Simple Monte Carlo Options Pricer In Python

Pre-Requisites:

Required:

  • Calculus
  • Probability and Statistics
  • Very basic programming

Recommended:

  • Stochastic Processes
  • Stochastic Calculus or an introductory asset pricing class

Understanding The Math

Assumptions

  • The price of our stock will generally increase with respect to time
  • The expected return is a fixed rate of the current share price
  • The stock follows a random-walk behavior

The Stock Price Evolution Model

Black-Scholes Option Pricing Model

Solving for Expectation

Final Expression

The Code

import math
import random

class SimpleMCPricer():
def __init__(self, expiry, strike, spot, vol, r, paths):
#The sigma value on the left side of the exponent
self.variance = vol**2 * expiry
#The sigma value on the right side of the e exponent
self.root_Variance = math.sqrt(self.variance)
#Corresponds to the (-1/2 * sigma^2)
self.itoCorr = -0.5*self.variance
##Corresponds to S0e^(rT - 1/2 sigma^2T)
self.movedSpot = spot*math.exp(r*expiry + self.itoCorr)
self.runningSum = 0
##Simulate for all paths
for i in range(0,paths):
thisGauss = random.randrange(0,1000,1)
thisGauss = thisGauss/1000
##Our rootVariance already has been multiplied by the expiry
thisSpot = self.movedSpot*math.exp(self.root_Variance*thisGauss)
#Determine payoff of this specific path
thisPayoff = thisSpot - strike
#Value of option is zero is our price is less than the strike
thisPayoff = thisPayoff if thisPayoff > 0 else 0
self.runningSum+=thisPayoff

self.mean = self.runningSum/paths
self.mean*= math.exp(-r * expiry)

def getMean(self):
return round(self.mean,2)

Run the Model!

model = SimpleMCPricer(2,32,30,.1,0.03,1000000)
model.getMean()
1.79

Sources and Further Reading

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