Caps, Floors, and Fixed Income Options (Chapter 18, Part 1)
Book: Fixed Income Securities: Tools for Today’s Markets | Author: Bruce Tuckman & Angel Serrat | ISBN: 978-0-470-89169-8
Previous: Arbitrage and Two-Curve Discounting (Chapter 17) | Next: Swaption Volatility and Skew (Chapter 18, Part 2)
Chapter 18 is the options chapter. Full term-structure models from Part Three can price everything here, but desks often reach for Black-Scholes variants because they need quick implied vols and hedge ratios between live quotes.
Caplets and Caps
A caplet pays max(0, LIBOR minus strike) times accrual on notional. The rate is fixed at the start of the period and paid at the end. It is not an option you exercise. The payoff is contractual.
Practitioners assume normal (not lognormal) forward LIBOR. The caplet price is discount factor times accrual times a Black-style normal formula on the forward rate and bp volatility.
A cap is a strip of caplets. Market quotes give one implied vol for all caplets in the strip (e.g., 77.22 bp for a 2-year ATM cap at 0.97% strike in Feb 2011). In reality each caplet has its own vol term structure. Extracting caplet vols from caps is messy because ATM cap strikes do not match each forward.
The first caplet after spot start is usually dropped because its payment is already known. Forward-starting caps (5x5, etc.) trade actively. USD caps use 3m LIBOR; EUR caps may use 6m LIBOR on longer structures.
Floors are the mirror image. Put-call parity links ATM caps and floors to forward-starting swaps.
Swaptions
A receiver swaption is the right to enter a swap receiving fixed. Payoff at expiry is annuity factor times max(strike minus par swap rate, 0). So a 5y5y receiver is a put on the forward 5-year par rate.
ATM swaptions quote in a matrix of normal vols (e.g., 2y10y at 116.4 bp in Feb 2011). U.S. swaptions are almost always cash-settled. Europe mixes cash and physical.
Swaptions that start ATM drift out of the money as the forward swap rate moves. Marking them requires interpolating across strike and expiry in the vol cube. That is part 2’s topic.
Callable Bonds and ED Options
Embedded calls in corporates are American options on the issuer’s side. Best practice is a short-rate tree with OAS. For simple cases (FNMA 2s called in six months) Black-Scholes on the forward noncallable bond price works. Implied OAS from swaption vol can back out the call value.
Callable bonds show negative convexity at low rates, like mortgages. Price caps out near call price.
Eurodollar futures options are American but treated European in practice. ED calls on price are puts on rate. Payoff scales with $25 per bp. Theoretical justification for ED options is weaker than for caplets. Desks use the formula anyway.
Euribor futures options are futures on options with daily settlement. No discount factor in the formula because the option is a futures contract.
Bond futures options use lognormal futures prices with an assumption that short discount factors are uncorrelated with long futures. Delivery option on the underlying futures breaks constant vol. Still used for quick deltas on JGB contracts and the like.
Summary Tables
Tuckman collects the BS mapping in Tables 18.8-18.10: normal vs lognormal, what S0 is (forward rate, forward bond price, futures price), and whether annuity or discount factors appear.
Personal Note
This half of the chapter is a toolkit chapter. It does not pretend BS is true. It documents what people actually do. The skew section in part 2 explains where the toolkit breaks and what replaces it.