def heston_calibration(df_option, ival=None):
"""
calibrate heston model
"""
# extract rates and div yields from the data set
df_tmp = DataFrame.filter(df_option, items=["dtExpiry", "iRate", "iDiv"])
grouped = df_tmp.groupby("dtExpiry")
df_rates = grouped.agg(lambda x: x[0])
dtTrade = df_option["dtTrade"][0]
# back out the spot from any forward
iRate = df_option["iRate"][0]
iDiv = df_option["iDiv"][0]
TTM = df_option["TTM"][0]
Fwd = df_option["Fwd"][0]
spot = SimpleQuote(Fwd * np.exp(-(iRate - iDiv) * TTM))
print("Spot: %f risk-free rate: %f div. yield: %f" % (spot.value, iRate, iDiv))
# build array of option helpers
hh = heston_helpers(spot, df_option, dtTrade, df_rates)
options = hh["options"]
spot = hh["spot"]
risk_free_ts = dfToZeroCurve(df_rates["iRate"], dtTrade)
dividend_ts = dfToZeroCurve(df_rates["iDiv"], dtTrade)
# initial values for parameters
if ival is None:
ival = {"v0": 0.1, "kappa": 1.0, "theta": 0.1, "sigma": 0.5, "rho": -0.5}
process = HestonProcess(
risk_free_ts, dividend_ts, spot, ival["v0"], ival["kappa"], ival["theta"], ival["sigma"], ival["rho"]
)
model = HestonModel(process)
engine = AnalyticHestonEngine(model, 64)
for option in options:
option.set_pricing_engine(engine)
om = LevenbergMarquardt(1e-8, 1e-8, 1e-8)
model.calibrate(options, om, EndCriteria(400, 40, 1.0e-8, 1.0e-8, 1.0e-8))
print("model calibration results:")
print("v0: %f kappa: %f theta: %f sigma: %f rho: %f" % (model.v0, model.kappa, model.theta, model.sigma, model.rho))
calib_error = (1.0 / len(options)) * sum([pow(o.calibration_error() * 100.0, 2) for o in options])
print("SSE: %f" % calib_error)
# merge the fitted volatility and the input data set
return merge_df(df_option, options, "Heston")
def heston_calibration(df_option, ival=None):
"""
calibrate heston model
"""
tmp = make_helpers(df_option)
risk_free_ts = tmp['risk_free_rate']
dividend_ts = tmp['dividend_rate']
spot = tmp['spot']
options = tmp['options']
# initial values for parameters
if ival is None:
ival = {
'v0': 0.1,
'kappa': 1.0,
'theta': 0.1,
'sigma': 0.5,
'rho': -.5
}
process = HestonProcess(risk_free_ts, dividend_ts, spot, ival['v0'],
ival['kappa'], ival['theta'], ival['sigma'],
ival['rho'])
model = HestonModel(process)
engine = AnalyticHestonEngine(model, 64)
for option in options:
option.set_pricing_engine(engine)
om = LevenbergMarquardt(1e-8, 1e-8, 1e-8)
model.calibrate(options, om, EndCriteria(400, 40, 1.0e-8, 1.0e-8, 1.0e-8))
print('model calibration results:')
print('v0: %f kappa: %f theta: %f sigma: %f rho: %f' %
(model.v0, model.kappa, model.theta, model.sigma, model.rho))
calib_error = (1.0 / len(options)) * sum(
[pow(o.calibration_error() * 100.0, 2) for o in options])
print('SSE: %f' % calib_error)
# merge the fitted volatility and the input data set
return merge_df(df_option, options, 'Heston')
def heston_calibration(df_option, dtTrade=None, df_rates=None, ival=None):
"""
calibrate heston model
"""
# array of option helpers
print df_option, df_rates, ival
hh = heston_helpers(df_option, dtTrade, df_rates, ival)
options = hh['options']
spot = hh['spot']
risk_free_ts = df_to_zero_curve(df_rates['R'], dtTrade)
dividend_ts = df_to_zero_curve(df_rates['D'], dtTrade)
if ival is None:
ival = {
'v0': 0.1,
'kappa': 1.0,
'theta': 0.1,
'sigma': 0.5,
'rho': -.5
}
process = HestonProcess(risk_free_ts, dividend_ts, spot, ival['v0'],
ival['kappa'], ival['theta'], ival['sigma'],
ival['rho'])
model = HestonModel(process)
