American options and the LSM algorithm: quasi-random sequences and Brownian bridges

From Continuous to Discrete: Studies on Continuity Corrections and Monte Carlo Simulation with Applications to Barrier Options and American Options Menghui Cao This dissertation 1) shows continuity corrections for first passage probabilities of Brownian bridge and barrier joint probabilities, which are applied to the pricing of two-dimensional barrier and partial barrier options, and 2) introduces new variance reduction techniques and computational improvements to Monte Carlo methods for pricing American options. The joint distribution of Brownian motion and its first passage time has found applications in many areas, including sequential analysis, pricing of barrier options, and credit risk modeling. There are, however, no simple closed-form solutions for these joint probabilities in a discrete-time setting. Chapter 2 shows that, discrete two-dimensional barrier and partial barrier joint probabilities can be approximated by their continuous-time probabilities with remarkable accuracy after shifting the barrier away from the underlying by a factor of βσ √ ∆t. We achieve this through a uniform continuity correction theorem on the first passage probabilities for Brownian bridge, extending relevant results in Siegmund (1985a). The continuity corrections are applied to the pricing of two-dimensional barrier and partial barrier options, extending the results in Broadie, Glasserman & Kou (1997) on one-dimensional barrier options. One interesting aspect is that for type B partial barrier options, the barrier correction cannot be applied throughout one pricing formula, but only to some barrier values and leaving the other unchanged, the direction of correction may also vary within one

High-dimensional integrals occur in a variety of areas, including mathematical finance. In the classical settings, multivariate integration problems suffer from the curse of dimensionality. To vanquish the curse of dimensionality, one may shrink the function class. Here, we use weighted function spaces, in which groups of variables are associated with weights, in order to capture the different importance of each group of variables. For practical applications, the principal difficulty is then in choosing the ‘right’ weights for a given problem or a class of problems. We work in weighted reproducing kernel Hilbert spaces (with ‘general’ rather than ‘product’ weights). We first present a principle to find the good weights for a given problem. This general approach is then applied to a simplified high-dimensional problem from finance, in which the dimension is the number of discrete timesteps for a price of a risky asset which follows geometric Brownian motion. The second focus of this paper is on the dimension reduction techniques, such as the Brownian bridge (BB) and the principal component analysis (PCA). It turns out that the behaviour of the model problem is dramatically improved when quasi-Monte Carlo method is used in conjunction with the dimension reduction techniques: if the right weights are used in every case, then the integration error can be bounded independently of the dimension, whereas without BB or PCA the error bound depends exponentially on the dimension. Finally, for this model problem we show how to construct shifted lattice rules which, when used in conjunction with BB or PCA, yield integration errors converging as O(n−3/4+δ) or O(n−1+δ) (for arbitrary small δ > 0), respectively, independently of the dimension. Thus, in both cases well-designed algorithms can avoid the curse of dimensionality, with PCA having an advantage over BB with respect to the proved order of convergence.

Bachvalov proved that the optimal order of convergence of a Monte Carlo method for numerical integration of functions with bounded kth order derivatives is \(\mathop{O}\left(N^{-\frac{k}{s}-\frac{1}{2}}\right)\), where s is the dimension. We construct a new Monte Carlo algorithm with such rate of convergence, which adapts to the variations of the sub-integral function and gains substantially in accuracy, when a low-discrepancy sequence is used instead of pseudo-random numbers.Theoretical estimates of the worst-case error of the method are obtained.Experimental results, showing the excellent parallelization properties of the algorithm and its applicability to problems of moderately high dimension, are also presented.

Simple or proxy production models are potentially very useful and tractable because they are computationally attractive whilst still providing insight to the decision at hand. Useful and tractable models are required for supporting high-quality decisions in uncertain, complex, and computationally demanding contexts. A key decision for development planning is: what is the optimal time to initiate an Improved-Oil-Recovery (IOR) process. We aim to illustrate the implementation and application of a useful and tractable approach for the analysis of the optimal IOR switch time using a two-factor production model and Least-Squares Monte Carlo (LSM) simulation. The two-factor production model contains only two parameters for each recovery phases. One parameter describes how much recovery efficiency a recovery mechanism can ultimately achieve whilst the other describes how fast the recovery efficiency increases. The simplicity of the model makes it computationally attractive. The LSM algorithm is an approximate dynamic programming approach, which allows for learning over time. It provides a near-optimal solution for the IOR switch time problem. The Value-Of-Information (VOI) framework—a powerful decision-analysis tool—provides an estimate of the value of learning. Closed-Loop Reservoir Management (CLRM) is considered to be a state-of-the-art approach to solving for the optimal IOR switch time. However, this approach can produce a suboptimal solution as the CLRM approach considers only uncertainties and actions reflecting currently available information but not those uncertainties and actions arising from future information. The dynamic programming approach used here considers both the impact of the information obtained before a decision is made and the impact of the information that might be obtained to support future decisions. We conclude that a dynamic programming approach, such as the LSM algorithm, can significantly improve both the timing and value of decisions, leading to a significant increase in a field's economic performance. Furthermore, the two-factor model combined with the LSM algorithm is tractable and provides useful insight into the IOR switch time problem. The novelties provided by this work are: developing and illustrating the structure of the IOR switch time problem in a decision tree, demonstrating and discussing the suboptimality of the CLRM solution, developing and illustrating the detailed steps of applying the LSM algorithm for the IOR switch time decision, and implementing the two-factor model combined with the LSM algorithm for analyzing the optimal IOR switch time.

