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 LT for a period from T to T + α. The payout at the end of
this period, at time T + α, is calculated using the following formula:

Payout = α * max(LT - K, 0)

In this formula:

- α is the day count fraction, which represents the portion of the year covered by the caplet (for example, 0.25 for a 3-month period).

- LT is the LIBOR rate observed at the beginning of the period for the duration of T to T + α.

- K is the strike rate of the caplet, which is the interest rate level that the caplet is designed to protect against.

The term max(LT - K, 0) means that if the LIBOR rate LT at time T is above the strike rate K, the payout will be the
difference between LT and K, multiplied by the day count fraction α. If LT is less than or equal to K, the payout is zero because the condition for payout is not met.

The caplet is called such 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 + α and owns a caplet with a strike rate of K on LT, then the company's borrowing cost will not exceed the rate K. If the LIBOR rate goes above K, the caplet pays out
and helps the company cover the difference. If the LIBOR rate is below K, the caplet does not pay out, but the company benefits from the lower borrowing cost anyway.

For clarity in notation:

- CK(t, T) is the price at time t of the caplet that expires at time T with a strike rate of K.

- Z(T, T + α) is the discount factor at time T for the period ending at T + α, which is used to calculate the present value of the caplet's expected payout.

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

CK(T, T) = α * (LT - K) * Z(T, T + α), when LT > K.

Both LT and Z(T, T + α) are considered random variables because their values are uncertain at the time the caplet is
purchased.

Let's go through an example:

- SONIA rate (LT) = 5.19%

- Strike rate (K) = 4%

- Day count fraction (α) for 3 months = 0.25

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

Difference = LT - K = 5.19% - 4% = 1.19%

Now, deannualize this difference for a 3-month period by multiplying by the day count fraction:

Deannualized Difference = 1.19% * 0.25 = 0.2975%

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

Payout = Principal * Deannualized Difference

Payout = £1,000,000 * 0.2975%

Payout = £1,000,000 * 0.002975

Payout = £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.

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