Optimized Python Code for CRR

The Python code below is optimized in a manner consistent with Broadie and Detemple (1996) and Haug (1997) who apply a one-dimensional dynamic binomial tree. This approach takes the option values at the last column and stores them in a dynamic vector Opt(j) for j = 0, 1, … , n. After moving one step back, the values in the re-dimensioned Opt(j) for j = 0, 1, …, n - 1 will be replaced by the option values of the corresponding nodes at the penultimate column. This dynamic use of memory continues until we reach the apex node in the lattice. The code below for Cox Ross and Rubinstein is adapted from https://github.com/tienusss/Option_Calculations

# adapted from Haug but need to adjust for greeks (this code originally developed for Leisen reimer)

# I think it is ok s*u*d = s for CRR that is the only difference

# https://github.com/tienusss/Option_Calculations

# Import packages

import numpy as np

# Defined functions

def CRRBinomial(OutputFlag, AmeEurFlag, CallPutFlag, S, X, T, r, c, v, n):

# This functions calculates CRR price, delta and gamma for of American and European options

# This code is based on "The complete guide to Option Pricing Formulas" by Espen Gaarder Haug (2007)

# Translated from a VBA code

# OutputFlag:

# "P" Returns the options price

# "d" Returns the options delta

# "a" Returns an array containing the option value, delta and gamma

# AmeEurFlag:

# "a" Returns the American option value

# "e" Returns the European option value

# CallPutFlag:

# "C" Returns the call value

# "P" Returns the put value

# S is the share price at time t

# X is the strike price

# T is the time to maturity in years (days/365)

# r is the risk-free interest rate

# c is the cost of carry rate

# v is the volatility

# n determines the stepsize

# Creates a list with values from 0 up to n (which will be used to determine to exercise or not)

n_list = np.arange(0, (n + 1), 1)

# Checks if the input option is a put or a call, if not it returns an error

if CallPutFlag == 'C':

z = 1

elif CallPutFlag == 'P':

z = -1

else:

return 'Call or put not defined'

# Calculates the stepsize in years

dt = T / n

# The up and down factors

u = np.exp(v*np.sqrt(dt))

d = 1./u

p = (np.exp((c)*dt)-d) / (u-d)

df = np.exp(-r * dt)

# Creates the most right column of the tree

max_pay_off_list = []

for i in n_list:

i = i.astype('int')

max_pay_off = np.maximum(0, z * (S * u ** i * d ** (n - i) - X))

max_pay_off_list.append(max_pay_off)

# The binominal tree

for j in np.arange(n - 1, 0 - 1, -1):

for i in np.arange(0, j + 1, 1):

i = i.astype(int) # Need to be converted to a integer

if AmeEurFlag == 'e':

max_pay_off_list[i] = (p * max_pay_off_list[i + 1] + (1 - p) * max_pay_off_list[i]) * df

elif AmeEurFlag == 'a':

max_pay_off_list[i] = np.maximum((z * (S * u ** i * d ** (j - i) - X)),

(p * max_pay_off_list[i + 1] + (1 - p) * max_pay_off_list[i]) * df)

if j == 2:

gamma = ((max_pay_off_list[2] - max_pay_off_list[1]) / (S * u ** 2 - S * u * d) - (

max_pay_off_list[1] - max_pay_off_list[0]) / (S * u * d - S * d ** 2)) / (

0.5 * (S * u ** 2 - S * d ** 2))

if j == 1:

delta = ((max_pay_off_list[1] - max_pay_off_list[0])) / (S * u - S * d)

price = max_pay_off_list[0]

# Put all the variables in the list

variable_list = [delta, gamma, price]

# Return values

if OutputFlag == 'P':

return price

elif OutputFlag == 'd':

return delta

elif OutputFlag == 'g':

return gamma

elif OutputFlag == 'a':

return variable_list

else:

return 'Indicate if you want to return P, d, g or a'

#Body of the script

# CRRBinomial(OutputFlag, AmeEurFlag, CallPutFlag, S, X, T, r, c, v, n)

S = 100

X = 100

T = 1.

r = 0.05

c = 0.05

v = 0.2

n = 100

Eur_call_result = CRRBinomial('P', 'e', 'C', S, X, T, r, c, v, n)

American_call_result = CRRBinomial('P', 'a', 'C', S, X, T, r, c, v, n)

Eur_put_result = CRRBinomial('P', 'e', 'P', S, X, T, r, c, v, n)

American_put_result = CRRBinomial('P', 'a', 'P', S, X, T, r, c, v, n)

#Print the output of the results

print('The price of the European call option is equal to ' +str(Eur_call_result))

print('The price of the American call option is equal to ' +str(American_call_result))

print('The price of the European put option is equal to ' +str(Eur_put_result))

print('The price of the American put option is equal to ' +str(American_put_result))