Valuation of American Basket Options using Quasi-Monte Carlo Methods d-fine GmbH Christ Church College University of Oxford A thesis submitted in partial fulfillment for the MSc in Mathematical Finance September 26, 2009 Abstract Title: Valuation of American Basket Options using Quasi-Monte Carlo Methods Author: d-fine GmbH Submitted for: MSc in Mathematical Finance Trinity Term 2009 The valuation of American basket options is normally done by using the Monte Carlo approach. This approach can easily deal with multiple ran- dom factors which are necessary due to the high number of state variables to describe the paths of the underlyings of basket options (e.g. the Ger- man Dax consists of 30 single stocks). In low-dimensional problems the convergence of the Monte Carlo valuation can be speed up by using low-discrepancy sequences instead of pseudo- random numbers. In high-dimensional problems, which is definitely the case for American basket options, this benefit is expected to diminish. This expectation was rebutted for different financial pricing problems in recent studies. In this thesis we investigate the effect of using different quasi random sequences (Sobol, Niederreiter, Halton) for path generation and compare the results to the path generation based on pseudo-random numbers, which is used as benchmark. American basket options incorporate two sources of high dimensional- ity, the underlying stocks and time to maturity. Consequently, different techniques can be used to reduce the effective dimension of the valuation problem. For the underlying stock dimension the principal component analysis (PCA) can be applied to reduce the effective dimension whereas for the time dimension the Brownian Bridge method can be used. We an- alyze the effect of using these techniques for effective dimension reduction on convergence behavior. To handle the early exercise feature of American (basket) options within the Monte Carlo framework we consider two common approaches: The Threshold approach proposed by Andersen (1999) and the Least-Squares Monte Carlo (LSM) approach suggested by Longstaff and Schwartz (2001). We investigate both pricing methods for the valuation of American (bas- ket) options in the equity market. Contents 1 Introduction 1 2 Basket Options 4 2.1 Introduction . .4 2.2 Different Types of Basket Options . .4 2.2.1 Plain Vanilla Basket Options . .4 2.2.2 Exchange Options . .5 2.2.3 Asian Basket Options . .6 2.3 American-style Basket Options . .7 3 Monte Carlo Simulation 8 3.1 Introduction . .8 3.2 Mathematical Foundations . .9 3.2.1 Strong Law of Large Numbers . .9 3.2.2 Error Estimation . 10 3.2.3 Central Limit Theorem . 10 3.3 Pseudo-Random Numbers . 11 3.4 Quasi-Random Numbers . 13 3.4.1 Low Discrepancy and Integration Error . 14 3.4.2 Halton Sequences . 16 3.4.3 Sobol Sequences . 18 3.4.4 Niederreiter Sequences . 19 3.5 The curse of dimensionality . 22 4 Effective Dimension Reduction 23 4.1 Introduction . 23 4.2 Principal Component Analysis . 23 4.3 Brownian Bridge . 25 i 5 Valuation of American Basket Options 28 5.1 Introduction . 28 5.2 Model Framework . 29 5.3 Monte Carlo Path Construction . 30 5.4 Valuation of Early Exercise Feature . 33 5.4.1 Least-Squares Approach . 33 5.4.2 Threshold Approach . 35 6 Comparison of Numerical Methods 37 6.1 Introduction . 37 6.2 Data . 38 6.3 Software Implementation . 38 6.4 Results . 40 6.4.1 Classical Put Options . 40 6.4.2 Exchange Options . 43 6.4.3 Basket Options . 45 7 Conclusions 49 A Data 51 Bibliography 51 ii List of Tables 6.1 Statistics for different paths generation methods for American Dax call valuation . 46 A.1 Description Dax constituents . 51 A.2 Correlation matrix for Dax constituents for April 30, 2009 . 52 iii List of Figures 3.1 Two-dimensional projections of pseudo-random sequences . 13 3.2 Two-dimensional projections of Halton sequences . 17 3.3 Two-dimensional projections of Sobol sequences . 20 3.4 Two-dimensional projections of Niederreiter sequences . 21 4.1 Brownian bridge construction . 26 6.1 Convergence properties plain-vanilla American put . 42 6.2 Convergence properties American exchange options using Underlying- Transformation ............................... 44 6.3 Convergence properties American Dax call for different random num- ber generators using Least-Squares approach . 48 A.1 Convergence properties of American put for different random number generators using Least-Squares approach . 53 A.2 Convergence properties of American put for different random number generators using Threshold approach . 54 A.3 Convergence properties of American exchange options for different ran- dom number generators . 55 A.4 Convergence properties American Dax call for pseudo-random num- bers and Sobol sequences . 56 A.5 Convergence properties American Dax call for Niederreiter and Halton sequences . 