Exchanging cash-flows: swaps
- Swaps are instruments based on the exchange of cash-flows;
- The most commons swaps are: interest rate swaps, currency swaps, commodity swaps and Credit Default swaps;
- Pricing vanilla interest rate swaps;
- Discussion on accreting, amortizing, and cancelable swaps;
- Swaptions are instruments that give the possibility to enter in a swap with predetermined specifications.
From my point of view, swaps are one of the smartest and most interesting instruments in finance. Its use might seem a bit complicated when analysed at a glance, but the applications of swaps are really worth the effort. Essentially, swaps are exchanges of cash-flows based on some predetermined criteria. The most liquid type of contract is the vanilla fixed-for-floating swap. In this contract, a regular fixed payment is exchanged for a floating one which follows a determined interest rate, usually Libor in some currency.
A swap has distinct applications (beyond the speculative one, of course). It can be used to change the nature of a liability. A company that has a floating paying liability, for instance, can enter into a vanilla payer swap contract to change its liability for a fixed rate one. The opposite is also possible, that is, change a fixed paying liability for a floating one. A currency swap can be used by a company with business abroad to hedge future operations. Also countries might employ currency swaps as a strategy for monetary policy. Commodity swaps can be also be used as a hedging strategy for future operations in this market. A much more riskier application could be enter into a swap in which the first cash-flows are expected to be positive aiming invest the money in some other operation, put in other words, one is betting on an arbitrage opportunity.
Another distinct category of swaps are the Credit Default Swaps that became even more famous after the movie "The big short" about the 2008 crisis. In this swap, the risk of default of a counterparty is transferred to a third party that assumes such a risk in exchange for a periodic payment.
Another distinct category of swaps are the Credit Default Swaps that became even more famous after the movie "The big short" about the 2008 crisis. In this swap, the risk of default of a counterparty is transferred to a third party that assumes such a risk in exchange for a periodic payment.
Pricing vanilla fixed-for-floating swaps
For convention, in this post a payer swap means the party paying the fixed rate, while in a receiver swap the investor is paying the floating rate. Therefore, receivers and payers have opposite expectations with respect to the future movements of the market. The fixed rate of a swap is determined in such a way that the initial value of the contract is null for both parties. With this feature in mind, it's easy to deduce that the contract is design to have a period in which one party has a positive cash-flow followed by a period with negative cash-flow, and vice versa.
In principle, vanilla fixed-for-floating swaps do not require a model for pricing, which in principle means that it is not necessary to specify a differential stochastic equation for the swap rate (explained later). I used the expression "in principle", because one needs a model that interpolates the forward rate as explained in the following.
Consider that the floating leg of the contract has a schedule of reset dates given by $\{T_n\}$, meaning that the value of the float interest rate is observed at each date $T_n$ to determine the value of the next net cash-flow. Let´s also assume that the fixed and floating legs have the same payment schedule. Moreover, the notional is a constant value along the contract. Therefore, the present value of a payer swap under these assumptions reads
$V(t)=R \sum_{n=q(t)}^{N-1}\tau_n \left[ L(t,T_n, T_{n+1}) - K \right]P(t,T_{n+1})$
where $\tau_n \equiv T_{n+1}-T_n$ is the accrual period, $R$ is the notional, $K$ is the fixed rate, $P(t, T)$ is the discount factor (see more on zero-coupon bonds), and $L(t, T_n, T_{n+1})$ is the forward rate as of time $t$. The index $q(t)$ indicates how many reset dates have passed since the beginning of the contract at $t_0$. An useful way to rewrite formula above is by doing
$V(t)=R A(t) \left[S(t) - K \right]$
with $A(t) \equiv \sum_{n=q(t)}^{N-1}\tau_n P(t,T_{n+1})$ and $S(t) \equiv \frac{\sum_{n=q(t)}^{N-1}\tau_n P(t,T_{n+1})L(t,T_n, T_{n+1})}{\sum_{n=q(t)}^{N-1}\tau_n P(t,T_{n+1})}$. The term $S(t)$ is the swap rate and the term $A(t)$ determines the numeraire of the called annuity measure. It is worth noticing that the dynamics of the swap rate is a Martingale under the annuity measure. This is an useful information when pricing swaptions (later in this post). The value of the fixed rate is determined in such a way that the initial value of the swap is null for both counterparties of the contract, hence $K = S(t_0)$.
Accreting, amortizing, and cancelable swaps
Vanilla interest rate swaps present a constant notional along the contract. Moreover, once the contract starts there is no way back (except for some legal reasons or default). Some other interesting possibilities for swaps also exist. An accreting swap, for instance, has the notional increasing periodically according to a predetermined specification. Conversely, in an amortizing swap the notional decreases. The difference in pricing this kind of swaps compared to vanillas is that the notional is not a common factor and cannot be taken out of the sum in $A(t)$ and $S(t)$.
Other interesting and much more complex swap flavor is the cancelable swap, in which one of the parties has the right but not the obligation to cancel the swap. The pricing in these cases involves the replication of the contract using vanilla swaps and swaptions, which are options whose underlying are swap contracts (see next section). The replication portfolio takes into account the number of dates in which the contract can be cancel. A cancelable swap with one exercise date can be replicate by a regular swap and a position in an European swaption with the opposite payoff. For instance, consider a callable swap (the payer fixed can cancel the contract) with five years expiration has an exercise date two years after the beginning of the contract. In this case, the callable swap could be replicate as a payer swap with five years tenor plus an European receiver swaption with two years of expiration whose underlying is a receiver swap with three years tenor.
The situation becomes much more complex if the cancelable swap has more than one exercise date, because the replication is done with a Bermudan swaption and not an European one. Bermudans present a intertemporal correlation which is more difficult to catch and models present a more complex calibration. For instance, the calibration of low-factor models for Bermudan swaptions is instrument-dependent and might result in serious inaccuracies. Ideally, more complex models should be used, like market models, for instance.
As easily deduced, the price of cancelable swaps has a strong dependency on the market volatility and on the future expectations of the yield curve. The relation between the slope of the curve and the exercise dates also plays an important role (which one? I´ll let as an interesting exercise for the reader).
Other interesting and much more complex swap flavor is the cancelable swap, in which one of the parties has the right but not the obligation to cancel the swap. The pricing in these cases involves the replication of the contract using vanilla swaps and swaptions, which are options whose underlying are swap contracts (see next section). The replication portfolio takes into account the number of dates in which the contract can be cancel. A cancelable swap with one exercise date can be replicate by a regular swap and a position in an European swaption with the opposite payoff. For instance, consider a callable swap (the payer fixed can cancel the contract) with five years expiration has an exercise date two years after the beginning of the contract. In this case, the callable swap could be replicate as a payer swap with five years tenor plus an European receiver swaption with two years of expiration whose underlying is a receiver swap with three years tenor.
The situation becomes much more complex if the cancelable swap has more than one exercise date, because the replication is done with a Bermudan swaption and not an European one. Bermudans present a intertemporal correlation which is more difficult to catch and models present a more complex calibration. For instance, the calibration of low-factor models for Bermudan swaptions is instrument-dependent and might result in serious inaccuracies. Ideally, more complex models should be used, like market models, for instance.
As easily deduced, the price of cancelable swaps has a strong dependency on the market volatility and on the future expectations of the yield curve. The relation between the slope of the curve and the exercise dates also plays an important role (which one? I´ll let as an interesting exercise for the reader).
Swaption: options on swaps
The pricing of swaptions is in the soul of interest rate models. There is a whole area of study on this topic which cannot be covered by this humble post.
European swaptions are the most liquid type of options on swaps. It has a single expiration and the underlying swap is usually a vanilla one. The payoff of an European payer swaption reads
I hope you enjoyed the post. Leave your comments and share!
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| Attention to important dates |
$V(T) = R \left \{A(T) [S(T) - K] \right \}^{+}$
where $T$ is the expiration of the swaption. Since the swap begins in a future time, one needs a model to estimate the expectation value of the floating rate (usually Libor) for periods in the future, that is, one estimates forward rate as seen at $t$ for a contract beginning at $T$ with a tenor $\delta$, with $t<T<T+\delta$.I hope you enjoyed the post. Leave your comments and share!
May the Force be with you.
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Diogo de Moura Pedroso
LinkedIn: www.linkedin.com/in/diogomourapedroso
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