def alpha_as_beta(beta, r=R, sigma2=SIGMA2, check_tol=1e-8):
'''alpha as beta so that hedge fund manager is indifferent'''
assert beta * T <= BETA_UPPER_BOUND
assert BETA0 == 0, "Formula only applies to the case where BETA0 is 0!"
_f_func = common._f_func_generator(mean=(r - 0.5 * sigma2) * T,
sigma2=sigma2 * T)
_g_func = common._g_func_generator(mean=(r - 0.5 * sigma2) * T,
sigma2=sigma2 * T)
deltat = DELTA * T
numerator = term2(BETA0, r=r, sigma2=sigma2) - \
term2(beta, r=r, sigma2=sigma2) + \
term3(BETA0, r=r, sigma2=sigma2) - \
term3(beta, r=r, sigma2=sigma2)
denominator = T * (_f_func(1 + deltat) - deltat * _g_func(1 + deltat))
alpha = ALPHA0 + numerator / denominator
assert alpha * T <= ALPHA_UPPER_BOUND
assert abs(option_px(alpha, beta, r=r, sigma2=sigma2)- \
option_px(ALPHA0, BETA0, r=r, sigma2=sigma2)) < check_tol
return alpha
def term3(beta, r=R, sigma2=SIGMA2):
'''Term 3'''
betat = beta * T
deltat = DELTA * T
_g_func = common._g_func_generator(mean=(r - 0.5 * sigma2) * T,
sigma2=sigma2 * T)
ksi = 1 - betat + deltat
return (deltat - betat) * (1 - _g_func(ksi))
def sterm2(beta, mu=MU, sigma2=SIGMA2):
'''Term 2 for second-order moment'''
betat = beta * T
deltat = DELTA * T
_g_func = common._g_func_generator(mean=(mu-0.5*sigma2)*T, sigma2=sigma2*T)
term2a = _g_func(1-betat+deltat)
term2b = _g_func(1+deltat)
return term2a - term2b
def eterm2(beta, mu=MU, sigma2=SIGMA2):
'''Term 2 for expectation'''
betat = beta * T
deltat = DELTA * T
_g_func = common._g_func_generator(mean=(mu-0.5*sigma2)*T, sigma2=sigma2*T)
term2a = _g_func(1-betat+deltat)
term2b = _g_func(1+deltat)
return term2a - term2b
def eterm1(alpha, mu=MU, sigma2=SIGMA2):
'''Term 1 for expectation'''
deltat = DELTA * T
alphat = alpha * T
_f_func = common._f_func_generator(mean=(mu-0.5*sigma2)*T, sigma2=sigma2*T)
_g_func = common._g_func_generator(mean=(mu-0.5*sigma2)*T, sigma2=sigma2*T)
term1a = (1-alphat)*_f_func(1+deltat)
term1b = deltat*(1-alphat)*_g_func(1+deltat)
return term1a - term1b
def eterm3(beta, mu=MU, sigma2=SIGMA2):
'''Term 3 for expectation'''
betat = beta * T
deltat = DELTA * T
_f_func = common._f_func_generator(mean=(mu-0.5*sigma2)*T, sigma2=sigma2*T)
_g_func = common._g_func_generator(mean=(mu-0.5*sigma2)*T, sigma2=sigma2*T)
ksi = 1-betat+deltat
term3a = np.exp(mu*T) - _f_func(ksi)
term3b = (betat-deltat)*(1-_g_func(ksi))
return term3a + term3b
def term2(beta, r=R, sigma2=SIGMA2):
'''Term 2'''
betat = beta * T
deltat = DELTA * T
_f_func = common._f_func_generator(mean=(r - 0.5 * sigma2) * T,
sigma2=sigma2 * T)
_g_func = common._g_func_generator(mean=(r - 0.5 * sigma2) * T,
sigma2=sigma2 * T)
term2a = _f_func(1 - betat + deltat) - _g_func(1 - betat + deltat)
term2b = _f_func(1 + deltat) - _g_func(1 + deltat)
return term2a - term2b
def term1(alpha, r=R, sigma2=SIGMA2):
'''Term 1'''
deltat = DELTA * T
alphat = alpha * T
_f_func = common._f_func_generator(mean=(r - 0.5 * sigma2) * T,
sigma2=sigma2 * T)
_g_func = common._g_func_generator(mean=(r - 0.5 * sigma2) * T,
sigma2=sigma2 * T)
term1a = alphat * (_f_func(1 + deltat) - deltat * _g_func(1 + deltat))
term1b = deltat * _g_func(1 + deltat)
return term1a + term1b
def sterm1(alpha, mu=MU, sigma2=SIGMA2):
'''Term 1 for second-order moment'''
deltat = DELTA * T
alphat = alpha * T
_f_func = common._f_func_generator(mean=(mu-0.5*sigma2)*T, sigma2=sigma2*T)
_g_func = common._g_func_generator(mean=(mu-0.5*sigma2)*T, sigma2=sigma2*T)
term1a = (1-alphat)*(1-alphat)*_h_func(1+deltat)
term1b = 2*deltat*(1-alphat)*(1-alphat)*_f_func(1+deltat)
term1c = deltat*deltat*(1-alphat)*(1-alphat)*_g_func(1+deltat)
return term1a - term1b + term1c
def sterm3(beta, mu=MU, sigma2=SIGMA2):
'''Term 3 for second-order moment'''
betat = beta * T
deltat = DELTA * T
_f_func = common._f_func_generator(mean=(mu-0.5*sigma2)*T, sigma2=sigma2*T)
_g_func = common._g_func_generator(mean=(mu-0.5*sigma2)*T, sigma2=sigma2*T)
_h_func = common._h_func_generator(mean=(mu-0.5*sigma2)*T, sigma2=sigma2*T)
ksi = 1-betat+deltat
term3a = np.exp(2*mu*T+SIGMA2*T) - _h_func(ksi)
term3b = 2*(betat-deltat)*(np.exp(mu*T)-_f_func(ksi))
term3c = (betat-deltat) * (betat-deltat) * (1-_g_func(ksi))
return term3a + term3b + term3c
"""
Created on Wed Nov 22 09:50:02 2017
@author: Yixian
"""
import numpy as np
import common
from common import (DELTA, T, X_0)
MU = 0.1 #10%
SIGMA2 = 0.4*0.4 # 40%
_f_func = common._f_func_generator(mean=(MU-0.5*SIGMA2)*T, sigma2=SIGMA2*T)
_g_func = common._g_func_generator(mean=(MU-0.5*SIGMA2)*T, sigma2=SIGMA2*T)
_h_func = common._h_func_generator(mean=(MU-0.5*SIGMA2)*T, sigma2=SIGMA2*T)
def eterm1(alpha, mu=MU, sigma2=SIGMA2):
'''Term 1 for expectation'''
deltat = DELTA * T
alphat = alpha * T
_f_func = common._f_func_generator(mean=(mu-0.5*sigma2)*T, sigma2=sigma2*T)
_g_func = common._g_func_generator(mean=(mu-0.5*sigma2)*T, sigma2=sigma2*T)
term1a = (1-alphat)*_f_func(1+deltat)
term1b = deltat*(1-alphat)*_g_func(1+deltat)
return term1a - term1b