57 iv Chapter 1 Introduction The pricing and optimal exercise of options with early exercise features is one of the most challenging problems in mathematical finance. The problem becomes even more complex, if the model for the economy has more than one state variable, i.e. more than one factor affecting the value of the option as well as the optimal exercise mo- ment. American basket options are a prominent example for derivatives with early exercise features and multiple state variables. Due to their construction principle bas- ket options generally represent an advantageous alternative to hedge risky positions of several assets, since trading only one basket option instead of several single-asset options decreases transaction costs. Usually lattice methods or finite differences will be used to price American or Bermudan style options. However, these techniques are inefficient for high dimen- sional problems (i.e. problems with more than three state variables) and they are very difficult to apply on path-dependent options. For higher dimensions it would be relatively cheap to use simulation techniques since its computational cost does not increase exponentially as compared to other methods and they are relatively simple to apply. However, for a long time, simulation techniques seemed unapplicable to options with early exercise features. This is due to their forward-construction prin- ciple and their path-by-path generation. Since contingent claims with early exercise features are traded in all important derivative markets, many suggestions have been made during the last years to price such options by simulation, and in particular to de- termine the optimal early-exercise strategy. Examples can be found in Tilley (1993), Barraquand and Martineau (1995), Carriere (1996), Broadie and Glasserman (1997), Andersen (1999), Longstaff and Schwartz (2001), or Ib´anezand Zapatero (2004). In this case the starting point will be the application of standard Monte Carlo simulation with pseudo-random numbers, eventually combined with some variance 1 reduction techniques such as antithetic variables. However, using pseudo-random numbers can be quite slow, due to the relatively poor convergence rate of O(M −1=2) for M sample paths. Here, the application of Quasi-Monte Carlo simulation (QMC), e.g. using Sobol, Niederreiter or Halton sequences, can improve the performance of Monte Carlo simulations dramatically, since in optimal cases the convergence rate of QMC is O(log(M)dM −1), where d represents the dimension of the integration problem. Therefore, for small values of d the convergence is due to the construc- tion principle of low-discrepancy sequences significantly higher as for standard Monte Carlo. However, for higher dimensional problems this advantage will diminish, since the convergence strongly depends on the dimension of the integration problem for which reason the application of QMC to price American basket option seems not ad- vantageous. Especially for American options it cannot be expected that QMC works, since due to regression or optimization over the sample paths the sample points have interactions. However, Chaudhary (2005) discovered a significant speedup in the nu- merical results for the Least-squares approach proposed by [25]. Therefore we also expect to find this improvement for the convergence of the valuation of American basket options when applying QMC. Moreover, earlier empirical studies for finance problems, see among others [30, 29, 12], have shown in fact significant improvements using QMC even in high dimensional financial pricing problems up to dimension 360. The main reason is that many practical problems in mathematical finance have a low effective dimension { a notion introduced by [12] { i.e. either the integrand depends only of several variables, or the integrand can be approximated by a sum of functions with each depending only on a small number of variables at the same time. Sev- eral techniques to reduce the effective dimension have been successfully applied for derivative pricing problems. The most prominent are the Brownian bridge method for path construction (BB) first applied by Moskowitz and Caflisch (1996) and Caflisch et al. (1997) and the principal component analysis (PCA) first used by Acworth et al. (1997) to identify the underlyings which contribute the largest part of the total variance. Besides this empirical studies Wang and Sloan (2007) provides a theoretical analysis of the possible convergence order of QMC used in conjunction with BB or PCA. However, all these studies only analyze the convergence properties of QMC with BB or with PCA, but, to the best of our knowledge, not the combined application of QMC with both { BB and PCA. In our analysis we consider the valuation of American basket options as a promi- nent example for a high-dimensional derivative pricing problem, whereby we reduce 2 the effective dimension by applying the Brownian bridge method (BB) and the Princi- pal Component Analysis (PCA) to reduce the effective time and underlying stock di- mension. We focus our analysis on the convergence properties and numerical stability by applying the different techniques as well as to develop a path construction method which takes both effective dimension reduction techniques into account.