# Black-Scholes-Merton (1973) European Call & Put Valuation
#
import math
import numpy as np
import matplotlib as mpl
import matplotlib.pyplot as plt
mpl.rcParams['font.family'] = 'serif'
from scipy.integrate import quad
#
# Helper Functions
#
def dN(x):
''' Probability density function of standard normal random variable x.'''
return math.exp(-0.5 * x ** 2) / math.sqrt(2 * math.pi)
def N(d):
''' Cumulative density function of standard normal random variable x. '''
return quad(lambda x: dN(x), -20, d, limit=50)[0]
def d1f(St, K, t, T, r, sigma):
''' Black-Scholes-Merton d1 function.
Parameters see e.g. BSM_call_value function. '''
d1 = (math.log(St / K) + (r + 0.5 * sigma ** 2)
* (T - t)) / (sigma * math.sqrt(T - t))
return d1
#
# Valuation Functions
#
def BSM_call_value(St, K, t, T, r, sigma):
''' Calculates Black-Scholes-Merton European call option value.
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86 DERIVATIVES ANALYTICS WITH PYTHON
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
=======
call_value: float
European call present value at t
'''
d1 = d1f(St, K, t, T, r, sigma)
d2 = d1 - sigma * math.sqrt(T - t)
call_value = St * N(d1) - math.exp(-r * (T - t)) * K * N(d2)
return call_value
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
www.it-ebooks.info
Complete Market Models 87
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
#
# Plotting European Option Values
#
def plot_values(function):
''' Plots European option values for different parameters c.p. '''
plt.figure(figsize=(10, 8.3))
points = 100
#
# Model Parameters
#
St = 100.0 # index level
K = 100.0 # option strike
t = 0.0 # valuation date
T = 1.0 # maturity date
r = 0.05 # risk-less short rate
sigma = 0.2 # volatility
# C(K) plot
plt.subplot(221)
klist = np.linspace(80, 120, points)
vlist = [function(St, K, t, T, r, sigma) for K in klist]
plt.plot(klist, vlist)
plt.grid()
plt.xlabel('strike $K$')
plt.ylabel('present value')
# C(T) plot
plt.subplot(222)
tlist = np.linspace(0.0001, 1, points)
vlist = [function(St, K, t, T, r, sigma) for T in tlist]
plt.plot(tlist, vlist)
plt.grid(True)
plt.xlabel('maturity $T$')
# C(r) plot
plt.subplot(223)
rlist = np.linspace(0, 0.1, points)
vlist = [function(St, K, t, T, r, sigma) for r in rlist]
plt.plot(tlist, vlist)
plt.grid(True)
plt.xlabel('short rate $r$')
plt.ylabel('present value')
plt.axis('tight')
# C(sigma) plot
plt.subplot(224)
slist = np.linspace(0.01, 0.5, points)
vlist = [function(St, K, t, T, r, sigma) for sigma in slist]
plt.plot(slist, vlist)
plt.grid(True)
plt.xlabel('volatility $\sigma$')
plt.tight_layout()
