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
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 < -0.6
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_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)
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)
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
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_jump_calculate_model_values(p0):
''' Calculates all model values given parameter vector p0. '''
lamb, mu, delta = p0
values = []
for row, option in options.iterrows():
T = (option['Maturity'] - option['Date']).days / 365.
B0T = B([kappa_r, theta_r, sigma_r, r0, T])
r = -math.log(B0T) / T
model_value = BCC_call_value(S0, option['Strike'], T, r, kappa_v,
theta_v, sigma_v, rho, v0, lamb, mu,
delta)
values.append(model_value)
return np.array(values)
def BCC_jump_calculate_model_values(p0):
''' Calculates all model values given parameter vector p0. '''
lamb, mu, delta = p0
values = []
for row, option in options.iterrows():
T = (option['Maturity'] - option['Date']).days / 365.
B0T = B([kappa_r, theta_r, sigma_r, r0, T])
r = -math.log(B0T) / T
model_value = BCC_call_value(S0, option['Strike'], T, r,
kappa_v, theta_v, sigma_v, rho, v0,
lamb, mu, delta)
values.append(model_value)
return np.array(values)
def BCC_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
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():
model_value = BCC_call_value(S0, option['Strike'], option['T'],
option['r'], kappa_v, theta_v, sigma_v,
rho, v0, lamb, mu, delta)
se.append((model_value - option['Call']) ** 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_error_function(p0):
''' Error function for parameter calibration in M76 Model via
Carr-Madan (1999) FFT approach.
Parameters
==========
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, local_opt, opt1
lamb, mu, delta = p0
if lamb < 0.0 or mu < -0.6 or mu > 0.0 or delta < 0.0:
return 5000.0
se = []
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)
se.append((model_value - option['Call']) ** 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
if local_opt:
penalty = np.sqrt(np.sum((p0 - opt1) ** 2)) * 1
return MSE + penalty
return MSE
def BCC_error_function(p0):
''' Error function for parameter calibration in M76 Model via
Carr-Madan (1999) FFT approach.
Parameters
==========
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, local_opt, opt1
lamb, mu, delta = p0
if lamb < 0.0 or mu < -0.6 or mu > 0.0 or delta < 0.0:
return 5000.0
se = []
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)
se.append((model_value - option['Call'])**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
if local_opt:
penalty = np.sqrt(np.sum((p0 - opt1)**2)) * 1
return MSE + penalty
return MSE
#
# 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 (Eonia 30.09.2014)
#
# Market Implied Volatilities
#
for row, option in options.iterrows():
call = call_option(S0, option['Strike'], option['Date'],
option['Maturity'], option['r'], 0.15)
options.loc[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.
#
# 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 (Eonia 30.09.2014)
#
# Market Implied Volatilities
#
for row, option in options.iterrows():
call = call_option(S0, option['Strike'], option['Date'],
option['Maturity'], option['r'], 0.15)
options.loc[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.