Interest rate currency and credit derivatives - Caps floors ...

Learning Outcomes

After reading this article, you will be able to describe the structure and mechanics of interest rate caps, floors, and swaptions, and apply arbitrage-free valuation methods in their pricing. You will learn to explain how interest rate volatility affects these derivatives, analyze the payoff profiles for each, and calculate values using both lattice and Black model approaches. You will also be able to interpret key terms and recognize common pitfalls on the CFA Level 2 exam.

CFA Level 2 Syllabus

For CFA Level 2, you are required to understand the valuation and application of interest rate derivatives including caps, floors, and swaptions. This article covers:

• Explaining the structure and payoff logic for interest rate caps, floors, and swaptions
• Applying arbitrage-free valuation and option-pricing models (binomial tree and Black model)
• Describing the effect of interest rate volatility on option values and effective duration
• Analyzing and interpreting practical worked examples for derivative pricing calculations
• Recognizing practical considerations in applying these derivatives for risk management

Test Your Knowledge

Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, remember to look more closely at that area during your revision.

1. Which derivative contract pays when a floating rate rises above a set level—a cap or a floor?
2. Outline the effect of higher interest rate volatility on the value of a swaption.
3. What is the put-call parity relationship for European swaptions, and why is it important?
4. How does the value of an interest rate cap relate to a portfolio of caplets?

Introduction

Interest rate caps, floors, and swaptions are widely used derivatives in fixed income and risk management. Caps and floors limit exposure to changes in floating interest rates, while swaptions confer the right but not the obligation to enter into an interest rate swap. Understanding their valuation is critical for fixed income, derivatives, and portfolio management questions on the CFA exam.

Key Term: interest rate cap
An agreement that sets an upper limit on the floating interest rate paid by the holder. A cap pays if the reference rate exceeds the strike in the future.

Key Term: interest rate floor
An agreement establishing a lower limit for a floating rate. The holder receives a payment if the reference rate falls below the strike rate.

Key Term: caplet and floorlet
Individual European call (caplet) or put (floorlet) options on a single floating rate period, typically comprising a series within an overall cap or floor.

Key Term: swaption
An option granting the right, but not the obligation, to enter into an interest rate swap at pre-specified terms on a future date.

Caps and Floors Structure and Payoff

Caps are series of caplets, each a call option on a forward interest rate for a future period. They protect floating-rate borrowers against rising rates. Conversely, a floor is a series of floorlets, each a put option on a future floating rate, typically used by floating-rate investors to guard against declining rates.

• Cap payoff per caplet at time $t_i$:
  $\max(0, r_{i} - X) \times \Delta \times N$
  where $r_{i}$ is the observed floating rate set at $t_{i-1}$ (effective for $[t_{i}, t_{i+1})$), $X$ is the strike, $\Delta$ is the accrual fraction, and $N$ is notional.
• Floor payoff per floorlet:
  $\max(0, X - r_{i}) \times \Delta \times N$

Key Term: notional amount
The principal amount used to calculate payments on a derivative, though not actually exchanged.

Caps, Floors, and Swaptions Valuation Approaches

Valuation Using the Lattice (Binomial Tree) Method

Caps and floors are valued as portfolios of separate options (caplets or floorlets), where each is a European option on an interest rate set at a previous date and paid at the next.

1. Build a short-rate tree reflecting the current term structure and volatility.
2. At each relevant node, calculate the option’s payoff.
3. Backward-induct: value each caplet or floorlet by working back through the tree, discounting expected payoffs using risk-neutral probabilities.
4. Sum present values of all caplets for a cap, or floorlets for a floor.

Key Term: arbitrage-free valuation
A pricing approach ensuring that the derivative value precludes riskless profit from buying and selling related instruments simultaneously.

Worked Example 1.1

A bank purchases a 1-year, $10m notional, semi-annual cap with a 4% strike. Forward rates are 4.5% at 6 months and 5.2% at 1 year. Day count is actual/360. Calculate the value of each caplet (assume 0.5 year periods) and the total cap value, ignoring volatility.

Answer:

• 1st caplet: payoff at time 1 (6 months from today) is $0$ (strike higher than reference).
• 2nd caplet: payoff at time 2 (1 year) = (5.2\% - 4\%) \times 0.5 \times \10m = $60,000$.
• Present value of $60,000$discounted at the 1-year spot rate ($\sim5.2%) = \57,035.
• Total cap value = $57,035.

Valuation Using the Black Model

The Black model is the industry standard for valuing European interest rate options and swaptions. Each caplet, floorlet, or swaption is priced as an option on a forward rate or swap rate, discounted to present value.

