Choosing the Discount Rate for Black–Scholes Options
Summary
The document explains how to select the interest rate input in Black–Scholes pricing. In the basic model, the rate represents short-term risk-free funding over the option’s life; a continuously compounded government bill yield with a maturity near the option expiry is a common practical proxy. The discussion also mentions interbank borrowing or lending rates, with the applicable rate depending on a dealer’s funding position.
For collateralized trades, the relevant discount rate may instead be the interest earned on collateral, or a blended rate for partial collateralization. The stock’s forward value must also be consistent with financing and carry, which may involve repo rates or market-implied forwards. These are practical conventions, not a single universal prescription: the appropriate rate depends on the trade’s collateral and funding terms, and a sovereign yield is only a proxy for a truly risk-free rate.
Key ideas
- The Black–Scholes rate represents the cost or return of short-term safe funding over the option horizon.
- A nearby-maturity government bill yield is a common practical proxy.
- Collateralized options may require discounting at the collateral remuneration rate.
- Funding rates and stock forward carry should be consistent with the trade’s actual market setup.
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# Which risk-free interest rate to use in Black-Scholes equation
# Which risk-free interest rate to use in Black-Scholes equation
Sorry but i'm new in quantitative finance. According to BS derivation the risk-free interest rate is the rate to wich the rate of a particular investment tends when the risk tends to zero. Suppose i want to buy on option with fixed strike price and maturity, which rate i have to put into the equation? And why?
## Answer by vonjd (score 7)
https://quant.stackexchange.com/a/38542
> In theory, $r$ is a short-term safe interest rate, and it is constant through time though the theory does goes through with $\bar{r}$ (average $r$ from $t$ to $T$) in place or $r$. In practice, you take the continuously compounded yield on a T-bill of maturity closest to that of your option. Eurocurrency rates work too, especially for currency options. In theory, you should choose whether to use a LIBOR or LIBID rate depending upon whether the option dealer who delta hedges your trade is going to be borrowing money (at the LIBOR rate) or lending money (at the LIBID rate).
Source: Basic Black-Scholes: Option Pricing and Trading (2'nd edition) by Timothy Falcon Crack, p. 143.
## Answer by q.t.f. (score 3)
https://quant.stackexchange.com/a/38557
Most option trades are collateralized. In that case, the correct rate to use for discounting is the rate earned by the collateral, or a mix of the collateral rate and risk-free rate for partial collateralization. You still need to pay attention that the stock forward level is priced correctly, so use a stock repo rate or similar backed out from call/put parity in the options market or data from futures or forwards.
See http://www.math.columbia.edu/~fts/What%20Rate%20to%20use%20v1.pdf
## Answer by Ashkar (score 1)
https://quant.stackexchange.com/a/32322
If you are doing this for fun then use Treasury/LIBOR rates. Otherwise the 'risk-free' rate in BS is the rate at which you can borrow/lend cash. If you have a brokerage account the broker should pay you an interest on any cash in your account or charge you interest for lending you cash.
## Answer by John Doe (score -1)
https://quant.stackexchange.com/a/32243
Conventionally you use the interest rate of a sovereign with same maturity, that is considered the virtually risk-free asset.
So for a call on AAPL (T = 6m), you would use 6m rate from t-bills and annualize it.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.