1}^{n-1,n})^{+}}{\mathcal{A}_% {n-1}}\biggm|T_{n-2}\bigg]\bigg)$ (12)

where the upper sign applies to the amortiser, and the lower sign applies to the accreter. Using:

 $\max(a+b,c)\leq\max(a,0)+\max(b,c)$

for the amortiser and:

 $\max(a+b,c)\geq-\max(-a,0)+\max(b,c)$

for the accreter, we obtain:

 $\displaystyle B^{\substack{\mathrm{amo}\\ \mathrm{acc}}}(T_{n-2})$ $\displaystyle\;\substack{\leq\\ \geq}\;N_{n-1}\max\bigg(V_{n-2}^{n-2,n},\mathcal{A}_{n-2}E\bigg[\frac{V_{n-1}^% {n-1,n}}{\mathcal{A}_{n-1}}\biggm|T_{n-2}\bigg]\bigg)$ $\displaystyle\qquad\pm|\delta N_{n-1}|(V_{n-2}^{n-2,n-1})^{+}$ (13)

Recognising the term proportional to $N_{n-1}$ as the standard Bermudan:

 $B(T_{n-2};n-2,n)\equiv B_{n-2}^{n-2,n}$

and the term proportional to $\delta N_{n-1}$ as the standard Bermudan $B_{n-2}^{n-2,n-1}$, we can write this as:

 $B^{\substack{\mathrm{amo}\\ \mathrm{acc}}}(T_{n-2})\;\substack{\leq\\ \geq}\;N_{n-1}B_{n-2}^{n-2,n}\pm|\delta N_{n-1}|B_{n-2}^{n-2,n-1}$ (14)

To complete the proof, one assumes that the inequalities in (14) hold at $T_{i}$ and shows they hold at $T_{i-1}$. This is done by repeatedly splitting the max operator, collecting terms proportional to the same notional and identifying them as standard Bermudans, as in the previous step.

### Co-terminal decomposition.

We now look at the co-terminal representation of the non-standard swap in (2), which will give us the lower bound for the amortising Bermudan and the upper bound for the accreting Bermudan. At $T_{n-1}$, the value of the ABerm is given by (11). At $T_{n-2}$, we have:

 $\displaystyle B^{\substack{\mathrm{amo}\\ \mathrm{acc}}}(T_{n-2})$ $\displaystyle=\max\bigg(N_{n-2}V_{n-2}^{n-2,n}\mp|\delta N_{n-1}|V_{n-2}^{n-1,% n},\mathcal{A}_{n-2}E\bigg[\frac{N_{n-1}(V_{n-1}^{n-1,n})^{+}}{\mathcal{A}_{n-% 1}}\biggm|T_{n-2}\bigg]\bigg)$ (15)

Using:

 $N_{n-1}=N_{n-2}\mp|\delta N_{n-1}|$

inside the expectation, and splitting the max operator as before to match terms with the same notional, we get the inequalities:

 $\displaystyle B^{\substack{\mathrm{amo}\\ \mathrm{acc}}}(T_{n-2})$ $\displaystyle\;\substack{\geq\\ \leq}\;N_{n-2}\max\bigg(V_{n-2}^{n-2,n},E\bigg[\frac{(V_{n-1}^{n-1,n})^{+}}{% \mathcal{A}_{n-1}}\biggm|T_{n-2}\bigg]\bigg)$ $\displaystyle\qquad\mp|\delta N_{n-1}|\max\bigg(V_{n-2}^{n-1,n},\mathcal{A}_{n% -2}E\bigg[\frac{(V_{n-1}^{n-1,n})^{+}}{\mathcal{A}_{n-1}}\biggm|T_{n-2}\bigg]\bigg)$ (16)

The second ‘max’ can be replaced with its second argument because:

 $V_{n-2}^{n-1,n}\;\leq B_{n-2}^{n-1,n}=\mathcal{A}_{n-2}E\bigg[\frac{(V_{n-1}^{% n-1,n})^{+}}{\mathcal{A}_{n-1}}\biggm|T_{n-2}\bigg]$

Since the first ‘max’ is simply $B_{n-2}^{n-2,n}$, we can write the inequalities as:

 $B^{\substack{\mathrm{amo}\\ \mathrm{acc}}}(T_{n-2})\;\substack{\geq\\ \leq}\;N_{n-2}B_{n-2}^{n-2,n}\mp|\delta N_{n-1}|B_{n-2}^{n-1,n}$ (17)

To complete the inductive proof, one shows these inequalities hold at $T_{i-1}$, assuming they hold at $T_{i}$. As above, this is done by repeatedly splitting the max operator and collecting terms with the same notional.

