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I understand this is very basic question but I still scramble to determine what would be right risk free rate to price a simple European call option using Black-scholes formula, with maturity is 5 years. This option is written on a USD based stock?

I have below choices

  1. US Treasury rate with maturity 5 years (because, this coincides with option maturity)
  2. US Treasury rate with maturity 3 years (because, people told me that this would be most liquid)
  3. Overnight Fed fund rate

What will be the most accurate choice for my case?

Also, should I convert the reported rate to Continuously compounded rate? Because, if I look into the Black scholes formula it considers continuously compounded rate.

Any pointer will be very helpful.

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Usually,government bond yields are not used when pricing derivates. Bloomberg for example does not even offer govy curves as a choice for the interest rate in all of their derivatives pricers (OVME, OVML, SWPM, DLIB etc.)

The RFR (for risk free rate) swap rates (SOFR for USD, ESTR for EUR for example) are used, and you have a choice for other swap curves like Euribor and Libor (legacy reasons) as well as other OIS swaps like the Fed funds swaps. It's also standard for clearing houses and exchanges like LCH and CME to use these RFR rates for discounting and Price Alignment Interest (PAI) calculations, which is the interest rate paid on the collateral that is held.

There is work on why treasuries are not a good proxy aside from the aforementioned point (usually based on convenience yield arguments). See for example Decomposing Swap Spreads by Feldhütter et al..

In terms of the tenor, you should use the exact term that coincides with the option term to maturity.

You should use continuous compounding. This answer for FX (almost identical if you swap one interest rate for dividends) shows how this works.

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