Callable Libor Exotics and TARNs

Interest Rate Modeling

[Quantitative Finance] · June 5, 2020 · 10 mins to read

Tutorial Contents

While standard Bermudan swaptions are Bermudan-style options giving the holder the right to enter into a regular fixed-for-floating swap, if the underlying swap is altered to an exotic swap then a callable Libor Exotic (CLE) is created. CLEs most often emerge as part of callable structured notes where the issuer receives principal from an investor and pays a structured coupon in return. As a callable structure, the issuer holds the right to to call the note on a schedule of dates - typically the call schedule coincides with the coupon fixing dates after some initial lock-out period. Should such a note be called by the issuer, the principal is returned to the investor and the contract is cancelled.

Risk Managing CLEs

Typically, once a CLE has been issued, it is passed through to an IR exotics trading desk to deal with its risk management. Meanwhile, the principal is invested and pays a Libor rate, plus or minus a spread (dependent on the cost of financing). From the perspective of the trading desk, what is left is an exotic swap paying structure coupons, while receiving Libor, plus a Bermudn-style right to call.

Another way of conceptualising a CLE is to consider it a straight exotic swap, plus a Bermudan-style option to enter a reverse swap. This makes valuation easier, while emphasising the fact that the cancelability of a CLE benefits the owner of the right (the structured note issuer). This benefit allows the issuer to offer a more attractive coupon structure to the investor.

Consider a CLE on an exotic swap with structured coupons $\{C_n\}_{n=0}^{N-1}$ . To the trading desk holding the optionality, they may exercise on date $T_n$ and enter into the exotic swap receiving the structured coupons and paying Libor. This Bermudan option thus has exercise value

U_n(T_n) = \beta(T_n) \sum_{i=n}^{N-1} \tau_i \mathbb{E}_{T_n}^{\mathbb{Q}}\left[\beta(T_{i+1})^{-1} (C_i - L_i(T_i)) \right].

The underlying exotic swap has value

V_e(T_n) = \beta(T_n) \sum_{i=n}^{N-1} \tau_i \mathbb{E}_{T_n}^{\mathbb{Q}}\left[\beta(T_{i+1})^{-1} (C_i - L_i(T_i)) \right]

Labelling the Bermudan exotic swap value at time $t$ by $V_b(t)$ , the CLE has value to the trading desk given by

V(t) = V_b(t) - V_e(t). \tag{1}

Since $V_b(t) > 0$ , we find that the desk is able to issue the CLE such that $V_e(t) = V_b(t) > 0$ and still maintain an contract with zero upfront fee to the client. This allows the desk to issue the exotic swap with artificially large coupons $C_i$ relative to their non-CLE equivalents.

Pricing CLEs

Clearly, the embedded optionality found in CLEs points to the use of PDE methods for valuation. However, since a CLE often exhibits path dependence, this makes the PDE approach impractical from a computational perspective. Hence, CLEs must generally utilise Monte Carlo methods.

Types of CLEs

Any exotic swap can be used as the underlying of a CLE. Observing equation (1) above, it is natural that most CLEs are issued such that the optionality has a large value to the desk. This allows the desk to issue a CLE with an underlying exotic swap with larger value and hence more favourable coupons for the client. Some of the most popular CLEs are given below.

Callable snowballs

Callable snowballs are CLEs with a snowball underlying. Notably, they are one of the first widely popular instruments that include both strong path-dependence and optionality. It is possible to incorporate path dependence into a PDE framework for snowball type products by introducing auxiliary variables. Alternatively, it is possible to introduce optionality into a Monte Carlo framework by using methods such as the Longstaff-Schwartz approach.

