A caplet is a financial derivative similar to a call option on an interest rate, such as the LIBOR rate when it was commonly used. The payout of a caplet is determined at the end of a specified period and is based on the interest rate during that period.

Let's say a caplet is struck at a rate of \( K \) on the LIBOR rate \( L_T \) for a period from \( T \) to \( T + \alpha \). The payout at the end of this period, at time \( T + \alpha \), is calculated using the following formula:

\[ \text{Payout} = \alpha \cdot \max(L_T - K, 0) \]

In this formula:

• \( \alpha \) is the day count fraction, representing the portion of the year covered by the caplet (e.g., 0.25 for a 3-month period).
• \( L_T \) is the LIBOR rate observed at the beginning of the period for the duration of \( T \) to \( T + \alpha \).
• \( K \) is the strike rate of the caplet, which sets the interest rate level that the caplet protects against.

The term \( \max(L_T - K, 0) \) ensures that if the LIBOR rate \( L_T \) at time \( T \) is above the strike rate \( K \), the payout will be the difference between \( L_T \) and \( K \), multiplied by \( \alpha \). If \( L_T \leq K \), the payout is zero because the condition for payout is not met.

The caplet is named so because it acts as a cap on the interest rate costs for the borrower. If a company borrows money at LIBOR from \( T \) to \( T + \alpha \) and owns a caplet with a strike rate of \( K \), then the company's borrowing cost will not exceed \( K \). If the LIBOR rate exceeds \( K \), the caplet pays out and helps cover the difference. If the LIBOR rate is below \( K \), the caplet does not pay out, but the company benefits from lower borrowing costs anyway.

The payout formula adjusted for the time value of money is:

\[ C_K(T, T) = \alpha \cdot (L_T - K) \cdot Z(T, T + \alpha), \quad \text{when } L_T > K. \]

Where:

• \( C_K(t, T) \) is the price at time \( t \) of the caplet expiring at \( T \) with a strike rate \( K \).
• \( Z(T, T + \alpha) \) is the discount factor at \( T \) for the period ending at \( T + \alpha \), used to calculate the present value of the caplet's expected payout.

Suppose the following values:

• SONIA rate (\( L_T \)) = 5.19%
• Strike rate (\( K \)) = 4%
• Day count fraction (\( \alpha \)) for 3 months = 0.25

First, calculate the difference between the SONIA rate and the strike rate, then adjust it for the 3-month period:

\[ \text{Difference} = L_T - K = 5.19\% - 4\% = 1.19\% \]

Deannualizing this difference for a 3-month period:

\[ \text{Deannualized Difference} = 1.19\% \times 0.25 = 0.2975\% \]

For a principal of £1,000,000, the caplet payout would be:

\[ \text{Payout} = \text{Principal} \times \text{Deannualized Difference} \] \[ = £1,000,000 \times 0.002975 = £2,975 \]

Thus, the caplet's payout of £2,975, once discounted to present value, offsets the company's higher interest costs from the SONIA rate rise.