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complyorexplain
Topic Author
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Joined: November 9th, 2015, 8:59 am

### Solving for interest rate

Hi all. If I am given the price of a European call option, plus the value of the underlying asset, the strike price, the imputed volatility and the time to expiry, I can mechanically solve for the implied interest rate (for example, by goal seek). Is there any closed form solution for this value?

Marsden
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### Re: Solving for interest rate

I don't think there is any closed form solution.

Alan
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### Re: Solving for interest rate

But, if you also have the put price, it's trivial (put-call parity).

Collector
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### Re: Solving for interest rate

continuous time rates

$r\approx \ln\left(\frac{CallPutPrice}{0.4F\sigma\sqrt{T}}\right)/T$

for non ATMF to do in your head you need my head, sorry it is not for hire. Go figure!

Collector
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### Re: Solving for interest rate

put a minus in front of formula, my head calculus was a bit fast.

if option a bit OTM or ITM perhaps good seed value for your goal seek

but interest rate do not change with strike, so just use ATMF option and u have it for rest. For what problem, challenge used?

Collector
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### Re: Solving for interest rate

is it something u plan to use for interview question?

Never ever hire traders that not can back out the interest from the ATMF option on back of an envelop, send them to the competitors!

Alan
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### Re: Solving for interest rate

A good one is estimate a 3-month ATM call price if the annual vol is, let's say 30%. You have 10 seconds.

complyorexplain
Topic Author
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Joined: November 9th, 2015, 8:59 am

### Re: Solving for interest rate

I agree if I have the put price, then I have the forward, hence I have the interest rate.

But if I don't have the put price (or the forward price) how do I solve?

As for ATMF, how do I know what the ATM is? Doesn't that require knowing F?

Collector
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### Re: Solving for interest rate

"As for ATMF, how do I know what the ATM is? Doesn't that require knowing F?"

depends on market etc. options on what? Assume on stocks,

Assume you trade the option at the Saudi Exchange (assume European style )

Now implied rate is zero is it not?  Islamic Banking rules (?)

So good interview question is: In what market can even quants back out the implied interest rate from the options in one a fraction of a second in their head for any strike price.

complyorexplain
Topic Author
Posts: 184
Joined: November 9th, 2015, 8:59 am

### Re: Solving for interest rate

The OP says the value (I meant price) of the underlying asset. I.e. S, not F. Thus, given the price of the call option, and S, K and T, how do we get a closed form value for r, the implied interest rate? I don't think there is a solution, although we can use numerical methods.

Clearly if we are told what the forward is, or the put price, or that it is ATM, then we can solve closed form. Otherwise not. Perhaps I am wrong.

Alan
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### Re: Solving for interest rate

Just out of curiosity, what is the actual situation here where you have a call price but not a put?

complyorexplain
Topic Author
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Joined: November 9th, 2015, 8:59 am

### Re: Solving for interest rate

Just out of curiosity, what is the actual situation here where you have a call price but not a put?
Purely theoretical!