#Calibration of BCC97 Model to Implied Volatilities
# Calibration of Bakshi, Cao and Chen (1997)
# Stoch Vol Jump Model to EURO STOXX Option Quotes
# Data Source: www.eurexchange.com
# via Numerical Integration
#
import sys
sys.path.append('09_gmm')
import math
import numpy as np
np.set_printoptions(suppress=True,
formatter={'all': lambda x: '%5.3f' % x})
import pandas as pd
from scipy.optimize import brute, fmin, minimize
import matplotlib as mpl
mpl.rcParams['font.family'] = 'serif'
from BSM_imp_vol import call_option
from BCC_option_valuation import BCC_call_value
from CIR_calibration import CIR_calibration, r_list
from CIR_zcb_valuation import B
from H93_calibration import options
#
# Calibrate Short Rate Model
#
kappa_r, theta_r, sigma_r = CIR_calibration()
#
# Market Data from www.eurexchange.com
# as of 30. September 2014
#
S0 = 3225.93 # EURO STOXX 50 level 30.09.2014
r0 = r_list[0] # initial short rate
#
# Market Implied Volatilities
#
for row, option in options.iterrows():
call = call_option(S0, option['Strike'], option['Date'],
option['Maturity'], option['r'], 0.15)
options.ix[row, 'Market_IV'] = call.imp_vol(option['Call'], 0.15)
#
# Calibration Functions
#
i = 0
min_MSE = 5000.0
def BCC_iv_error_function(p0):
''' Error function for parameter calibration in BCC97 model via
Lewis (2001) Fourier approach.
Parameters
==========
kappa_v: float
mean-reversion factor
theta_v: float
long-run mean of variance
sigma_v: float
volatility of variance
rho: float
correlation between variance and stock/index level
www.it-ebooks.info
260 DERIVATIVES ANALYTICS WITH PYTHON
v0: float
initial, instantaneous variance
lamb: float
jump intensity
mu: float
expected jump size
delta: float
standard deviation of jump
Returns
=======
MSE: float
mean squared error
'''
global i, min_MSE
kappa_v, theta_v, sigma_v, rho, v0, lamb, mu, delta = p0
if kappa_v < 0.0 or theta_v < 0.005 or sigma_v < 0.0 or \
rho < -1.0 or rho > 1.0 or v0 < 0.0 or lamb < 0.0 or \
mu < -.6 or mu > 0.0 or delta < 0.0:
return 5000.0
if 2 * kappa_v * theta_v < sigma_v ** 2:
return 5000.0
se = []
for row, option in options.iterrows():
call = call_option(S0, option['Strike'], option['Date'],
option['Maturity'], option['r'],
option['Market_IV'])
model_value = BCC_call_value(S0, option['Strike'], option['T'],
option['r'], kappa_v, theta_v, sigma_v, rho, v0,
lamb, mu, delta)
model_iv = call.imp_vol(model_value, 0.15)
se.append(((model_iv - option['Market_IV']) * call.vega()) ** 2)
MSE = sum(se) / len(se)
min_MSE = min(min_MSE, MSE)
if i % 25 == 0:
print '%4d |' % i, np.array(p0), '| %7.3f | %7.3f' % (MSE, min_MSE)
i += 1
return MSE
def BCC_iv_calibration_full():
''' Calibrates complete BCC97 model to market implied volatilities. '''
p0 = np.load('11_cal/opt_full.npy')
# local, convex minimization
opt = fmin(BCC_iv_error_function, p0,
xtol=0.000001, ftol=0.000001,
maxiter=450, maxfun=650)
np.save('11_cal/opt_iv', opt)
return opt
def BCC_calculate_model_values(p0):
''' Calculates all model values given parameter vector p0. '''
kappa_v, theta_v, sigma_v, rho, v0, lamb, mu, delta = p0
values = []
for row, option in options.iterrows():
model_value = BCC_call_value(S0, option['Strike'], option['T'],
option['r'], kappa_v, theta_v, sigma_v, rho, v0,
lamb, mu, delta)
values.append(model_value)
return np.array(values)
