Black-Scholes-Merton Option Pricing, Analytical Greeks & Newton-Raphson Implied Volatility Engine
For an at-the-money European call option with spot S = $100.00, strike K = $100.00, time to expiry T = 0.25 years (91.25 days), risk-free rate r = 4.50%, continuous dividend yield q = 1.50%, and implied volatility σ = 20.00%, the exact Black-Scholes-Merton theoretical premium is $4.3584 with Delta = +0.5299, Gamma = 0.0396, annualized Vega = $0.1978 per 1% vol, and daily Theta = -$0.0241.