def test_pandas(self):
"""Test with pandas input."""
count = int(1e3)
maturity = 30 / 365
call = True
sigma = np.random.uniform(.05, .8, count)
moneyness = np.random.uniform(-.1, .1, count)
premium = blackscholes_norm(moneyness, maturity, sigma, call)
table = pd.DataFrame({'premium': premium, 'moneyness': moneyness})
table['maturity'] = maturity
table['call'] = call
table['imp_vol'] = impvol_table(table)
self.assertIsInstance(table, pd.DataFrame)
np.testing.assert_array_almost_equal(table['imp_vol'].values, sigma, 3)
dct = {
'premium': premium,
'moneyness': moneyness,
'maturity': maturity,
'call': call
}
dct['imp_vol'] = impvol_table(dct)
self.assertIsInstance(dct, dict)
np.testing.assert_array_almost_equal(dct['imp_vol'], sigma, 3)
def test_gbm(self):
"""Test GBM model."""
price, strike = 100, 110
riskfree, maturity = .01, 30/365
call = True
put = np.logical_not(call)
moneyness = lfmoneyness(price, strike, riskfree, maturity)
sigma = .15
model = GBM(GBMParam(sigma=sigma), riskfree, maturity)
premium = cosmethod(model, moneyness=moneyness, call=call)
premium_true = blackscholes_norm(moneyness, maturity, sigma, call)
impvol_model = impvol_bisection(moneyness, maturity, premium, call)
self.assertEqual(premium.shape, (1,))
np.testing.assert_array_almost_equal(premium, premium_true, 3)
np.testing.assert_array_almost_equal(impvol_model, sigma, 2)
moneyness = np.linspace(0, .1, 10)
premium = cosmethod(model, moneyness=moneyness, call=call)
premium_true = blackscholes_norm(moneyness, maturity, sigma, call)
impvol_model = impvol_bisection(moneyness, maturity, premium, call)
impvol_true = np.ones_like(impvol_model) * sigma
self.assertEqual(premium.shape, moneyness.shape)
np.testing.assert_array_almost_equal(premium, premium_true, 2)
np.testing.assert_array_almost_equal(impvol_model, impvol_true, 2)
riskfree = np.zeros_like(moneyness)
premium = cosmethod(model, moneyness=moneyness, call=call)
premium_true = blackscholes_norm(moneyness, maturity, sigma, call)
impvol_model = impvol_bisection(moneyness, maturity, premium, call)
self.assertEqual(premium.shape, moneyness.shape)
np.testing.assert_array_almost_equal(premium, premium_true, 3)
np.testing.assert_array_almost_equal(impvol_model, sigma, 2)
moneyness = np.linspace(-.1, 0, 10)
premium = cosmethod(model, moneyness=moneyness, call=put)
premium_true = blackscholes_norm(moneyness, maturity, sigma, put)
impvol_model = impvol_bisection(moneyness, maturity, premium, put)
np.testing.assert_array_almost_equal(premium, premium_true, 2)
np.testing.assert_array_almost_equal(impvol_model, impvol_true, 2)
def test_gbm(self):
"""Test GBM model."""
price, strike = 100, 110
riskfree, maturity = .01, 30 / 365
call = True
put = np.logical_not(call)
moneyness = lfmoneyness(price, strike, riskfree, maturity)
sigma = .15
model = GBM(GBMParam(sigma=sigma), riskfree, maturity)
premium = cosmethod(model, moneyness=moneyness, call=call)
premium_true = blackscholes_norm(moneyness, maturity, sigma, call)
impvol_model = impvol_bisection(moneyness, maturity, premium, call)
self.assertEqual(premium.shape, (1, ))
np.testing.assert_array_almost_equal(premium, premium_true, 3)
np.testing.assert_array_almost_equal(impvol_model, sigma, 2)
moneyness = np.linspace(0, .1, 10)
premium = cosmethod(model, moneyness=moneyness, call=call)
premium_true = blackscholes_norm(moneyness, maturity, sigma, call)
impvol_model = impvol_bisection(moneyness, maturity, premium, call)
impvol_true = np.ones_like(impvol_model) * sigma
self.assertEqual(premium.shape, moneyness.shape)
np.testing.assert_array_almost_equal(premium, premium_true, 2)
np.testing.assert_array_almost_equal(impvol_model, impvol_true, 2)
riskfree = np.zeros_like(moneyness)
premium = cosmethod(model, moneyness=moneyness, call=call)
premium_true = blackscholes_norm(moneyness, maturity, sigma, call)
impvol_model = impvol_bisection(moneyness, maturity, premium, call)
self.assertEqual(premium.shape, moneyness.shape)
np.testing.assert_array_almost_equal(premium, premium_true, 3)
np.testing.assert_array_almost_equal(impvol_model, sigma, 2)
moneyness = np.linspace(-.1, 0, 10)
premium = cosmethod(model, moneyness=moneyness, call=put)
premium_true = blackscholes_norm(moneyness, maturity, sigma, put)
impvol_model = impvol_bisection(moneyness, maturity, premium, put)
np.testing.assert_array_almost_equal(premium, premium_true, 2)
np.testing.assert_array_almost_equal(impvol_model, impvol_true, 2)
def impvol_bisection(moneyness, maturity, premium, call, tol=1e-5, fcount=1e3):
"""Function to find BS Implied Vol using Bisection Method.
