# 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.

I believe, they are testing two things here:

1. That you know the Put-Call Parity (with dividends)
2. 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?

• They also mentioned the rate $r$. Maybe the original interviewer distinguished between a treasury (i.e. unsecured) rate $r$, and a repo rate $h$. But I agree with the gist of your answer. In an interview I think something along these lines should probably be enough. – Daneel Olivaw Jul 8 at 17:42