# Binomial Valuation
#
# Valuation of American Options
# with the Cox-Ross-Rubinstein Model
# Primal Algorithm
# Case 1: American Put Option (APO)
# Case 2: Short Condor Spread (SCS)
#
import math
import numpy as np
# General Parameters and Option Values
def set_parameters(otype, M):
''' Sets parameters depending on valuation case.
Parameters
==========
otype: int
option type
1 = American put option
2 = Short Condor Spread
'''
if otype == 1:
# Parameters -- American Put Option
S0 = 36. # initial stock level
T = 1.0 # time-to-maturity
r = 0.06 # short rate
sigma = 0.2 # volatility
elif otype == 2:
# Parameters -- Short Condor Spread
S0 = 100. # initial stock level
T = 1.0 # time-to-maturity
r = 0.05 # short rate
sigma = 0.5 # volatility
else:
raise ValueError('Option type not known.')
# Numerical Parameters
dt = T / M # time interval
df = math.exp(-r * dt) # discount factor
u = math.exp(sigma * math.sqrt(dt)) # up-movement
d = 1 / u # down-movement
q = (math.exp(r * dt) - d) / (u - d) # martingale probability
return S0, T, r, sigma, M, dt, df, u, d, q
def inner_value(S, otype):
''' Inner value functions for American put option and short condor spread
option with American exercise.
Parameters
==========
otype: int
option type
1 = American put option
2 = Short Condor Spread
'''
if otype == 1:
return np.maximum(40. - S, 0)
elif otype == 2:
return np.minimum(40., np.maximum(90. - S, 0)
+ np.maximum(S - 110., 0))
else:
raise ValueError('Option type not known.')
def CRR_option_valuation(otype, M=500):
S0, T, r, sigma, M, dt, df, u, d, q = set_parameters(otype, M)
# Array Generation for Stock Prices
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
# Valuation by Backwards Induction
h = inner_value(S, otype) # innver value matrix
V = inner_value(S, otype) # value matrix
C = np.zeros((M + 1, M + 1), dtype=np.float) # continuation values
ex = np.zeros((M + 1, M + 1), dtype=np.float) # exercise matrix
z = 0
for i in range(M - 1, -1, -1):
C[0:M - z, i] = (q * V[0:M - z, i + 1]
+ (1 - q) * V[1:M - z + 1, i + 1]) * df
V[0:M - z, i] = np.where(h[0:M - z, i] > C[0:M - z, i],
h[0:M - z, i], C[0:M - z, i])
ex[0:M - z, i] = np.where(h[0:M - z, i] > C[0:M - z, i], 1, 0)
z += 1
return V[0, 0]
