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

High dimensional American options

Pricing single asset American options is a hard problem in mathematical finance. There are no closed form solutions available (apart from in the case of the perpetual option), so many approximations and numerical techniques have been developed. Pricing multi–asset (high dimensional) American options is still more difficult. We extend the method proposed theoretically by Glasserman and Yu (2004) by employing regression basis functions that are martingales under geometric Brownian motion. This results in more accurate Monte Carlo simulations, and computationally cheap lower and upper bounds to the American option price. We have implemented these models in QuantLib, the open–source derivatives pricing library. The code for many of the models discussed in this thesis can be downloaded from quantlib.org as part of a practical pricing and risk management library. We propose a new type of multi–asset option, the “Radial Barrier Option” for which we find analytic solutions. This is a barrier style option that pays out when a barrier, which is a function of the assets and their correlations, is hit. This is a useful benchmark test case for Monte Carlo simulations and may be of use in approximating multi–asset American options. We use Laplace transforms in this analysis which can be applied to give analytic results for the hitting times of Bessel processes. We investigate the asymptotic solution of the single asset Black–Scholes–Merton equation in the case of low volatility. This analysis explains the success of some American option approximations, and has the potential to be extended to basket options

Pricing of early-exercise Asian options under L\'evy processes based on Fourier cosine expansions

In this article, we propose a pricing method for Asian options with early-exercise features. It is based on a two-dimensional integration and a backward recursion of the Fourier coefficients, in which several numerical techniques, like Fourier cosine expansions, Clenshaw–Curtis quadrature and the Fast Fourier Transform (FFT) are employed. Rapid convergence of the pricing method is illustrated by an error analysis. Its performance is further demonstrated by various numerical examples, where we also show the power of an implementation on Graphics Processing Units (GPUs)

The History of the Quantitative Methods in Finance Conference Series. 1992-2007

This report charts the history of the Quantitative Methods in Finance (QMF) conference from its beginning in 1993 to the 15th conference in 2007. It lists alphabetically the 1037 speakers who presented at all 15 conferences and the titles of their papers.

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

Three Essays on Stochastic Optimization Applied in Financial Engineering and Inventory Management

Stochastic optimization methods are now being widely used in a multitude of applications. This dissertation includes three essays on applying stochastic optimization methods to solve problems in inventory management and financial engineering. Essay one addresses the problem of simultaneous price determination and inventory management. Demand depends explicitly on the product price p, and the inventory control system operates under a periodic review (s, S) ordering policy. To minimize the long-run average loss, we derive sample path derivatives that can be used in a gradient-based algorithm for determining the optimal values of the three parameters (s, S, p) in a simulation-based optimization procedure. Numerical results for several optimization examples via different stochastic algorithms are presented, and consistency proofs for the estimators are provided. Essay two considers the application of stochastic optimization methods to American-style option pricing. We apply a randomized optimization algorithm called Model Reference Adaptive Search (MRAS) to pricing American-style options through parameterizing the early exercise boundary. Numerical results are provided for pricing American-style call and put options written on underlying assets following geometric Brownian motion and Merton jump-diffusion processes. We also price American-style Asian options written on underlying assets following geometric Brownian motion. The results from the MRAS algorithm are compared with the cross-entropy (CE) method, and MRAS is found to be an efficient method. Essay three addresses the problem of finding the optimal importance sampling measure when simulating portfolios of credit risky assets. We apply a gradient-based stochastic approximation method to find the parameters in the minimum variance problem when importance sampling is used. The gradient estimator is obtained under the original measure. We also employ the CE method to solve the same variance minimization problem. Numerical results illustrating the variance reduction are presented for the estimation of the portfolios' expected loss, unexpected loss and quantiles

Multilevel Monte Carlo methods

The author's presentation of multilevel Monte Carlo path simulation at the MCQMC 2006 conference stimulated a lot of research into multilevel Monte Carlo methods. This paper reviews the progress since then, emphasising the simplicity, flexibility and generality of the multilevel Monte Carlo approach. It also offers a few original ideas and suggests areas for future research

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