Parameters
----------
moneyness : float
Log-forward moneyness
maturity : float
Fraction of the year
premium : float
Option premium normalized by current asset price
call : bool
Call/put flag. True for call, False for put
Returns
-------
float
Implied volatilities.
Shape of the array is according to broadcasting rules.
"""
sig, sig_u, sig_d = .2, 1, 1e-3
count = 0
err = blackscholes_norm(moneyness, maturity, sig, call) - premium
# repeat until error is sufficiently small
# or counter hits fcount
while abs(err) > tol and count < fcount:
if err < 0:
sig_d = sig
sig = (sig_u + sig)/2
else:
sig_u = sig
sig = (sig_d + sig)/2
err = blackscholes_norm(moneyness, maturity, sig, call) - premium
count += 1
# return NA if counter hit 1000
if count == fcount:
return -1
else:
return sig
def impvol_bisection(moneyness, maturity, premium, call, tol=1e-5, fcount=1e3):
"""Function to find BS Implied Vol using Bisection Method.
Parameters
----------
moneyness : float
Log-forward moneyness
maturity : float
Fraction of the year
premium : float
Option premium normalized by current asset price
call : bool
Call/put flag. True for call, False for put
Returns
-------
float
Implied volatilities.
Shape of the array is according to broadcasting rules.
"""
sig, sig_u, sig_d = .2, 1, 1e-3
count = 0
err = blackscholes_norm(moneyness, maturity, sig, call) - premium
# repeat until error is sufficiently small
# or counter hits fcount
while abs(err) > tol and count < fcount:
if err < 0:
sig_d = sig
sig = (sig_u + sig) / 2
else:
sig_u = sig
sig = (sig_d + sig) / 2
err = blackscholes_norm(moneyness, maturity, sig, call) - premium
count += 1
# return NA if counter hit 1000
if count == fcount:
return -1
else:
return sig
def test_bs_prices(self):
"""Test accuracy of back and forth BS conversion."""
count = int(1e3)
maturity = 30/365
call = True
sigma = np.random.uniform(.05, .8, count)
moneyness = np.random.uniform(-.1, .1, count)
premium = blackscholes_norm(moneyness, maturity, sigma, call)
vol = impvol_bisection(moneyness, maturity, premium, call)
np.testing.assert_array_almost_equal(vol, sigma, 3)
def test_bs_prices(self):
"""Test accuracy of back and forth BS conversion."""
count = int(1e3)
maturity = 30 / 365
call = True
sigma = np.random.uniform(.05, .8, count)
moneyness = np.random.uniform(-.1, .1, count)
premium = blackscholes_norm(moneyness, maturity, sigma, call)
vol = impvol_bisection(moneyness, maturity, premium, call)
np.testing.assert_array_almost_equal(vol, sigma, 3)
def premium(self):
"""Get price given implied volatility.
Returns
-------
float array
Option premium normalized by current underlying price
"""
return blackscholes_norm(self.data['moneyness'], self.data['maturity'],
self.impvol(), self.data['call'])
def premium(self):
"""Get price given implied volatility.
Returns
-------
float array
Option premium normalized by current underlying price
"""
return blackscholes_norm(self.data['moneyness'],
self.data['maturity'],
self.impvol(), self.data['call'])
def premium(self):
"""Black-Scholes option price formula.
Parameters
----------
moneyness : float array
Log-forward moneyness
call : bool array
Call/put flag. True for call, False for put
Returns
-------
float array
Option premium normalized by current underlying price
"""
return blackscholes_norm(self.data['moneyness'], self.data['maturity'],
self.sigma, self.data['call'])
def premium(self):
"""Black-Scholes option price formula.
