Caps, Floors and Swaptions

Interest Rate Modeling

[Quantitative Finance] · August 28, 2020 · 10 mins to read

Tutorial Contents

In the previous section we covered the most basic rates derivatives. In the world of fixed income, these products are known as linear instruments in the sense that their payoff functions are generally linear in their respective underlyings. A consequence of this linearity is that they are generally simple to price and do not require any stochastic model to do so. In general, this ease of pricing and the liquidity of these products makes them ideal candidates for calibrating the parameters used for models required to price more advanced products. For example, swaps are used by major institutions to build their in-house curves for various different rates, including swap curves and zero curves.

In this section, we move on to the most basic non-linear products, specifically swaptions, caps and floors. While these products still exhibit great liquidity and their prices are readily observed in the market, technically speaking a model is required to arrive at these prices. This comes as a consequence of the non-linearity of their payoffs in their underlying, otherwise known as 'convexity'. Another important way of separating non-linear products from their linear cousins is by noting their non-zero gamma (or equivalently their varying delta). This simply means that the rate at which the value of these products change as a function of their underlying is not constant. In practice, a dealer holding one of these product who desires to hedge their exposure to movements in the underlying is required to update their position in the underlying whenever the market moves.

Caps and Floors

A firm with liabilities funded at a floating rate (e.g. Libor) is naturally concerned by the possibility that rates may rise in the future. Of course, the firm could hedge this risk by paying fixed in a vanilla IR swap on the same floating rate. In doing so, they could turn their floating payments into fixed ones and guarantee a certain fixed rate on their payments. However, you can't have your cake and eat it. By hedging this floating rate risk they forgo the possibilty of benefitting from a possible drop in floating rates.

An interest rate cap is a security that allows the holder to benefit from low floating rates yet still remain protected from high rates. Similarly, an interest rate floor is suitable for an investor receiving a floating rate on their assets. A floor allows the investor to benefit from high floating rates but puts a floor on how low the rate they receive is able to fall.

Formally, a cap consists of a strip of caplets, call options on successive Libor rates. Likewise a floor is a strip of floorlets, put options on successive Libor rates. Each caplet is given mathematically by

\tau_n(L_n(T_n) - K)^+

per unit notional at time $T_{n+1}$ . Similarly, a floor is given by

\tau_n(K - L_n(T_n))^+.

It should now be simple to see where the names of both products come from. For a liability holder owing Libor set at time $T_n$ at time $T_{n+1}$ , were they to purchase a cap, their payoff at each $T_n$ would become

\tau_n(L_n(T_n) - K)^+ - \tau_n L_n(T_n) = -\tau_n\text{min}(L_n(T_n), K),

i.e. their payoff is capped by the fixed rate $K$ . Likewise, a floor holder receives payoffs floored by the fixed rate $K$ .

Setting $\omega = \pm 1$ , we can mathematically encode both an $N$ period cap and an $N$ period floor with strike $K$ via

V(t; \omega) = \beta(t) \sum_{n=0}^{N-1} \tau_n \mathbb{E}_t^{\mathbb{Q}} \left[ \beta(T_{n+1})^{-1} \left(\omega (L_n(T_n) - K) \right)^+ \right],

where $V_{\text{cap}}(t) = V(t; 1)$ and $V_{\text{floor}}(t) = V(t; -1)$ .

As when pricing swaps in the previous section, by switching to the $T_{n+1}$ -forward measure for the $n$ th cashflow, the above valuation formulas can be more conveniently expressed as

V(t; \omega) = \sum_{n=0}^{N-1} \tau_n P(t,T_{n+1}) \mathbb{E}_t^{T_{n+1}} \left[ \left(\omega (L_n(T_n) - K) \right)^+ \right].

Since $L_n(\cdot)$ is a martingale under $\mathbb{Q}^{T_{n+1}}$ (see section 2), caplets/floorlets can each be priced via a vanilla model, such as the log-normal Black model, a modification of the classic Black-Scholes model.

While the OTC market for caps/floors is liquid, individual caplets/floorlets are not themselves traded. By considering caps/floors of various maturities it is possible to back out volatility curves (volatilities vs strike) for individual forward Libor rates making up caps/floors. Once extracted, these volatilities combined with those from swaption quotes can form calibration targets for more advanced IR models used for pricing exotic products.

European Swaptions

While caps and floors give investors a way of hedging cashflow downsides on an individual level through the lifetime of an asset/liability while benefitting from upswings, European swaptions give the holder a means of entering into a swap on a future date if the strike swap rate at the time of exercise is more favourable than the current spot swap rate. Put differently, they allow the holder to enter into a swap with a positive PV. European swaptions give the holder the right, but not the obligation, to enter into a swap at a future date at a given fixed rate. A payer swaption is an option to pay the fixed leg on a fixed-floating swap, while a receiver swaption is an option to receive the fixed leg.

Assuming the underlying swap begins at the expiry date $T_0$ (this is typical), the payoff for a payer swaption at $T_0$ is given by

V(T_0) = (V_{\text{swap}}(T_0))^+ = \left(\sum_{n=0}^{N-1}\tau_n P(T_0, T_{n+1})(L_n(T_0) - K) \right)^+.

