In my previous article, price of european option (put and call) is estimated by using binomial model (Cox-Ross-Rubinstein (CRR) approach [1]). In this article, i provide java and Scilab (similar to Matlab) source code to estimate these option prices by Monte Carlo simulation.
In binomial model, intrinsic value of an asset (S_T) at expiry time (T) is estimated with a sequence of discrete time steps, at each step, stock price is estimated with a probability (either down or up probability. In monte carlo simulation, intrinsic value of an asset (S_T) at expiry time (T) is obtained from a normal distribution such as [2]:
where, r is annual interest rate S_t asset price at time t and sigma is volatility and x is normal distribution variable.Having estimated S_T, option's payoff can be estimated easily (max (0, X-S_T) or max (0, S_T-X) for call and put options respectively, where X is options strike, exercise price). Based on expected values of pay off functions which generated with monte carlo simulation, option prices can be calculated as follows for european call option:
Source code of this estimation in java and Scilab is listed
1: // SciLab (Matlab) Source code
2: // Estimate European Option price by Monte Carlo Simulation
3: // denizstij (http://denizstij.blogspot.com/),
4: // Aug-2009
5: asset=230; //S
6: strike=210;//X
7: volatility=0.25; //sigma
8: r=0.04545;
9: time=0.5;
10: num_sims=10000;
11: R= (r-0.5*volatility^2)*time;
12: SD=volatility*sqrt(time);
13: sum_call_payoffs=0.0;
14: sum_put_payoffs=0.0;
15: for i=0:num_sims,
16: S_T= asset*exp(R+SD*rand(1,'normal'));
17: sum_call_payoffs=sum_call_payoffs+max([0,S_T-strike]);
18: sum_put_payoffs=sum_put_payoffs+max([0,strike-S_T]);
19: end
20: eu_call_option_price= exp(-r*time)*(sum_call_payoffs/double(num_sims))
21: eu_put_option_price= exp(-r*time)*(sum_put_payoffs/double(num_sims))
1: package com.denizstij.finance.pricing;
2: import java.util.ArrayList;
3: import java.util.List;
4: import java.util.Random;
5: /**
6: *
7: * Estimate European Option price by Monte Carlo Simulation
8: * @author denizstij (http://denizstij.blogspot.com/)
9: * Aug-2009
10: *
11: */
12: public class EuropeanOptionPricingByMonteCarlo {
13: /**
14: * Estimate European Option price by Monte Carlo Simulation
15: *
16: * @param asset Current Asset Price
17: * @param strike Exercise Price
18: * @param volatility Annual volatility
19: * @param intRate Annual interest rate
20: * @param expiry: Time to maturity (in terms of year)
21: * @param num_sim : Number of simulations
22: * @return Put and call price of european options based on
23: * Monte Carlo Simulation
24: */
25: public strictfp List<Double> estimatePrice(double asset,
26: double strike,
27: double volatility,
28: double intRate,
29: double time,
30: int num_sim) {
31: List<Double> results = new ArrayList<Double>();
32: double R = (intRate - 0.5 *Math.pow(volatility,2))*time;
33: double SD = volatility * Math.sqrt(time);
34: double dF = Math.exp(-intRate*time); // discount Factor
35: double sumCallPayoffs=0.0;
36: double sumPutPayoffs=0.0;
37: Random random= new Random();
38: for (int i = 0; i <= num_sim; i++) {
39: double nextGaussian = random.nextGaussian();
40: double S_T= asset*Math.exp(R+SD*nextGaussian);
41: sumCallPayoffs+=callPayOff(S_T,strike);
42: sumPutPayoffs+=putPayOff(S_T,strike);
43: }
44: double callOptionPrices= dF*sumCallPayoffs/num_sim;
45: double putOptionPrices= dF*sumPutPayoffs/num_sim;
46: results.add(callOptionPrices);
47: results.add(putOptionPrices);
48: return results;
49: }
50: // Pay off method for put options
51: private double putPayOff(double stockPrice, double strike) {
52: return Math.max(strike - stockPrice, 0);
53: }
54: // Pay off method for call options
55: private double callPayOff(double stockPrice, double strike) {
56: return Math.max(stockPrice - strike, 0);
57: }
58: public static void main(String args[]) {
59: EuropeanOptionPricingByMonteCarlo euOptionPricing = new EuropeanOptionPricingByMonteCarlo();
60: List<Double> results = euOptionPricing.estimatePrice(
61: 230,
62: 210,
63: 0.25,
64: 0.04545,
65: 0.5, // In terms of year
66: 10000);
67: Double callOptionPrice = results.get(0);
68: Double putOptionPrice = results.get(1);
69: System.out.println("call Option Price:" + callOptionPrice);
70: System.out.println("put Option Price:" + putOptionPrice);
71: }
72: }
