# One period binomial option pricing
from math import *
import numpy as np
u = 1.02
d = 0.95
r = 0.01
T = 2
So = 100
K = 100
# produce the stock price vector at the nth period where the 0th period has one price So
def stock(So, u, d, n):
s = np.zeros((n+1,1))
for i in range(n+1):
s[i] = So*(d**i)*(u**(n-i))
return s
S = stock(So,u,d,1)
print("The stock price vector at period 1 is \n", S)
print("...\n\n")
# produce the risk neutral probability vector at the nth period
def risk_neutral_prob(u, d, r, T, n):
delta_T = T/n
q = (exp(r*delta_T) - d)/(u-d)
Q = np.zeros((n+1,1))
for i in range(n+1):
Q[i] = ((1-q)**i)*(q**(n-i))
if Q[i] < 0 or Q[i] > 1 :
print("Arbitrage condition not satisfied !")
return np.zeros((n+1,1))
return Q
Q = risk_neutral_prob(u, d, r, T, 1)
# produce the call option payoff function with strike K at the nth period
def Euro_Call(S_o, u, d, K, n):
S = stock(So, u,d, n)
C = np.zeros((n+1,1))
for i in range(n+1):
C[i] = max(S[i]-K, 0)
return C
# produce the put option payoff function with strike K at the nth period
def Euro_Put(S_o, u, d, K, n):
S = stock(So, u,d, n)
P = np.zeros((n+1,1))
for i in range(n+1):
P[i] = max(K - S[i], 0)
return P
# produce the butterfly spread payoff function with strike K at the nth period
def Butterfly(S_o, u, d, K1, K2, K3, n):
S = stock(So, u,d, n)
B = np.zeros((n+1,1))
for i in range(n+1):
if K1 <= S[i] < K2:
B[i]= S[i] - K1
if K2 <= S[i] < K3:
B[i] = K3 - S[i]
return B
# produce the butterfly spread payoff function with strike K at the nth period
def Strangle(S_o, u, d, K1, K2, n):
S = stock(So, u,d, n)
Strangle = np.zeros((n+1,1))
for i in range(n+1):
if S[i] <= K1:
Strangle[i]= K1 - S[i]
if S[i] >= K2:
Strangle[i] = S[i] - K2
return Strangle
# risk neutral pricing for an option with a given pay_off function with expiry T in a n period model
def risk_neutral_pricing(So, u, d, n, T, pay_off):
v = 0
S = stock(So, u,d, n)
Q = risk_neutral_prob(u,d,r,T,n)
delta_T = T/n
for i in range(n+1):
v += exp(-r*delta_T)* pay_off[i]*Q[i]
return v
#Only produce output if no arbitrage condition is satisfied
if Q.any() :
print("The risk neutral probability vector at period 1 is \n", Q)
print("...\n\n")
call_pay_off = Euro_Call(So, u, d, K, 1)
print("The payoff of Euro call in the one period model with strike ", K, " is \n", call_pay_off)
print("...\n\n")
put_pay_off = Euro_Put(So, u, d, K, 1)
print("The payoff of Euro put in the one period model with strike ", K, " is \n", put_pay_off)
print("...\n\n")
butterfly_pay_off = Butterfly(So, u, d, 90, 100, 110, 1)
print("The payoff of butterfly spread in the one period model with strikes 90, 100, 110 is \n", butterfly_pay_off)
print("...\n\n")
strangle_pay_off = Strangle(So, u, d, 90, 110, 1)
print("The payoff of the strangle in the one period model with strikes 90, 110 is \n", strangle_pay_off)
print("...\n\n")
call_0 = risk_neutral_pricing(So, u, d, 1, 1, call_pay_off)
print("The price of Euro call in the one period model with strike ", K, " is \n", call_0)
print("...\n\n")
put_0 = risk_neutral_pricing(So, u, d, 1, 1, put_pay_off)
print("The price of Euro put in the one period model with strike ", K, " is \n", put_0)
print("...\n\n")
butterfly_0 = risk_neutral_pricing(So, u, d, 1, 1, butterfly_pay_off)
print("The price of butterfly spread in the one period model with strikes 90, 100, 110 is \n", butterfly_0)
print("...\n\n")
strangle_0 = risk_neutral_pricing(So, u, d, 1, 1, strangle_pay_off)
print("The price of the strangle in the one period model with strikes 90, 110 is \n", strangle_0)
print("...\n\n")