I am trying to calculate the price of a european call option using the Monte Carlo approach. I coded the algorithm in C++ and Python. As far as I know the implementation is correct and as N (the number of trials) gets bigger, the price should converge to a similar value in both the programs.
My problem is that as N gets bigger, say just from 1000 to 10000 trials the prices converge to two different values. In C++ the price converges towards the value of 3.30 while with Python it converges towards 3.70.
I think that gap of 0.40 is too wide, I should get more similar resutls. Why is this gap so big? What did I do wrong? I cannot seem to find my mistake.
Here is the code I used:
Python
import numpy as np
import matplotlib.pyplot as plt
def stoc_walk(p,dr,vol,periods):
w = np.random.normal(0,1,size=periods)
for i in range(periods):
p += dr*p + w[i]*vol*p
return p
s0 = 10;
drift = 0.001502
volatility = 0.026
r = 0.02
days = 255
N = 10000
zero_trials = 0
k=12
payoffs = []
for i in range(N):
temp = stoc_walk(s0,drift,volatility,days)
if temp > k:
payoff = temp-k
payoffs.append(payoff*np.exp(-r))
else:
payoffs.append(0)
zero_trials += 1
payoffs = np.array(payoffs)
avg = payoffs.mean()
print("MONTE CARLO PLAIN VANILLA CALL OPTION PRICING")
print("Option price: ",avg)
print("Initial price: ",s0)
print("Strike price: ",k)
print("Daily expected drift: ",drift)
print("Daily expected volatility: ",volatility)
print("Total trials: ",N)
print("Zero trials: ",zero_trials)
print("Percentage of total trials: ",zero_trials/N)
C++
//Call option Monte Carlo evaluation;
#include <iostream>
#include <random>
#include <math.h>
#include <chrono>
using namespace std;
/* double stoc_walk: returns simulated price after periods
p = price at t=t0
dr = drift
vol = volatility
periods (days)
*/
double stoc_walk(double p,double dr,double vol,int periods)
{
double mean = 0.0;
double stdv = 1.0;
/* initialize random seed: */
int seed = rand() %1000 + 1;
//unsigned seed = std::chrono::system_clock::now().time_since_epoch().count();
std::default_random_engine generator(seed);
std::normal_distribution<double> distribution(mean,stdv);
for(int i=0; i < periods; i++)
{
double w = distribution(generator);
p += dr*p + w*vol*p;
}
return p;
}
int main()
{
//Initialize variables
double s0 = 10; //Initial price
double drift = 0.001502; //daily drift
double volatility = 0.026; //volatility (daily)
double r = 0.02; //Risk free yearly rate
int days = 255; //Days
int N = 10000; //Number of Monte Carlo trials
double zero_trials = 0;
double k = 12; //Strike price
int temp = 0; //Temporary variable
double payoffs[N]; //Payoff vector
double payoff = 0;
srand (time(NULL)); //Initialize random number generator
//Calculate N payoffs
for(int j=0; j < N; j++)
{
temp = stoc_walk(s0,drift,volatility,days);
if(temp > k)
{
payoff = temp - k;
payoffs[j] = payoff * exp(-r);
}
else
{
payoffs[j] = 0;
zero_trials += 1;
}
}
//Average the results
double sum_ = 0;
double avg_ = 0;
for(int i=0; i<N; i++)
{
sum_ += payoffs[i];
}
avg_ = sum_/N;
//Print results
cout << "MONTE CARLO PLAIN VANILLA CALL OPTION PRICING" << endl;
cout << "Option price: " << avg_ << endl;
cout << "Initial price: " << s0 << endl;
cout << "Strike price: " << k << endl;
cout << "Daily expected drift: " << drift*100 << "%" << endl;
cout << "Daily volatility: " << volatility*100 << "%" << endl;
cout << "Total trials: " << N << endl;
cout << "Zero trials: " << zero_trials << endl;
cout << "Percentage of total trials: " << zero_trials/N*100 << "%";
return 0;
}
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