# Black 1976 caplet value

I've seen from two sources different formulas for the caplet value (Black 1976):

• $$Caplet_1 = N\cdot DiscountFactor_{0,k}\cdot yrFrcn_{k,k+1}\cdot [F_{k,k+1}\cdot N(d_1) - R_k\cdot N(d_2)]$$
• $$Caplet_2 = N\cdot \frac{DiscountFactor_{0,k}\cdot yrFrcn_{k,k+1}}{1+F_{k,k+1}\cdot yrFrcn_{k,k+1}}\cdot [F_{k,k+1}\cdot N(d_1) - R_k\cdot N(d_2)]$$

With $F_{k,k+1}$ as the forward rate, $R_k$ as the strike and $yrFrcn_{k,k+1}$ as year fraction between date $k$ and $k+1$

I'd like to know which one is correct.

The sources:

• @noob2 DiscountFactor = $e^{-rT}$ in $Caplet_2$ source, while it's equal to $P(0,t_{k+1})$ in $Caplet_1$ source. – Oliver Mohr Bonometti Feb 22 '18 at 17:54
• @What is the $T$ above? Is it the $t_k$? – Gordon Feb 23 '18 at 14:54
• @Gordon in DiscountFactor = $e^{-rT}$ you mean? I undertand it as $t_{k+1}$, but I'm not sure, it could be $t_k$. But the main difference is in the denominator $1+F_{k,k+1}\cdot yrFrcn_{k,k+1}$ of the $Caplet_2$ formula. – Oliver Mohr Bonometti Feb 23 '18 at 15:04

Note that \begin{align*} F_{k, k+1} = \frac{1}{yrFrcn_{k,k+1}}\left(\frac{P(0, t_k)}{P(0, t_{k+1})}-1 \right). \end{align*} Then, in $Caplet_2$, \begin{align*} \frac{DiscountFactor_{0,k}\cdot yrFrcn_{k,k+1}}{1+F_{k,k+1}\cdot yrFrcn_{k,k+1}}&=\frac{P(0, t_k)\cdot yrFrcn_{k,k+1}}{1+F_{k,k+1}\cdot yrFrcn_{k,k+1}} \\ &=P(0, t_{k+1})\cdot yrFrcn_{k,k+1}, \end{align*} which is consistent with $Caplet_1$.