# Black-Scholes-Merton (1973) European Call Option Greeks
# 05_com/BSM_call_greeks.py
#
# (c) Dr. Yves J. Hilpisch
# Derivatives Analytics with Python
#
import math
import numpy as np
import matplotlib as mpl
import matplotlib.pyplot as plt
mpl.rcParams['font.family'] = 'serif'
import mpl_toolkits.mplot3d.axes3d as p3
from BSM_option_valuation import d1f, N, dN
#
# Functions for Greeks
#
def BSM_delta(St, K, t, T, r, sigma):
''' Black-Scholes-Merton DELTA of European call option.
Parameters
==========
St: float
stock/index level at time t
www.it-ebooks.info
Complete Market Models 89
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
=======
delta: float
European call option DELTA
'''
d1 = d1f(St, K, t, T, r, sigma)
delta = N(d1)
return delta
def BSM_gamma(St, K, t, T, r, sigma):
''' Black-Scholes-Merton GAMMA of European call option.
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
=======
gamma: float
European call option GAMMA
'''
d1 = d1f(St, K, t, T, r, sigma)
gamma = dN(d1) / (St * sigma * math.sqrt(T - t))
return gamma
def BSM_theta(St, K, t, T, r, sigma):
''' Black-Scholes-Merton THETA of European call option.
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
=======
theta: float
European call option THETA
'''
d1 = d1f(St, K, t, T, r, sigma)
d2 = d1 - sigma * math.sqrt(T - t)
theta = -(St * dN(d1) * sigma / (2 * math.sqrt(T - t))
+ r * K * math.exp(-r * (T - t)) * N(d2))
return theta
def BSM_rho(St, K, t, T, r, sigma):
''' Black-Scholes-Merton RHO of European call option.
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
www.it-ebooks.info
Complete Market Models 91
Returns
=======
rho: float
European call option RHO
'''
d1 = d1f(St, K, t, T, r, sigma)
d2 = d1 - sigma * math.sqrt(T - t)
rho = K * (T - t) * math.exp(-r * (T - t)) * N(d2)
return rho
def BSM_vega(St, K, t, T, r, sigma):
''' Black-Scholes-Merton VEGA of European call option.
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
=======
vega: float
European call option VEGA
'''
d1 = d1f(St, K, t, T, r, sigma)
vega = St * dN(d1) * math.sqrt(T - t)
return vega
#
# Plotting the Greeks
#
def plot_greeks(function, greek):
# 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
# Greek Calculations
tlist = np.linspace(0.01, 1, 25)
klist = np.linspace(80, 120, 25)
V = np.zeros((len(tlist), len(klist)), dtype=np.float)
for j in range(len(klist)):
for i in range(len(tlist)):
V[i, j] = function(St, klist[j], t, tlist[i], r, sigma)
# 3D Plotting
x, y = np.meshgrid(klist, tlist)
fig = plt.figure(figsize=(9, 5))
plot = p3.Axes3D(fig)
plot.plot_wireframe(x, y, V)
plot.set_xlabel('strike $K$')
plot.set_ylabel('maturity $T$')
plot.set_zlabel('%s(K, T)' % greek)
