Can you break down a single caplet valuation step-by-step?

Question

AcadiFi · Accepted Answer

Consider a caplet on 6-month SOFR, strike K=5.00%, forward rate F=5.30%, vol sigma=18%, time to option expiry T=1.5 years, tau=0.5 years (accrual), notional $10M, discount factor from today to payment date DF(2.0)=0.905.

Step 1: Compute d1 and d2.
d1 = [ln(F/K) + 0.5 x sigma^2 x T] / (sigma x sqrt(T))
d1 = [ln(0.053/0.050) + 0.5 x 0.18^2 x 1.5] / (0.18 x sqrt(1.5))
d1 = [0.0583 + 0.0243] / 0.2205 = 0.375
d2 = d1 - sigma x sqrt(T) = 0.375 - 0.2205 = 0.154

Step 2: Standard normal CDFs.
N(0.375) ~ 0.646
N(0.154) ~ 0.561

Step 3: Apply Black's caplet formula.
Caplet = notional x tau x DF x [F x N(d1) - K x N(d2)]
= 10,000,000 x 0.5 x 0.905 x [0.053 x 0.646 - 0.050 x 0.561]
= 4,525,000 x [0.03424 - 0.02805]
= 4,525,000 x 0.00619
= $28,010

Key nuance: the discount factor is to the payment date T+tau=2.0, not to the fixing date T=1.5. The fixing occurs at 1.5 years, but cash settles in arrears at 2.0 years. Practitioners often use forward vol from a caplet vol surface rather than flat cap vol, because cap vol is a complex weighted average of caplet vols.

Put-call parity: Caplet - Floorlet = tau x DF x (F - K), so Floorlet = 28,010 - 4,525,000 x 0.003 = $14,435.

Can you break down a single caplet valuation step-by-step?

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