By exploiting the representation of swaps in terms of annuitities and swap rates developed in the previous section, we find that

V(T_0) = A(T_0)(S(T_0) - K)^+.

Hence, the time $t \le T_0$ European payer swaption value is given by

V(t) = \beta(t) \mathbb{E}_t^{\mathbb{Q}}\left[\beta(T_0)^{-1} A(T_0)(S(T_0) - K)^+ \right].

At this point, naively pricing under the risk-neutral measure requires joint dynamics of the bank account, swap rate and the annuity to be prescribed. Instead, by switching to the annuity measure $\mathbb{Q}^A$ of section 2, the swaption value collapses nicely to the expression

V(t) = A(t)\mathbb{E}_t^A\left[(S(T_0) - K)^+ \right],

with the forward swap rate $S(\cdot)$ being a martingale under $\mathbb{Q}^A$ .

This observation reduces European payer/receiver swaption pricing back to pricing of a simple European call/put option, for which we can utilise the Black model once again. Conversely, and more typically, market swaption prices can be mapped to market implied distributional characteristics of forward swap rates. In particular, it is universal practice across the industry to quote swaption prices in terms of implied Black or Normal volatilities.

The swaptions market is very liquid with many different maturities and swap underlyings traded actively. We proceed to fully characterise the set of all traded swaptions. Given a tenor structure

0 \le T_0 < T_1 \cdots < T_N, \quad \tau_n = T_{n+1} - T_n,

consider swaptions of expiries $\{T_n\}_{n=0}^{N-1}$ that can be exercised into swaps beginning at $T_n$ and covering $m$ periods, i.e. with last payment dates $T_{n+m}$ . We denote the corresponding underlying swap rates and annuity factors by

\begin{aligned} A_{n,m}(t) & = \sum_{i=n}^{n+m-1} \tau_i P(t, T_{i+1}), \\ S_{n,m}(t) & = \frac{\sum_{i=n}^{n+m-1}\tau_i P(t,T_{i+1})L_i(t)}{\sum_{i=n}^{n+m-1}\tau_i P(t,T_{i+1})}, \end{aligned}

for $n=0,\ldots,N-1, m=1,\ldots, N-n$ . Then the value of the $(n, m)$ -swaption is equal to

A_{n,m}(t) \mathbb{E}_t^{n,m}\left[(S_{n,m}(T_n) - K)^+ \right],

where $\mathbb{E}_t^{n,m}[\cdot] \equiv \mathbb{E}_t^{\mathbb{Q}^{n,m}}[\cdot]$ and $\mathbb{Q}^{n,m}$ represents the $A_{n,m}(\cdot)$ annuity measure. In market language, a $T_n$ -maturity European swaption on a swap running from $T_n$ to $T_{m+n}$ is called a $T_n$ y( $T_{m+n}-T_n$ )y swaption. For instance, a 1 year swaption on a 3 year swap is called the 1y3y swaption.

Evidently, if $m = 1$ , we just have an option to enter into a caplet or floorlet. Hence, any general discussion on swaptions implicitly includes caplets/floorlets. Together, the set of $(n,m)$ -swaptions constitutes the swaption grid. Due to the broad range of swaptions observable in the market, market quotes on swaptions in the form of implied volatilities provide the most readily-available information on the volatility structure of interest rates. However, since different swaptions in the grid cover overlapping sections of the interest rate term structure, extracting clean volatility information from the market is not a simple feat.

Options on vanilla swaps comprise the bulk of the liquid IR non-linear derivatives market, but there are also traded options on general swaps. It is often the case that general swaps can be decomposed (mathematically) into a basket of standard swaps. In this case, options on these swaps become basket options which require information on the joint distribution of the constituent underlyings.

Cash-Settled Swaptions

The above swaption contracts assume physical settlement. This means, upon exercise, an actual swap contract is entered into. An economically equivalent swaption contract is one where the parties agree to make a cash payment equal to the PV of the swap observed at exercise. Indeed, both types give a payoff of

A(T_0)(S(T_0) - K)^+, \tag{6.1}

at time $T_0$ . The final variety of swaption is known as a cash-settled swaption. This type of swaption is also very common. Here, rather than entering into a swap at exercise, the option holder again receives a cash payout. The settlement amount is calculated in a manner similar to (6.1), except the annuity factor $A(\cdot)$ $ is calculated by discounting fixed rate payments at the swap rate $S(T_0)$ . Specifically, we find that

V_{\text{css}}(T_0) = a(S(T_0))(S(T_0) - K)^+,

where

a(x) := \sum_{n = 0}^{N-1} \frac{\tau_n}{\prod_{i=0}^n (1+\tau_i x)}.

The benefit of this method is that, while physically settled swaptions require knowledge of a strip of discount factors (which may be estimated differently by different dealers due to differences in their curve building methodologies), cash settlement only requires that the swap rate $S(T_0)$ is observable in the market, an unambiguous quantity for all market participants.

From a technical perspective, cash settlement introduces difficulties in valuation of these contracts, since under the measure associated with the numeraire $X(t) := a(S(t))$ , the swap rate $S(\cdot)$ is not a martingale, so certain drift assumptions are required.