Lecture 4 - Monte Carlo improvements via variance reduction techniques: antithetic sampling Antithetic variates: for any one path obtained by a gaussian variate vector draw {zi}, we generate a mirror image by changing the sign of all random numbers {zi}. Then we compute the pay-off along the two paths. The second mirrored path has the same probability of the original one. So that we can combine the two pay-offs in a new estimator: The Monte Carlo error for the improved estimate is: 2/25/2019

Monte Carlo improvements via variance reduction techniques: control variates Control variates: the Monte Carlo simulation is carried out both for the original problem as well as a similar problem for which we have a closed form solution. Being f : the option value we want to estimate and y : the analytical exact value of the auxiliary option an improved estimator of f is: As a consequence of the above relations, control variates become more and more efficient as the auxiliary option is more correlated (or anti-correlated) with the original option, i.e. when the two problems are similar. 2/25/2019

Monte Carlo improvements via variance reduction techniques: control variates and p Let us back to the problem of estimate p. A good choice for a control variable is a polygon with n edges inscribed in the circle. Indeed: a closed formula exists for any value of n it has an high superposition with the circle Stochastic term The error scales again as 1/sqrt(N) but with a smaller proportional constant : 2/25/2019

Monte Carlo improvements: low discrepancy sequences – Quasi Monte Carlo In the standard Monte Carlo the error decreases slowly, as 1/sqrt(N), with number of samples, N, because draws do not fill in the space in a regular way. Indeed some gaps are present (clustering effects). In low discrepancy sequences the points are chosen in order to fill in the space more regularly and uniformly, without inhomogeneities. As a result the function to be integrated converges not as one over the square root of the number of samples (N) but much more closely as one over N (Quasi Monte Carlo). 2/25/2019

Monte Carlo improvements: low discrepancy sequences – definition and results The most famous algorithms to generate low discrepancy numbers are the Sobol and Halton sequences. The discrepancy is a measure of how inhomogeneous a set of D-dimensional vectors of random numbers fills in a unit hypercube. By definition, in a low discrepancy sequence (in D dimension), the discrepancy scales with the number of draws, N, as: 2/25/2019

Monte Carlo improvements: low discrepancy sequences – high dimensional behavior Pros: For a given precision, a lower number of scenarios are needed. Cons: The convergence speed depends on problem dimension (making the method inefficient in very high dimensions). Quasi Monte carlo simulation are not reliable when high dimensions are involved, with a breakdown of homogeneity along some hyper-planes. 2/25/2019

Monte Carlo pros and cons Can be used in high dimensional problems. Easy to implement Easily extensible to any type of pay-off Cons: Heavy from a computational point of view. 2/25/2019

Conclusion We have presented a powerful numerical technique to price exotic options: the Monte Carlo method. Numerical methods in finance will become more and more important due to the rapid growth in financial markets of the exotic products, with a clear trend to increase the complexity embedded in the exotic options. References: P. Jackel - Monte Carlo Methods in Finance, Wiley Finance, (2002). 2/25/2019