from math import log, sqrt, erf
from numba import jit, float64, boolean
import math

def d1(F, K, sigma, T):
    return (log(F / K) + sigma**2 * T / 2) / (sigma * math.sqrt(T))

def d2(F, K, sigma, T):
    return d1(F, K, sigma, T) - sigma * math.sqrt(T)

@jit(cache=True,nopython=True)
def d12(F, K, sigma, T):
    sigmaT = sigma * sqrt(T)
    d1 = log(F / K) / sigmaT
    d2 = d1
    d1 += 0.5 * sigmaT
    d2 -= 0.5 * sigmaT
    return d1, d2

@jit(float64(float64),cache=True,nopython=True)
def cnd_erf(d):
    RSQRT2 = 0.7071067811865475
    return 1 + erf(RSQRT2 * d)

@jit(float64(float64,float64,float64,float64,boolean),cache=True,nopython=True)
def black(F, K, T, sigma, payer=True):
    d1, d2 = d12(F, K, sigma, T)
    if payer:
        return 0.5 * (F * cnd_erf(d1) - K * cnd_erf(d2))
    else:
       return 0.5 * (K * cnd_erf(-d2) - F * cnd_erf(-d1))