How do I price an interest rate cap and floor using the Black model?

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AcadiFi · Accepted Answer

An interest rate cap is a portfolio of European call options on a reference rate (say, 3-month SOFR), with each option called a caplet. A floor is the analogous portfolio of put options (floorlets). To price a 2-year semi-annual cap with strike 4.5% on a $25M notional, you value four caplets (excluding the first period, which is typically known).

Each caplet pays max(SOFR - 4.5%, 0) x notional x (days/360), discounted back. Under Black's model, caplet value = notional x tau x DF(T+tau) x [F x N(d1) - K x N(d2)], where F is the forward rate for period [T, T+tau], DF is the discount factor from today to payment date T+tau, and d1 = [ln(F/K) + 0.5 x sigma^2 x T] / (sigma x sqrt(T)).

Assume forwards from the SOFR curve are 4.2%, 4.6%, 4.9%, 5.1% and caplet vol is 22%. Caplet 3 (T=1 year, F=4.9%, K=4.5%, sigma=0.22): d1 = [ln(0.049/0.045) + 0.5 x 0.22^2 x 1]/(0.22 x 1) = (0.0851 + 0.0242)/0.22 = 0.497, d2 = 0.277. N(d1)=0.690, N(d2)=0.609. Value per unit = 0.5 x DF x (0.049 x 0.690 - 0.045 x 0.609) = 0.5 x 0.92 x 0.00644 = 0.00296, so about $74,000 on $25M.

Sum all four caplet values for the cap price. Put-call parity holds: Cap - Floor = Swap (pay-fixed at strike K).

How do I price an interest rate cap and floor using the Black model?

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