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# cython: language_level=3, cdivision=True
from libc.math cimport log, sqrt, erf
from scipy.stats import norm
import cython
cpdef double cnd_erf(double d):
""" 2 * Phi where Phi is the cdf of a Normal """
cdef double RSQRT2 = 0.7071067811865475
return 1 + erf(RSQRT2 * d)
cpdef double black(double F, double K, double T, double sigma, bint payer=True):
cdef:
double x = log(F / K)
double sigmaT = sigma * sqrt(T)
double d1 = (x + 0.5 * sigmaT * sigmaT) / sigmaT
double d2 = (x - 0.5 * sigmaT * sigmaT) / sigmaT
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))
cpdef double Nx(double F, double K, double sigma, double T):
return cnd_erf((log(F / K) - sigma ** 2 * T / 2) / (sigma * sqrt(T))) / 2
cpdef double bachelier(double F, double K, double T, double sigma):
""" Bachelier formula for normal dynamics
need to multiply by discount factor
"""
cdef double d1 = (F - K) / (sigma * sqrt(T))
return 0.5 * (F - K) * cnd_erf(d1) + sigma * sqrt(T) * norm.pdf(d1)
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