Quasi-Monte Carlo Methods in Financial Risk management

1. Quasi-Monte Carlo Methods in Financial Risk management Maurizio Mondello and Maurizio Ferconi ©1999 Tech Hackers, Inc.

2. Overview • Uses of Monte Carlo in financial risk management • Improving on Monte Carlo convergence: quasi-random sequences • Quasi-random points in high dimensions • Dimensional reduction techniques • In-sample error estimation: when to stop? • Conclusions

3. Financial Risk Managementand Monte Carlo • In a trading environment, one can identify at least two levels of Risk management where MC can be useful: • High Level: VaR/Scenario Analysis at the Desk (Book) Level or Higher (Firm wide). • Low Level : sensitivities of individual portfolios/deals (as part of the marking-to-market/hedging process).

4. Risk Aggregation • The two levels need to be integrated: • Sensitivities at the deal/portfolio level provide inputs for Delta-Gamma VaR. • Recalculation of individual PV (marking-to-market of portfolio/deal) is required as part of a stress test/scenario analysis.

5. Monte Carlo for Pricing and VaR • Monte Carlo simulation in the pricing of individual deals is based on the use of a risk-neutraldistribution (returns on all assets are equal to the riskless interest rate, which is also used for present value discounting). • Monte Carlo simulation for VaR requires the use of the“real world” (historical) distribution of returns. Typically, we are interested in the future value (no discounting) of a static portfolio (no dynamic hedging).

6. Integration/Quantile Determination • In deal pricing we are interested in the average value with respect to the risk-neutral distribution (Monte Carlo integration under the risk-neutral measure over the life of the deal). • In VaR we are interested in determining a quantile in the distribution of (future) values for the “book.” (Monte Carlo is used to map the multivariate distribution of risk factors into the one-dimensional distribution of “book” values.)

7. Determining the dimensionality of a (simulation) problem • Two main factors to consider: • The number ofrisk factors (underlyings) involved: e.g., valuation of basket option, yield-curve sensitivities of an interest rate option, VaR. • The number of time steps required (for each factor): e.g., valuation of an Asian option or knock-out swap (path dependence). • When both factors are present, the dimension is the product of the two numbers.

8. Convergence of MC: Can we do better? • Theconvergence of MC is controlled by the total variance of the problem, sV2, and the number of simulations, N. The error decreases as ~sV/N0.5. The dimension of the problem, D ,does not appear. • For low-dimensional problems we can easily do better using a uniform distribution of sample point. In this case the error varies as ~N-2/D, where N is the number of sampled point. Note, however, that the rate of convergence decreases rapidly with increasing D (“curse of dimensionality”).

9. Quasi-random points • Quasi-random points provide a higher degree of uniformity (low-discrepancy) than random points, leading to better low-dimensional performance, while exhibiting a milder dimensional dependence than uniform points. The asymptotic convergence rateis~(Log(N))D/N. • It should be noticed that, contrary to MC, this convergence rate can only be guaranteed for smooth integrals.

10. Random vs. Sobol points in 2D 1024 points

11. Random vs. Sobol points in 3D 512 points

12. Pricing Simulation Convergence (10D) Asian Put (10 observations/year) S=100, K=105, r=0.5, d=0.4, s=10, T=1 Price vs. Log2(N) Number of simulation points N varies from 64 to 262144

13. Quasi-Monte Carlo Heuristics • Skip the first “few” numbers (warm up sequence length depends on generator). • Choose your initial point (coordinates of directions) carefully. • In our experience these choices DO have some effect, but recommendations found in the literature are not always consistent. Systematic improvement is partly hampered by the lack of open research in this area.

14. High Dimensional Limitations • The performance ofallquasi-random sequences deteriorates in higher dimensions. Sobol sequences seem comparatively robust. • Moment matching may be of help. Bias introduced is more than compensated by variance reduction.