Parameters
----------
moneyness : float array
Log-forward moneyness
call : bool array
Call/put flag. True for call, False for put
Returns
-------
float array
Option premium normalized by current underlying price
"""
return blackscholes_norm(self.data['moneyness'], self.data['maturity'],
self.sigma, self.data['call'])
def test_pandas(self):
"""Test with pandas input."""
count = int(1e3)
maturity = 30/365
call = True
sigma = np.random.uniform(.05, .8, count)
moneyness = np.random.uniform(-.1, .1, count)
premium = blackscholes_norm(moneyness, maturity, sigma, call)
table = pd.DataFrame({'premium': premium, 'moneyness': moneyness})
table['maturity'] = maturity
table['call'] = call
table['imp_vol'] = impvol_table(table)
self.assertIsInstance(table, pd.DataFrame)
np.testing.assert_array_almost_equal(table['imp_vol'].values, sigma, 3)
dct = {'premium': premium, 'moneyness': moneyness,
'maturity': maturity, 'call': call}
dct['imp_vol'] = impvol_table(dct)
self.assertIsInstance(dct, dict)
np.testing.assert_array_almost_equal(dct['imp_vol'], sigma, 3)
def premium(self):
"""Mixture of log-normals option price formula.
Parameters
----------
moneyness : float array
Log-forward moneyness
call : bool array
Call/put flag. True for call, False for put
Returns
-------
float array
Option premium normalized by current underlying price
"""
premium = 0
for weight, mean, std in zip(self.weights, self.means, self.stds):
shift = (self.data['riskfree'] - mean) * self.data['maturity']
moneyness = np.array(self.data['moneyness']) + shift
premium += weight * blackscholes_norm(
moneyness, self.data['maturity'], std, self.data['call'])
return premium
def premium(self):
"""Mixture of log-normals option price formula.
Parameters
----------
moneyness : float array
Log-forward moneyness
call : bool array
Call/put flag. True for call, False for put
Returns
-------
float array
Option premium normalized by current underlying price
"""
premium = 0
for weight, mean, std in zip(self.weights, self.means, self.stds):
shift = (self.data['riskfree'] - mean) * self.data['maturity']
moneyness = np.array(self.data['moneyness']) + shift
premium += weight * blackscholes_norm(moneyness,
self.data['maturity'],
std, self.data['call'])
return premium
if __name__ == '__main__':
price = 1
strike = 1
riskfree = .02
maturity = 30/365
premium = .057
call = True
moneyness = lfmoneyness(price, strike, riskfree, maturity)
vol = impvol_bisection(moneyness, maturity, premium, call)
count = int(1e2)
sigma = np.random.uniform(.05, .8, count)
moneyness = np.random.uniform(-.1, .1, count)
premium = blackscholes_norm(moneyness, maturity, sigma, call)
text = 'Time elapsed: %.2f seconds'
time_start = time.time()
# Based on SciPy root method
vol = imp_vol(moneyness, maturity, premium, call)
print(np.allclose(sigma, vol))
print(text % (time.time() - time_start))
time_start = time.time()
# Based on bisection method
vol = impvol_bisection(moneyness, maturity, premium, call)
print('Relative difference (percent) = ', ((sigma/vol-1).max()*100))
print(np.allclose(vol, sigma, rtol=1e-3))
print(text % (time.time() - time_start))
if __name__ == '__main__':
price = 1
strike = 1
riskfree = .02
maturity = 30 / 365
premium = .057
call = True
moneyness = lfmoneyness(price, strike, riskfree, maturity)
vol = impvol_bisection(moneyness, maturity, premium, call)
count = int(1e2)
sigma = np.random.uniform(.05, .8, count)
moneyness = np.random.uniform(-.1, .1, count)
premium = blackscholes_norm(moneyness, maturity, sigma, call)
text = 'Time elapsed: %.2f seconds'
time_start = time.time()
# Based on SciPy root method
vol = imp_vol(moneyness, maturity, premium, call)
print(np.allclose(sigma, vol))
print(text % (time.time() - time_start))
time_start = time.time()
# Based on bisection method
vol = impvol_bisection(moneyness, maturity, premium, call)
print('Relative difference (percent) = ', ((sigma / vol - 1).max() * 100))
print(np.allclose(vol, sigma, rtol=1e-3))
print(text % (time.time() - time_start))
