Interest Rate Caps and Pricing Caplets Using the Black Formula: Practice Problems and Solutions

A collection of caplets is called an interest rate cap. A at each time t_i+1, a cap makes the following payment: Cap payment at time t_i+1 = max[0, R_{t_i}(t_i, t_i+1) - K_R].

A cap on a floating rate loan that makes interest payments at integer times t = 1 through t = n would have a value equal to the sum of the individual caplet values.

The Black formula can be used to price caplets. The Black formula for the price of a caplet which expires in T years and pays in T + s years is

C[F, P(0, T+s), σ, T+s] = P(0, T+s)[FN(d₁) - KN(d₂)], where

d₁ = [ln(F/K) + 0.5σ²T]/[σ√(T)] and

d₂ = d₁ - σ√(T)

But, in the case of a caplet, what are the relevant values of P(0, T+s) and F?

P(0, T+s) is the (T+s)-year discount factor or the value at t = 0 of a zero-coupon bond paying $1 at time T+s.

F is P(0, T)/P(0, T + s) - 1 or the non-annualized forward rate from time T to time T + s.

Sources: "Interest Rate Cap and Floor." Wikipedia, the Free Encyclopedia.

McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 24, p. 792.

Original Practice Problems and Solutions from the Actuary's Free Study Guide:

Problem IRCPCUBF1. An interest rate cap has a strike price of 0.0345. The interest rate cap makes payments at times t = 3, t = 4, and t = 5.

The forward rates for these time periods are as follows:

Forward rate from t = 2 to t = 3: 0.04343

Forward rate from t = 3 to t = 4: 0.09661

Forward rate from t = 4 to t = 5: 0.01342

Forward rate from t = 5 to t = 6: 0.04361

Okonkwo owns such an interest rate cap. What will be the sum of the payments he receives from it?

Solution IRCPCUBF1. We use the formula Cap payment at time t_i+1 = max[0, R_{t_i}(t_i, t_i+1) - K_R].

Since the cap makes its first payment at t = 3 and its last payment at t = 4, we need to consider forward rates from t = 2 to t = 3, from t = 3 to t = 4, and from t = 4 to t = 5. We are given that

K_R = 0.0345. Thus, the total sum Okonkwo will receive from this cap is

max[0, 0.04343 - 0.0345] + max[0, 0.09661 - 0.0345] + max[0, 0.01342 - 0.0345] =

0.00893 + 0.06211 + 0 = 0.07104

Problem IRCPCUBF2. Alcibiades has bought a caplet which expires in 2 years and pays in 4 years and has a strike rate of 0.2334. A zero-coupon bond paying $1 in 4 years has a price of 0.5523 today. A zero-coupon bond paying $1 in 2 years has a price of 0.8868 today. The annual interest rate volatility is 0.24. In the Black formula for the price of a caplet, what is the value of F?

Solution IRCPCUBF2. F = P(0, T)/P(0, T + s) - 1, where T = 2 and s = 2. We are given that P(0, 2) = 0.8868 and P(0, 4) = 0.5523, so F = 0.8868/0.5523 - 1 = F = 0.6056491037

Problem IRCPCUBF3. Alcibiades has bought a caplet which expires in 4 years and has a strike rate of 0.2334. A zero-coupon bond paying $1 in 4 years has a price of 0.5523 today. A zero-coupon bond paying $1 in 2 years has a price of 0.8868 today. The annual interest rate volatility is 0.24. In the Black formula for the price of a caplet, what is the value of d₁?

Solution IRCPCUBF3. We use the formula d₁ = [ln(F/K) + 0.5σ²T]/[σ√(T)], where, from Solution IRCPCUBF2, F = 0.6056491037. We are also given that K = 0.2334, σ = 0.24, and

T = 2. Thus, d₁ = [ln(0.6056491037/0.2334) + 0.5*0.24²*2]/[0.24√(2)] = d₁ = 2.979120601

Problem IRCPCUBF4. Alcibiades has bought a caplet which expires in 4 years and has a strike rate of 0.2334. A zero-coupon bond paying $1 in 4 years has a price of 0.5523 today. A zero-coupon bond paying $1 in 2 years has a price of 0.8868 today. The annual interest rate volatility is 0.24. In the Black formula for the price of a caplet, what is the value of d₂?

Solution IRCPCUBF4. We use the formula d₂ = d₁ - σ√(T), where, from Solution IRCPCUBF3, d₁ = 2.979120601. Thus, d₂ = 2.979120601- 0.24√(2) = d₂ = 2.639709346

Problem IRCPCUBF5. Alcibiades has bought a caplet which expires in 4 years and has a strike rate of 0.2334. A zero-coupon bond paying $1 in 4 years has a price of 0.5523 today. A zero-coupon bond paying $1 in 2 years has a price of 0.8868 today. The annual interest rate volatility is 0.24. Use the Black formula to find the price of this caplet.

Solution IRCPCUBF5. From Solutions IRCPCUBF2-4, we know that F = 0.6056491037,

d₁ = 2.979120601, and d₂ = 2.639709346. Furthermore, K = 0.2334 and P(0, T + s) = P(0, 4) = 0.5523.

In MS Excel, using the input "=NormSDist(2.979120601)", we find that N(d₁) = 0.998554546

In MS Excel, using the input "=NormSDist(2.639709346)", we find that N(d₂) = 0.995851102

We use the formula C[F, P(0, T+s), σ, T+s] = P(0, T+s)[FN(d₁) - KN(d₂)] =

0.5523[0.6056491037*0.998554546 - 0.2334*0.995851102] = C = 0.2056444969

See other sections of The Actuary's Free Study Guide for Exam 3F / Exam MFE.