Converting Quarterly LIBOR Quotes to Continuously Compounded Rates

Summary

The document explains why a cap option example presents both a 7% LIBOR or swap zero curve and a 6.9395% continuously compounded zero rate. The rates use different quoting conventions: a 7% annual LIBOR rate with quarterly payments corresponds to four quarterly periods at 1.75% each. Compounding those quarterly amounts produces an effective annual factor, whose natural logarithm gives the continuously compounded rate used in the example.

The explanation shows how to reconcile the two stated rates rather than choosing one simply because the calculation uses a forward price. It is a focused clarification of rate conversion, not a full treatment of cap valuation; the discussion does not develop the option-pricing steps or explain how discounting conventions may vary across models and instruments.

Key ideas

• Quoted interest rates can differ numerically when they use different compounding conventions.
• A 7% annual rate with quarterly payments implies a 1.75% rate per quarter.
• The equivalent continuously compounded rate is found by taking the logarithm of the compounded annual factor.
• The rate conversion resolves why the cap example gives both 7% and 6.9395%.

Tags

# What's discounted rate used in the cap option


# What's discounted rate used in the cap option












Here is a example of `Cap option` in John Holl's book `Options, Futures and Other Derivatives 9th` `page 681.`

One thing I am confused that there are two discounted rate, one is `LIBOR/swap zero curve is flat at 7%` and the other is `The continuously compounded zero rate for all maturities is 6.9395%.`

Here we use forward price $F_k,$ it seems we should use the `LIBOR` as the a discounted rate.

## Answer by Lliane (score 1, accepted)

https://quant.stackexchange.com/a/35615

7% annual LIBOR with quarterly payment = 4 * 1.75% coupons

1.0175 ^ 4 = 1.071859

ln(1.071859) = 6.9395%, the continuously compounded rate.

This is the second question you ask about converting yields with different quoting conventions, I suggest you read some literature about this before attempting more complex equations. By the way it's John Hull, John Holl is writing beer guides.

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