Here, we derive a simple model for amortising and accreting European swaptions. Prices obtained from this model are used as calibration targets for the Hull-White based ABerm PDE pricer discussed in the text. The payout at expiry $T_{0}$ is:

 $V(T_{0})=(\kappa(S^{\mathrm{a}}(T_{0})-K))^{+}A^{\mathrm{a}}(T_{0})$ (18)

where:

 $\displaystyle S^{\mathrm{a}}(t)$ $\displaystyle=\frac{\sum_{k=0}^{n-1}{N_{k}L_{k}(t)\tau_{k}P(t,T_{k+1})}}{A^{% \mathrm{a}}(t)}$ and: $\displaystyle A^{\mathrm{a}}(t)$ $\displaystyle=\sum_{k=0}^{n-1}{N_{k}\tau_{k}P(t,T_{k+1})}$

are the amortising/accreting swap rate and fixed leg annuity, respectively, and $\kappa=+1$ for a payer and $-1$ for a receiver (see footnote 2). Choosing $A^{\mathrm{a}}(t)$ as the numeraire, the value today is:

 $V(0)=A^{\mathrm{a}}(0)E_{A^{\mathrm{a}}}[(\kappa(S^{\mathrm{a}}(T_{0})-K))^{+}]$ (19)

We also know that $E_{A^{\mathrm{a}}}[S^{\mathrm{a}}(T_{0})]=S^{\mathrm{a}}(0)$, since $S^{\mathrm{a}}(t)$ is a martingale in the measure induced by $A^{\mathrm{a}}(t)$. In order to evaluate (19), it therefore remains to determine the volatility of $S^{\mathrm{a}}(T_{0})$. To this end, we use (1) to write:

 $S^{\mathrm{a}}(T_{0})=\sum_{k=1}^{n}{w_{k}S(T_{0};T_{0},T_{k})}$ (20)

where:

 $w_{k}=-\delta N_{k}A(T_{0};T_{0},T_{k})/A^{\mathrm{a}}(T_{0})$

and $S(T_{0};T_{0},T_{k})$ and $A(T_{0};T_{0},T_{k})$ are the standard swap rate and annuity spanning $(T_{0},T_{k})$, respectively. If we now freeze the relatively slow-moving weights $w_{k}$ at today’s values and assume that $S(T_{0};T_{0},T_{k})$ is normally distributed with volatility $\sigma_{k}$, then the volatility of $S^{\mathrm{a}}(T_{0})$ is:

 $\sigma^{\mathrm{a}}=\sqrt{\sum_{j=1}^{n}\sum_{k=1}^{n}{w_{j}w_{k}\rho_{ij}% \sigma_{j}\sigma_{k}}}$ (21)

where $\rho_{ij}$ is the correlation between rates $S(T_{0};T_{0},T_{i})$ and $S(T_{0};T_{0},T_{j})$. By (19), today’s value of the swaption is then given by Bachelier’s formula:

 $V(0)=A^{\mathrm{a}}(0)\sigma^{\mathrm{a}}\sqrt{T_{0}}[zN(z)+n(z)]$ (22)

with:

 $z=\frac{\kappa(S^{\mathrm{a}}(0)-K)}{\sigma^{\mathrm{a}}\sqrt{T_{0}}}$

and $N(z)$ and $n(z)$ being the normal cumulative and probability distribution functions, respectively. To complete the specification of the model, it remains to specify the volatilities $\smash{\sigma_{j}}$ and the correlations $\smash{\rho_{ij}}$. We set $\smash{\sigma_{j}}$ equal to the market-implied swaption volatility of rate $\smash{S(T_{0};T_{0},T_{j})}$ for strike $K$. The correlation specification is more difficult. Ideally, one would use market-implied correlations obtained, for example, from spread option prices. However, these trade only for a limited number of pairs; they are also often illiquid, and their correlations are strike dependent. Given the absence of suitable hedge instruments and market-implied values, one can expect the correlation parameters to be marked conservatively for risk management and to incur substantial bid/offer spreads when quoting. In this context, we note the correlation dependence of amortisers and accreters is generally opposite: for amortisers, all weights ${w_{k}}$ are positive, so the volatility $\sigma^{\mathrm{a}}$ in (21), and therefore the amortising swaption price, are long correlation; for accreters, only the final weight $w_{n}$ is positive (recall $N_{n}\equiv 0$), with all other weights being negative. The accreter is therefore a weighted spread option between a rate and a basket, and one can show that for reasonable volatility and correlation structures, the accreter swaption price is short correlation.

Thomas Roos is an independent consultant specialising in derivatives, located in London. He was previously global head of interest rate quantitative analytics at Barclays. The author would like to thank Vladimir Piterbarg for stimulating discussions.

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