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CRR_JarrowRudd_Tian_option_valuation.py
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CRR_JarrowRudd_Tian_option_valuation.py
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# Cox-Ross-Rubinstein, Jarrow Rudd and Tian's Binomial Model
# European Option Valuation
import math
import numpy as np
import matplotlib as mpl
import matplotlib.pyplot as plt
from scipy.integrate import quad
mpl.rcParams['font.family'] = 'serif'
# Model Parameters
#
S0 = 100.0 # index level
K = 100.0 # option strike
T = 1.0 # maturity date
r = 0.05 # risk-less short rate
sigma = 0.2 # volatility
# Valuation Function
def CRR_option_value(S0, K, T, r, sigma, otype, M=4):
''' Cox-Ross-Rubinstein European option valuation.
Parameters
==========
S0 : float
stock/index level at time 0
K : float
strike price
T : float
date of maturity
r : float
constant, risk-less short rate
sigma : float
volatility
otype : string
either 'call' or 'put'
M : int
number of time intervals
'''
# Time Parameters
dt = T / M # length of time interval
df = math.exp(-r * dt) # discount per interval
# Binomial Parameters
u = math.exp(sigma * math.sqrt(dt)) # up movement
d = 1 / u # down movement
q = (math.exp(r * dt) - d) / (u - d) # martingale branch probability
# Array Initialization for Index Levels
mu = np.arange(M + 1)
mu = np.resize(mu, (M + 1, M + 1))
md = np.transpose(mu)
mu = u ** (mu - md)
md = d ** md
S = S0 * mu * md
# Inner Values
if otype == 'call':
V = np.maximum(S - K, 0) # inner values for European call option
else:
V = np.maximum(K - S, 0) # inner values for European put option
z = 0
for t in range(M - 1, -1, -1): # backwards iteration
V[0:M - z, t] = (q * V[0:M - z, t + 1] +
(1 - q) * V[1:M - z + 1, t + 1]) * df
z += 1
return V[0, 0]
def Jarrow_Rudd_option_value(S0, K, T, r, sigma, otype, M=4):
''' Jarrow and Rudd's modified Cox-Ross-Rubinstein European option valuation.
Parameters
==========
S0 : float
stock/index level at time 0
K : float
strike price
T : float
date of maturity
r : float
constant, risk-less short rate
sigma : float
volatility
otype : string
either 'call' or 'put'
M : int
number of time intervals
'''
# Time Parameters
dt = T / M # length of time interval
df = math.exp(-r * dt) # discount per interval
# Binomial Parameters
u = math.exp((r-(sigma**2)/2)*dt + sigma * math.sqrt(dt)) # up movement
d = math.exp((r-(sigma**2)/2)*dt - sigma * math.sqrt(dt)) # down movement
q = (math.exp(r * dt) - d) / (u - d) # martingale branch probability
# Array Initialization for Index Levels
mu = np.arange(M + 1)
mu = np.resize(mu, (M + 1, M + 1))
md = np.transpose(mu)
mu = u ** (mu - md)
md = d ** md
S = S0 * mu * md
# Inner Values
if otype == 'call':
V = np.maximum(S - K, 0) # inner values for European call option
else:
V = np.maximum(K - S, 0) # inner values for European put option
z = 0
for t in range(M - 1, -1, -1): # backwards iteration
V[0:M - z, t] = (q * V[0:M - z, t + 1] +
(1 - q) * V[1:M - z + 1, t + 1]) * df
z += 1
return V[0, 0]
def Tian_option_value(S0, K, T, r, sigma, otype, M=4):
''' Tian's modfied Cox-Ross-Rubinstein European option valuation.
Parameters
==========
S0 : float
stock/index level at time 0
K : float
strike price
T : float
date of maturity
r : float
constant, risk-less short rate
sigma : float
volatility
otype : string
either 'call' or 'put'
M : int
number of time intervals
'''
# Time Parameters
dt = T / M # length of time interval
df = math.exp(-r * dt) # discount per interval
# Binomial Parameters
v = math.exp((sigma**2) * dt)
u = 0.5 * math.exp(r * dt) * v * (v + 1 + math.sqrt(v**2 + 2*v -3)) # up movement
d = 0.5 * math.exp(r * dt) * v * (v + 1 - math.sqrt(v**2 + 2*v -3)) # down movement
q = (math.exp(r * dt) - d) / (u - d) # martingale branch probability
# Array Initialization for Index Levels
mu = np.arange(M + 1)
mu = np.resize(mu, (M + 1, M + 1))
md = np.transpose(mu)
mu = u ** (mu - md)
md = d ** md
S = S0 * mu * md
# Inner Values
if otype == 'call':
V = np.maximum(S - K, 0) # inner values for European call option
else:
V = np.maximum(K - S, 0) # inner values for European put option
z = 0
for t in range(M - 1, -1, -1): # backwards iteration
V[0:M - z, t] = (q * V[0:M - z, t + 1] +
(1 - q) * V[1:M - z + 1, t + 1]) * df
z += 1
return V[0, 0]
def plot_convergence(mmin=1, mmax=30, step_size=1):
''' 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]
JR_values = [Jarrow_Rudd_option_value(S0, K, T, r, sigma, 'call', M) for M in m]
Tian_values = [Tian_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.plot(m, JR_values, label='JR values')
plt.plot(m, Tian_values, label='Tian 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)
### 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
def BSM_call_value(St, K, t, T, r, sigma):
''' Calculates Black-Scholes-Merton European call 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
=======
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
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