High Dimensionality And Discontinuity:New Challenges In QMC For Quantitative Finance

Many finance problems cannot be solved analytically due to the complexity of financial model and the diversity of financial products. This thesis develops quasi-Monte Carlo(QMC) methods for solving these problems approximately. However, the e?cency of QMC methods depends highly on the dimension and smoothness of the target functions. High dimensionality and discontinuity in finance problems are new challenges for QMC methods. In order to improve the e?ciency of QMC for finance problems, this thesis proposes some new methods for overcoming the two di?culties.Path generations of underlying assets are crucial for the implementation of QMC methods. Di?erent path generation methods could have an impact on the e?cency of QMC. With a special concern of discontinuity structures involved in finance problems(such as the pricing and hedging of financial derivatives), this thesis develops a new path genearation method– QR method. The method transforms the discontinuity structures such that the discontinuity surfaces are parallel to as many as coordinate axes as possible.In doing so, the integrands become “QMC-friendly”. Additionally, we propose a measurement for quantifying the importance of di?erent discontinuity structures. Numerical results show that the QR method can improve significantly the e?cency of QMC for prcing some exotic options.Due to the impact of discontinuty structures on QMC methods, this thesis proposes a new smoothing method for removing the discontinuities. In order to make the smoothing method applicable for some common finance problems, we develop a modified QR method. The combination of the modified QR method and the smoothing method can weaken the adverse e?ects of high dimensionality and discontinuity simultaneously.Numerical results show that the combined method can reduce significantly the e?ective dimension, and improve markedly the e?cency of QMC.This thesis also studies the convergence rate of randomized quasi-Monte Carlo(RQMC) methods for discontinuous functions. For cerntain discontinuous functions,we prove that for arbitrary > 0, the root mean squared error(RMSE) of RQMC is O(n-1/2-1/(4d-2)+), where d is the dimension, and n is the sample size. It was previously known that the rate is only o(n-1/2). If the discontinuity surface is parallel to some coordinate axes, we can get a faster rate O(n-1/2-1/(4du-2)+), where du is the “irregular dimension”(the number of axes to which discontinuity surface is parallel). Numercial results show better estimated rates than the theoretical rates, especially for low-dimensional discontinuous functions. Moreover, the estimated rates deteriorate greatly with increasing dimension d or irregular dimension du. These insights are in line with the theoretical rates.Finally, this thesis makes use of Hilbert’s space-filling curve to transform ddimensiol quadrature rules into one-dimensional quadrature rules. We prove that the randomized quadrature rules based on Hilbert’s space-filling curve yield an RMSE of O(n-1/2-1/d) for Lipschitz functions, when d ≥ 3. For certain discontinuous functions,the RMSE turns out to be O(n-1/2-1/(2d)), which is better than the aforementioned rate of RQMC.