Many important practical problems connected to energy efficiency in buildings, ecology, metallurgy, the development of wireless communication systems, the optimization of radar technology, quantum computing, pharmacology, and seismology are described by large-scale mathematical models that are typically represented by systems of partial differential equations. Such systems often involve numerous input parameters. It is crucial to understand how susceptible the solutions are to uncontrolled variations or uncertainties within these input parameters. This knowledge helps in identifying critical factors that significantly influence the model’s outcomes and can guide efforts to improve the accuracy and reliability of predictions. Sensitivity analysis (SA) is a method used efficiently to assess the sensitivity of the output results from large-scale mathematical models to uncertainties in their input data. By performing SA, we can better manage risks associated with uncertain inputs and make more informed decisions based on the model’s outputs. In recent years, researchers have developed advanced algorithms based on the analysis of variance (ANOVA) technique for computing numerical sensitivity indicators. These methods have also incorporated computationally efficient Monte Carlo integration techniques. This paper presents a comprehensive theoretical and experimental investigation of Monte Carlo algorithms based on “symmetrized shaking” of Sobol’s quasi-random sequences. The theoretical proof demonstrates that these algorithms exhibit an optimal rate of convergence for functions with continuous and bounded first derivatives and for functions with continuous and bounded second derivatives, respectively, both in terms of probability and mean square error. For the purposes of numerical study, these approaches were successfully applied to a particular problem. A specialized software tool for the global sensitivity analysis of an air pollution mathematical model was developed. Sensitivity analyses were conducted regarding some important air pollutant levels, calculated using a large-scale mathematical model describing the long-distance transport of air pollutants—the Unified Danish Eulerian Model (UNI-DEM). The sensitivity of the model was explored focusing on two distinct categories of key input parameters: chemical reaction rates and input emissions. To validate the theoretical findings and study the applicability of the algorithms across diverse problem classes, extensive numerical experiments were conducted to calculate the main sensitivity indicators—Sobol’ global sensitivity indices. Various numerical integration algorithms were employed to meet this goal—Monte Carlo, quasi-Monte Carlo (QMC), scrambled quasi-Monte Carlo methods based on Sobol’s sequences, and a sensitivity analysis approach implemented in the SIMLAB software for sensitivity analysis. During the study, an essential task arose that is small in value sensitivity measures. It required numerical integration approaches with higher accuracy to ensure reliable predictions based on a specific mathematical model, defining a vital role for small sensitivity measures. Both the analysis and numerical results highlight the advantages of one of the proposed approaches in terms of accuracy and efficiency, particularly for relatively small sensitivity indices.

We present quasi-Monte Carlo analogs of Monte Carlo methods for some linear algebra problems: solving systems of linear equations, computing extreme eigenvalues, and matrix inversion. Reformulating the problems as solving integral equations with a special kernels and domains permits us to analyze the quasi-Monte Carlo methods with bounds from numerical integration. Standard Monte Carlo methods for integration provide a convergence rate of O(N^(−1/2)) using N samples. Quasi-Monte Carlo methods use quasirandom sequences with the resulting convergence rate for numerical integration as good as O((logN)^k)N^(−1)). We have shown theoretically and through numerical tests that the use of quasirandom sequences improves both the magnitude of the error and the convergence rate of the considered Monte Carlo methods. We also analyze the complexity of considered quasi-Monte Carlo algorithms and compare them to the complexity of the analogous Monte Carlo and deterministic algorithms.

Front propagation is a challenge in numerical modeling, particularly for multi-fluid or multi-material systems requiring clear material separation. The GRU (Glimm Random Update) scheme, inspired by Glimm’s method, has been developed to handle sharp fronts on unstructured 2D/3D meshes. It achieves convergence rates numerically measured between 0 . 8 and 0 . 9 , which are higher than those of classical first-order schemes, typically limited to 0 . 5 in the presence of contact discontinuities. Previous work established convergence in probability for planar fronts with uniform velocity but relied on random sequences, whereas practical implementations use low-discrepancy sequences. This study aims to establish new theoretical convergence results for the GRU scheme for 1D non-uniform meshes and uniform front velocity conditions, extending beyond uniform meshes considered previously. More precisely, we prove convergence in probability of order close to one-half when using random sequences on non-uniform meshes, and exact first-order convergence in probability when using low-discrepancy sequences (deterministic) on uniform meshes. Partial results are also obtained for non-uniform meshes, which are of significant practical interest. These findings provide insights into the scheme’s behavior with both random and low-discrepancy sequences. Numerical tests are presented to illustrate the theoretical results, with applications to non-uniform meshes, highlighting the scheme’s practical relevance.