• Caplet price (time $t=0$): $$C = \Delta N e^{-r_0 t_j} \left[ F_j N(d_1) - X N(d_2) \right]$$
• $F_j$ = forward rate for $[t_j, t_{j+1})$; $N(\cdot)$ = standard normal cdf; $\sigma$ = volatility; $d_1 = [\ln(F_j/X) + 0.5\sigma^2 T]/(\sigma\sqrt{T})$; $d_2 = d_1 - \sigma\sqrt{T}$.

Key Term: Black model
An option valuation formula for derivatives on forwards and swaps, assuming lognormal distribution of reference rates.

Worked Example 1.2

A $5m$ notional caplet expires in 6 months, on a 6-month LIBOR. Cap rate is 5%, forward 6-month LIBOR is 4.8%, volatility is 20%, and the discount factor to payment is 0.985. What is the caplet's price?

Answer:

• $X = 0.05$, $F = 0.048$, $\sigma = 0.20$, $T = 0.5$.
• $d_1 = [\ln(0.048/0.05) + (0.2^2/2) \times 0.5] / (0.20 \times \sqrt{0.5}) \approx -0.116$
• $d_2 = -0.116 - 0.2 \times 0.707 = -0.257$
• $N(d_1) \approx 0.454$, $N(d_2) \approx 0.398$
• Value = 0.5 \times 5{,}000{,}000 \times 0.985 [0.048 \times 0.454 - 0.05 \times 0.398] = \2,299$

Valuing Swaptions

A swaption is an option to enter into an interest rate swap. There are two main types:

• Payer swaption: right to pay fixed, receive float (analogous to a call on the swap rate).
• Receiver swaption: right to receive fixed, pay float (analogous to a put).

Swaptions are typically valued using the Black model, treating the referenced rate as a forward swap rate.

Payer Swaption Value:

Where $S$ is the forward swap rate, $X$ is the strike, and "Annuity PV" is the present value of $1$ per period for the referenced swap.

Key Term: annuity present value (Annuity PV)
The sum of present values of all future fixed swap payments, discounted at the relevant rates.

Worked Example 1.3

A 1-year into 5-year $20m$ payer swaption on a swap with annual fixed payments at 4.2%. The forward swap rate is 4.5%, volatility is 19%, and the discount factor to option expiry is 0.98; Annuity PV over swap's fixed legs is 4.3. What is the price?

Answer:

• $S=0.045$, $X=0.042$, $\sigma=0.19$, $T=1$.
• $d_1 = [\ln(0.045/0.042) + 0.5 \times 0.19^2 \times 1] / (0.19 \times 1) \approx 0.258$
• $d_2 = 0.258 - 0.19 = 0.068$
• $N(d_1) \approx 0.601$, $N(d_2) \approx 0.527$
• Value = 20{,}000{,}000 \times 4.3 \times 0.98 \times [0.045 \times 0.601 - 0.042 \times 0.527] = \1,172,532$

Exam Warning

Not adjusting for the present value of future payments when using the Black model for caps, floors, or swaptions is a common error. Always multiply the option value by the annuity present value (for swaptions), and use the correct discount factor for caplets/floorlets.

Effect of Volatility on Option Values

The value of caps, floors, and swaptions increases with higher interest rate volatility, as the probability for extreme outcomes rises. For caps, higher volatility increases the chance of floating rates exceeding the cap strike (and thus positive payoffs); for floors, it increases the chance of rates falling below the floor; for swaptions, the probability of entering an in-the-money swap increases.

Relationship Between Caps/Floors and Swaptions

A long interest rate cap (series of caplets) is similar in risk profile to a long position in payer swaptions over the same series of periods. A long floor is analogous to a series of receiver swaptions.

Summary Table: Cap, Floor, and Swaption Overview

Key Point Checklist

This article has covered the following key knowledge points:

• Structure, payoff, and typical users of interest rate caps, floors, and swaptions
• Arbitrage-free valuation of caplets, floorlets, and swaptions (lattice and Black model)
• Impact of volatility changes on derivative values
• Practical calculation of cap, floor, and swaption values using standard formulas
• Parity relationship: long cap + short floor = plain vanilla swap
• Difference in valuation approaches and the importance of discounting correctly

Key Terms and Concepts

• interest rate cap
• interest rate floor
• caplet and floorlet
• swaption
• notional amount
• arbitrage-free valuation
• Black model
• annuity present value (Annuity PV)