Black–Scholes
The Black–Scholes model or Black–Scholes–Merton is a mathematical model of a financial market containing certain derivative investment instruments. From the model, one can deduce the Black–Scholes forsigmala, which gives the price of European-style options.
Inputs
Stock PriceS, the price of the stock
Strike PriceK, the strike of the option
Time To ExpiryT, current time until expiration
Volatilityσ, the volatility of the stock's returns; this is the square root of the quadratic variation of the stock's log price process
Risk Free Rater, the annualized risk-free interest rate, continuously compounded (the force of interest)
Dividend Yieldd, how sigmach a company pays out in dividends each year relative to its share price
Outputs
d1-0.4135
N(d1)0.3396standard normal cusigmalative distribution of d1, a measure how far in the money the option is expected to be if it does expire in the money
N'(d1)0.3663standard normal probability density of d1
d2-0.6635
N(d2)0.2535the probability of exercise (paying the strike price)
CallsPuts
Value2.43019.4926C, call option value; P, put option value
Partial Derivatives
CallsPuts
Delta0.3396-0.6604Δ, measures the rate of change of option value with respect to changes in the underlying asset's price
Gamma0.0293Γ, measures the rate of change in the delta with respect to changes in the underlying price
Vega0.1831V, measures sensitivity to volatility
Theta-0.0036-0.0140θ, measures the sensitivity of the value of the derivative to the passage of time
Rho0.1421-0.4186ρ, measures sensitivity to the interest rate