Finance/Mathematics A First Course in Financial Mathematics Option Valuation: A First Course in Financial Mathematics provides a straightforward introduction to the mathematics and models used in the valuation of financial derivatives It examines the principles of option pricing in detail via standard binomial and stochastic calculus models Developing the requisite mathematical background as needed, the text introduces probability theory and stochastic calculus at an undergraduate level Hugo D Junghenn Option Valuation A First Course in Financial Mathematics Junghenn Largely self-contained, this classroom-tested text offers a sound introduction to applied probability through a mathematical finance perspective Numerous examples and exercises help readers gain expertise with financial calculus methods and increase their general mathematical sophistication The exercises range from routine applications to spreadsheet projects to the pricing of a variety of complex financial instruments Hints and solutions to odd-numbered problems are given in an appendix A First Course in Financial Mathematics The first nine chapters of the book describe option valuation techniques in discrete time, focusing on the binomial model The author shows how the binomial model offers a practical method for pricing options using relatively elementary mathematical tools The binomial model also enables a clear, concrete exposition of fundamental principles of finance, such as arbitrage and hedging, without the distraction of complex mathematical constructs The remaining chapters illustrate the theory in continuous time, with an emphasis on the more mathematically sophisticated Black– Scholes–Merton model Option Valuation Option Valuation K14090 K14090_Cover.indd 10/7/11 11:23 AM Option Valuation A First Course in Financial Mathematics CHAPMAN & HALL/CRC Financial Mathematics Series Aims and scope: The field of financial mathematics forms an ever-expanding slice of the financial sector This series aims to capture new developments and summarize what is known over the whole spectrum of this field It will include a broad range of textbooks, reference works and handbooks that are meant to appeal to both academics and practitioners The inclusion of numerical code and concrete realworld examples is highly encouraged Series Editors M.A.H Dempster Dilip B Madan Rama Cont Centre for Financial Research Department of Pure Mathematics and Statistics University of Cambridge Robert H Smith School of Business University of Maryland Center for Financial Engineering Columbia University New York Published Titles American-Style Derivatives; Valuation and Computation, Jerome Detemple Analysis, Geometry, and Modeling in Finance: Advanced Methods in Option Pricing,  Pierre Henry-Labordère Credit Risk: Models, Derivatives, and Management, Niklas Wagner Engineering BGM, Alan Brace Financial Modelling with Jump Processes, Rama Cont and Peter Tankov Interest Rate Modeling: Theory and Practice, Lixin Wu Introduction to Credit Risk Modeling, Second Edition, Christian Bluhm, Ludger Overbeck, and  Christoph Wagner Introduction to Stochastic Calculus Applied to Finance, Second Edition,  Damien Lamberton and Bernard Lapeyre Monte Carlo Methods and Models in Finance and Insurance, Ralf Korn, Elke Korn,  and Gerald Kroisandt Numerical Methods for Finance, John A D Appleby, David C Edelman, and John J H Miller Option Valuation: A First Course in Financial Mathematics, Hugo D Junghenn Portfolio Optimization and Performance Analysis, Jean-Luc Prigent Quantitative Fund Management, M A H Dempster, Georg Pflug, and Gautam Mitra Risk Analysis in Finance and Insurance, Second Edition, Alexander Melnikov Robust Libor Modelling and Pricing of Derivative Products, John Schoenmakers Stochastic Finance: A Numeraire Approach, Jan Vecer Stochastic Financial Models, Douglas Kennedy Structured Credit Portfolio Analysis, Baskets & CDOs, Christian Bluhm and Ludger Overbeck Understanding Risk: The Theory and Practice of Financial Risk Management, David Murphy Unravelling the Credit Crunch, David Murphy Proposals for the series should be submitted to one of the series editors above or directly to: CRC Press, Taylor & Francis Group 4th, Floor, Albert House 1-4 Singer Street London EC2A 4BQ UK Option Valuation A First Course in Financial Mathematics Hugo D Junghenn CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2011 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S Government works Version Date: 20150312 International Standard Book Number-13: 978-1-4398-8912-1 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use 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✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✷ ❉✐s❝r❡t❡ Pr♦❜❛❜✐❧✐t② ❙♣❛❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✸ ●❡♥❡r❛❧ Pr♦❜❛❜✐❧✐t② ❙♣❛❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✹ ❈♦♥❞✐t✐♦♥❛❧ Pr♦❜❛❜✐❧✐t② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✺ ■♥❞❡♣❡♥❞❡♥❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✻ ❊①❡r❝✐s❡s ✷✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ❘❛♥❞♦♠ ❱❛r✐❛❜❧❡s ✷✼ ✸✳✶ ❉❡✜♥✐t✐♦♥ ❛♥❞ ●❡♥❡r❛❧ Pr♦♣❡rt✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✷ ❉✐s❝r❡t❡ ❘❛♥❞♦♠ ❱❛r✐❛❜❧❡s ✸✳✸ ❈♦♥t✐♥✉♦✉s ❘❛♥❞♦♠ ❱❛r✐❛❜❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✹ ❏♦✐♥t ❉✐str✐❜✉t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✳✺ ■♥❞❡♣❡♥❞❡♥t ❘❛♥❞♦♠ ❱❛r✐❛❜❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✻ ❙✉♠s ♦❢ ■♥❞❡♣❡♥❞❡♥t ❘❛♥❞♦♠ ❱❛r✐❛❜❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✸✳✼ ❊①❡r❝✐s❡s ✹✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ❖♣t✐♦♥s ❛♥❞ ❆r❜✐tr❛❣❡ ✷✼ ✷✾ ✹✸ ✹✳✶ ❆r❜✐tr❛❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✷ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❉❡r✐✈❛t✐✈❡s ✹✹ ✹✳✸ ❋♦r✇❛r❞s ✹✳✹ ❈✉rr❡♥❝② ❋♦r✇❛r❞s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✹✳✺ ❋✉t✉r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✹✳✻ ❖♣t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✼ Pr♦♣❡rt✐❡s ♦❢ ❖♣t✐♦♥s ✹✳✽ ❉✐✈✐❞❡♥❞✲P❛②✐♥❣ ❙t♦❝❦s ✹✳✾ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✺✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✈✐✐ ✈✐✐✐ ✺ ❉✐s❝r❡t❡✲❚✐♠❡ P♦rt❢♦❧✐♦ Pr♦❝❡ss❡s ✺✾ ✺✳✶ ❉✐s❝r❡t❡✲❚✐♠❡ ❙t♦❝❤❛st✐❝ Pr♦❝❡ss❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✷ ❙❡❧❢✲❋✐♥❛♥❝✐♥❣ P♦rt❢♦❧✐♦s ✺✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✳✸ ❖♣t✐♦♥ ❱❛❧✉❛t✐♦♥ ❜② P♦rt❢♦❧✐♦s ✻✶ ✺✳✹ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✻ ❊①♣❡❝t❛t✐♦♥ ♦❢ ❛ ❘❛♥❞♦♠ ❱❛r✐❛❜❧❡ ✻✼ ✻✳✶ ❉✐s❝r❡t❡ ❈❛s❡✿ ❉❡✜♥✐t✐♦♥ ❛♥❞ ❊①❛♠♣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✳✷ ❈♦♥t✐♥✉♦✉s ❈❛s❡✿ ❉❡✜♥✐t✐♦♥ ❛♥❞ ❊①❛♠♣❧❡s ✻✳✸ Pr♦♣❡rt✐❡s ♦❢ ❊①♣❡❝t❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✻✳✹ ❱❛r✐❛♥❝❡ ♦❢ ❛ ❘❛♥❞♦♠ ❱❛r✐❛❜❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ✻✳✺ ❚❤❡ ❈❡♥tr❛❧ ▲✐♠✐t ❚❤❡♦r❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✻✳✻ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ❚❤❡ ❇✐♥♦♠✐❛❧ ▼♦❞❡❧ ✻✼ ✻✽ ✼✼ ✼✳✶ ❈♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ❇✐♥♦♠✐❛❧ ▼♦❞❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✳✷ Pr✐❝✐♥❣ ❛ ❈❧❛✐♠ ✐♥ t❤❡ ❇✐♥♦♠✐❛❧ ▼♦❞❡❧ ✼✳✸ ❚❤❡ ❈♦①✲❘♦ss✲❘✉❜✐♥st❡✐♥ ❋♦r♠✉❧❛ ✼✳✹ ❊①❡r❝✐s❡s ✼✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✻ ✽ ❈♦♥❞✐t✐♦♥❛❧ ❊①♣❡❝t❛t✐♦♥ ❛♥❞ ❉✐s❝r❡t❡✲❚✐♠❡ ▼❛rt✐♥❣❛❧❡s ✽✾ ✽✳✶ ❉❡✜♥✐t✐♦♥ ♦❢ ❈♦♥❞✐t✐♦♥❛❧ ❊①♣❡❝t❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✳✷ ❊①❛♠♣❧❡s ♦❢ ❈♦♥❞✐t✐♦♥❛❧ ❊①♣❡❝t❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✷ ✽✳✸ Pr♦♣❡rt✐❡s ♦❢ ❈♦♥❞✐t✐♦♥❛❧ ❊①♣❡❝t❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✹ ✽✳✹ ❉✐s❝r❡t❡✲❚✐♠❡ ▼❛rt✐♥❣❛❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✻ ✽✳✺ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✽ ✾ ❚❤❡ ❇✐♥♦♠✐❛❧ ▼♦❞❡❧ ❘❡✈✐s✐t❡❞ ✽✾ ✶✵✶ ✾✳✶ ▼❛rt✐♥❣❛❧❡s ✐♥ t❤❡ ❇✐♥♦♠✐❛❧ ▼♦❞❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✳✷ ❈❤❛♥❣❡ ♦❢ Pr♦❜❛❜✐❧✐t② ✾✳✸ ❆♠❡r✐❝❛♥ ❈❧❛✐♠s ✐♥ t❤❡ ❇✐♥♦♠✐❛❧ ▼♦❞❡❧ ✾✳✹ ❙t♦♣♣✐♥❣ ❚✐♠❡s ✾✳✺ ❖♣t✐♠❛❧ ❊①❡r❝✐s❡ ♦❢ ❛♥ ❆♠❡r✐❝❛♥ ❈❧❛✐♠ ✾✳✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✶ ✶✵✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✶ ❉✐✈✐❞❡♥❞s ✐♥ t❤❡ ❇✐♥♦♠✐❛❧ ▼♦❞❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✹ ✾✳✼ ❚❤❡ ●❡♥❡r❛❧ ❋✐♥✐t❡ ▼❛r❦❡t ▼♦❞❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✺ ✾✳✽ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✼ ✶✵ ❙t♦❝❤❛st✐❝ ❈❛❧❝✉❧✉s ✶✶✾ ✶✵✳✶ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✾ ✶✵✳✷ ❈♦♥t✐♥✉♦✉s✲❚✐♠❡ ❙t♦❝❤❛st✐❝ Pr♦❝❡ss❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✵ ✶✵✳✸ ❇r♦✇♥✐❛♥ ▼♦t✐♦♥ ✶✷✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✳✹ ❱❛r✐❛t✐♦♥ ♦❢ ❇r♦✇♥✐❛♥ P❛t❤s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✻ ✶✵✳✺ ❘✐❡♠❛♥♥✲❙t✐❡❧t❥❡s ■♥t❡❣r❛❧s ✶✵✳✻ ❙t♦❝❤❛st✐❝ ■♥t❡❣r❛❧s ✶✵✳✼ ❚❤❡ ■t♦✲❉♦❡❜❧✐♥ ❋♦r♠✉❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✳✽ ❙t♦❝❤❛st✐❝ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✶ ✶✸✻ ✐① ✶✵✳✾ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ❚❤❡ ❇❧❛❝❦✲❙❝❤♦❧❡s✲▼❡rt♦♥ ▼♦❞❡❧ ✶✶✳✶ ❚❤❡ ❙t♦❝❦ Pr✐❝❡ ❙❉❊ ✶✹✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✳✷ ❈♦♥t✐♥✉♦✉s✲❚✐♠❡ P♦rt❢♦❧✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✳✸ ❚❤❡ ❇❧❛❝❦✲❙❝❤♦❧❡s✲▼❡rt♦♥ P❉❊ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✳✹ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❇❙▼ ❈❛❧❧ ❋✉♥❝t✐♦♥ ✶✶✳✺ ❊①❡r❝✐s❡s ✶✸✾ ✶✹✶ ✶✹✷ ✶✹✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✾ ✶✷ ❈♦♥t✐♥✉♦✉s✲❚✐♠❡ ▼❛rt✐♥❣❛❧❡s ✶✷✳✶ ❈♦♥❞✐t✐♦♥❛❧ ❊①♣❡❝t❛t✐♦♥ ✶✺✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✳✷ ▼❛rt✐♥❣❛❧❡s✿ ❉❡✜♥✐t✐♦♥ ❛♥❞ ❊①❛♠♣❧❡s ✶✺✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✻ ✶✷✳✸ ▼❛rt✐♥❣❛❧❡ ❘❡♣r❡s❡♥t❛t✐♦♥ ❚❤❡♦r❡♠ ✶✷✳✹ ▼♦♠❡♥t ●❡♥❡r❛t✐♥❣ ❋✉♥❝t✐♦♥s ✶✺✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✳✺ ❈❤❛♥❣❡ ♦❢ Pr♦❜❛❜✐❧✐t② ❛♥❞ ●✐rs❛♥♦✈✬s ❚❤❡♦r❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✽ ✶✷✳✻ ❊①❡r❝✐s❡s ✶✻✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ❚❤❡ ❇❙▼ ▼♦❞❡❧ ❘❡✈✐s✐t❡❞ ✶✻✸ ✶✸✳✶ ❘✐s❦✲◆❡✉tr❛❧ ❱❛❧✉❛t✐♦♥ ♦❢ ❛ ❉❡r✐✈❛t✐✈❡ ✶✸✳✷ Pr♦♦❢s ♦❢ t❤❡ ❱❛❧✉❛t✐♦♥ ❋♦r♠✉❧❛s ✶✸✳✸ ❱❛❧✉❛t✐♦♥ ✉♥❞❡r P ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✼ ✶✸✳✹ ❚❤❡ ❋❡②♥♠❛♥✲❑❛❝ ❘❡♣r❡s❡♥t❛t✐♦♥ ❚❤❡♦r❡♠ ✶✸✳✺ ❊①❡r❝✐s❡s ✶✻✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✶ ✶✹ ❖t❤❡r ❖♣t✐♦♥s ✶✼✸ ✶✹✳✶ ❈✉rr❡♥❝② ❖♣t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✳✷ ❋♦r✇❛r❞ ