Interest Rate Derivatives: Caps, Floors, and Swaptions

If you thought equity options were complex, welcome to the world of interest rate derivatives. Chapter 32 of Wilmott’s book takes everything we learned about modeling bonds and the yield curve and applies it to actual products that traders buy and sell every day. Caps, floors, swaptions, callable bonds, and a whole zoo of exotic contracts.

The key idea here is that there are two approaches to pricing these things. One is the “proper” way using partial differential equations and a consistent spot rate model. The other is the “market” way where you squeeze the contract into the Black-Scholes framework, pretend interest rates behave like stock prices, and pray it works out. Surprisingly, the second approach is often more practical.

Callable Bonds: A Gentle Start

A callable bond is a regular coupon-bearing bond, but the company that issued it can buy it back on certain dates for a certain price. This is bad for the investor. If rates drop and the bond becomes valuable, the issuer calls it back. You lose the upside.

From a math perspective, this is very similar to an American option. The bond satisfies the same pricing equation as a regular bond, but with an extra constraint: the bond value can never exceed the call price. At the boundary where the issuer calls, we need the value and its derivative with respect to the interest rate to be continuous. If you have ever worked through American option pricing, this should feel familiar.

Bond Options

A bond option is exactly what it sounds like. It is an option, but instead of a stock being the underlying, it is a bond. European and American versions exist, just like with equity options.

Here is where things get tricky. To price a bond option “properly,” you first solve the bond pricing equation to get the bond value, then solve again with the option payoff as the final condition. But any error in your bond model gets amplified when you price the option on top of it. The contract is “second order” in the sense that it sits on top of another derivative.

This is why practitioners often skip the fancy models and just use Black-Scholes directly. They treat the bond as if it were a lognormal asset, estimate a bond volatility, and plug it into the formula. As long as the option expires well before the bond matures, this works reasonably. The bond behaves stochastically with measurable volatility during that period.

But watch out for the pull to par. As a bond gets close to maturity, we know exactly what it will be worth: the principal plus the last coupon. This is completely different from a stock, whose future value is never known. Near maturity, a bond cannot possibly follow a lognormal random walk because its destination is fixed. So the Black-Scholes approach fails for bond options where the option life is close to the bond maturity.

Caps and Floors

A cap protects you from rising interest rates. Say you have a floating rate loan and you are worried rates will spike. A cap guarantees that you will never pay more than a specified rate, the cap rate. Each individual payment in a cap is called a caplet, and the cap is just the sum of many caplets.

Each caplet has a payoff that looks like max(floating_rate - cap_rate, 0). Sound familiar? It is basically a call option on the interest rate.

A floor does the opposite. It ensures the rate you receive is never below a certain level. A floorlet payoff looks like max(floor_rate - floating_rate, 0), which is a put on the interest rate.

There is a beautiful no-arbitrage result called cap/floor parity: a long caplet plus a short floorlet (with the same strike) equals one payment of a swap. This is model-independent, meaning it holds regardless of which interest rate model you use.

Another important relationship: a caplet is equivalent to a put option on a bond. This connection between interest rate caps and bond options is not just academic. It links two different market segments and lets you check for consistency.

In practice, people price caps and floors using Black-Scholes formulas. Each caplet is treated as a call on a lognormally distributed interest rate. You need a forward rate (playing the role of the stock price), a discount rate, and a volatility of the interest rate. Simple, not entirely consistent, but it works.

A collar combines a cap and a floor, bounding the rate from both above and below.

Range Notes

A range note pays interest for every day that the rate stays within a specified band. If rates wander outside the band, you get nothing for those days. This is like a barrier-style product but applied to interest rates.

Swaptions

A swaption is an option to enter into a swap. In a payer swaption, you get the right to pay the fixed rate and receive floating. In a receiver swaption, you get the right to receive fixed and pay floating.

Market practitioners price European swaptions using Black-Scholes as well. They model the par swap rate as a lognormal variable. If the swap rate at exercise exceeds the strike, the payer swaption is in the money. The formula looks like a standard call option formula, adjusted with proper discount factors.

Captions and floortions are options on caps and floors. Yes, it is derivatives on derivatives. These are even more “second order” than bond options, making them prone to pricing errors since inaccuracies compound across the layers.

Spread Options

Spread options have payoffs that depend on the difference between two interest rates. This is where one-factor models completely fail. In a one-factor world, all rates are perfectly correlated, so there is no randomness in the spread. You need at least a two-factor model (which we will see in Chapter 35) or you model the spread directly as a random variable using Black-Scholes-style formulas.

Index Amortizing Rate Swaps

The index amortizing rate (IAR) swap is one of the more interesting path-dependent products. It is a swap, but the principal decreases over time based on where interest rates go.

Here is how it works. You start with some principal, say $10 million. Every quarter, the two parties exchange interest payments on the current principal. But at each quarter, the principal may shrink according to an amortization schedule linked to the spot rate. If rates are very low (say below 3%), the principal gets wiped out completely. If rates are moderate, the principal drops by some percentage. If rates are very high, nothing happens.

This makes the contract strongly path dependent. The principal at any point depends on the entire history of rates.

The person receiving the fixed rate suffers in both directions. If rates rise, they pay more floating on a principal that does not decrease. If rates fall, the principal shrinks, so their lower floating payments happen on a smaller base. The fixed rate receiver is essentially selling volatility. They want rates to stay put.

The Spot Rate Approximation

Contracts specify real rates like three-month LIBOR, but our models use the spot rate, an instantaneous rate nobody can observe. Wilmott gives a rough guide: use the spot rate when contract rates are six months or less, the contract is highly nonlinear or path dependent, and no liquid hedging instruments are available. For short maturities, a Taylor series expansion relates the spot rate to finite maturity rates, with a correction term involving the risk-adjusted drift.

The Two Approaches

Wilmott wraps up the chapter by summarizing the two philosophies. The PDE approach is theoretically consistent across all instruments, but it is computationally expensive and dangerous for liquid, higher-order contracts. The Black-Scholes “squeeze-it-until-it-fits” approach is admittedly a fudge, but it is simple, fast, and less likely to produce catastrophically wrong answers.

For complex, illiquid, path-dependent contracts, use the PDE framework. For simpler liquid products, use the market formulas. And always be aware that both approaches have weaknesses.

Key Takeaways

Interest rate derivatives are the bread and butter of fixed-income markets. Caps protect against rising rates. Floors protect against falling rates. Swaptions give optionality on swaps. Path-dependent products like IAR swaps add complexity by making the contract depend on the entire history of rates. In practice, most of these products are priced using Black-Scholes formulas with interest rates playing the role of stock prices, because the “proper” approach is too fragile for liquid markets. But for exotic products, there is no shortcut: you need the full PDE framework.

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