Choosing and Constructing Risk-Free Curves for Option Pricing
Summary
The document explains how to choose a discount curve for Black–Scholes option valuation and how to turn quoted market rates into continuously compounded rates for different maturities. It describes building a money-market curve from short-term interbank rates, interest-rate futures, and swaps, first converting quoted rates to discount factors and then deriving tenor-specific rates. Interpolation can estimate rates between quoted maturities.
It also contrasts this approach with using Treasury yields. One response recommends Fed Funds discounting and says Treasury rates may be close for shorter expirations but can diverge at longer maturities as market supply and demand change. The discussion is an introductory outline rather than a complete curve-building treatment: it does not address detailed conventions, bootstrapping, or how rate benchmarks have evolved, so the appropriate curve depends on the instrument and market context.
Key ideas
- Option valuation requires a maturity-matched discount curve rather than a single rate for all expirations.
- A money-market curve can combine short-term rates, interest-rate futures, and swaps across maturities.
- Quoted rates can be converted to discount factors and then expressed as continuously compounded rates.
- Treasury yields may approximate other funding rates at shorter maturities but can diverge at longer terms.
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Full text
# Risk-free interest rate for option pricing from treasury yield curve rates
# Risk-free interest rate for option pricing from treasury yield curve rates
I am experimenting with an implementation of the Black-Scholes valuation for call options, and ran into the following questions:
- Black-Scholes pricing requires a risk-free interest rate. What is 'best practice', i.e. if I were writing a finance paper, where exactly would I get these rates over different time periods? Is this LIBOR/SOFR?
- Suppose I only had access to the Daily Treasury Yield Curve rates. Can I extract a reasonable, if crude, risk-free interest rate for options with different maturities? Any help, or references, would be appreciated.
## Answer by Carson McKee (score 0, accepted)
https://quant.stackexchange.com/a/61187
My understanding is that technically, B-S uses the 'short rate' which is the instantaneous rate of borrow/lending for term T, denoted $r_t(T)$. I.e. at time $t=0$, if you invest £1 risk-free for term T, at T your investment will be worth $1\times e^{r_{t=0}(T)*T}$. Now, to obtain values for $r_{t}(T)$ you need to construct a yield curve for varying T. Note that $r_t(T)$ isn't directly observable, so we have to calculate discount factors first then convert these to $r_t(T)$.
The first step is to choose which instruments to use in order to do this, addressing your first point. When large banks engage in buying/selling options, shares etc. they will fund this by lending/depositing from other banks at the LIBOR (soon to be SOFR/SONIA) rate, not the treasury rate. Therefore, you should construct what's called the 'money market' curve, i.e. use LIBOR rates for short term (t<3m), interest rate futures for the medium term (3m<t<1y) and interest rate swaps for the long end (1y up to 20/30 years).
So let's say you want the short rate, $r_t(T)$ for term 3 months (T=0.25). First, look up the Libor rate for 3 months, denoted $L_{3m}$. Now if you invest, £1 at $L_{3m}$, in 3 months you will get back $1 + L_{3m}*0.25$. So the discount factor for this period is, $$ \delta_{3m} = \frac{1}{1 + L_{3m}*0.25} $$ so now all we have to do is convert this to the short rate $r_t(T)$. This is simply, $$ r_t(T) = \frac{-1}{T} * ln(\delta_{3m}) $$ (assume now is t=0)
Now by repeating this for the various other LIBOR tenors and doing similar things for the Futures/Swaps you will get a set of values for $r_t(T)$ which you can then interpolate to get an estimate for values in between. At this point you have your yield curve and you can just pick off the $r_t(T)$ for the tenor you are pricing your option for. This is a rather naive approach to constructing the money market curve in general, however, yield curve construction is a whole field in itself and for your purposes this should be more than enough.
*Note LIBOR/Futures/Swap rates are published daily for various currencies on the CME Group website.
## Answer by dm63 (score 2)
https://quant.stackexchange.com/a/61818
Best practice nowadays is to use Fed Funds rates to discount. If you only have Treasury rates , this will be quite close for 0-3 year expirations, since those Treasuries trade quite close to FedFunds. However, the Treasury rates diverge more from Fed Funds at longer maturities. Eg in 10yrs, where Treasuries are currently about 18bp higher than Fed Funds. Of course these differences can vary a lot based on supply and demand in the market place.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.