Example 3-2. Monte Carlo valuation of European call option with pure Python

Monte Carlo valuation of European call options with pure Python

mcs_pure_python.py

from time import time
from math import exp, sqrt, log
from random import gauss, seed

seed( 20000 )
t0 = time()

Parameters

S0 = 100. # initial value
K = 105. # strike price
T = 1.0 # maturity
r = 0.05 # riskless short rate
sigma = 0.2 # volatility
M = 50 # number of time steps
dt = T / M # length of time interval
I = 250000 # number of paths

Simulating I paths with M time steps

S = []
for i in range(I):
path = []
for t in range(M + 1 ):
if t == 0 :
path.append(S0)
else:
z = gauss(0.0, 1.0)
St = path[t - 1 ] exp((r - 0.5 sigma * 2 ) dt

• sigma sqrt(dt) z)
  path.append(St)
  S.append(path)

Calculating the Monte Carlo estimator

C0 = exp(-r T) sum([max(path[- 1 ] - K, 0 ) for path in S]) / I

Results output

tpy = time() - t0
print “European Option Value %7.3f” % C0
print “Duration in Seconds %7.3f” % tpy

Running the script yields the following output:

In  [ 20 ]: %run mcs_pure_python.py

Out[ 20 ]:  European Option Value 7.999

Duration in Seconds 34.258

Note that the estimated option value itself depends on the pseudorandom numbers

generated while the time needed is influenced by the hardware the script is executed on.

The major part of the code in Example 3-2 consists of a nested loop that generates step-

by-step single values of an index level path in the inner loop and adds completed paths to

a list object with the outer loop. The Monte Carlo estimator is calculated using Python’s

list comprehension syntax. The estimator could also be calculated by a for loop:

In  [ 21 ]: sum_val =   0.0

for path in S:

#   C-like  iteration   for comparison

sum_val +=  max(path[- 1 ]  -   K,   0 )

C0  =   exp(-r  *   T)  *   sum_val /   I

round(C0,    3 )

Out[21]:    7.999

Although this loop yields the same result, the list comprehension syntax is more compact

and closer to the mathematical notation of the Monte Carlo estimator.