Implied Repo Rate from European Option Put-Call Parity

Summary

The document derives an implied financing rate for an underlying asset from the prices of European call and put options with the same strike and expiry. It uses put-call parity with continuously compounded risk-free discounting and a continuous dividend yield. Rearranging parity isolates the risk-free rate implied by the observed spot, option prices, strike, dividend yield, and time to expiry.

The result is a direct algebraic formula, rather than a simple annualization of a present-value/future-value ratio. Its use depends on the parity assumptions: matching European options, consistent expiry and strike, and the stated continuous-compounding conventions. The document gives no market example or discussion of transaction costs, funding spreads, or quote quality, so practical estimates may require adjustments beyond the derivation.

Key ideas

• Put-call parity with continuous dividends links call and put prices to spot and discounted strike values.
• Rearranging parity yields the implied continuously compounded financing rate from option and underlying prices.
• The calculation assumes European options with matching strike and expiry and requires a specified dividend yield and discounting convention.
• The derivation does not account for transaction costs, funding spreads, or market quote imperfections.

Tags

# Calculate borrow/loan or repo rate


# Calculate borrow/loan or repo rate












I was given this question on interview and couldn't find an answer in time (it is a software developer job in a place that deals with options). Can someone explain how to do this or point me to a good source of material?

Given two European options - a call $C$ and a put $P$ - struck at $K$, expiring at time $T$, on an underlying asset priced at $S_t$ today, derive the formula for the implied asset financing rate, i.e., the asset borrow/loan or repo rate. Assume the discount rate is $r$ and the dividend yield on the asset is $q$, both annual continuously compounded.

My answer was: Interest rate = [(future value/present value) – 1] x year/number of days, that's what I was able to find online at the time.

## Answer by jmh (score 4, accepted)

https://quant.stackexchange.com/a/55520

I believe, they are testing two things here:

- That you know the Put-Call Parity (with dividends)

- That you can successfully rearrange an equation

The Put-Call Parity with continuously compounded dividends is:

$$ C-P=Se^{-qT}-Ke^{-rT}$$

The second part of the question is to rearrange the above for r.

Which gives:

$$ r = \frac{-1}{T}\ln\left( \frac{Se^{-qT}-C+P}{K}\right) $$

I hope that this helps.

Let me know if you would like me to break down the rearranging?

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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