# How to solve for the implied stock lending rate given equity options prices?

When market makers price options on hard-to-borrow equities, they include the cost to borrow the underlying equity that their broker is going to charge them to sell the security short to hedge. I'm trying to back-out this cost. I'm guessing it is similar to implied volatility but I'm solving for the interest rate. Can anyone point me in the right direction?

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Sorry for not being clear. See my edits. –  unclepaul84 Sep 2 '11 at 14:00

The cost of the hedge does not appear directly in the price for any one option, but rather will appear as an apparent violation of put-call parity. However, due to differing demand for puts and calls, this merely widens the arbitrage bounds ordinarily set by put-call parity, but does not imply a single implied borrow rate. In other words, the borrow rate is an input to establishing the put-call parity bounds. One can reverse the relationship to obtain a lower limit on the borrow rate, but an upper limit cannot be inferred.

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As 'sheegaon' suggested, you can solve for an implied interest rate -- which is not necessarily the cost of borrowing the underlying stock -- using put-call parity.

As you probably know, an implied volatility algorithm increases and decreases its implied volatility guess until the theoretical price and market prices of an option converge. Similarly, to imply options' interest rate and implied volatility, take a put and call pair with the same strike and expiration, and increase or decrease their interest rate and implied volatility inputs until put-call parity holds and the put and call theoretical and market prices converge. In building such algorithms, I take advantage of the property that puts have positive and calls have negative sensitivity to interest rates, while both calls and puts have positive sensitivity to implied volatility.

Note that put-call pairs of different strikes or expirations often imply different rates.

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Please be aware when you back out implied interest rate for stocks with uncertain dividend payout.

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