# 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:

1. 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?
2. 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.

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.

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.

• By Fed Funds, you mean OIS swaps indexed to EFFR? To turn these into discount factors, I suppose for maturities > 1-year, the swaps need to be bootstrapped, right (I suppose maturities greater than 1y would not be single-period swaps)? PS: unrelated to my question: as of a few months ago, discounting has now switched to SOFR, rather than EFFR (i.e. at LCH and other clearing houses). Mar 18, 2021 at 9:44
• Yes, I mean ois linked to Fed Funds. You’re right, Clearing houses have switched to SOFR for cleared instruments. I was assuming the option in question was bilateral with a Fed funds based CSA, but if it is cleared then SOFR could be appropriate. Those two curves are pretty close.
– dm63
Mar 18, 2021 at 13:29
• Thank you. And just to make sure I am on the same page: for (say) three year option, the 3y OIS swap (assuming annual fixed coupons) would have to be bootsrtapped anyway, to derive a zero-coupon discounting curve, right? Mar 18, 2021 at 14:06
• Yes it would have to be bootstrapped to create discount factors
– dm63
Mar 19, 2021 at 3:21