Lesson 5 · Derivatives PricingCFA L2

Interest Rate Derivative Instruments

Interest-rate derivatives are instruments whose payoffs depend on a bond or a reference rate: futures, options, swaps, caps and floors. This lesson covers the mechanics of Treasury bond and note futures (conversion factors, the cheapest-to-deliver issue, the implied repo rate), options on futures, the risk profile of interest-rate swaps, and how caps, floors and collars are simply packages of interest-rate options.

On this page

1 · Introduction 2 · Interest-rate futures 3 · Interest-rate options 4 · Interest-rate swaps 5 · Caps & floors 5.3 · Collars

Interest-rate derivative instruments are financial contracts whose payoffs depend on a cash-market instrument or a reference interest rate. Relative to trading the underlying bonds directly, they offer three advantages: cost, speed, and liquidity.

Learning outcomes

Explain Treasury bond futures: conversion factors, the invoice price, the cheapest-to-deliver issue, and the implied repo rate.
Describe interest-rate options and the payoffs of exercising options on bond futures.
Interpret an interest-rate swap as a package of forwards/futures and as a package of cash-market instruments, and identify how a swap's value changes with rates.
Demonstrate that a cap and a floor are packages of options (caplets/floorlets) on rates and on fixed-income instruments, and compute their payoffs.
Explain how a collar is created.

1 · Introduction

The family includes futures, forwards, options, swaps, caps and floors. Recall the futures vs forward distinction: futures are standardised, exchange-traded, clearinghouse-guaranteed, marked to market, margined, and free of counterparty credit risk; forwards are customised OTC contracts with bilateral counterparty risk and no required margin. Because only initial margin is posted, futures create a leveraged position useful for controlling interest-rate risk.

2 · Interest-rate futures

A futures contract obliges the buyer (seller) to take (make) delivery at a set price on a set date, guaranteed by the clearinghouse. Margin terms: initial margin (minimum per contract), maintenance margin (the floor the account may fall to before a top-up), and variation margin (the deposit restoring the account to initial margin).

Treasury bond futures (CBOT)

The underlying is $100,000 par of a hypothetical 20-year coupon bond; par = 100; quotes are in 32nds of 1%. The short may deliver any of several eligible Treasury bonds, each with a CBOT-published conversion factor. The amount the long pays is the invoice price:

Invoice price

$$ \text{Invoice} = \text{contract size}\times\text{futures settlement price}\times\text{conversion factor} + \text{accrued interest} $$

Settlement quotes like "105-16" mean $105+\tfrac{16}{32}=105.5\%$ of par, i.e. 1.055 per $1 par.

Worked example 1

Invoice price

June T-bond futures settle at 97-24; delivered issue's conversion factor 1.17; accrued interest $3,800 per $100,000 par.

Show solution

97-24 $=97+\tfrac{24}{32}=97.75$ → 0.9775 per $1 par.

$$ \$100{,}000\times0.9775\times1.17 + \$3{,}800 = \$118{,}167.50 $$ $118,167.50

Cheapest-to-deliver & the implied repo rate

The short profits by delivering the bond that is cheapest to buy-and-deliver. The cheapest-to-deliver (CTD) issue is the one with the highest implied repo rate — the financing rate at which buying the bond and delivering it into the futures breaks even:

Implied repo rate

$$ \text{Implied repo rate} = \frac{\text{Dollar return}}{\text{Cost of the investment}}\times\frac{360}{\text{days}_1} $$

Dollar return = Proceeds received − Cost. Proceeds = converted price (futures price × CF) + accrued interest received + interim coupon + interest on reinvesting that coupon (coupon × repo × days₂/360). Cost = purchase price + accrued interest paid. days₁ = days to settlement; days₂ = days from interim coupon to delivery.

Worked example 2

Implied repo rate

Futures price 97, days₁ = 62. Deliverable: price 95, accrued paid 3.0110, coupon 7%, interim coupon 3.50 after 25 days, days₂ = 37, CF 0.9710, accrued received 0.7096, 37-day repo 4.7%.

Show solution

Converted price $=97\times0.9710=94.1870$. Reinvestment interest $=3.50\times0.047\times\tfrac{37}{360}=0.0169$.

Proceeds $=94.1870+0.7096+0.0169+3.5000=98.4135$; Cost $=95+3.0110=98.0110$.

$$ \text{Implied repo} = \frac{98.4135-98.0110}{98.0110}\times\frac{360}{62}=0.0238=2.38\% $$ 2.38%

The short also holds delivery options: the quality (swap) option (which issue), the timing option (when in the month), and the wild-card option (deliver after the close is set). Delivery runs over three days — position day, notice day, delivery day. Treasury note futures (2-, 5-, 10-year) and agency note futures (Fannie/Freddie, 6.5% notional) are modelled the same way.

