Reconciling Black 1976 Caplet Formulas with Forward Rates

Summary

The document compares two expressions for a Black 1976 caplet price. One discounts the option payoff using the discount factor to the payment date; the other uses the discount factor to the rate period's start date and divides by one plus the forward rate times the accrual year fraction. The apparent discrepancy is resolved by the relationship between the simple forward rate and discount factors for the two period dates.

Substituting that forward-rate identity into the second expression converts its discounting term into the payment-date discount factor multiplied by the accrual fraction. The two formulas are therefore equivalent when the forward rate, discount factors, and year fraction use consistent conventions. This reconciliation addresses the discounting difference shown, but relies on the stated simple-compounding forward-rate relationship; conventions or rate definitions that differ may require a corresponding adjustment.

Key ideas

• The simple forward rate can be expressed using discount factors at the period's start and end dates.
• The second formula's adjusted start-date discount factor equals the end-date discount factor times the accrual fraction.
• The two caplet formulas are equivalent under consistent discounting and accrual conventions.
• Confirm the rate definition and year-fraction convention before applying either expression.

Tags

# Black 1976 caplet value


# 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:

- Caplet 1: https://www.slideshare.net/uichong/chap-26 slide 6.

- Caplet 2: http://slideplayer.com/slide/3558788/ slide 3.

## Answer by Gordon (score 2)

https://quant.stackexchange.com/a/38431

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 $Caplet_1$.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.