engine = AnalyticHestonEngine(model, 64)
for option in options:
option.set_pricing_engine(engine)
om = LevenbergMarquardt(1e-8, 1e-8, 1e-8)
model.calibrate(options, om, EndCriteria(400, 40, 1.0e-8, 1.0e-8, 1.0e-8))
print('model calibration results:')
print('v0: %f kappa: %f theta: %f sigma: %f rho: %f' %
(model.v0, model.kappa, model.theta, model.sigma, model.rho))
calib_error = (1.0 / len(options)) * sum(
[pow(o.calibration_error() * 100.0, 2) for o in options])
print('SSE: %f' % calib_error)
return merge_df(df_option, options, 'Heston')
def heston_calibration(df_option, dtTrade=None, df_rates=None, ival=None):
"""
calibrate heston model
"""
# array of option helpers
print df_option, df_rates, ival
hh = heston_helpers(df_option, dtTrade, df_rates, ival)
options = hh['options']
spot = hh['spot']
risk_free_ts = dfToZeroCurve(df_rates['R'], dtTrade)
dividend_ts = dfToZeroCurve(df_rates['D'], dtTrade)
if ival is None:
ival = {'v0': 0.1, 'kappa': 1.0, 'theta': 0.1,
'sigma': 0.5, 'rho': -.5}
process = HestonProcess(
risk_free_ts, dividend_ts, spot, ival['v0'], ival['kappa'],
ival['theta'], ival['sigma'], ival['rho'])
model = HestonModel(process)
engine = AnalyticHestonEngine(model, 64)
for option in options:
option.set_pricing_engine(engine)
om = LevenbergMarquardt(1e-8, 1e-8, 1e-8)
model.calibrate(
options, om, EndCriteria(400, 40, 1.0e-8, 1.0e-8, 1.0e-8)
)
print('model calibration results:')
print('v0: %f kappa: %f theta: %f sigma: %f rho: %f' %
(model.v0, model.kappa, model.theta, model.sigma,
model.rho))
calib_error = (1.0/len(options)) * sum(
[pow(o.calibration_error()*100.0,2) for o in options])
print('SSE: %f' % calib_error)
return merge_df(df_option, options, 'Heston')
def heston_calibration(df_option, ival=None):
"""
calibrate heston model
"""
tmp = make_helpers(df_option)
risk_free_ts = tmp['risk_free_rate']
dividend_ts = tmp['dividend_rate']
spot = tmp['spot']
options = tmp['options']
# initial values for parameters
if ival is None:
ival = {'v0': 0.1, 'kappa': 1.0, 'theta': 0.1,
'sigma': 0.5, 'rho': -.5}
process = HestonProcess(
risk_free_ts, dividend_ts, spot, ival['v0'], ival['kappa'],
ival['theta'], ival['sigma'], ival['rho'])
model = HestonModel(process)
engine = AnalyticHestonEngine(model, 64)
for option in options:
option.set_pricing_engine(engine)
om = LevenbergMarquardt(1e-8, 1e-8, 1e-8)
model.calibrate(
options, om, EndCriteria(400, 40, 1.0e-8, 1.0e-8, 1.0e-8)
)
print('model calibration results:')
print('v0: %f kappa: %f theta: %f sigma: %f rho: %f' %
(model.v0, model.kappa, model.theta, model.sigma,
model.rho))
calib_error = (1.0 / len(options)) * sum(
[pow(o.calibration_error() * 100.0, 2) for o in options])
print('SSE: %f' % calib_error)
# merge the fitted volatility and the input data set
return merge_df(df_option, options, 'Heston')
def test_heston_hw_calibration(self):
"""
From Quantlib test suite
"""
print("Testing Heston Hull-White calibration...")
## Calibration of a hybrid Heston-Hull-White model using
## the finite difference HestonHullWhite pricing engine
## Input surface is based on a Heston-Hull-White model with
## Hull-White: a = 0.00883, \sigma = 0.00631
## Heston : \nu = 0.12, \kappa = 2.0,
## \theta = 0.09, \sigma = 0.5, \rho=-0.75
## Equity Short rate correlation: -0.5
dc = Actual365Fixed()
calendar = TARGET()
todays_date = Date(28, March, 2004)
settings = Settings()
settings.evaluation_date = todays_date
r_ts = flat_rate(todays_date, 0.05, dc)
## assuming, that the Hull-White process is already calibrated
## on a given set of pure interest rate calibration instruments.
hw_process = HullWhiteProcess(r_ts, a=0.00883, sigma=0.00631)
q_ts = flat_rate(todays_date, 0.02, dc)
s0 = SimpleQuote(100.0)
# vol surface
strikes = [50, 75, 90, 100, 110, 125, 150, 200]
maturities = [1 / 12., 3 / 12., 0.5, 1.0, 2.0, 3.0, 5.0, 7.5, 10]
vol = [
0.482627, 0.407617, 0.366682, 0.340110, 0.314266, 0.280241,
0.252471, 0.325552, 0.464811, 0.393336, 0.354664, 0.329758,
0.305668, 0.273563, 0.244024, 0.244886, 0.441864, 0.375618,
0.340464, 0.318249, 0.297127, 0.268839, 0.237972, 0.225553,
0.407506, 0.351125, 0.322571, 0.305173, 0.289034, 0.267361,
0.239315, 0.213761, 0.366761, 0.326166, 0.306764, 0.295279,
0.284765, 0.270592, 0.250702, 0.222928, 0.345671, 0.314748,
0.300259, 0.291744, 0.283971, 0.273475, 0.258503, 0.235683,
0.324512, 0.303631, 0.293981, 0.288338, 0.283193, 0.276248,
0.266271, 0.250506, 0.311278, 0.296340, 0.289481, 0.285482,
0.281840, 0.276924, 0.269856, 0.258609, 0.303219, 0.291534,
0.286187, 0.283073, 0.280239, 0.276414, 0.270926, 0.262173
]
start_v0 = 0.2 * 0.2
start_theta = start_v0
start_kappa = 0.5
start_sigma = 0.25
start_rho = -0.5
equityShortRateCorr = -0.5
corrConstraint = HestonHullWhiteCorrelationConstraint(
equityShortRateCorr)
heston_process = HestonProcess(r_ts, q_ts, s0, start_v0, start_kappa,
start_theta, start_sigma, start_rho)
h_model = HestonModel(heston_process)
h_engine = AnalyticHestonEngine(h_model)
options = []
# first calibrate a heston model to get good initial
# parameters
for i in range(len(maturities)):
maturity = Period(int(maturities[i] * 12.0 + 0.5), Months)
for j, s in enumerate(strikes):
v = SimpleQuote(vol[i * len(strikes) + j])
helper = HestonModelHelper(maturity, calendar, s0.value, s, v,
r_ts, q_ts, PriceError)
helper.set_pricing_engine(h_engine)
options.append(helper)
om = LevenbergMarquardt(1e-6, 1e-8, 1e-8)
# Heston model
h_model.calibrate(options, om,
EndCriteria(400, 40, 1.0e-8, 1.0e-4, 1.0e-8))
print("Heston calibration")
print("v0: %f" % h_model.v0)
print("theta: %f" % h_model.theta)
print("kappa: %f" % h_model.kappa)
print("sigma: %f" % h_model.sigma)
print("rho: %f" % h_model.rho)
h_process_2 = HestonProcess(r_ts, q_ts, s0, h_model.v0, h_model.kappa,
h_model.theta, h_model.sigma, h_model.rho)
hhw_model = HestonModel(h_process_2)
options = []
for i in range(len(maturities)):
tGrid = np.max((10.0, maturities[i] * 10.0))
hhw_engine = FdHestonHullWhiteVanillaEngine(
hhw_model, hw_process, equityShortRateCorr, tGrid, 61, 13, 9,
0, True, FdmSchemeDesc.Hundsdorfer())
hhw_engine.enable_multiple_strikes_caching(strikes)
maturity = Period(int(maturities[i] * 12.0 + 0.5), Months)
# multiple strikes engine works best if the first option
# per maturity has the average strike (because the first
# option is priced first during the calibration and
# the first pricing is used to calculate the prices
# for all strikes
# list of strikes by distance from moneyness
indx = np.argsort(np.abs(np.array(strikes) - s0.value))
for j, tmp in enumerate(indx):
js = indx[j]
s = strikes[js]
v = SimpleQuote(vol[i * len(strikes) + js])
helper = HestonModelHelper(maturity, calendar, s0.value,
strikes[js], v, r_ts, q_ts,
PriceError)
helper.set_pricing_engine(hhw_engine)
options.append(helper)
vm = LevenbergMarquardt(1e-6, 1e-2, 1e-2)
hhw_model.calibrate(options, vm,
EndCriteria(400, 40, 1.0e-8, 1.0e-4, 1.0e-8),
corrConstraint)
print("Heston HW calibration with FD engine")
print("v0: %f" % hhw_model.v0)
print("theta: %f" % hhw_model.theta)
print("kappa: %f" % hhw_model.kappa)
print("sigma: %f" % hhw_model.sigma)
print("rho: %f" % hhw_model.rho)
relTol = 0.05
expected_v0 = 0.12
expected_kappa = 2.0
expected_theta = 0.09
expected_sigma = 0.5
expected_rho = -0.75
self.assertAlmostEquals(np.abs(hhw_model.v0 - expected_v0) /
expected_v0,
0,
delta=relTol)
self.assertAlmostEquals(np.abs(hhw_model.theta - expected_theta) /
expected_theta,
0,
delta=relTol)
self.assertAlmostEquals(np.abs(hhw_model.kappa - expected_kappa) /
expected_kappa,
0,
delta=relTol)
self.assertAlmostEquals(np.abs(hhw_model.sigma - expected_sigma) /
expected_sigma,
0,
delta=relTol)
self.assertAlmostEquals(np.abs(hhw_model.rho - expected_rho) /
expected_rho,
0,
delta=relTol)
def test_black_calibration(self):
# calibrate a Heston model to a constant volatility surface without
# smile. expected result is a vanishing volatility of the volatility.
# In addition theta and v0 should be equal to the constant variance
todays_date = today()
self.settings.evaluation_date = todays_date
daycounter = Actual360()
calendar = NullCalendar()
risk_free_ts = flat_rate(0.04, daycounter)
dividend_ts = flat_rate(0.50, daycounter)
option_maturities = [
Period(1, Months),
Period(2, Months),
Period(3, Months),
Period(6, Months),
Period(9, Months),
Period(1, Years),
Period(2, Years)
]
options = []
s0 = SimpleQuote(1.0)
vol = SimpleQuote(0.1)
volatility = vol.value
for maturity in option_maturities:
for moneyness in np.arange(-1.0, 2.0, 1.):
tau = daycounter.year_fraction(
risk_free_ts.reference_date,
calendar.advance(
risk_free_ts.reference_date,
period=maturity)
)
forward_price = s0.value * dividend_ts.discount(tau) / \
risk_free_ts.discount(tau)
strike_price = forward_price * np.exp(
-moneyness * volatility * np.sqrt(tau)
)
options.append(
HestonModelHelper(
maturity, calendar, s0.value, strike_price, vol,
risk_free_ts, dividend_ts
)
)
for sigma in np.arange(0.1, 0.7, 0.2):
v0 = 0.01
kappa = 0.2
theta = 0.02
rho = -0.75
process = HestonProcess(
risk_free_ts, dividend_ts, s0, v0, kappa, theta, sigma, rho
)
self.assertEqual(v0, process.v0)
self.assertEqual(kappa, process.kappa)
self.assertEqual(theta, process.theta)
self.assertEqual(sigma, process.sigma)
self.assertEqual(rho, process.rho)
self.assertEqual(1.0, process.s0().value)
model = HestonModel(process)
engine = AnalyticHestonEngine(model, 96)
for option in options:
option.set_pricing_engine(engine)
optimisation_method = LevenbergMarquardt(1e-8, 1e-8, 1e-8)
end_criteria = EndCriteria(400, 40, 1.0e-8, 1.0e-8, 1.0e-8)
model.calibrate(options, optimisation_method, end_criteria)
tolerance = 3.0e-3
self.assertFalse(model.sigma > tolerance)
self.assertAlmostEqual(
model.kappa * model.theta,
model.kappa * volatility ** 2,
delta=tolerance
)
self.assertAlmostEqual(model.v0, volatility ** 2, delta=tolerance)
def test_DAX_calibration(self):
# this example is taken from A. Sepp
# Pricing European-Style Options under Jump Diffusion Processes
# with Stochstic Volatility: Applications of Fourier Transform
# http://math.ut.ee/~spartak/papers/stochjumpvols.pdf
settlement_date = Date(5, July, 2002)
self.settings.evaluation_date = settlement_date
daycounter = Actual365Fixed()
calendar = TARGET()
t = [13, 41, 75, 165, 256, 345, 524, 703]
r = [0.0357,0.0349,0.0341,0.0355,0.0359,0.0368,0.0386,0.0401]
dates = [settlement_date] + [settlement_date + val for val in t]
rates = [0.0357] + r
risk_free_ts = ZeroCurve(dates, rates, daycounter)
dividend_ts = FlatForward(
settlement_date, forward=0.0, daycounter=daycounter
)
v = [
0.6625,0.4875,0.4204,0.3667,0.3431,0.3267,0.3121,0.3121,
0.6007,0.4543,0.3967,0.3511,0.3279,0.3154,0.2984,0.2921,
0.5084,0.4221,0.3718,0.3327,0.3155,0.3027,0.2919,0.2889,
0.4541,0.3869,0.3492,0.3149,0.2963,0.2926,0.2819,0.2800,
0.4060,0.3607,0.3330,0.2999,0.2887,0.2811,0.2751,0.2775,
0.3726,0.3396,0.3108,0.2781,0.2788,0.2722,0.2661,0.2686,
0.3550,0.3277,0.3012,0.2781,0.2781,0.2661,0.2661,0.2681,
0.3428,0.3209,0.2958,0.2740,0.2688,0.2627,0.2580,0.2620,
0.3302,0.3062,0.2799,0.2631,0.2573,0.2533,0.2504,0.2544,
0.3343,0.2959,0.2705,0.2540,0.2504,0.2464,0.2448,0.2462,
0.3460,0.2845,0.2624,0.2463,0.2425,0.2385,0.2373,0.2422,
0.3857,0.2860,0.2578,0.2399,0.2357,0.2327,0.2312,0.2351,
0.3976,0.2860,0.2607,0.2356,0.2297,0.2268,0.2241,0.2320
]
s0 = SimpleQuote(4468.17)
strikes = [
3400, 3600, 3800, 4000, 4200, 4400, 4500, 4600, 4800, 5000, 5200,
5400, 5600
]
options = []
for s, strike in enumerate(strikes):
for m in range(len(t)):
vol = SimpleQuote(v[s * 8 + m])
# round to weeks
maturity = Period((int)((t[m] + 3) / 7.), Weeks)
options.append(
HestonModelHelper(
maturity, calendar, s0.value, strike, vol,
risk_free_ts, dividend_ts,
ImpliedVolError
)
)
v0 = 0.1
kappa = 1.0
theta = 0.1
sigma = 0.5
rho = -0.5
process = HestonProcess(
risk_free_ts, dividend_ts, s0, v0, kappa, theta, sigma, rho
)
model = HestonModel(process)
engine = AnalyticHestonEngine(model, 64)
for option in options:
option.set_pricing_engine(engine)
om = LevenbergMarquardt(1e-8, 1e-8, 1e-8)
model.calibrate(
options, om, EndCriteria(400, 40, 1.0e-8, 1.0e-8, 1.0e-8)
)
sse = 0
for i in range(len(strikes) * len(t)):
diff = options[i].calibration_error() * 100.0
sse += diff * diff
expected = 177.2 # see article by A. Sepp.
self.assertAlmostEqual(expected, sse, delta=1.0)
def heston_calibration(df_option, ival=None):
"""
calibrate heston model
"""
# extract rates and div yields from the data set
df_tmp = DataFrame.filter(df_option, items=['dtExpiry', 'iRate', 'iDiv'])
grouped = df_tmp.groupby('dtExpiry')
df_rates = grouped.agg(lambda x: x[0])
dtTrade = df_option['dtTrade'][0]
# back out the spot from any forward
iRate = df_option['iRate'][0]
iDiv = df_option['iDiv'][0]
TTM = df_option['TTM'][0]
Fwd = df_option['Fwd'][0]
spot = SimpleQuote(Fwd * np.exp(-(iRate - iDiv) * TTM))
print('Spot: %f risk-free rate: %f div. yield: %f' %
(spot.value, iRate, iDiv))
# build array of option helpers
hh = heston_helpers(spot, df_option, dtTrade, df_rates)
options = hh['options']
spot = hh['spot']
risk_free_ts = dfToZeroCurve(df_rates['iRate'], dtTrade)
dividend_ts = dfToZeroCurve(df_rates['iDiv'], dtTrade)
# initial values for parameters
if ival is None:
ival = {
'v0': 0.1,
'kappa': 1.0,
'theta': 0.1,
'sigma': 0.5,
'rho': -.5
}
process = HestonProcess(risk_free_ts, dividend_ts, spot, ival['v0'],
ival['kappa'], ival['theta'], ival['sigma'],
ival['rho'])
model = HestonModel(process)
engine = AnalyticHestonEngine(model, 64)
for option in options:
option.set_pricing_engine(engine)
om = LevenbergMarquardt(1e-8, 1e-8, 1e-8)
model.calibrate(options, om, EndCriteria(400, 40, 1.0e-8, 1.0e-8, 1.0e-8))
print('model calibration results:')
print('v0: %f kappa: %f theta: %f sigma: %f rho: %f' %
(model.v0, model.kappa, model.theta, model.sigma, model.rho))
calib_error = (1.0 / len(options)) * sum(
[pow(o.calibration_error() * 100.0, 2) for o in options])
print('SSE: %f' % calib_error)
# merge the fitted volatility and the input data set
return merge_df(df_option, options, 'Heston')
def test_black_calibration(self):
# calibrate a Heston model to a constant volatility surface without
# smile. expected result is a vanishing volatility of the volatility.
# In addition theta and v0 should be equal to the constant variance
todays_date = today()
self.settings.evaluation_date = todays_date
daycounter = Actual360()
calendar = NullCalendar()
risk_free_ts = flat_rate(0.04, daycounter)
dividend_ts = flat_rate(0.50, daycounter)
option_maturities = [
Period(1, Months),
Period(2, Months),
Period(3, Months),
Period(6, Months),
Period(9, Months),
Period(1, Years),
Period(2, Years)
]
options = []
s0 = SimpleQuote(1.0)
vol = SimpleQuote(0.1)
volatility = vol.value
for maturity in option_maturities:
for moneyness in np.arange(-1.0, 2.0, 1.):
tau = daycounter.year_fraction(
risk_free_ts.reference_date,
calendar.advance(
risk_free_ts.reference_date,
period=maturity)
)
forward_price = s0.value * dividend_ts.discount(tau) / \
risk_free_ts.discount(tau)
strike_price = forward_price * np.exp(
-moneyness * volatility * np.sqrt(tau)
)
options.append(
HestonModelHelper(
maturity, calendar, s0.value, strike_price, vol,
risk_free_ts, dividend_ts
)
)
for sigma in np.arange(0.1, 0.7, 0.2):
v0 = 0.01
kappa = 0.2
theta = 0.02
rho = -0.75
process = HestonProcess(
risk_free_ts, dividend_ts, s0, v0, kappa, theta, sigma, rho
)
self.assertEqual(v0, process.v0)
self.assertEqual(kappa, process.kappa)
self.assertEqual(theta, process.theta)
self.assertEqual(sigma, process.sigma)
self.assertEqual(rho, process.rho)
self.assertEqual(1.0, process.s0.value)
model = HestonModel(process)
engine = AnalyticHestonEngine(model, 96)
for option in options:
option.set_pricing_engine(engine)
optimisation_method = LevenbergMarquardt(1e-8, 1e-8, 1e-8)
end_criteria = EndCriteria(400, 40, 1.0e-8, 1.0e-8, 1.0e-8)
model.calibrate(options, optimisation_method, end_criteria)
tolerance = 3.0e-3
self.assertFalse(model.sigma > tolerance)
self.assertAlmostEqual(
model.kappa * model.theta,
model.kappa * volatility ** 2,
delta=tolerance
)
self.assertAlmostEqual(model.v0, volatility ** 2, delta=tolerance)
def test_DAX_calibration(self):
# this example is taken from A. Sepp
# Pricing European-Style Options under Jump Diffusion Processes
# with Stochstic Volatility: Applications of Fourier Transform
# http://math.ut.ee/~spartak/papers/stochjumpvols.pdf
settlement_date = Date(5, July, 2002)
self.settings.evaluation_date = settlement_date
daycounter = Actual365Fixed()
calendar = TARGET()
t = [13, 41, 75, 165, 256, 345, 524, 703]
r = [0.0357,0.0349,0.0341,0.0355,0.0359,0.0368,0.0386,0.0401]
dates = [settlement_date] + [settlement_date + val for val in t]
rates = [0.0357] + r
risk_free_ts = ZeroCurve(dates, rates, daycounter)
dividend_ts = FlatForward(
settlement_date, forward=0.0, daycounter=daycounter
)
v = [
0.6625,0.4875,0.4204,0.3667,0.3431,0.3267,0.3121,0.3121,
0.6007,0.4543,0.3967,0.3511,0.3279,0.3154,0.2984,0.2921,
0.5084,0.4221,0.3718,0.3327,0.3155,0.3027,0.2919,0.2889,
0.4541,0.3869,0.3492,0.3149,0.2963,0.2926,0.2819,0.2800,
0.4060,0.3607,0.3330,0.2999,0.2887,0.2811,0.2751,0.2775,
0.3726,0.3396,0.3108,0.2781,0.2788,0.2722,0.2661,0.2686,
0.3550,0.3277,0.3012,0.2781,0.2781,0.2661,0.2661,0.2681,
0.3428,0.3209,0.2958,0.2740,0.2688,0.2627,0.2580,0.2620,
0.3302,0.3062,0.2799,0.2631,0.2573,0.2533,0.2504,0.2544,
0.3343,0.2959,0.2705,0.2540,0.2504,0.2464,0.2448,0.2462,
0.3460,0.2845,0.2624,0.2463,0.2425,0.2385,0.2373,0.2422,
0.3857,0.2860,0.2578,0.2399,0.2357,0.2327,0.2312,0.2351,
0.3976,0.2860,0.2607,0.2356,0.2297,0.2268,0.2241,0.2320
]
s0 = SimpleQuote(4468.17)
strikes = [
3400, 3600, 3800, 4000, 4200, 4400, 4500, 4600, 4800, 5000, 5200,
5400, 5600
]
options = []
for s, strike in enumerate(strikes):
for m in range(len(t)):
vol = SimpleQuote(v[s * 8 + m])
# round to weeks
maturity = Period((int)((t[m] + 3) / 7.), Weeks)
options.append(
HestonModelHelper(
maturity, calendar, s0.value, strike, vol,
risk_free_ts, dividend_ts,
ImpliedVolError
)
)
v0 = 0.1
kappa = 1.0
theta = 0.1
sigma = 0.5
rho = -0.5
process = HestonProcess(
risk_free_ts, dividend_ts, s0, v0, kappa, theta, sigma, rho
)
model = HestonModel(process)
engine = AnalyticHestonEngine(model, 64)
for option in options:
option.set_pricing_engine(engine)
om = LevenbergMarquardt(1e-8, 1e-8, 1e-8)
model.calibrate(
options, om, EndCriteria(400, 40, 1.0e-8, 1.0e-8, 1.0e-8)
)
sse = 0
for i in range(len(strikes) * len(t)):
diff = options[i].calibration_error() * 100.0
sse += diff * diff
expected = 177.2 # see article by A. Sepp.
self.assertAlmostEqual(expected, sse, delta=1.0)
def test_heston_hw_calibration(self):
"""
From Quantlib test suite
"""
print("Testing Heston Hull-White calibration...")
## Calibration of a hybrid Heston-Hull-White model using
## the finite difference HestonHullWhite pricing engine
## Input surface is based on a Heston-Hull-White model with
## Hull-White: a = 0.00883, \sigma = 0.00631
## Heston : \nu = 0.12, \kappa = 2.0,
## \theta = 0.09, \sigma = 0.5, \rho=-0.75
## Equity Short rate correlation: -0.5
dc = Actual365Fixed()
calendar = TARGET()
todays_date = Date(28, March, 2004)
settings = Settings()
settings.evaluation_date = todays_date
r_ts = flat_rate(todays_date, 0.05, dc)
## assuming, that the Hull-White process is already calibrated
## on a given set of pure interest rate calibration instruments.
hw_process = HullWhiteProcess(r_ts, a=0.00883, sigma=0.00631)
q_ts = flat_rate(todays_date, 0.02, dc)
s0 = SimpleQuote(100.0)
# vol surface
strikes = [50, 75, 90, 100, 110, 125, 150, 200]
maturities = [1 / 12., 3 / 12., 0.5, 1.0, 2.0, 3.0, 5.0, 7.5, 10]
vol = [
0.482627,0.407617,0.366682,0.340110,0.314266,0.280241,0.252471,0.325552,
0.464811,0.393336,0.354664,0.329758,0.305668,0.273563,0.244024,0.244886,
0.441864,0.375618,0.340464,0.318249,0.297127,0.268839,0.237972,0.225553,
0.407506,0.351125,0.322571,0.305173,0.289034,0.267361,0.239315,0.213761,
0.366761,0.326166,0.306764,0.295279,0.284765,0.270592,0.250702,0.222928,
0.345671,0.314748,0.300259,0.291744,0.283971,0.273475,0.258503,0.235683,
0.324512,0.303631,0.293981,0.288338,0.283193,0.276248,0.266271,0.250506,
0.311278,0.296340,0.289481,0.285482,0.281840,0.276924,0.269856,0.258609,
0.303219,0.291534,0.286187,0.283073,0.280239,0.276414,0.270926,0.262173]
start_v0 = 0.2 * 0.2
start_theta = start_v0
start_kappa = 0.5
start_sigma = 0.25
start_rho = -0.5
equityShortRateCorr = -0.5
corrConstraint = HestonHullWhiteCorrelationConstraint(
equityShortRateCorr)
heston_process = HestonProcess(r_ts, q_ts, s0, start_v0, start_kappa,
start_theta, start_sigma, start_rho)
h_model = HestonModel(heston_process)
h_engine = AnalyticHestonEngine(h_model)
options = []
# first calibrate a heston model to get good initial
# parameters
for i in range(len(maturities)):
maturity = Period(int(maturities[i] * 12.0 + 0.5), Months)
for j, s in enumerate(strikes):
v = SimpleQuote(vol[i * len(strikes) + j])
helper = HestonModelHelper(maturity, calendar, s0.value,
s, v, r_ts, q_ts,
PriceError)
helper.set_pricing_engine(h_engine)
options.append(helper)
om = LevenbergMarquardt(1e-6, 1e-8, 1e-8)
# Heston model
h_model.calibrate(options, om,
EndCriteria(400, 40, 1.0e-8,
1.0e-4, 1.0e-8))
print("Heston calibration")
print("v0: %f" % h_model.v0)
print("theta: %f" % h_model.theta)
print("kappa: %f" % h_model.kappa)
print("sigma: %f" % h_model.sigma)
print("rho: %f" % h_model.rho)
h_process_2 = HestonProcess(r_ts, q_ts, s0, h_model.v0,
h_model.kappa,
h_model.theta,
h_model.sigma,
h_model.rho)
hhw_model = HestonModel(h_process_2)
options = []
for i in range(len(maturities)):
tGrid = np.max((10.0, maturities[i] * 10.0))
hhw_engine = FdHestonHullWhiteVanillaEngine(
hhw_model, hw_process,
equityShortRateCorr,
tGrid, 61, 13, 9, 0, True, FdmSchemeDesc.Hundsdorfer())
hhw_engine.enable_multiple_strikes_caching(strikes)
maturity = Period(int(maturities[i] * 12.0 + 0.5), Months)
# multiple strikes engine works best if the first option
# per maturity has the average strike (because the first
# option is priced first during the calibration and
# the first pricing is used to calculate the prices
# for all strikes
# list of strikes by distance from moneyness
indx = np.argsort(np.abs(np.array(strikes) - s0.value))
for j, tmp in enumerate(indx):
js = indx[j]
s = strikes[js]
v = SimpleQuote(vol[i * len(strikes) + js])
helper = HestonModelHelper(maturity,
calendar, s0.value,
strikes[js], v, r_ts, q_ts,
PriceError)
helper.set_pricing_engine(hhw_engine)
options.append(helper)
vm = LevenbergMarquardt(1e-6, 1e-2, 1e-2)
hhw_model.calibrate(options, vm,
EndCriteria(400, 40, 1.0e-8, 1.0e-4, 1.0e-8),
corrConstraint)
print("Heston HW calibration with FD engine")
print("v0: %f" % hhw_model.v0)
print("theta: %f" % hhw_model.theta)
print("kappa: %f" % hhw_model.kappa)
print("sigma: %f" % hhw_model.sigma)
print("rho: %f" % hhw_model.rho)
relTol = 0.05
expected_v0 = 0.12
expected_kappa = 2.0
expected_theta = 0.09
expected_sigma = 0.5
expected_rho = -0.75
self.assertAlmostEqual(
np.abs(hhw_model.v0 - expected_v0) / expected_v0, 0,
delta=relTol)
self.assertAlmostEqual(
np.abs(hhw_model.theta - expected_theta) / expected_theta, 0,
delta=relTol)
self.assertAlmostEqual(
np.abs(hhw_model.kappa - expected_kappa) / expected_kappa, 0,
delta=relTol)
self.assertAlmostEqual(
np.abs(hhw_model.sigma - expected_sigma) / expected_sigma, 0,
delta=relTol)
self.assertAlmostEqual(
np.abs(hhw_model.rho - expected_rho) / expected_rho, 0,
delta=relTol)
def heston_calibration(df_option, ival=None):
"""
calibrate heston model
"""
# extract rates and div yields from the data set
df_tmp = DataFrame.filter(df_option, items=['dtExpiry', 'iRate', 'iDiv'])
grouped = df_tmp.groupby('dtExpiry')
def aggregate(serie):
return serie[serie.index[0]]
df_rates = grouped.agg(aggregate)
# Get first index:
first_index = 0
dtTrade = df_option['dtTrade'][first_index]
# back out the spot from any forward
iRate = df_option['iRate'][first_index]
iDiv = df_option['iDiv'][first_index]
TTM = df_option['TTM'][first_index]
Fwd = df_option['Fwd'][first_index]
spot = SimpleQuote(Fwd*np.exp(-(iRate-iDiv)*TTM))
print('Spot: %f risk-free rate: %f div. yield: %f' % (spot.value, iRate, iDiv))
# build array of option helpers
hh = heston_helpers(spot, df_option, dtTrade, df_rates)
options = hh['options']
spot = hh['spot']
risk_free_ts = dfToZeroCurve(df_rates['iRate'], dtTrade)
dividend_ts = dfToZeroCurve(df_rates['iDiv'], dtTrade)
# initial values for parameters
if ival is None:
ival = {'v0': 0.1, 'kappa': 1.0, 'theta': 0.1,
'sigma': 0.5, 'rho': -.5}
process = HestonProcess(
risk_free_ts, dividend_ts, spot, ival['v0'], ival['kappa'],
ival['theta'], ival['sigma'], ival['rho'])
model = HestonModel(process)
engine = AnalyticHestonEngine(model, 64)
for option in options:
option.set_pricing_engine(engine)
om = LevenbergMarquardt(1e-8, 1e-8, 1e-8)
model.calibrate(
options, om, EndCriteria(400, 40, 1.0e-8, 1.0e-8, 1.0e-8)
)
print('model calibration results:')
print('v0: %f kappa: %f theta: %f sigma: %f rho: %f' %
(model.v0, model.kappa, model.theta, model.sigma,
model.rho))
calib_error = (1.0/len(options)) * sum(
[pow(o.calibration_error()*100.0,2) for o in options])
print('SSE: %f' % calib_error)
# merge the fitted volatility and the input data set
return merge_df(df_option, options, 'Heston')