Digital net properties of a polynomial analogue of Frolov's construction

In this paper, we investigate the feasibility of using low-discrepancy sequences to allow complex derivatives, such as mortgage-backed securities (MBSs) and exotic options, to be calculated considerably faster than is possible by using conventional Monte Carlo methods. In our experiments, we examine classical classes of low-discrepancy sequences, such as Halton, Sobol', and Faure sequences, as well as the very recent class called generalized Niederreiter sequences, in the light of the actual convergence rate of numerical integration with practical numbers of dimensions. Our results show that for the problems of pricing financial derivatives that we tested: (1) generalized Niederreiter sequences perform markedly better than both classical sequences and Monte Carlo methods; and (2) classical low-discrepancy sequences often perform worse than Monte Carlo methods. Finally, we discuss several important research issues from both practical and theoretical viewpoints.

PurposeThe purpose of this paper is to study the importance of business time, and market opening/closing times and days, for American option pricing.Design/methodology/approachA Bermudan pricing approach is employed whereby the option can be exercised only during the times and days the market is open. The authors apply the approach to the S&P 100 options market.FindingsIt was found that the potential biases that can arise from ignoring the non‐continuous operation of the market are not negligible.Research limitations/implicationsFor expositional purposes, the authors assume that the price of the underlying follows a Geometric Brownian motion. This assumption could be relaxed by future research and more complex price dynamics models could be considered.Practical implicationsThe findings in this paper could be used in correcting observed option prices, prior to investigating the rationality of early exercise decisions, or in measuring the size of early exercise premia.Originality/valueThis is the first study to examine the effects of business time, and market opening/closing times and days, to American option prices.

Arbitrage, Hedging and Utility Maximization Using Semi-Static Trading Strategies with American Options

Fix an integer s. Let \(f:[0,1)^s \rightarrow {\mathbb {R}}\) be an integrable function. Let \({\mathscr {P}}\subset [0, 1]^s\) be a finite point set. Quasi-Monte Carlo integration of f by \({\mathscr {P}}\) is the average value of f over \({\mathscr {P}}\) that approximates the integration of f over the s-dimensional cube. Koksma–Hlawka inequality tells that, by a smart choice of \({\mathscr {P}}\), one may expect that the error decreases roughly \(O(N^{-1}(\log N)^s)\). For any \(\alpha \ge 1\), J. Dick gave a construction of point sets such that for \(\alpha \)-smooth f, convergence rate \(O(N^{-\alpha }(\log N)^{s\alpha })\) is assured. As a coarse version of his theory, M-Saito-Matoba introduced Walsh figure of Merit (WAFOM), which gives the convergence rate \(O(N^{-C\log N/s})\). WAFOM is efficiently computable. By a brute-force search of low WAFOM point sets, we observe a convergence rate of order \(N^{-\alpha }\) with \(\alpha >1\), for several test integrands for \(s=4\) and 8.

In this article, we evaluate a quasi-Monte Carlo (QMC) method with various low-discrepancy sequences (LDS) in illumination optical systems which are adopted in some commercial products, and clarify the method's effectiveness quantitatively. We assumed the evaluated systems were an illumination optical system with a perfectly diffusing surface, and we compared them against the theoretical irradiance distribution. The evaluation results indicate that the QMC method delivers higher asymptotic convergence rate than the MC method does, and there is little difference between each LDS. In evaluation of simple optical systems that can be boiled down to low-dimensional numerical integration problems, the QMC method was found to be extremely effective.

A Brownian bridge is a stochastic process derived from standard Brownian motion by requiring an extra constraint. This gives Brownian bridges unique mathematical properties, fascinating, itself, and useful in statistical and mathematical modeling. Here, we define a Brownian bridge and explain some of its unique mathematical properties. These properties make it useful as a mathematical model for real‐world phenomena and as a tool for statistical analysis. We then give some examples of applications. We will see many mathematical ideas that touch Brownian bridges, such as differential equations, linear spaces, topology, metric spaces of functions, measure theory, stochastic integrals, estimation, and hypothesis testing. Copyright © 2009 Wiley Periodicals, Inc.This article is categorized under: Statistical Models > Time Series Models Data: Types and Structure > Time Series, Stochastic Processes, and Functional Data

Image based probabilistic slope stability analysis of soil layer interface fluctuations with Brownian bridges