Black and Scholes Formula

This tool calculates European options Prices (Call and Put) including on stocks with dividend. It also calculates the inverse black and scholes formula to estimate implied volatilities.

Input x in 'Option price' field to price the option.
Input x in 'Volatility' field to estimate implied volatility.

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We denote,

`sigma` : Volatility
S : Spot (or underlying) price
K : Strike price
T : Time to maturity (in years)
r : Risk free rate
d : Dividend yield

Then, Black and Scholes equations are written as follows,

Call price

`C = S*N(d_1)- K*e^(-r*T)N(d_2)`

Put Price

`C = -S*N(-d_1)+ K*e^(-r*T)N(-d_2)`

with,

N(x) is Cumulative density funtion of the standard normal distribution (normal distribution of mean and standard deviation equal to 0 and 1 respectively).

`N(x) = 1/(sqrt(2pi))*\int_-oo^xe^(-1/2*t^2)\ dt`

`d_1 = 1/(sigma*sqrt(T))[ln(S/K)+(r-d+1/2*sigma^2)*T]`

`d_2 = d_1 - sigma*sqrt(T)`

Black and Scholes Formula

See also