CPU-GPU hybrid parallel binomial American option pricing

We present in this paper a novel parallel binomial algorithm that computes the price of an American option. The algorithm partitions a binomial tree constructed for the pricing into blocks of multiple levels of nodes, and assigns each such block to multiple processors. Each of the processors then computes the option's values at its assigned nodes in two phases. The algorithm is implemented and tested on a heterogeneous system consisting of an Intel multi-core processor and a NVIDIA GPU. The whole task is split and divided over and the CPU and GPU so that the computations are performed on the two processors simultaneously. In the hybrid processing, the GPU is always assigned the last part of a block, and makes use of a couple of buffers in the on-chip shared memory to reduce the number of accesses to the off-chip device memory. The performance of the hybrid processing is compared with an optimised CPU serial code, a CPU parallel implementation and a GPU standalone program.published_or_final_versio

Automatic generation of high-throughput systolic tree-based solvers for modern FPGAs

Tree-based models are a class of numerical methods widely used in financial option pricing, which have a computational complexity that is quadratic with respect to the solution accuracy. Previous research has employed reconfigurable computing with small degrees of parallelism to provide faster hardware solutions compared with general-purpose processing software designs. However, due to the nature of their vector hardware architectures, they cannot scale their compute resources efficiently, leaving them with pricing latency figures which are quadratic with respect to the problem size, and hence to the solution accuracy. Also, their solutions are not productive as they require hardware engineering effort, and can only solve one type of tree problems, known as the standard American option. This thesis presents a novel methodology in the form of a high-level design framework which can capture any common tree-based problem, and automatically generates high-throughput field-programmable gate array (FPGA) solvers based on proposed scalable hardware architectures. The thesis has made three main contributions. First, systolic architectures were proposed for solving binomial and trinomial trees, which due to their custom systolic data-movement mechanisms, can scale their compute resources efficiently to provide linear latency scaling for medium-size trees and improved quadratic latency scaling for large trees. Using the proposed systolic architectures, throughput speed-ups of up to 5.6X and 12X were achieved for modern FPGAs, compared to previous vector designs, for medium and large trees, respectively. Second, a productive high-level design framework was proposed, that can capture any common binomial and trinomial tree problem, and a methodology was suggested to generate high-throughput systolic solvers with custom data precision, where the methodology requires no hardware design effort from the end user. Third, a fully-automated tool-chain methodology was proposed that, compared to previous tree-based solvers, improves user productivity by removing the manual engineering effort of applying the design framework to option pricing problems. Using the productive design framework, high-throughput systolic FPGA solvers have been automatically generated from simple end-user C descriptions for several tree problems, such as American, Bermudan, and barrier options.Open Acces

A study in the financial valuation of a topping oil refinery

Oil refineries underpin modern day economics, finance and engineering – without their refined products the world would stand still, as vehicles would not have petrol, planes grounded without kerosene and homes not heated, without heating oil. In this thesis I study the refinery as a financial asset; it is not too dissimilar to a chemical plant, in this respect. There are a number of reasons for this research; over recent years there have been legal disputes based on a refiner's value, investors and entrepreneurs are interested in purchasing refineries, and finally the research in this arena is sparse. In this thesis I utilise knowledge and techniques within finance, optimisation, stochastic mathematics and commodities to build programs that obtain a financial value for an oil refinery. In chapter one I introduce the background of crude oil and the significance of the refinery in the oil value chain. In chapter two I construct a traditional discounted cash flow valuation often applied within practical finance. In chapter three I program an extensive piecewise non linear optimisation solution on the entire state space, leveraging off a simulation of the refined products using a set of single factor Schwartz (1997) stochastic equations often applied to commodities. In chapter four I program an optimisation using an approximation on crack spread option data with the aim of lowering the duration of solution found in chapter three; this is achieved by utilising a two-factor Hull & White sub-trinomial tree based numerical scheme; see Hull & White (1994) articles I & II for a thorough description. I obtain realistic and accurate numbers for a topping oil refinery using financial market contracts and other real data for the Vadinar refinery based in Gujurat India

Valuation of Multiple Exercise Options

Multiple exercise options may be considered as generalizations of American-style options as they provide the holder more than one exercise right. Examples of financial derivatives and real options with these properties have become more prevalent over the past decade and appear in sectors ranging from insurance to energy industries. Throughout the thesis particular attention is paid to swing options although we note that the methods described are equally applicable to other types of multiple exercise options. This thesis presents two novel methods for pricing multiple exercise option by simulation; the forest of stochastic trees and the forest of stochastic meshes. The proposed methods are of particular use in cases where there are potentially a large number (3 or more) of assets underlying the contract and/or if a number of risk factors are desirable for modelling the underlying price process. These valuation methods result in positively- and negatively-biased estimators for the true option value. We prove the sign of the estimator bias and show that these estimators are consistent for the true option value. A confidence interval for the true option value is easily constructed. Examples confirm that the implementation of these methods is correct and consistent with the theoretical properties of the estimators. This thesis also explores in detail a number of methods meant to enhance the effectiveness of the proposed simulation methods. These include using high performance computing techniques which include both parallel computing techniques on CPU-clusters and General purpose Graphics Processing Units (GPGPU) that take advantage of relatively inexpensive processors. Additionally we explore bias-corrected estimators for the option values which attempt to estimate the bias introduced at each time step by the estimator and then subtract this result. These improvements are desirable due to the computationally intensive nature of both methods

