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 Price 50 S, the price of the stock Strike Price 60 K, the strike of the option Time To Expiry 1 T, current time until expiration Volatility σ, 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 Rate r, the annualized risk-free interest rate, continuously compounded (the force of interest) Dividend Yield d, how sigmach a company pays out in dividends each year relative to its share price Outputs d1 -0.4135 N(d1) 0.3396 standard 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) 0.3663 standard normal probability density of d1 d2 -0.6635 N(d2) 0.2535 the probability of exercise (paying the strike price) Calls Puts Value 2.4301 9.4926 C, call option value; P, put option value Partial Derivatives Calls Puts Delta 0.3396 -0.6604 Δ, measures the rate of change of option value with respect to changes in the underlying asset's price Gamma 0.0293 Γ, measures the rate of change in the delta with respect to changes in the underlying price Vega 0.1831 V, measures sensitivity to volatility Theta -0.0036 -0.0140 θ, measures the sensitivity of the value of the derivative to the passage of time Rho 0.1421 -0.4186 ρ, measures sensitivity to the interest rate