Variance Reduction for Multi-Variable Monte Carlo Simulation

無The Monte Carlo Simulation is the most popular and widely used numerical method on option pricing. However, it often takes too much time to converge, especially when the number of underlying assets increases. In this thesis, new methods are introduced to govern the correlations between random variables, combined with traditional variance reduction techniques to reduce the variance effectively. The numerical experiments show that the present method to regulate the correlations between random variables is more efficient than the crude Monte Carlo method. European call options on maximum of several underlying assets are provided as examples in this paper.1 Introduction 5 1.1 Fundamental of Options Pricing . . . . . . . . . . . . . . . . . . . 5 1.2 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Layout of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Background 9 2.1 Pricing European Option using simulation . . . . . . . . . . . . . . . . . . 9 2.2 Variance Reduction Techniques . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.1 Antithetic Variates . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.2 Control Variates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.3 Moment Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Cholesky Decomposition . . . . . . . . . . . . . . . . . . . . . . 13 2.4 Eigenvalue Decomposition . . . . . . . . . . . . . . . . . . . . . 15 3 The new method 17 3.1 Motive of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Our new method . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2.1 The extension of antithetic sampling approach . . . . . . . . . . . . 19 3.2.2 Inverse Cholesky (or Eigenvalue) Decomposition Method . . . . . . 22 4 Numerical Results 24 4.1 European maximum call options of five underlying assets. . . . . . . . . . . 25 4.2 European maximum call options of five underlying assets v.s. ten underlying assets.. . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.3 The comparison of matching the inverse Cholesky (or Eigenvalue) Decomposition with Cholesky (or Eigenvalue) Decomposition. . . . . . . . . . . . 28 4.4 Comparisons of low-discrepancy sequence . . . . . . . . . . . . . . . . . . . 28 5 Conclusion