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Beating Monte Carlo

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Beating Monte Carlo SIMULATION BEATING MONTE CARLO Simulation methods using low-discrepancy point sets beat Monte Carlo hands down when valuing complex financial derivatives, report Anargyros Papageorgiou and Joseph Traub onte Carlo simulation is widely used to ods with basic Monte Carlo in the valuation of alised Faure achieves accuracy of 10"2 with 170 M price complex financial instruments, and a collateraliscd mortgage obligation (CMO). points, while modified Sobol uses 600 points. much time and money have been invested in We found that deterministic methods beat The Monte Carlo method, on the other hand, re• the hope of improving its performance. How• Monte Carlo: quires 2,700 points for the same accuracy, ever, recent theoretical results and extensive • by a wide margin. In particular: (iii) Monte Carlo tends to waste points due to computer testing indicate that deterministic (i) Both the generalised Faure and modified clustering, which severely compromises its per• methods, such as simulations using Sobol or Sobol methods converge significantly faster formance when the sample size is small. Faure points, may be superior in both speed than Monte Carlo. • as the sample size and the accuracy de• and accuracy. (ii) The generalised Faure method always con• mands grow. In particular: Tn this paper, we refer to a deterministic verges at least as fast as the modified Sobol (i) Deterministic methods are 20 to 50 times method by the name of the sequence of points method and often faster. faster than Monte Carlo (the speed-up factor) it uses, eg, the Sobol method. Wc tested the gen• (iii) The Monte Carlo method is sensitive to the even with moderate sample sizes (2,000 de• eralised Faure sequence due to Tezuka (1995) initial seed. terministic points or more). and a modified Sobol method which includes • for a small number of sample points: (ii) When high accuracy is desired, determin• additional improvements to those document• (i) Deterministic methods achieve a low error istic methods can be as much as 1,000 times ed in Paskov & Traub (1995). We compared level with a small number of points. faster than Monte Carlo. these two low-discrepancy deterministic meth• (ii) For the most difficult CMO tranche, gener• There are two ways of valuing financial de• rivatives: via paths or as a high-dimensional integral. For simplicity, we will restrict our• 1. Distribution of 512 pseudo-random points selves to a discussion of the integral formula• 1.0 tion and, without loss of generality (Paskov, 1996), the integral over the unit cube in d di• 0.9 - mensions. For most finance problems, this in• 0.8 - tegral cannot be analytically computed; we 0.7 - have to settle for a numerical approximation. 0.6 - The basic Monte Carlo method obtains this approximation by computing the arithmetic 0.5 - mean of the integrand evaluated at randomly 0.4 - chosen points. More precisely, only pseudo• 0.3 - random points can be generated on a digital 0.2 - computer and these are used in lieu of random 0.1 - points. There are sophisticated variations of this method but we refer only to the basic ver• 0.0 -J • . ^ r— -, , • —^ ^— , sion in this paper. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 If pseudo-random points from a flat dis• tribution are plotted on the unit square in two dimensions (see figure 1), there are some re• 2. Distribution of 512 low-discrepancy points gions where no sample points occur and oth• ers where the points are more concentrated. 1.0 n This is clearly undesirable. Random point sam• 0.9 - ples are wasted due to clustering. Indeed, 0.8 - Monte Carlo simulations with very small sam• 0.7 - ple sizes camiot be trusted. It would be better to place our sample points as uniformly as pos• 0.6 - sible, which is the idea behind low-discrepan• 0.5 - cy sequences. Discrepancy is a measure of de• 0.4 - viation from uniformity; hence low7-discrep- 0.3 - ancy points are desirable. Figure 2 shows a plot of certain low-discrepancy points on the unit 0.2 - square in two dimensions. 0.1 - A low-discrepancy method approximates 0 0 - the integral by computing the arithmetic mean 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 of the integrand evaluated at low-discrepan• cy points. Low-discrepancy sequences have RISK VOL9/NO 6/JUNE 1996 SIMULATION been extensively studied.1 In contrast to the In September 1995, IBM announced a prod• of points a method needs to achieve and main• Monte Carlo method, these use deterministic uct called the Deterministic Simulation Blaster, tain a relative error below a specified level say, points. They are sometimes said to be quasi- which uses a low-discrepancv deterministic 10~2. This is gauged by the speed-up of one random. method (Risk Technology Supplement, August method relative to another, which we define In 1992, the conventional wisdom was that 1995, pages 23-24; K/'s£ November 1995, page as the ratio of the least number of points re• although theory suggested that low-discrep• 47). The company claimed a very large im• quired by one method, divided by the least ancy methods were sometimes superior to provement over Monte Carlo. However, IBM number of points required by the other Monte Carlo, this theoretical advantage was has not revealed the method for choosing the method. This ensures that both methods main• not seen for high-dimensional problems. sample points and the methodology for calcu• tain the same level of accuracy. Joseph Traub and a PhD student, Spassimir lating the speed-up. IBM acknowledges that Col• This definition of speed-up is new: we study Paskov, decided to compare the efficacy of low- umbia University pioneered the use of low-dis• the convergence and the error of a method discrepancy and Monte Carlo methods in valu• crepancy methods to price financial derivatives. throughout a simulation. We believe that this ing financial derivatives. has advantages over speed-up calculations They used a CMO (Fannie Mae REMIC Results which are based only on the error values at the Trust 1989-2023) provided by Goldman Sachs, We have built a software system called Find• end of a simulation. Note that our definition with 10 tranches requiring the evaluation of 10 er - for computing high-dimensional integrals of speed-up is a more rigorous requirement integrals, each over 360 dimensions, and a par• which has modules for generating generalised than only computing the confidence level of ticular low-discrepancy sequence due to Sobol. Faure points and modified Sobol points. It in• Monte Carlo. The values of the tranches depended on the in• cludes major improvements. Indeed, a num• Our extensive testing has shown that fixed terest rate and prepayment models used. ber of financial institutions have informed us accuracy requires different tranches to be treat• Paskov & Traub made major improvements in that they could not replicate our results using, ed with different numbers of points. Again, we the Sobol points, which led to a more uniform for example, the Sobol point generator found emphasise that deterministic methods beat distribution. The improved Sobol method con• in Press et al (1992). Monte Carlo for every tranche. Our results are sistently outperformed Monte Carlo (Paskov We used Finder to price the CMO and to reported using the residual tranche of the & Traub, 1995, and Paskov, 1996). compare low-discrepancy methods with Monte CMO, which depends on all 360 monthly in• Software construction and testing of low-dis• Carlo simulation. As both types of method terest rates, as the reference point, since it is crepancy deterministic methods for pricing fi• compute the arithmetic mean of the integrand the most difficult to price. If this tranche can nancial derivatives began at Columbia Univer• evaluated at a number of points, the difference be priced with a given accuracy using a cer• sity in autumn 1992. Preliminary results were in performance depends on the number of tain number of samples, the same number of shared with a number of New York City finan• points that each method uses to achieve the. samples will yield at least the same accuracy cial houses in autumn 1993 and spring 1994. The same accuracy. We observe the least number for the rest of the tranches. first published announcement was Traub and Since pricing models for complicated de• Wozniakowski's January 1994 article in Scien• 1 See Paskov (1996) for the formal definition of rivatives are subject to uncertainty, financial tific American. A more detailed history is given discrepancy and an extensive bibliography houses are often content with relative errors of in Paskov & Traub (1995). 2 Finder may be obtained from Columbia University one part in a hundred. Furthermore, if they THE AXIOM mit EiTr£^FSi^E-mij£ MSX MAMA€'£'M'BTI • VAR; MARKET, CREDIT AND LIQUIDITY RISK • GLOBAL FINANCIAL DATA WAREHOUSE • BUSINESS RULES FOR DECISION SUPPORT • BIS & IOSCO, CAD, REGULATORY COMPLIANCE • FINANCIAL ENGINEERING & CONSULTING SERVICES GLOBALIZATION OF CUENI RELATIONSHIP MANAGEMENT AND SERVICES £^ I |^ j| TOR MORE INFORMATION PLEASE CONTACT US AT: ICVMKUIOIMMIES'.wc. 63 WALL STREET NEW YORK NY 10005 USA • TEL: (212) 248-4188 • FAX: (212) 248-4354 • E MAIL: [email protected] RISK VOL9/NO 6/JUNE 1996 3. Generalised Faure and Monte Carlo errors 0.03 wish to price a book of instruments, it is criti• Generalised Faure cal to use a small number of samples. Deter• Monte Carlo ministic methods achieve a relative error of one 0.02 A part in a hundred using a small number of points.
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