Structured Notes and Exotic Swaps

Interest Rate Modeling

[Quantitative Finance] · June 5, 2020 · 8 mins to read

Tutorial Contents

Investors have grown familiar with the swap or bond format of payout on fixed income securities. Sophisticated market participants seeking complex payoffs to express their market views fueled the growth of so called exotic swaps and structured notes. It should be noted that, post-crisis, demand for these kind of products has fallen significantly on pre-crisis levels. That said, these products do still exists and proper modelling still remains an important task.

Exotic swaps

An exotic swap consists of a regular floating Libor leg versus a structured leg that pays a coupon based on an observed interest rate. This coupon may be any arbitrary function of the underlying rate. A standard fixed-floating vanilla swap is a trivial example where the structured coupon is simply a fixed rate. Likewise, a cap/floor can be seen as another example, by noting that

(L - K)^+ = L - \left[L - (L-K)^+\right] = L - \text{min}(K, L).

Exotic swaps often start their lives as bonds sold by banks to investors. In a structured note, the investor pays an up-front principal to the issuer of the note (the bank), who in turn pays the investor a structured coupon, and repays the principal at the maturity of the note. The principal is invested by the issuing bank, and pays the Libor rate plus or minus a spread. From the perspective of the bank, then, the net cash flows on the note are equivalent to those of an exotic swap.

In terms of valuation, if $C_n$ is the structured coupon for the $n$ th period, the value of the exotic swap is equal to

V_{\text{exotic}}(t) = \beta(t) N \sum_{n=0}^{N-1} \tau_n \mathbb{E}_t^\mathbb{Q}\left[\beta(T_{n-1})^{-1} (C_n - L_n(T_n)) \right],

where $N$ is the notional of the swap. From now on, we will normalise our swap so that $N=1$ . For simplicity, we have assumed that the two legs of the swap pay out on the same payment schedule. As mentioned, the coupon $C_n$ can be an arbitrary function of interest rates, structured to reflect investors views on the market.

In the following, we distinguish between exotic swaps based on different features, such as Libor rates, CMS rates, multi-rates, range accruals and general path dependence.

Libor-based exotic swaps

In a Libor-based exotic swap, the structured coupon is a function of a Libor rate, i.e.

C_n \equiv C_n(L_n(T_n)).

A large variety of structured coupons $C_n(\cdot)$ can be used. Examples include:

Standard swaps: $C_n = K.$
Capped and floored floaters. For a given strike $K$ , gearing $G$ , cap $C$ and floor $F$ , $\text{max}(\text{min}(G \cdot x - K, C), F).$
Capped and floored inverse floaters. For a given strike $K$ , gearing $G$ , cap $C$ and floor $F$ , $C_n(x) = \text{max}(\text{min}(K - G \cdot x, C), F).$
Digitals. For a given strike $K$ and coupon $C$ , $C_n(x) = C \cdot 1_{\{x > K\}}$
Flip-flops. For a given strike $K$ and two coupons, $C_1$ and $C_2$ , $C_n(x) = \begin{cases} C_1, & x \le K, \\ C_2, & x > K. \end{cases}$ A libor-based exotic swap can usually be decomposed further into a sum of simpler instruments, such as ordinary swap floating legs, fixed legs, caps/floors and digital caps/floors. Hence, if the prices of these simpler contracts are observable in the market, Libor-based exotics can be perfectly replicated via the static hedging strategy of purchasing the constituent underlying instruments.

CMS-based exotic swaps

By setting $x = S_{n,m}(\cdot)$ in $C_n(x)$ , all of the above payoffs can be applied to CMS rates, rather than Libor rates. If an $m$ -period rate is used, a structured coupon for the period $n$ can be defined by

C_n = C_n(S_{n,m}(T_n)).

CMS-based exotics can be decomposed into linear combinations of CMS swaps and CMS caps/floors. Hence, CMS-based exotics rarely present any significant modelling challenges beyond those present for pricing CMS swaps/caps/floors.

Multi-rate exotic swaps

Unlike the previous two types of exotic swaps, multi-rate exotic swaps reference multiple market rates for their coupon calculations. The most common form of multi-rate exotic swap is the CMS spread coupon, which references two CMS swap rates for differing periods, $a$ and $b$ . For example, let both rates start at the time $T_n$ , $n=0,\ldots,N-1$ and cover $a$ and $b$ periods, respectively. The CMS spread is given by $X_{n,a,b}(T_n) := S_{n,a}(T_n) - S_{n,b}(T_n)$ . A CMS spread coupon with gearing $G$ , spread $S$ , cap $C$ , and floor $F$ is then given by

C_{n} := \text{max}(\text{min}(G \cdot X_{n,a,b}(T_n) + S, C), F).

