# 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?

• If one of those answers were helpful it would be great if you could upvote and accept it - Thank you :-) – vonjd Mar 2 '18 at 7:38

## 4 Answers

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.

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.

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.

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.

• Thanks! If there are two or more sovereigns with same maturity but different rate which is the right one? – ab94 Feb 4 '17 at 11:31
• I would say that your example violates the no-arbitrage condition, which is a base assumption of BS model. Can you be more specific? Also I edited my answer as r must be expressed in annualized terms. – John Doe Feb 4 '17 at 11:46
• I was imagining something but actually it violates no arbitrage condition. Thank you again – ab94 Feb 4 '17 at 12:00
• Using the T-Bill rate has been quite standard for a long time (in fact it is what the old textbooks recommend). In recent years some financial institutions have begun to use the OIS rate of the applicable maturity, on the idea that it better approximates the funding rate of the financial institution. – Alex C Feb 4 '17 at 16:29
• Why does Treasury Bill make sense ? Theoretically speaking, isn't replicating portfolio approach require borrowing or lending? At the very least, the rate won't be the same because of bid/ask spread. More importantly, I cannot for sure borrow anything with the same rate as US Treasury. – nyan314sn Jul 1 '20 at 3:48