Explain how the Black model values European swaptions and European interest rate options
The idea
The Black model extends to rates. A caplet, a call on a future reference rate, is worth c = notional × accrual fraction × discount factor to the payment date × [FRA × N(d1) − X × N(d2)], paid one period after the rate is set. Floorlets use the put form, and caps and floors are sums of these. A payer swaption, the right to pay fixed, is a call on the forward swap rate: notional × annuity factor (the sum of discount factors over the underlying swap) × [FS × N(d1) − X × N(d2)]. A receiver swaption is the matching put. The trap is leaving out the accrual fraction or the annuity factor.
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