def M76_error_function_FFT(p0):
''' Error Function for parameter calibration in M76 Model via Carr-Madan (1999) FFT approach.
Parameters
==========
sigma: float
volatility factor in diffusion term
lamb: float
jump intensity
mu: float
expected jump size
delta: float
standard deviation of jump
Returns
=======
RMSE: float
root mean squared error
'''
global i, min_RMSE
sigma, lamb, mu, delta = p0
if sigma < 0.0 or delta < 0.0 or lamb < 0.0:
return 500.0
se = []
for row, option in options.iterrows():
T = (option['Maturity'] - option['Date']).days / 365.
model_value = M76_value_call_FFT(S0, option['Strike'], T, r, sigma,
lamb, mu, delta)
se.append((model_value - option['Call'])**2)
RMSE = math.sqrt(sum(se) / len(se))
min_RMSE = min(min_RMSE, RMSE)
if i % 50 == 0:
print('%4d |' % i, np.array(p0), '| %7.3f | %7.3f' % (RMSE, min_RMSE))
i += 1
return RMSE
def generate_plot(opt, options):
#
# Calculating Model Prices
#
sigma, lamb, mu, delta = opt
options['Model'] = 0.0
for row, option in options.iterrows():
T = (option['Maturity'] - option['Date']).days / 365.
options.loc[row,
'Model'] = M76_value_call_FFT(S0, option['Strike'], T, r,
sigma, lamb, mu, delta)
#
# Plotting
#
options = options.set_index('Strike')
fig, ax = plt.subplots(2, sharex=True, figsize=(8, 7))
options[['Call', 'Model']].plot(style=['b-', 'ro'],
title='%s' % str(option['Maturity'])[:10],
ax=ax[0])
ax[0].set_ylabel('option values')
ax[0].grid(True)
xv = options.index.values
ax[1] = plt.bar(xv - 5 / 2., options['Model'] - options['Call'], width=5)
plt.ylabel('difference')
plt.xlim(min(xv) - 10, max(xv) + 10)
plt.tight_layout()
plt.grid(True)
def generate_plot(opt, options):
#
# Calculating Model Prices
#
sigma, lamb, mu, delta = opt
options['Model'] = 0.0
for row, option in options.iterrows():
T = (option['Maturity'] - option['Date']).days / 365.
options.loc[row, 'Model'] = M76_value_call_FFT(S0, option['Strike'],
T, r, sigma, lamb, mu, delta)
#
# Plotting
#
mats = sorted(set(options['Maturity']))
options = options.set_index('Strike')
for i, mat in enumerate(mats):
options[options['Maturity'] == mat][['Call', 'Model']].\
plot(style=['b-', 'ro'], title='%s' % str(mat)[:10],
grid=True)
plt.ylabel('option value')
plt.savefig('../images/08_m76/M76_calibration_3_%s.pdf' % i)
if __name__ == '__main__':
# Single Valuation
S = M76_generate_paths(S0, T, r, sigma, lamb, mu, delta, M, I)
print("Value of Call Option %8.3f" % M76_value_call_MCS(S0))
# Value Comparisons
strikes = np.arange(3000, 3601, 50)
values = np.zeros((3, len(strikes)), dtype=np.float)
z = 0
for k in strikes:
print("CALL STRIKE %10.3f" % k)
print("----------------------------")
values[0, z] = M76_value_call_INT(S0, k, T, r, sigma, lamb, mu, delta)
print("Call Value by Int %10.3f" % values[0, z])
values[1, z] = M76_value_call_FFT(S0, k, T, r, sigma, lamb, mu, delta)
print("Call Value by FFT %10.3f" % values[1, z])
print("Difference FFT/Int%10.3f" % (values[1, z] - values[0, z]))
values[2, z] = M76_value_call_MCS(k)
print("Call Value by MCS %10.3f" % values[2, z])
print("Difference MCS/Int%10.3f" % (values[2, z] - values[0, z]))
print("----------------------------")
z = z + 1
results = pd.DataFrame(values.T,
index=strikes,
columns=['INT', 'FFT', 'MCS'])
results.index.name = 'Strike'
