Question

An exchange rate is currently $2.00. The volatility of the exchange rate is 20% and interest rates in the two countries are the same. Using the Black-Scholes model, estimate the probability that the exchange rate in one year will be between $1.70 and $2.30.

Answer 1

Black-Scholes Model	The Black–Scholes model is a mathematical model simulating the dynamics of a financial market containing derivative financial instruments.In the model, N(d1​) is roughly looking for the probability that the future price will be above the strike price on the expiration date. Similarly, N(d2​) is the percentage of probabilities that the option will expire in the money. Therefore, in order to calculate the probabilty of exchange rate to be between $ 1.7 and $2.30, we can calculate N(d1) for both the target exchange rates. Probability of exchange rate between $1.7 to $2.3 is 54% as per the below calculations.
Current Exchange Rate	2.00	2.00
Target Exchange Rate	1.70	2.30
Rate %	0%	0%
Sigma	20%	20%
Time	1	1
Formula Used in Excel
d_1	0.9126	-0.5988	(LN(Current Exchange Rate/Target Exchange Rate)+(Rate+(Sigma^2)/2)*Time)/(Sigma*SQRT(Time))
d_2	0.7126	-0.7988	(LN(Current Exchange Rate/Target Exchange Rate)+(Rate-(Sigma^2)/2)*Time)/(Sigma*SQRT(Time))
Nd_1	0.8193	0.2746	NORM.S.DIST(d_1,TRUE)
Nd_2	0.7620	0.2122	NORM.S.DIST(d_2,TRUE)
Probability Above $1.70	82%
Probability below $2.30		73%	1-Nd_1
Not above $2.30		27%	1-73%
Between $1.70 and $2.30	54%	82% - 27%