15. 1024 Niederreiter points in 50D First 2 dimensions Last 2 dimensions

16. Pricing Simulation Convergence (50D) Asian Put (10 observations/year) S=100, K=105, r=0.5, d=0.4, s=0.35, T=5 Price vs. Log2(N) Number of simulation points N varies from 64 to 262144

17. What is dimensional reduction? • It is a way to characterize the minimal number of risk factors that will explain a given percentage (e.g., 95%) ofthe variance of our valuation problem. While all factors will, in general, be necessary to explain 100% of this variance, a few factors may already explain a large fraction of it. • The size of the variance in the distribution of values (sV2) controls the convergence rate of our simulation.

18. How to achieve dimensional reduction • The optimal method is some form of spectral or principal-component analysis. This is the only method available, in general, when the dimensionality of the problem is determined by the number n of correlated risk factors. It requires a O(n2) algorithm, which can be slow for large n. • When a time path-dependence controls the dimensionality of the problem, an effective and fasterO(n)algorithmis given by the Brownian Bridge path generation.

19. Brownian-Bridge Construction A C B D E In the sequential Brownian pathconstruction, point C is obtained conditional on the position of B. In the Brownian-Bridge construction, point C is obtained conditional on the positions of A and E.

20. Comparison of effective dimensionality (variance distribution) for eight-step path At the 95% level, the sequentially generated and the Brownian-Bridge path have effective dimensions of 7 and 5, respectively.

21. When does dimensional reduction help? • Remember that the error of the sample average of a Monte Carlo simulation varies as ~sV/N0.5. The rate of convergence of the simulation only depends on the total variance and the number of simulations, N. • For Quasi-Monte Carlo the situation is different, because the relative efficiency ofvariance reduction (with respect to MC) is highest for low-dimensional problems.

22. How (effective) dimensionality is handled by different generators • Dimensional reduction is optimal for Sobol and Niederreiter sequences, where each dimension is generated independently, but not for Faure. In this last case it may be worth combining a low-dimensional Faure generator and sample the remaining dimensions using a (pseudo-)random number generator. • This also allows the flexibility to generate a single high-dimensional (Sobol or Niederreiter) sequence to be (kept in memory and) used for problems of different dimensionality.

23. 512 Faure points in 52D First 3 dimensions Last 3 dimensions

24. Error Determination • A strength of standard MC method is the ability to perform in-sample statistical error analysis (the error is itself a random quantity). • Quasi-Monte Carlo methods offer, in principle, the ability to obtain absolute error bound. In practice, this is very difficult to calculate and may grossly overestimate the actual error. • (Note that this leaves open the possibility of a “catastrophic failure” of convergence, for sample sizes of practical interest. However remote, such a possibility is worrisome, if no error estimation procedure is in place.)

25. Practical solution • In our experience, averaging over subsequences is an effective strategy to obtain heuristic error estimates for Quasi-Monte Carlo simulations. It should not be used for automated dynamic control of the simulation convergence without a preliminary product and generator specific study.

26. Comparison of VaR Calculation Portfolio includes a stock index and foreign zero (three correlated risk factors, sampled 4096 times). Random Generator (2) Sobol Generator (1) P&L distribution of the portfolio over a 1 day horizon. Grid lines indicate position of 0.05 quantile (95% VaR).

27. Error in VaR calculation Comparison of 16 VaR calculations (4096 samples each) for 7 generators

28. Reintroducing Randomness • It is possible to reintroduce randomness in a quasi-random sequence without destroying the low-discrepancy property. This offers a number of advantages: • Applicability of Central Limit Theorem for in-sample error estimation. • More stringent theoretical bounds on “worst case” performance: “soft-failure”. • Faster convergence (~Log(N)(D-1)/2 N-1.5) for smooth integrands.

29. Conclusions • In our search for general (robust) and fast simulation methods, Quasi-Monte Carlo offers a good compromise. It can be easily implemented “on top” of existing Monte Carlo simulation engines and it has broad applicability. • We must, however, be mindful of the possible pitfalls, particularly for high dimensional problems. Convergence tests for each (new) class of products is recommended. • The use of randomized (scrambled) quasi-random sequences offers the important guarantee of “soft-failure.”

30. Contact Information Dr. Maurizio Mondello TechHackers 5 Hanover Square New York, NY 10004 USA phone 1.212.344.9500 fax 1.212.344.9519 mmondello@thi.com http://www.thi.com/