A company borrows $100m from a bank. It’s a floating rate loan tied to SOFR, the risk-free rate (RFR) index for USD. Interest is calculated quarterly in-arrears by compounding the daily SOFR rate over each 3M period. From an accounting perspective, the loan is booked in the bank’s banking book with interest P&L straight-lined accrued to maturity.
The company also purchases protection from the bank against an int rate rise. If rates rise above the cap rate K in the diagram below, the company will receive an int amt equivalent to its increased int cost on the loan.
The bank receives a cap premium from the company. It books the short cap position in its trading book. The cap premium is a realized asset that is offset by the unrealized liability to the company that the short cap position creates. With the realized cap asset & unrealized cap liability exactly offsetting each other, there is no P&L recorded against the cap on Day 1. The bank uses the Black 76 model to calculate the cap premium.
The primary risk factor for the cap position is a term structure of SOFR index swap rates. The SOFR curve impacts the cap valuation in two ways. 1) the observable SOFR rates determine the cap CFs throughout its life. 2) The SOFR rates are the observable inputs into the USD yc building process. The yc is built using a bootstrapping routine. The bootstrapped yc contains a set of DFs that are multiplied by the future expected CFs of the cap, to arrive at the cap PV.
On Day 2 of the cap, SOFR rates change, the Black 76 model calculates a cap price that is slightly higher than the premium received, and an unrealized gain is booked.
Black 76 values the cap as a series of caplets. Each caplet is an option on a fwd SOFR rate with strike K. Caplet 1 in the diagram below is an option on the 3M forward rate F1 which starts in 3M and expires in 6M. As F1 is lower than K, caplet 1 is out of the money (OTM). Its valuation is represented by the first yellow bar in the cap PV bar chart. The fwd rates F2 and F3 are closer to K. Caplets 2 and 3 are therefore less OTM and have higher vals. Caplet 4 is at-the-money and has the highest val of the four caplets. Its fwd rate F4 is equal to the cap strike, K. There are no in-the-money caplets in the diagram.
The valuation of each caplet can be visualized its fwd rate less the strike (i.e., F – K). But if the fwd rate is below the strike in all four caplets, why then does each caplet have +ve valuations? The answer is: because F and K are adjusted using probabilities that the fwd rates finish ITM. These probability adjustments ensure that each caplet, no matter how far OTM its fwd rate is, always has some value.
This +ve rate, output by Term 1 below, is multiplied by a notional CF of $1 to generate the expected future CF of the caplet in unit $ terms. This expected CF is in turn multiplied the DF derived from the yc to generate the PV of the caplet. The individual caplet PVs are summed to obtain the cap price.