Numerical Methods for Finance

Numerical methods form an important part of options pricing and especially in cases where there is no closed form analytic formula. We discuss two of the primary numerical methods that are currently used by financial professionals for determining the price of an options namely Monte Carlo method and finite difference method. Then we compare the convergence of the two methods to the analytic Black-Scholes price of European option. Monte Carlo method is good for pricing exotic options while Crank Nicolson finite difference method is unconditionally stable, more accurate and converges faster than Monte Carlo method when pricing standard options.

1996

We discuss the numerical solution of a Cree boundary parabolic partial differential problem based on the Black-Scholes model for the pricing of derivative securities. We present the formulation and numerical behavior of tIte linear complementarity, and froni-tracking solutions to the option pricing problem for American style options. We show experimentally that the class of front-tracking methods produce efficient and accurate approximations to the pricing oC derivative securities compared to the binomial pricing method.

Numerical Methods form an important part of options valuation and especially in cases where there is no closed form analytic formula. We discuss three numerical methods for options valuation namely Binomial model, Finite difference methods and Monte Carlo simulation method. Then we compare the convergence of these methods to the analytic Black-Scholes price of the options. Among the methods considered, Crank Nicolson finite difference method is unconditionally stable, more accurate and converges faster than binomial model and Monte Carlo Method when pricing vanilla options, while Monte carlo simulation method is good for pricing path dependent options.

Numerical methods form an important part of options pricing and especially in cases where there is no closed form analytic formula. We discuss two of the primary numerical methods that are currently used by financial professionals for determining the price of an options namely Monte Carlo method and finite difference method. Then we compare the convergence of the two methods to the analytic Black-Scholes price of European option. Monte Carlo method is good for pricing exotic options while Crank Nicolson finite difference method is unconditionally stable, more accurate and converges faster than Monte Carlo method when pricing standard options.

Worcester Polytechnic Institute, 2007

This project is about the pricing of options by some finite difference methods in C++. European call and put options and also American call and put options will be priced by the Explicit and Implicit finite difference methods in this project. Even though C++ is being used to price options in the financial field, it is not known who has done this project in this manner before. This project provides a direct means to price the options. The user need not have a knowledge of C++ in order to make use of the software that will be attached. This may be the difference between this project and existing ones where a user needs to have a working knowledge of C++ in addition to knowing how to price derivatives. The work may be extended to price derivatives using the Crank-Nicolson method. The goals of this project are the following: Compute Asset prices at maturity for call and put options for both European and American type options, compute option prices at maturity, and last but not the least to back-track into the mesh to compute option prices today. These goals were achieved using abstract factory console programs written in C++ for each option type(call or put) for both European and American type options. The results will be displayed in arrays and matrix / mesh forms. The client-server interaction is self-directional. The simplicity of the source codes will enable clients to use it without having to be trained by professional programmers. Of course, the impact of this is the elimination of training time and the complexities of programming. In view of this, the source codes/software that will be attached may compete with some of the existing software on the market. The outline presented here will provide the reader with a bit of training on derivatives pricing if the individual is not already familiar with pricing methods, or it may provide an already experienced pricer with information or insight into derivatives pricing theory. A roadmap will be provided shortly to serve as a guide to the reader.

This is a book about Monte Carlo methods from the perspective of financial engineering. Monte Carlo simulation has become an essential tool in the pricing of derivative securities and in risk management; these applications have, in turn, stimulated research into new Monte Carlo techniques and renewed interest in some old techniques. This is also a book about financial engineering from the perspective of Monte Carlo methods. One of the best ways to develop an understanding of a model of, say, the term structure of interest rates is to implement a simulation of the model; and finding ways to improve the efficiency of a simulation motivates a deeper investigation into properties of a model. My intended audience is a mix of graduate students in financial engineering, researchers interested in the application of Monte Carlo methods in finance, and practitioners implementing models in industry. This book has grown out of lecture notes I have used over several years at Columbia, for a semester at Princeton, and for a short course at Aarhus University. These classes have been attended by masters and doctoral students in engineering, the mathematical and physical sciences, and finance. The selection of topics has also been influenced by my experiences in developing and delivering professional training courses with Mark Broadie, often in collaboration with Leif Andersen and Phelim Boyle. The opportunity to discuss the use of Monte Carlo methods in the derivatives industry with practitioners and colleagues has helped shaped my thinking about the methods and their application. Students and practitioners come to the area of financial engineering from diverse academic fields and with widely ranging levels of training in mathematics, statistics, finance, and computing. This presents a challenge in setting the appropriate level for discourse. The most important prerequisite for reading this book is familiarity with the mathematical tools routinely used to specify and analyze continuous-time models in finance. Prior exposure to the basic principles of option pricing is useful but less essential. The tools of mathematical finance include Itˆo calculus, stochastic differential equations, and martingales. Perhaps the most advanced idea used in many places in vi this book is the concept of a change of measure. This idea is so central both to derivatives pricing and to Monte Carlo methods that there is simply no avoiding it. The prerequisites to understanding the statement of the Girsanov theorem should suffice for reading this book. Whereas the language of mathematical finance is......

2018

Option pricing is a very attractive issue of financial engineering and optimization. The problem of determining the fair price of an option arises from the assumptions made under a given financial market model. The increasing complexity of these market assumptions contributes to the popularity of the numerical treatment of option valuation. Therefore, the pricing and hedging of plain vanilla options under the Black-Scholes model usually serve as a benchmark for the development of new numerical pricing approaches and methods designed for advanced option pricing models. The objective of the paper is to present and compare the methodological concepts for the valuation of simple vanilla options using the relatively modern numerical techniques in this issue which arise from the discontinuous Galerkin method, the wavelet approach and the fuzzy transform technique. A theoretical comparison is accompanied by an empirical study based on the numerical verification of simple vanilla option prices. The resulting numerical schemes represent a particularly effective option pricing tool that enables some features of options that are dependent on the discretization of the computational domain as well as the order of the polynomial approximation to be captured better.

This paper presents finite difference methods for options pricing. These methods are useful to solve partial differential equations and provide a general numerical solution to the valuation problems, as well as an optimal early exercise strategy and other physical sciences. The methods considered are the basic implicit and Crank Nicolson finite difference methods. The stability and accuracy of each of the methods were considered. Crank Nicolson method is more accurate and converges faster than implicit method.

2007