Manual Monte Carlo and Quasi-Monte Carlo Methods 2004

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We feel however that this is due to the much greater attention quasi-Monte Carlo has received in academic circles than in the markets. Quasi-Monte Carlo is mentioned, but no road map is offered as far as implementation and situations where the technique might or not be efficient. This is obviously due to the problems faced by quasi-Monte Carlo when dealing with problems of high dimensions 50 could be an appropriate subjective threshold.

The failure of the method in treating such cases has generated a feeling of unsafety that has so far mitigated market practitioner's response to it. It is thus not surprising that much research has been devoted to solving this shortcoming. Various solutions have been put forward but none seems to have been adopted by a wide range of users.

Although it could be argued that this is due to the state of infancy in which quasi-Monte Carlo methods lie, we believe it is due to a combination of two factors. One of these is the case by case approach, where it is shown that a variant of a deterministic method works for a particular financial instrument, reducing the scope of its applications. The other factor is the implementation issue.

In many cases the complexity of the simulation makes it lose some of its attractiveness. Therefore, the aim of this paper is to introduce a technique, at the same time easy to implement and of general use, which is unaffected by the dimensionality of the problem. We hope to catch the attention of some practitioners with the applicability of this procedure and contribute to the popularization of deterministic methods. The rest of this paper is organized as follows. The next section offers a review of the different approaches that have been adopted for the treatment of the problem of dimensionality.

We outline the main trends and comment on the results obtained. We then proceed to introduce our simulation procedure.

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We show how to construct a modification of the Sobol sequence and how to apply it to the pricing of derivatives. In the third section the results of our experiments, the pricing of Asian options of dimension up to , 1 are presented. We will then be able to judge its effectiveness as we compare it to traditional Monte Carlo and to a purely deterministic method.

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We finally conclude with suggestions as to what could be done to refine this procedure. The use of Quasi-Monte Carlo in Finance is relatively recent. The pioneering contributions of Paskov and Traub , Galanti and Jung , Joy, Boyle and Tan were all published in the latter half of the nineties. However, in other fields, notably in Physics, researchers have long been aware of the advantages and disadvantages of quasi-Monte Carlo.

Quasi-Monte Carlo Methods Applied to Tau-Leaping in Stochastic Biological Systems

Interestingly the authors conclude that the " error reduction for quasi-Monte Carlo is limited as the spatial dimension increases. However, the results obtained in finance seemed at first more appealing.

(ML 18.1) Markov chain Monte Carlo (MCMC) introduction

Indeed, Paskov and Traub price a CMO with dimension and find that quasi-random methods outperform standard Monte Carlo even with the use of variance reduction methods. But, in Galanti and Jung the quasi-random sequences cease to be effective for certain ranges of dimensionality and appear to converge again for other dimensions.

Quasi-Monte Carlo Methods Applied to Tau-Leaping in Stochastic Biological Systems

Sometimes the Faure sequence is more efficient, sometimes it is the Sobol sequence. Joy, Boyle and Tan don't venture in these territories since the higher dimension tested is 52 for an Asian option. They used the Faure sequence and found it more efficient than standard Monte Carlo methods. A later research by Boyle et al.

The sequences of Faure and Sobol are used. At that point Faure sequences ceased to be appropriate for the pricing of the Asian option. Sobol was still close to the theoretical result but it was clear that, as expected, the pricing error increased with the dimension.

These somewhat conflicting results may be explained by what has been coined the "effective dimension" of the problem as shown by Caflisch, Morokoff and Owen In this paper the authors try to explain how such surprising results could be obtained and conclude that the dimension that "matters" for the simulation in the case for example of the CMO used by Paskov and Traub is much lower than They emphasize that "quasi-Monte Carlo methods provide significant improvements in accuracy and computational speed for problems of small to moderate dimension".

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We feel that one of the main reasons that has kept market professionals reluctant to embrace deterministic methods, even for problems of small to moderate dimensions, is the confusion generated by the initial results. Some sequences fare better than others, depending on the problem at hand. Some sequences cease to be effective for some dimensions, and then converge again.

Why is it that one can use the sequence for a mortgage-backed security of dimension and not use it for an Asian option of dimension ? Nonetheless, it seems now that what will be called "crude quasi-Monte Carlo" has been accepted as non-operational for high dimensional problems. Much of the research is now addressing this problem. We will now see what has been done in order to circumvent this problem. It is noteworthy that another interesting line of work resides, following the steps of Cranley and Patterson , in an attempt to define confidence intervals for quasi-Monte Carlo simulations.

The absence of these estimates may also be a hindrance for practitioners.

Researchers have tried to extend quasi-Monte Carlo in many ways. We can classify the approaches in two categories, those that try to produce new sequences better adapted to the problems at hand and those that try to modify the problems to render them compatible with quasi-Monte Carlo. The first trend is best represented by Niederreiter and Xing who have produced a number of sequences which improve substantially on Sobol for a comparison see for example Niederreiter and Xing, Another approach for the generation of sequences that could be used for quasi-Monte Carlo simulations is a scrambling of the deterministic sequence.

In an attempt to derive the variance of quasi-Monte Carlo simulations Owen a, b showed that his method of randomization of deterministic sequences, which maintains their main characteristics, improves accuracy over high dimensions. These ideas have been very well adapted, with some modifications for greater ease of use, by Tan and Boyle to the pricing of financial derivatives. Results are shown for an Asian option with dimensions 50, and The scrambled sequence outperforms crude quasi-Monte Carlo and standard Monte Carlo. The second trend has focused on the modification of the problem being treated.

Indeed, when dealing with the pricing of options for which sample paths of the underlying asset have to be generated, it is possible to "reduce" the dimensionality of the problem through a carefully chosen discretization. This is shown in a very nice exposition by Moskowitz and Caflisch where the authors use the Brownian bridge to evaluate high dimensional Feynman Kac path integrals.

The results are very promising but the dimensions tested don't exceed Although this line of work is very useful we understand that the first approach is more promising for it seeks greater generality, through a procedure that can be applied to all problems.

Indeed many problems cannot benefit from the treatment suggested. An example would be the calculation of VaR. Incidently this has been tested by Papageorgiou and Paskov but with only 34 risk factors. The method we present has two objectives: the first is to attain greater coverage of financial instruments for which quasi-Monte Carlo remains operational and the second is the ease of implementation, an important factor for use in non-academic environments.

We show our construction below. We first review the standard algorithm for the generation of uniform numbers in the unit hypercube. This will facilitate the understanding of the innovation proposed. Press et al.

Quasi-Monte Carlo Methods Applied to Tau-Leaping in Stochastic Biological Systems

We follow a sequential approach. Initially we construct a vector of numbers, known as direction numbers, of length w that will serve as a base for the calculation of the Sobol numbers. We need a direction number for each digit, in base 2, of the numbers that will be used in the sequence. The construction of these direction numbers is quite complicated and will be sketched in what follows. Given a series of integers a 1 , a 2 , It is necessary, of course, to supply the initial values of in each dimension.

These can, in principle, be chosen amongst any of the odd integers inferior to , with. The next step is to transform in binary fractions:. This is tantamount to transforming a decimal integer m ik to its equivalent in base 2 and then shifting the position of the fractional point by i positions to the left. Table 1 displays some examples of direction numbers.

Each dimension k of the Sobol sequence is created with the use of a different primitive polynomial. There are several tables of these polynomials available in the literature. These should suffice for nearly all practical problems in Finance. Construction of the initial term of the Sobol sequence for each dimension k.