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This study describes and tests a simple, efficient method of approximating stochastic volatility call option prices that is lattice based, and can easily accommodate the value of early exercise for American options. The proposed model exploits the fact that stochastic volatility call option prices are approximately linear in the degree of correlation between the asset price and the volatility. By calculating binomial prices for perfect and zero correlations, a simple linear function of the binomial prices can be used to accurately estimate call option prices for any other degree of correlation. A distinct advantage of the model is that it can be easily modified to price American puts.
This article develops a method for valuing contingent payoffs for a non-constant volatility process via a simple recombining binomial tree. The direct application of the technology provides a way to price, for example, American calls or puts governed by a stock price process with stochastic volatility. The stock price and volatility diffusions may have non-zero correlations. This feature allows model prices consistent with the volatility smile. Numerical estimates of the hedge statistics (delta, gamma, and vega) are obtained directly from the tree.
The early exercise opportunity of an American option makes it challenging to price and an array of approaches have been proposed in the vast literature on this topic. In The Numerical Solution of the American Option Pricing Problem, Carl Chiarella, Boda Kang and Gunter Meyer focus on two numerical approaches that have proved useful for finding all prices, hedge ratios and early exercise boundaries of an American option. One is a finite difference approach which is based on the numerical solution of the partial differential equations with the free boundary problem arising in American option pricing, including the method of lines, the component wise splitting and the finite difference with PSOR. The other approach is the integral transform approach which includes Fourier or Fourier Cosine transforms. Written in a concise and systematic manner, Chiarella, Kang and Meyer explain and demonstrate the advantages and limitations of each of them based on their and their co-workers'' experiences with these approaches over the years. Contents: Introduction; The Merton and Heston Model for a Call; American Call Options under Jump-Diffusion Processes; American Option Prices under Stochastic Volatility and Jump-Diffusion Dynamics OCo The Transform Approach; Representation and Numerical Approximation of American Option Prices under Heston; Fourier Cosine Expansion Approach; A Numerical Approach to Pricing American Call Options under SVJD; Conclusion; Bibliography; Index; About the Authors. Readership: Post-graduates/ Researchers in finance and applied mathematics with interest in numerical methods for American option pricing; mathematicians/physicists doing applied research in option pricing. Key Features: Complete discussion of different numerical methods for American options; Able to handle stochastic volatility and/or jump diffusion dynamics; Able to produce hedge ratios efficiently and accurately"
The book gives a systematical presentation of stochastic approximation methods for discrete time Markov price processes. Advanced methods combining backward recurrence algorithms for computing of option rewards and general results on convergence of stochastic space skeleton and tree approximations for option rewards are applied to a variety of models of multivariate modulated Markov price processes. The principal novelty of presented results is based on consideration of multivariate modulated Markov price processes and general pay-off functions, which can depend not only on price but also an additional stochastic modulating index component, and use of minimal conditions of smoothness for transition probabilities and pay-off functions, compactness conditions for log-price processes and rate of growth conditions for pay-off functions. The volume presents results on structural studies of optimal stopping domains, Monte Carlo based approximation reward algorithms, and convergence of American-type options for autoregressive and continuous time models, as well as results of the corresponding experimental studies.
This book introduces some advanced topics in probability theories ? both pure and applied ? is divided into two parts. The first part deals with the analysis of stochastic dynamical systems, in terms of Gaussian processes, white noise theory, and diffusion processes. The second part of the book discusses some up-to-date applications of optimization theories, martingale measure theories, reliability theories, stochastic filtering theories and stochastic algorithms towards mathematical finance issues such as option pricing and hedging, bond market analysis, volatility studies and asset trading modeling.
Modern option pricing techniques are often considered among the most mathematically complex of all applied areas of finance. Financial analyst has reached a point where they are able to calculate with alarming accuracy, the value of an option. In this book we study binomial approximation methods for European as well as American options. We study options on stocks, with as well as without dividends. We also include a chapter on how to derive Black-Scholes equation from a binomial model. Our study shows how versatile the binomial method is, both from a theoretical and a practical point of view.
Derivatives and Risk Management provides readers with a thorough knowledge of the functions of derivatives and the many risks associated with their use. It covers particular derivative instruments available in India and the four types of derivatives. It is useful for postgraduate students of commerce, finance and management, fund managers, risk-management specialists, treasury managers, students taking the CFA examinations and anyone who wants to understand the derivatives market in India.
An unprecedented book on option pricing! For the first time, the basics on modern option pricing are explained ``from scratch'' using only minimal mathematics. Market practitioners and students alike will learn how and why the Black-Scholes equation works, and what other new methods have been developed that build on the success of Black-Shcoles. The Cox-Ross-Rubinstein binomial trees are discussed, as well as two recent theories of option pricing: the Derman-Kani theory on implied volatility trees and Mark Rubinstein's implied binomial trees. Black-Scholes and Beyond will not only help the reader gain a solid understanding of the Balck-Scholes formula, but will also bring the reader up to date by detailing current theoretical developments from Wall Street. Furthermore, the author expands upon existing research and adds his own new approaches to modern option pricing theory. Among the topics covered in Black-Scholes and Beyond: detailed discussions of pricing and hedging options; volatility smiles and how to price options ``in the presence of the smile''; complete explanation on pricing barrier options.
Through the incorporation of real-life examples from Indian organizations, Derivatives and Risk Management provides cutting-edge material comprising new and unique study tools and fresh, thought-provoking content. The organization of the text is designed to conceptually link a firm’s actions to its value as determined in the derivatives market. It addresses the specific needs of Indian students and managers by successfully blending the best global derivatives and risk management practices with an in-depth coverage of the Indian environment.