CLEs accreting at coupon rate

Usually, the notional on the underlying exotic swap in a CLE is fixed throughout the life of the deal, but this does not have to be the case. It is possible to have accreting or amortising notionals. If the rate of accretion/ amortisation is deterministic, this does not provide much of a modelling challenge. However, some contracts specify that the notional accretes at the structured coupon rate, in which case the accretion rate also contains randomness. Specifically, if $q_i$ is the notional value for the period $[T_i, T_{i+1}]$ , then $q_i$ is obtained from the previous by multiplication with the structured coupon over the previous period. Specifically, we have

\begin{aligned} U_n(T_n) & = \beta(T_n) \sum_{i=n}^{N-1} \tau_i \mathbb{E}_{T_n}^{\mathbb{Q}}\left[\beta(T_{i+1})^{-1} q_i (C_i - L_i(T_i)) \right], \\ q_i & = q_{i-1} (1 + \tau_{i-1}C_{i-1}), \quad i = n, \ldots, N-1, \\ q_0 & = N. \end{aligned}

Multi-tranches

Multi-tranches give the issuer the right to increase the size of the note on a set of predetermined dates if they see fit. The times when the issuer has the right to increase the notional typically come before the times when the note can be cancelled by the issuer and callability generally applies to all tranches of the note.

Generally speaking, the more optionality the issuer can retain, the better a coupon they are able to offer to the investor.

TARNs and Other Trade-Level Features

It is often helpful to view a Libor exotic as a definition of a coupon (i.e. a formula converting rates into a payment to the investor), along with a collection of trade-level features (features that don't act upon the coupons but upon the trade as a whole).

Knock-out swaps

a knock-out swap is an exotic swap that cancels on the first fixing date upon which the specified reference rate is above/below a specified barrier. Denoting the reference rate for the period $n$ at time $t$ by $X_n(t)$ , the coupon by $C_n$ , the Libor rate $L_n$ , the knock-out barrier by $R$ , the value of a down-and-out knock-out swap from the perspective of a desk paying the structured coupon is given by

V(t) = \beta(t)\mathbb{E}_t^{\mathbb{Q}}\left[\sum_{n=0}^{N-1}\tau_n \beta(T_{n+1})^{-1}(L_n(T_n) - C_n) \prod_{i=0}^n 1(X_i(T_i) > R) \right].

TARNs

While Libor exotics have proven popular among investors, it is clear that investors require complex modelling tools to deduce when an issuer will call the note. One way to reduce this issue is through the Target Redemption Note (TARN). TARNs track the total investor return, defined as the sum of all structured coupons paid to the current date. When this total return exceeds a pre-specified target, the note is terminated and the principal returned to the investor.

As with callable notes, issuers do not typically keep TARNs on their books, but pass them through to a trading desk while investing the principal. The trading desk then receives Libor payments from the issuer in exchange for paying the structured leg back to the issuer, who passes this on to the investor. Again, from the perspective of the trading desk, we can decompose the TARN into an exotic swap with an offsetting callable swap that cancels the position once the threshold return has been reached.

Formally, let the structured coupon for the period $[T_n, T_{n+1}]$ be $C_n$ . The coupon over the period $[T_n, T_{n+1}]$ is only paid if the sum of structured coupons up to time $T_n$ is below a total return $R$ . Thus, the value of the TARN at time $0$ from the desk's point of view is given by

\begin{aligned} V(0) & = \beta(0)\mathbb{E}^{\mathbb{Q}}\left[ \sum_{n=1}^{N-1} \tau_n \beta(T_{n+1})^{-1} (L_n(T_n) - C_n) 1_{\{Q_n < R \}} \right], \\ Q_n & := \sum_{i=1}^{n-1} \tau_i C_i, \\ Q_1 & = 0. \end{aligned}

Note:

TARNs will typically pay some fixed coupons to the investor before the knock-out features start - these payments can be valued statically off of an interest rate curve separately.

Global caps

One form of TARN has the feature that the total return is restricted to be exactly $R$ . It achieves this by ensuring that if a coupon that will be paid crosses the total return threshold, it pays exactly $R - Q_n$ so that the total return at the end of the TARN is $R$ . This feature is called a global cap. This feature can be decoupled from the TARN and applied to generic exotic swaps. Note that a swap typically will not terminate at this point - the trading desk will continue to receive Libor until the maturity of the trade or, from the investors perspective, their principal remains invested with the issuer until the trades maturity, while they receive no coupons.

Global floors

Analogously to the global cap, a global floor ensures that the investors cumulative return reaches a minimum level of $F$ by the maturity of the note. If the total return is less than $F$ by the final coupon, the final coupon will pay $F - Q_n$ to ensure that this level is met.