Efficient methods for modelling one extra contract

We now describe two numerical methods for the efficient modelling of one extra contract on a portfolio. For both these methods we make the following assumptions:

• n is the original number of indices in the portfolio;

• m is the number of years of historical data;

• k is the number of simulations;

• Xh is the original historical data in an n by m matrix;

• Zh is the same data transformed to standard normal distributions;

• we have simulated these n indices for k years using the rank correlation method;

• Zs is the n by k matrix of the normally distributed simulations;

• Xs are the final simulations with correct marginals;

• xh is the 1 by m vector of the new historical data;

• zh is the same when transformed to a standard normal distribution;

• ph is the 1 by m vector of historical pay-offs for the portfolio;

• qh is the same when transformed to a standard normal distribution using the empirical CDF;

• ps is the 1 by k vector of simulated pay-offs for the portfolio;

• qs is the same when transformed to a standard normal distribution using the empirical CDF.