A typical example would be the 10 year - 2 year (10y2y) CMS call spread option, where $a = 40$ and $b = 8$ (where $a$ and $b$ are measured in quarters), with quarterly coupon given by

C_n = X_{n,40,8}(T_n)^+.

By setting $x := X_{n,a,b}(T_n)$ in the examples for Libor/CMS rate exotic swaps above, more general examples of multi-rate swaps can be obtained.

While multi-rate exotic swaps cannot typically be decomposed into standard swaps and a valuation model is generally required, such a model does not always need to be a fully fledged term structure model - some spread-linked payoffs can be valued by vanilla models.

Range accruals

Range accruals have coupons defined as a given rate that only accrues while a reference rate is within a given range. Let $R_n(t)$ denote the payment rate, $X_n(t)$ denote the reference rate and let $L$ and $U$ be the lower and upper bounds respectively. A range accrual coupon then pays

C_n := R_n(T_n) \left(\frac{d(t \in [T_n, T_{n+1}] : X_n(t) \in [L,U])}{d(t \in [T_n, T_{n+1}])} \right),

where $d$ counts the number of days that the given criteria is met for. Hence, here, the coupon $C_n$ clearly represents the payment rate multiplied by the proportion of dates where the reference rate lies within the given lower and upper bounds.

The most common choice of payment rate $R_n(t)$ is either a Libor rate or simply a constant rate. The reference rate $X_n(t)$ is free to be any observable rate, such as the Libor rate fixed at $t$ , a CMS rate fixing at $t$ , or even a CMS spread rate.

Note that range accruals can always be decomposed into simple digital payoffs, because

d(t \in [T_n, T_{n+1}] : X_n(t) \in [L,U]) = \sum_{t \in [T_n, T_{n+1}]} 1_{\{X_n(t) \in [L,U] \}}.

This decomposition is particularly useful when $R_n \equiv K$ , a fixed rate, because simple digital options can be priced directly from the market. For floating rates, the above decomposition requires further work to be converted into a valuation formula.

More advanced extensions of range accruals can depend on two or more different reference rates, where each of the two rates must lie within their own specified ranges. Another generalisation is a product-of-ranges range accrual, where the range accrual multiplies up all the previous range accrual factors to define the multiplier used for the current coupon - for example

C_n = R_n(T_n) \cdot Y_n,

where

\begin{aligned} Y_n & = Y_{n-1} \cdot \left(\frac{d(t \in [T_n, T_{n+1}] : X_n(t) \in [L,U])}{d(t \in [T_n, T_{n+1}])} \right), \\ Y_{-1} & = 1. \end{aligned}

Notice that the dependence of $Y_n$ on $Y_{n-1}$ implies that $C_n$ depends on all previous factors $Y_0, \ldots, Y_{n-1}$ which in turn depend on $X_0,\ldots,X_{n-1}$ . Hence, product-of-ranges range accruals are path-dependent.

Path-dependent swaps

The product-of-ranges swap is one example of a path-dependent swap - a swap where one coupon depends on rate observations from previous coupons. Most commonly, path-dependence in exotic swaps is introduced by linking the structured coupon to both current rates but also to previous coupons themselves. This is often known as a snowball feature. The archetypal snowball structure involves a coupon of inverse floater type, with the $n$ th coupon given by

C_n = (C_{n-1} + S_n - G_n \cdot L_n(T_n))^+,

where $\{S_n\}_{n=0}^{N}$ , $\{G_n\}_{n=0}^N$ are deterministic sequences of spreads and gearings specified in the contract.

The term snowball comes from the analogy of a rolling snowball quickly picking up size/momentum. Clearly, if $C_0$ begins as a very large payment, this is likely to continue into subsequent coupons. It is clear to see that if all terms other than the previous coupon term $C_{n-1}$ remain equal, a positive coupon implies a linearly increasing sequence of subsequent coupons. This highly leveraged exposure to previous coupons makes snowballs attractive to some investors, while simultaneously makes them difficult to risk manage.