Module bs_analytic 

Analytical Black-Scholes formulas for European options and Greeks

Mathematical Foundation

Under the Black-Scholes model, the underlying asset follows:

dS_t = r S_t dt + σ S_t dW_t

The risk-neutral pricing formula gives:

V(S,t) = e^(-r(T-t)) * E^Q[payoff(S_T) | S_t = S]

For European options, this has closed-form solutions involving the cumulative normal distribution function Φ(x).

Functions

bs_call_delta

Black-Scholes Delta (∂V/∂S) for European call

bs_call_gamma

Black-Scholes Gamma (∂²V/∂S²) for European call

bs_call_price

Black-Scholes European call option price

bs_call_rho

Black-Scholes Rho (∂V/∂r) for European call

bs_call_theta

Black-Scholes Theta (∂V/∂t) for European call

bs_call_vega

Black-Scholes Vega (∂V/∂σ) for European call

bs_put_price

Black-Scholes European put option price