Fast American Basket Option Pricing on a multi-GPU Cluster

8 pagesInternational audienceThis article presents a multi-GPU adaptation of a specific Monte Carlo and classification based method for pricing American basket options, due to Picazo. The first part relates how to combine fine and coarse-grained parallelization to price American basket options. A dynamic strategy of kernel calibration is proposed. Doing so, our implementation on a reasonable size (18) GPU cluster achieves the pricing of a high dimensional (40) option in less than one hour against almost 8 as observed for runs we conducted in the past, using a 64-core cluster (composed of quad-core AMD Opteron 2356). In order to benefit from different GPU device types, we detail the dynamic strategy we have used to load balance GPU calculus which greatly improves the overall pricing time we obtained. An analysis of possible bottleneck effects demonstrates that there is a sequential bottleneck due to the training phase that relies upon the AdaBoost classification method, which prevents the implementation to be fully scalable, and so prevents to envision further decreasing pricing time down to handful of minutes. For this we propose to consider using Random Forests classification method: it is naturally dividable over a cluster, and available like AdaBoost as a black box from the popular Weka machine learning library. However our experimental tests will show that its use is costly

American Options Based on Malliavin Calculus and Nonparametric Variance Reduction Methods

This paper is devoted to pricing American options using Monte Carlo and the Malliavin calculus. Unlike the majority of articles related to this topic, in this work we will not use localization fonctions to reduce the variance. Our method is based on expressing the conditional expectation E[f(St)/Ss] using the Malliavin calculus without localization. Then the variance of the estimator of E[f(St)/Ss] is reduced using closed formulas, techniques based on a conditioning and a judicious choice of the number of simulated paths. Finally, we perform the stopping times version of the dynamic programming algorithm to decrease the bias. On the one hand, we will develop the Malliavin calculus tools for exponential multi-dimensional diffusions that have deterministic and no constant coefficients. On the other hand, we will detail various nonparametric technics to reduce the variance. Moreover, we will test the numerical efficiency of our method on a heterogeneous CPU/GPU multi-core machine

Lower Precision calculation for option pricing

The problem of options pricing is one of the most critical issues and fundamental building blocks in mathematical finance. The research includes deployment of lower precision type in two options pricing algorithms: Black-Scholes and Monte Carlo simulation. We make an assumption that the shorter the number used for calculations is (in bits), the more operations we are able to perform in the same time. The results are examined by a comparison to the outputs of single and double precision types. The major goal of the study is to indicate whether the lower precision types can be used in financial mathematics. The findings indicate that Black-Scholes provided more precise outputs than the basic implementation of Monte Carlo simulation. Modification of the Monte Carlo algorithm is also proposed. The research shows the limitations and opportunities of the lower precision type usage. In order to benefit from the application in terms of the time of calculation improved algorithms can be implemented on GPU or FPGA. We conclude that under particular restrictions the lower precision calculation can be used in mathematical finance.

Numerical methods for option pricing.

This thesis aims to introduce some fundamental concepts underlying option valuation theory including implementation of computational tools. In many cases analytical solution for option pricing does not exist, thus the following numerical methods are used: binomial trees, Monte Carlo simulations and finite difference methods. First, an algorithm based on Hull and Wilmott is written for every method. Then these algorithms are improved in different ways. For the binomial tree both speed and memory usage is significantly improved by using only one vector instead of a whole price storing matrix. Computational time in Monte Carlo simulations is reduced by implementing a parallel algorithm (in C) which is capable of improving speed by a factor which equals the number of processors used. Furthermore, MatLab code for Monte Carlo was made faster by vectorizing simulation process. Finally, obtained option values are compared to those obtained with popular finite difference methods, and it is discussed which of the algorithms is more appropriate for which purpose

Pricing Higher-Dimensional American Options Using The Stochastic Grid Method

This paper considers the problem of pricing options with early-exercise features whose payo depends on several sources of uncertainty. We propose a stochastic grid method for estimating the upper and lower bound values of high-dimensional American options. The method is a hybrid of the least squares method of Longsta and Schwartz (2001) [22], the stochastic mesh method of Broadie and Glasserman (2004) [11], and stratified state aggregation along the pay-off method of Barraquand and Martineau (1995) [3]. Numerical results are given for single asset Bermudan options, Bermudan max options, Bermudan options on the arithmetic mean of a collection of stocks

Accelerating Reconfigurable Financial Computing

This thesis proposes novel approaches to the design, optimisation, and management of reconfigurable computer accelerators for financial computing. There are three contributions. First, we propose novel reconfigurable designs for derivative pricing using both Monte-Carlo and quadrature methods. Such designs involve exploring techniques such as control variate optimisation for Monte-Carlo, and multi-dimensional analysis for quadrature methods. Significant speedups and energy savings are achieved using our Field-Programmable Gate Array (FPGA) designs over both Central Processing Unit (CPU) and Graphical Processing Unit (GPU) designs. Second, we propose a framework for distributing computing tasks on multi-accelerator heterogeneous clusters. In this framework, different computational devices including FPGAs, GPUs and CPUs work collaboratively on the same financial problem based on a dynamic scheduling policy. The trade-off in speed and in energy consumption of different accelerator allocations is investigated. Third, we propose a mixed precision methodology for optimising Monte-Carlo designs, and a reduced precision methodology for optimising quadrature designs. These methodologies enable us to optimise throughput of reconfigurable designs by using datapaths with minimised precision, while maintaining the same accuracy of the results as in the original designs

The Evaluation of Gas Sales Agreements

A gas sales agreement, also called a gas swing contract, is an agreement between a supplier and a purchaser for the delivery of variable daily quantities of gas, between specified minimum and maximum daily limits, over a certain number of years at a strike price. The main constraint of such an agreement is that there is a minimum volume of gas for which the buyer will be charged at the end of the year, regardless of the actual quantity of gas taken. For multiple year contracts, there are also features called the make-up and carry-forward banks which add another level of complexity to the analysis. We propose a framework for pricing such multiple year contracts where both the gas price and strike price are stochastic processes. With the help of a two-dimensional trinomial tree, we are able to price such swing contracts with both make-up and carry-forward banks, and find the optimal daily decisions and the optimal yearly usage of the make-up and carry-forward banks. We also provide a detailed analysis of the different features that these contracts possess. Furthermore, another feature, called the indexation principle, is popular in real markets, under which the strike price is called the index. In each month, the value of the index is determined by the weighted average price of some energy products in the previous month. We design a lattice-based algorithm to price such swing contracts and find optimal daily decisions by using graphics processing units. Since the least-squares Monte Carlo method is well-known to handle sophisticated models, such as multi-factor models, models with regime-switching, or models with jumps, we build this method for the pricing of gas sales agreements and analyze the performance of it, especially the impacts of explanatory variables. With the help of concrete numerical examples, various features of such contracts with indexation are demonstrated