finance c++ programming course [part 5] – basket options with monte carlo c++

Part 5

c++ finance course

Monte-Carlo c++ – basket options

Objective

– price basket options

based on this you’ll be able to price Autocallable, Himalaya, Spread and similar basket options

Basket option

Here let’s see how to price an option on 2 stocks with payoff
$$ ( w_1S_1+w_2S_2-K)^+ $$
where $$ w_i $$ are basket weights

let’s consider zero interest rate environment and n non-dividend paying stocks with dynamics: (for interest rates and dividends just add $$(r-q)dt$$ terms to this dynamics

$$dS_1=S_1 \sigma_1 dW_1(t)$$
$$dS_2=S_2 \sigma_2 dW_2(t)$$

$$ dW_i dW_j=\rho_{i,j} dt$$

where $$\rho_{i,j}$$ is a stock correlation matrix

given the correlation matrix we’ll construct it’s Cholesky decomposition.
Cholesky decomposition can be done with boost but it’s easier to do it with QuantLib implementation.
to store correlation and cholesky matrices we’ll use QuantLib::Matrix class.
we’ll store vector of independent gaussians in the QuantLib::Array class.
we don’t store in std::vector class because to be able to multiply matrix by vector they need to come from the same library.

to obtain correlated gaussians we’ll generate vector of independent gaussians $$\vec{\epsilon}= (\epsilon_1, \epsilon_2)$$ first.
then vector of correlated gaussians is $$M_{cholesky} \vec{\epsilon}$$

here’s the code for wights =1

#include <boost/random/variate_generator.hpp>
#include <boost/random/mersenne_twister.hpp>
#include <boost/random/normal_distribution.hpp>
#include <iostream>
#include <ql/quantlib.hpp>


double pp(double x) //positive part   x+   
{
if(x>0) return x;
return 0;
}

void main()
{

	boost::mt19937 mt; 
    boost::normal_distribution<> distribution;
    boost::variate_generator<boost::mt19937&,boost::normal_distribution<>>* normal(new boost::variate_generator<boost::mt19937&,boost::normal_distribution<>>(mt,distribution));
	
	int numberSimulations=10000;

	std::vector<double> S0(2);
	S0[0]=100.0;// S1(0) - spot price of first stock
	S0[1]=100.0;//S2(0) - spot price of second stock

	std::vector<double> sigma(2);
	sigma[0]=0.2; //volatility of S1
	sigma[1]=0.2;//volatility of S2

	double T=1.0;//maturity in years
	double K=100.0;//strike of option
	double sum=0;//for monte-carlo averaging
	double payoff=0;
	double NT=100;//number of time intervals
	double dt=T/NT;
	
	QuantLib::Matrix rho(2,2);//correlation matrix

	rho[0][0]=1;rho[0][1]=0.5;
	rho[1][0]=0.5;rho[1][1]=1;

	QuantLib::Matrix cholesky=QuantLib::CholeskyDecomposition(rho);
	
	for(int iSim=0;iSim<numberSimulations;iSim++) //outer loop for simulations
	{
	
		std::vector<double> Sprev(S0); //create a vector equal to S0 with its values
		std::vector<double> Snext(2);//create vector with 2 elemens

		for(double t=0;t<T;t+=dt) //inner loop for constructing one stock price from time 0 to T
		{
			QuantLib::Array epsilon(2);//create a vector with 2 elements .we'll put here independent gaussians
			epsilon[0]=(*normal)();
			epsilon[1]=(*normal)();

			QuantLib::Array x(2);
			x=cholesky*epsilon; //correlated gaussians

			for(int iStock=0;iStock<2;iStock++) //loop over all stocks 
			{
				Snext[iStock]=Sprev[iStock]*std::exp(-0.5*sigma[iStock]*sigma[iStock]*dt+sigma[iStock]*std::sqrt(dt)*x[iStock]);
				Sprev[iStock]=Snext[iStock];
			}
		}

		payoff=pp(Snext[0]+Snext[1]-K);

		sum+=payoff;
	}
	
	
	std::cout<<"\nBasket Call="<<sum/numberSimulations<<std::endl;
	
	int dummy;std::cin>>dummy;
}

Exercises

– modify the program to calculate asian basket payoff
– modify the program to N stocks
– modify program to include interest rates and dividend yield