# How to use the Black-Scholes formula with LIBOR rates?

I want to price an FX option using the Black-Scholes model, but I don't know the risk free rate, nor the volatility. I only know the LIBOR rates, the strike, and that the expiration day is 87 days from today. I also know the historical values of the exchange rate.

I am not sure how to use the LIBOR rate and how to calculate the volatility. Do I use the 3 months LIBOR as a risk free rate? Do I have to convert the LIBOR to countinously compounded rate?

You simply required 2 things: 1) Risk free rate, and 2) Standard Deviation.

For the interest rate you can use LIBOR of nearest maturity. Convert your LIBOR rate into continuous compound rate by taking log. Additional: VIX also uses LIBOR as an proxy for risk free interest rate and they also select LIBOR of nearest maturity of option contract.

Standard deviation can be easily computed from past historical data. You can also use GARCH model to forecast volatility for next 87 days and then take it average.

• Thank you. One question: time to maturity should be in years. Do I divide 87 with 365 or with the number of trading days: 252? Feb 20 '16 at 15:04
• @Lanza always use no. of trading days in a year (also make adjustment for holidays). Feb 20 '16 at 16:38
• @Neeraj Actually it depends on how you count the numerator. If it's actual number of days between now and expiry, you should divide by 365 ("calendar vol"). If it's number of biz days between now and expiry, then divide it by the # of biz days in a year (usually 252) ("business vol"). More advanced models would overweight event days ("event vol"). Most data vendors use Actual/365 (e.g., Bloomberg). Feb 20 '16 at 19:15
• @haginile If I use trading days, that is only ~40 days. Do I still use the continous 6 months Libor? Feb 21 '16 at 8:55
• Hi @Neeraj It's really just a "reporting" convention. You pick one and stick w/ it. The shop I worked at used to do biz/251.3 (until we replaced it with event vol). It was just a number a trader picked and we always used it for consistency. Feb 21 '16 at 19:36

Correct me if I am mistaken but since you are trying to price a forex option, wouldn't it be more appropriate to use the Garman and Kohlhagen extended Black-Scholes model since it better copes with the presence of the two interest rates associated with each currency respectively?

"Also, LIBOR rates were considered a useful measure of the risk free rate (the rate taken closest to option expiration) due to its proximity to the overnight indexed swap (OIS) rates. Since 2007 however (and as a result of the financial crisis), the LIBOR-OIS spread has spiked and become unstable . John Hull has a very popular paper on this topic." http://www-2.rotman.utoronto.ca/~hull/DownloadablePublications/LIBORvsOIS.pdf

John Hull said: "Most derivatives dealers now use interest rates based on overnight indexed swap (OIS) rates rather than LIBOR when valuing collateralized derivatives. For non-collateralized transactions, most dealers continue to use LIBOR rates for valuation."

Most of this information was taken from a thread with the link below: Risk Free Rate vs LIBOR