Quantitative Analyses on Non-Linearities in Financial Markets

"The brief market plunge was just a small indicator of how complex and chaotic, in the formal sense, these systems have become. Our nancial system is so complicated and so interactive [...]. What happened in the stock market is just a little example of how things can cascade or how technology can interact with market panic" (Ben Bernanke, IHT, May 17, 2010) One of the most important issues in economics is modeling and fore- casting the uctuations that characterize both nancial and real mar- kets, such as interest rates, commodities and stock prices, output growth, unemployment, or exchange rate. There are mainly two op- posite views concerning these economic uctuations. According to the rst one, which was the predominant thought in the 1930s, the economic system is mainly linear and stable, only randomly hit by exogenous shocks. Ragnar Frisch, Eugen Slutsky and Jan Tinbergen, to cite a few, are important exponents of this view, and they demon- strated that the uctuations observed in the real business cycle may be produced in a stable linear system subject to an external sequence of random shocks. This view has been criticized starting from the 1940s and the 1950s, since it was not able to provide a strong eco- nomic explanation of observed uctuations. Richard Goodwin,John Hicks and Nicholas Kaldor introduced a nonlinear view of the econ- omy, showing that even in absence of external shocks, uctuations might arise. The economists then suggested an alternative within the exogenous approach, at rst by using the stochastic real busi- ness cycle models (Finn E. Kidland and Edward C. Prescott, 1982) and, more recently, by the adoption of the New Keynesian Dynamic Stochastic General Equilibrium (DSGE) models, very adopted from the most important institutions and central banks. These models, however, have also been criticized for the assumption of the rational- ity of agents' behaviour, since rational expectations have been found to be systematically wrong in the business cycle. Expectations are of fundamental importance in economics and nance, since the agents' decisions about the future depends upon their expectations and their beliefs. It is in fact very unlikely that agents are perfect foresighters with rational expectations in a complex world, characterized by an irregular pattern of prices and quantities dealt in nancial markets, in which sophisticated nancial instruments are widespread. In the rst chapter of this dissertation, I will face the machine learn- ing technique, which is a nonlinear tool used for a better tting, fore- casting and clustering of dierent nancial time series and existing information in nancial markets. In particular, I will present a collec- tion of three dierent applications of these techniques, adapted from three dierent joint works: "Yield curve estimation under extreme conditions: do RBF net- works perform better?, joint with Pier Giuseppe Giribone, Marco Neelli, Marina Resta, published Anna Esposito, Marcos Faundez- Zanuy, Carlo Francesco Morabito, Eros Pasero Edrs, Multidisci- plinary Approaches to Neural Computing/Vol. 69/ WIRN 2017 and Chapter 22 in book "Neural Advances in Processing Non- linear Dynamic Signals", Springer; Interest rates term structure models and their impact on actuarial forecasting, joint with Pier Giuseppe Giribone and Marina Resta, presented at XVIII Quantitative Finance Workshop, University of Roma 3, January 2018; Applications of Kohonen Maps in financial markets: design of an automatic system for the detection of pricing anomalies, joint with Pier Giuseppe Giribone and published on Risk Management Magazine, 3-2017. In the second chapter, I will present the study A nancial market model with conrmation bias, in which nonlinearity is present as a result of the formation of heterogeneous expectations. This work is joint with Fabio Tramontana and it has been presented during the X MDEF (Dynamic Models in Economics and Finance) Workshop at University of Urbino Carlo Bo. Finally, the third chapter is a rielaboration of another joint paper, "The eects of negative nominal risk rates on the pricing of American Calls: some theoretical and numerical insights", with Pier Giuseppe Giribone and Marina Resta, published on Modern Economy 8(7), July 2017, pp 878-887. The problem of quantifying the value of early ex- ercise in an option written on equity is a complex mathematical issue that deals with continuous optimal control. In order to solve the con- tinuous dynamic optimization problem that involves high non linearity in the state variables, we have adopted a discretization scheme based on a stochastic trinomial tree. This methodology reveals a higher reliability and exibility than the traditional approaches based on approximated quasi-closed formulas in a context where financial markets are characterized by strong anomalies such as negative interest rates

Evaluating Multicore Algorithms on the Unified Memory Model

One of the challenges to achieving good performance on multicore architectures is the effective utilization of the underlying memory hierarchy. While this is an issue for single-core architectures, it is a critical problem for multicore chips. In this paper, we formulate the unified multicore model (UMM) to help understand the fundamental limits on cache performance on these architectures. The UMM seamlessly handles different types of multiple-core processors with varying degrees of cache sharing at different levels. We demonstrate that our model can be used to study a variety of multicore architectures on a variety of applications. In particular, we use it to analyze an option pricing problem using the trinomial model and develop an algorithm for it that has near-optimal memory traffic between cache levels. We have implemented the algorithm on a two Quad-Core Intel Xeon 5310 1.6 GHz processors (8 cores). It achieves a peak performance of 19.5 GFLOPs, which is 38% of the theoretical peak of the multicore system. We demonstrate that our algorithm outperforms compiler-optimized and auto-parallelized code by a factor of up to 7.5

Valuation of Multiple Exercise Option Using a Modified Longstaff and Schwartz Approach

In this work we study the problem of pricing multiple exercise options, a class of early exercise options that are traded in the energy market, using a modified Longstaff and Schwartz approach. Recent work by Letourneau and Stentoft (2014) shows American option price estimator bias is reduced by imposing additional structure on the regressions used in Monte Carlo pricing algorithms. We extend their methodology to the Monte Carlo valuation of multiple exercise options by requiring additional structure on the regressions used to estimate continuation values. The resulting price estimators have reduced bias, particularly for small sample sizes, and results hold across a variety of option types, maturities and moneyness. A comparison of the original Longstaff and Schwartz approach to the modified Longstaff and Schwartz approach demonstrates the strengths of the developed numerical technique

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