Choosing the Discount Curve and Rate for Black–Scholes Options
Summary
The document addresses which interest rate to use when pricing a long-dated European option on a USD stock with Black–Scholes. It recommends using a risk-free rate swap curve, such as the USD SOFR curve, rather than choosing a Treasury yield or the overnight Fed funds rate by default. It also says to select the curve tenor that matches the option’s time to maturity and to express the rate with continuous compounding for the formula.
The rationale is that derivatives desks, clearing houses, and exchanges commonly use risk-free-rate swap curves for discounting and collateral interest calculations. The answer notes that government yields can differ from suitable discount rates, including because of convenience-yield effects, and points to research on swap spreads. These are general market conventions, not a worked valuation or proof that one curve is appropriate in every contract. The precise curve should reflect the instrument’s collateral and market conventions.
Key ideas
- Risk-free-rate swap curves such as SOFR are commonly used to discount derivatives.
- Match the rate curve tenor to the option’s time to maturity.
- Black–Scholes uses a continuously compounded rate, so quoted rates need consistent conversion.
- Government bond yields may not be suitable proxies for derivative discount rates.
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# Right risk free rate to price an Option using BS formula # Right risk free rate to price an Option using BS formula 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 - US Treasury rate with maturity 5 years (because, this coincides with option maturity) - US Treasury rate with maturity 3 years (because, people told me that this would be most liquid) - 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. ## Answer by AKdemy (score 4, accepted) https://quant.stackexchange.com/a/74098 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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