def BSM_put_value(St, K, t, T, r, sigma):
''' Calculates Black-Scholes-Merton European put option value.
Parameters
==========
St : float
stock/index level at time t
K : float
strike price
t : float
valuation date
T : float
date of maturity/time-to-maturity if t = 0; T > t
r : float
constant, risk-less short rate
sigma : float
volatility
Returns
=======
put_value : float
European put present value at t
'''
put_value = BSM_call_value(St, K, t, T, r, sigma) \
- St + math.exp(-r * (T - t)) * K
return put_value
def BSM_error_function(p0):
''' Error Function for parameter calibration in BSM Model via
Parameters
==========
sigma: float
volatility factor in diffusion term
Returns
=======
RMSE: float
root mean squared error
'''
global i, min_RMSE
sigma = p0
if sigma < 0.0:
return 500.0
se = []
for row, option in options.iterrows():
T = (option['Maturity'] - option['Date']).days / 365.
model_value = BSM_call_value(S0, option['Strike'], 0, T, r, sigma)
se.append((model_value - option['Call']) ** 2)
RMSE = math.sqrt(sum(se) / len(se))
min_RMSE = min(min_RMSE, RMSE)
if i % 5 == 0:
print('%4d |' % i, np.array(p0), '| %7.3f | %7.3f' % (RMSE, min_RMSE))
i += 1
return RMSE
def plot_convergence(mmin, mmax, step_size):
''' Plots the CRR option values for increasing number of time
intervals M against the Black-Scholes-Merton benchmark value.'''
BSM_benchmark = BSM_call_value(S0, K, 0, T, r, sigma)
m = range(mmin, mmax, step_size)
CRR_values = [CRR_option_value(S0, K, T, r, sigma, 'call', M) for M in m]
plt.figure(figsize=(9, 5))
plt.plot(m, CRR_values, label='CRR values')
plt.axhline(BSM_benchmark, color='r', ls='dashed', lw=1.5,
label='BSM benchmark')
plt.xlabel('# of binomial steps $M$')
plt.ylabel('European call option value')
plt.legend(loc=4)
plt.xlim(0, mmax)
def IVolBsm(S0, K, T, r, P0):
"""
Inputs:
S0: spot price
K: strike
T: time to maturity in year
r: rate
P0: market price
Outputs:
Implied volatility
"""
InitVol = .3
error = lambda sigma: (BSM_call_value(S0, K, 0, T, r, sigma) - P0)**2
opt = optimize.fmin(error, InitVol);
return opt[0]
def generate_plot(opt, options):
#
# Calculating Model Prices
#
sigma = opt
options['Model'] = 0.0
for row, option in options.iterrows():
T = (option['Maturity'] - option['Date']).days / 365.
options.loc[row, 'Model'] = BSM_call_value(S0, option['Strike'], 0, T, r, sigma)
#
# 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('BSM_calibration_3_%s.pdf' % i)
mpl.rcParams['font.family'] = 'serif'
#import the defined function
import sys
sys.path.append('05_com')
from BSM_option_valuation import BSM_call_value
#Model and option Parameters
K = 8000
T = 1.0
r = 0.025
vol = 0.2
#sample generation
S = np.linspace(4000, 12000, 150)
h = np.maximum(S - K, 0)
C = [BSM_call_value(S0, K, 0, T, r, vol) for S0 in S]
plt.figure()
plt.plot(S, h, 'b-.', lw=2.5, label='inner value')
#plot inner value at maturity
plt.plot(S, C, 'r', lw=2.5, label='present value')
#plot option present value
plt.grid(True)
plt.legend(loc=0)
plt.xlabel('index level $S_0$ ')
plt.ylabel('present value $C(t=0)$')
plt.show()