### Caplet/Floorlet Price Formula

Recall that a Caplet provides the buyer protection against an increase in a forward interest rate, whereas a Floorlet provides the buyer protection against a decrease in a forward interest rate. Let \( F (t, T_1, T_2) \) be the forward rate from time \( T_1 \) to \( T_2 \) as seen at time t (the forward rate will be stochastic until time \( T_1 \), which is when it fixes and is no longer stochastic), and let K be the agreed strike rate.

- If the forward rate \( F (T_1, T_1, T_2) \) turns out to be higher than K, then the buyer of the caplet will receive, at time \(T_2\), the difference, \( F (T_1, T_1, T_2) - K \), times the notional times the accrual factor (which will be the tenor of the forward rate - i.e.,\(T_2-T_1\), whereas the buyer of the Floorlet will receive nothing.
- On the other hand, if the forward rate \( F (T_1, T_1, T_2) \) turns out to be lower than K, then the buyer of the Caplet will receive nothing, whereas the buyer of the Floorlet will receive the difference, \( F (T_1, T_1, T_2) - K \), scaled by the notional and the accrual factor.

To simplify the presentation, we derive the formula for the price of a Caplet, which can then be easliy generalised to cater for both Caplets and Floorlets by inserting the \( \phi \) symbol we have been using at the appropriate places. Now the payoff of a Caplet at maturity of the contract \( T_2 \) can be written as:

$$ Payoff(T_2) = {(F (T_1, T_1, T_2) - K)}^{+} \; Notional \; (T_2-T_1) $$

Where \( F (T_1, T_1, T_2) \) represents the forward rate from \(T_1\) to \(T_2\) as at time \(T_1\), \(T_1\) the start time of of the forward rate, \(T_2\) the end time of the forward rate, \( (T_2-T_1)\) the accrual factor, and K the strike rate.

The price of the Caplet at any time t before the Maturity T is just the present value of its expectation, where the expectation in this case is taken under the measure associated with the \(T_2\) bond numeraire:

$$ Caplet_0 = P (0,T_2) E^{P_{T2}} \left[ \frac{1}{P (T_2,T_2)} Payoff(T_2) \right]$$ $$ \quad = P (0,T_2) E^{P_{T2}} \left[ \frac{1}{P (T_2,T_2)} {(F (T_1, T_1, T_2) - K)}^{+} Notional \; (T_2-T_1) \right]$$ $$ \quad = P (0,T_2) E^{P_{T2}} \left[ {(F (T_1, T_1, T_2) - K)}^{+} \; Notional \; (T_2-T_1) \right]$$ $$ \quad = P (0,T_2) \; Notional \; (T_2-T_1) E^{P_{T2}} \left[ {(F (T_1, T_1, T_2) - K)}^{+} \right]$$

As the forward rate is martingale under the measure associated with the \(T_2\) bond numeraire, its dynamic can be written as lognormal with zero drift (as per Black's model), and the price of the Caplet can be expressed in terms of Black's price formula:

$$ Caplet_0 = P (0,T_2) \; Notional \; (T_2-T_1) E^{P_{T2}} \left[ {(F (T_1, T_1, T_2) - K)}^{+} \right]$$ $$ \quad \quad = \frac{P (0,T_2)}{e^{-r T_1}} \; Notional \; (T_2-T_1) e^{-r T_1} E^{P_{T2}} \left[ {(F (T_1, T_1, T_2) - K)}^{+} \right]$$ $$ \quad \quad = \frac{P (0,T_2)}{P (0,T_1)} \; Notional \; (T_2-T_1) BlackCallPrice $$ $$ \quad \quad = \frac{1}{1+ (T_2-T_1) F(0, T_1, T_2)} \; Notional \; (T_2-T_1) BlackCallPrice $$ $$ \quad \quad = A \; Notional \; (T_2-T_1) BlackCallPrice $$

Where we have used the relationship between zero coupon bond prices and forward rate: \( P(0,T_2) =\frac{P (0,T_1)}{1+ (T_2-T_1) F(0, T_1, T_2)}\). We also let \( \frac{1}{1+ (T_2-T_1) F(0, T_1, T_2)}=A \).

Similarly, the Floorlet price can be shown to be:

$$ Floorlet = \frac{1}{1+ (T_2-T_1) F(0, T_1, T_2)} \; Notional \; (T_2-T_1) BlackPutPrice $$ $$ \quad \quad = A \; Notional \; (T_2-T_1) BlackPutPrice $$

As a Cap/Floor is just the sum of Caplets/Floorlets, we can compute their prices as linear sum of the Caplets/Floorlets prices over the period spanned by the Cap/Floor.