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from math import log, sqrt, erf
from numba import jit, float64, boolean
from scipy.stats import norm
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):
""" 2 * Phi where Phi is the cdf of a Normal """
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))
@jit(float64(float64, float64, float64, float64), cache=True, nopython=True)
def Nx(F, K, sigma, T):
return cnd_erf((log(F / K) - sigma ** 2 * T / 2) / (sigma * sqrt(T))) / 2
def bachelier(F, K, T, sigma):
""" Bachelier formula for normal dynamics
need to multiply by discount factor
"""
d1 = (F - K) / (sigma * sqrt(T))
return 0.5 * (F - K) * cnd_erf(d1) + sigma * sqrt(T) * norm.pdf(d1)
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