Monte Carlo and Quasi-Monte Carlo Methods 2004

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Real world complications that can easily be introduced into a simulation often make a problem too hard for exact mathematical treatment. Quasi-Monte Carlo methods can be used to drive a simulation byt replacing the random number sequence by very carefully chosen numbers that are much more balanced than random numbers are. The result is often a tremendous speedup for a given level of accuracy, or a tremendous increase in accuracy for a given computation time. This project pushes quasi-Monte Carlo methods into simulations that had hitherto been thought incapable of benefiting from them.

Those simulation techniques, known as Markov chain Monte Carlo, are used in many areas including materials science, analysis of educational testing data sets, biomedical research, robotics, computer graphics, and marketing. This project is also looking at other ways to improve Monte Carlo methods with similarly broad potential for benefit.

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Owen, A. Zahn et al.. Tribble, S. Please report errors in award information by writing to: awardsearch nsf. Construction of the following numbers. The most efficient way to construct the following numbers is to use the algorithm suggested by Antonov-Saleev and described in Press et al.

The second number in the sequence will be:. Therefore XOR the third direction number. The third number in the sequence is:. The fourth number depends on and on the first direction number, because the rightmost zero in the binary representation of 4 is in the first position:. And so forth. It is worth noticing that the numbers "fill" the gaps in the interval 0,1 looking for empty spaces, as if the procedure knew where the positions of all prior numbers were.

The Gray Code of an integer j is defined as. Table 2 displays some examples of Gray codes. Notice that G j-1 and G j differ, in their binary representations, only in the digit relative to the rightmost zero in the binary representation of j Therefore, in the construction of the jth Sobol number, the induction is the same as XORing all the direction numbers associated to the unit bits of G j.

It seemed that quasi-random sequences, Sobol's construction in particular, would revolutionize the use of Monte Carlo simulation in Finance thanks to the economy of time allowed by the method. But, as mentioned before, for problems of high dimension it is necessary to simultaneously use many primitive polynomials and the results become unsatisfactory. The high degree of dependence among dimensions arises because of the initial values freely chosen for the construction of the m ik in each dimension k.

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Theoretically, the only restriction on these numbers is that m ik is both odd and inferior to 2 i. Usually a deterministic rule is used for choosing them. These deterministic rules, although valid from a theoretical standpoint, generate surprisingly bad results empirically. Sobol presents some mathematical conditions that should be met in order to avoid such distortions. The equations that appear are complicated and Paskov and Traub have already solved them for problems involving up to dimensions.

It amounts to a very simple randomization of the choice process of the initial values m ik. For each dimension k, using a pseudo-random generator, generate d k uniform numbers such that. Unfortunately, we didn't succeed in reproducing such results.

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On the contrary, the dimensions can be quite correlated, especially when they are very high. The process of random generation of the initial direction numbers should have been more efficient for high dimensions; since more initial direction numbers were necessary, it should have been easier to break the regularities. Alternative random method of generation of the initial direction numbers. The procedure is as follows:. For each dimension k, using a pseudo-random generator, draw uniform numbers such that for.

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Generate randomly a seed for each dimension. It is preferable to choose large numbers as seeds; this will force the algorithm to use direction numbers of higher orders in the w-bit long array for each dimension this enhances the power of randomization of the algorithm.

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The method was used by Silva to generate dimensional Sobol sequences with points each. Table 4 , at the end, shows one example of the first 10 direction numbers for the first 50 dimensions using the method proposed here. The analysis of unit square plots of the sequences shows that even in high dimensions the correlation between consecutive dimensions was low and the generated points covered the unit square in a very uniform way.

This result can be improved with a shuffling of the generated points.

Monte Carlo and Quasi-Monte Carlo Methods

It has been possible to decrease the absolute value of all correlations between consecutive and non-consecutive dimensions to less than 20 percent. The number of dimensions that can be tackled with this method depends only on the amount of available primitive polynomials, currently around 8 million. Thus it seems that the curse of dimensionality has been broken in practice. The mathematical solution of the uniformity conditions proposed by Sobol for problems with dimension superior to is not available yet. This is carried out in the next section. We have applied our procedure to the pricing of Asian options.

We work initially with an option that depends on the geometric mean of the underlying asset for which there is a closed formula. This enables us to compare our results with a bulletproof benchmark. Then, we use an arithmetic mean option for which the theoretical price is obtained averaging a large number of Monte Carlo simulations.

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In turn these will provide us with the confidence intervals that will allow a comparison between Monte Carlo and the methods here outlined. Finally, in order to test our procedure with correlated assets, we show the results obtained when pricing a Asian basket option. We present first the results obtained with the pricing of an option whose price depends on the geometric average of prices, including the spot price. The dimension of this problem is therefore Parameters of interest are: spot price is In this case an analytic solution for the price of the option is available.

Following Kemna and Vorst , we obtain This provides us with a benchmark against which our simulations can be compared. Four pricing methods are used: crude Monte Carlo with the Cholesky decomposition, crude Monte Carlo with Schur decomposition, modified Sobol with Cholesky decomposition and modified Sobol with Schur decomposition. It is widely accepted in the literature. We varied the number of sample paths N generated by our simulations. These range from 1, to 50, with increments of 1, We don't show the results obtained with crude quasi-Monte Carlo because errors are too big for this dimension.

We have chosen two figures to help visualize the magnitude of the errors and the convergence patterns of the methods applied. In Figure 1 we can see the price obtained with each method. Focusing first on the quasi-Monte Carlo method we can see that the "optimized" Sobol sequence with the Cholesky decomposition does not diverge substantially from the theoretical price in a manner similar to a crude quasi-Monte Carlo. In fact it fares quite well when compared to a rea-lization of a crude Monte Carlo simulation.

This is already very encouraging.

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It seems indeed that the dimension has ceased to bear an effect on the results. However, in order to fully exploit the sequence we apply the Schur decomposition. The results are extremely encouraging. We can see that for a number as modest as 5, sample paths we are already very close to the theoretical price and convergence is extremely stable. As far as crude Monte Carlo is concerned, as expected, the Schur decomposition does not improve on the Cholesky decomposition.