# Mathematical Modeling And Methods Of Option Pricing Pdf

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## Analysis of Options Pricing using Mathematical Models

Skip to search form Skip to main content You are currently offline. Some features of the site may not work correctly. DOI: We consider a model of linear market impact, and address the problem of replicating a contingent claim in this framework. We derive a non-linear Black-Scholes Equation that provides an exact replication strategy. This equation is fully non-linear and singular, but we show that it is well posed, and we prove existence of smooth solutions for a large class of final payoffs, both for constant and local volatility.

A wide range of financial derivatives commonly traded in the equity and fixed income markets are. Starting from the renowned Black-Scholes-Merton formulation of option pricing model, readers are guided through the text on the new advances on the state-of-the-art derivative pricing models and interest rate models. Both analytic techniques and numerical methods for solving various types of derivative pricing models are emphasized. The second edition presents a substantial revision of the first edition. The continuous-time martingale pricing theory is motivated through analysis of the underlying financial economics principles within a discrete-time framework. A large collection of closed-form formulas of various forms of exotic equity and fixed income derivatives are documented.

## Option Pricing with Linear Market Impact and Non-Linear Black-Scholes Equations

From the partial differential equation in the model, known as the Black—Scholes equation , one can deduce the Black—Scholes formula , which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price given the risk of the security and its expected return instead replacing the security's expected return with the risk-neutral rate. The formula led to a boom in options trading and provided mathematical legitimacy to the activities of the Chicago Board Options Exchange and other options markets around the world. Based on works previously developed by market researchers and practitioners, such as Louis Bachelier , Sheen Kassouf and Ed Thorp among others, Fischer Black and Myron Scholes demonstrated in that a dynamic revision of a portfolio removes the expected return of the security, thus inventing the risk neutral argument.

### Option Pricing Theory

Working knowledge of calculus, statistics and probability, and interest in the use of mathematical modeling. Please go to Unit 0 in the Course Outline to take the prerequisites assessment. This is an introductory course on options and other financial derivatives, and their applications to risk management. We will start with discrete-time, binomial trees models, but most of the course will be in the framework of continuous-time, Brownian Motion driven models. A basic introduction to Stochastic, Ito Calculus will be given. The benchmark model will be the Black-Scholes-Merton pricing model, but we will also discuss more general models, such as stochastic volatility models. We will discuss both the Partial Differential Equations approach, and the probabilistic, martingale approach.

First published in Chinese in by Higher Education Press. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc. In this case permission to photocopy is not required from the publisher. In this course, I intended to present a systematic and in-depth introduction to the Black- Scholes-Merton's option pricing theory from the perspective of partial differential equation theory.

Risk Management and Financial Derivatives; Arbitrage-Free Principle; Binomial Tree Methods — Discrete Models of Option Pricing; Brownian Motion and Itô.

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