Pricing and Risk Management with High-Dimensional Quasi Monte Carlo and Global Sensitivity Analysis

We review and apply Quasi Monte Carlo (QMC) and Global Sensitivity Analysis (GSA) techniques to pricing and risk management (greeks) of representative financial instruments of increasing complexity. We compare QMC vs standard Monte Carlo (MC) results in great detail, using high-dimensional Sobol' low discrepancy sequences, different discretization methods, and specific analyses of convergence, performance, speed up, stability, and error optimization for finite differences greeks. We find that our QMC outperforms MC in most cases, including the highest-dimensional simulations and greeks calculations, showing faster and more stable convergence to exact or almost exact results. Using GSA, we are able to fully explain our findings in terms of reduced effective dimension of our QMC simulation, allowed in most cases, but not always, by Brownian bridge discretization. We conclude that, beyond pricing, QMC is a very promising technique also for computing risk figures, greeks in particular, as it allows to reduce the computational effort of high-dimensional Monte Carlo simulations typical of modern risk management.Comment: 43 pages, 21 figures, 6 table

Different numerical estimators for main effect global sensitivity indices

The variance-based method of global sensitivity indices based on Sobol sensitivity indices became very popular among practitioners due to its easiness of interpretation. For complex practical problems computation of Sobol indices generally requires a large number of function evaluations to achieve reasonable convergence. Four different direct formulas for computing Sobol main effect sensitivity indices are compared on a set of test problems for which there are analytical results. These formulas are based on high-dimensional integrals which are evaluated using MC and QMC techniques. Direct formulas are also compared with a different approach based on the so-called double loop reordering formula. It is found that the double loop reordering (DLR) approach shows a superior performance among all methods both for models with independent and dependent variables

Application of Quasi Monte Carlo and Global Sensitivity Analysis to Option Pricing and Greeks

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Quasi-Monte Carlo methods for calculating derivatives sensitivities on the GPU

The calculation of option Greeks is vital for risk management. Traditional pathwise and finite-difference methods work poorly for higher-order Greeks and options with discontinuous payoff functions. The Quasi-Monte Carlo-based conditional pathwise method (QMC-CPW) for options Greeks allows the payoff function of options to be effectively smoothed, allowing for increased efficiency when calculating sensitivities. Also demonstrated in literature is the increased computational speed gained by applying GPUs to highly parallelisable finance problems such as calculating Greeks. We pair QMC-CPW with simulation on the GPU using the CUDA platform. We estimate the delta, vega and gamma Greeks of three exotic options: arithmetic Asian, binary Asian, and lookback. Not only are the benefits of QMC-CPW shown through variance reduction factors of up to $1.0 \times 10^{18}$, but the increased computational speed through usage of the GPU is shown as we achieve speedups over sequential CPU implementations of more than $200$x for our most accurate method.Comment: 26 pages, 12 figure

The future of sensitivity analysis: an essential discipline for systems modeling and policy support

Sensitivity analysis (SA) is en route to becoming an integral part of mathematical modeling. The tremendous potential benefits of SA are, however, yet to be fully realized, both for advancing mechanistic and data-driven modeling of human and natural systems, and in support of decision making. In this perspective paper, a multidisciplinary group of researchers and practitioners revisit the current status of SA, and outline research challenges in regard to both theoretical frameworks and their applications to solve real-world problems. Six areas are discussed that warrant further attention, including (1) structuring and standardizing SA as a discipline, (2) realizing the untapped potential of SA for systems modeling, (3) addressing the computational burden of SA, (4) progressing SA in the context of machine learning, (5) clarifying the relationship and role of SA to uncertainty quantification, and (6) evolving the use of SA in support of decision making. An outlook for the future of SA is provided that underlines how SA must underpin a wide variety of activities to better serve science and society

Application of Quasi Monte Carlo and Global Sensitivity Analysis to Option Pricing and Greeks: Finite Differences vs. AAD

Quasi Monte Carlo (QMC) and Global Sensitivity Analysis (GSA) techniques are applied to pricing and hedging of representative financial instruments with increasing complexity. We compare standard Monte Carlo (MC) vs. QMC results using Sobol' low-discrepancy sequences, different sampling strategies, and various analyses of performance. We find that QMC outperforms MC in most cases, including the highest-dimensional simulations, showing faster and more stable convergence. Regarding greeks computation, we compare standard approaches, based on finite difference (FD) approximations, with Adjoint Algorithmic Differentiation (AAD) methods providing evidence that, when the number of greeks is small, switching from MC to QMC simulation, the FD approach can lead to the same accuracy as AAD, thanks to increased convergence rate and stability, thus saving a lot of implementation effort while keeping low computational cost. Using GSA, we are able to fully explain our findings in terms of reduced effective dimension of QMC simulation, allowed in most cases by Brownian bridge discretization or Principal Component Analysis (PCA) construction. We conclude that, beyond pricing, QMC is a very efficient technique also for computing risk measures, greeks in particular, as it allows us to reduce the computational effort of high-dimensional Monte Carlo simulations typical of modern risk management