3 · Interest-rate options

An interest-rate option grants the right, not the obligation, to transact a fixed-income instrument or rate. The buyer's loss is capped at the premium; the writer's gain is capped at the premium received. Unlike futures, only the writer is obligated, the payoff is asymmetric, and there is no buyer margin. Exchange-traded options are standardised and cleared; OTC options are tailor-made, less liquid, and costlier.

Worked example 3

Options on T-bond futures

A. Call, strike $98; futures at expiration $103.

Show solution

Exercise (103 > 98). The buyer gets a long futures at the strike $98; marking to market to $103 means the writer pays the buyer $5. Result: buyer holds a long future at $103 plus $5 cash; writer holds a short future at $103 having paid $5.

Exercised; writer pays $5

B. Put, strike $105; futures at expiration $96.

Show solution

Exercise (96 < 105). The buyer gets a short futures at $105; the writer pays the buyer $9 (105 − 96). Buyer: short future at $105 + $9 cash; writer: long future at $105 having paid $9.

Exercised; writer pays $9

4 · Interest-rate swaps

Two parties exchange periodic interest payments on a notional principal (only interest changes hands, never the notional). The fixed-rate payer pays the swap rate — a spread (the swap spread) over the matched-maturity Treasury yield — and receives the floating reference rate; the fixed-rate receiver takes the other side.

Worked example 4

Swap payments

4-year swap, $100m notional, quarterly, reference 3-month LIBOR, swap rate 4.4%.

Show solution

A. Fixed payment: $\$100\text{m}\times\tfrac{0.044}{4}=\$1.1\text{m}$ per quarter.

B. First floating (3-month LIBOR 7.2%): $\$100\text{m}\times\tfrac{0.072}{4}=\$1.8\text{m}$.

Fixed $1.1m; first floating $1.8m

Risk/return: how the swap's value moves

When rates rise, a fixed-rate payer who locked in the old (lower) rate is advantaged — the swap's value to the fixed-rate payer increases and to the receiver decreases. When rates fall, the reverse holds.

Two ways to read a swap position

As a package of forwards/futures: the fixed-rate payer behaves like a short interest-rate-futures (long FRA) position — gains when rates rise; the floating-rate payer is the mirror. As a package of cash-market instruments: being the fixed-rate payer is identical to buying a floating-rate bond financed by issuing (shorting) a fixed-rate bond — the net cash flow $(\text{LIBOR}/2)\times\text{notional} - \text{fixed}$ is exactly the swap. So the fixed-rate payer is short the bond market (long a swap); the fixed-rate receiver is long the bond market (short a swap).

5 · Interest-rate caps and floors

A cap pays the buyer when the reference rate rises above the strike (the cap rate); a floor pays when it falls below the strike (the floor rate). The buyer pays an up-front premium — its maximum loss — and only the writer is obligated (unilateral counterparty risk, like an option).

Cap / floor payoff per settlement

$$ \text{Cap: } N\times\max(0,\,\text{rate}-\text{cap rate})\times\tfrac{1}{\text{periods}}, \qquad \text{Floor: } N\times\max(0,\,\text{floor rate}-\text{rate})\times\tfrac{1}{\text{periods}} $$

Worked example 5

Cap payoffs

4-year cap, cap rate 7%, $100m notional, quarterly, on 3-month LIBOR. LIBOR by quarter: 6.7%, 7.0%, 7.4%, 7.6%.

Show solution

Payoff $=\$100\text{m}\times\max(0,\text{LIBOR}-7\%)/4$:

Q1 (6.7%): 0 · Q2 (7.0%): 0
Q3 (7.4%): $\$100\text{m}\times0.004/4=\$100{,}000$
Q4 (7.6%): $\$100\text{m}\times0.006/4=\$150{,}000$

0, 0, $100,000, $150,000

Caps and floors are packages of options

A cap is a string of caplets; a floor is a string of floorlets — one option per settlement date. There are two equivalent ways to see them:

As options on the rate: a long cap = package of call options on the interest rate (calls gain as rates rise); a long floor = package of put options on the interest rate.
As options on a fixed-income instrument: because bond prices move inversely to rates, a long cap = package of puts on the bond and a long floor = package of calls on the bond.

5.3 · Interest-rate collar

A collar = buy a cap + sell a floor. A borrower who buys a 7% cap and sells a 4% floor receives a payment if rates exceed 7% and makes a payment if rates fall below 4% — so the effective borrowing cost is bounded to a 4%–7% band (adjusted by the net premium). Selling the floor finances the cap, cheapening the protection at the cost of giving up the benefit of very low rates.

Key takeaway

Interest-rate derivatives are the bond-market toolkit. Futures on Treasuries carry delivery options that make the conversion factor and the cheapest-to-deliver (highest implied repo) issue central to pricing. A swap is two things at once — a strip of forwards and a long-floating/short-fixed bond pair — so the fixed-rate payer gains when rates rise. And caps, floors and collars are nothing more than packages of interest-rate options: a long cap = calls on the rate (puts on the bond), a long floor = puts on the rate (calls on the bond), and a collar bounds a borrower's cost between the floor and cap strikes.

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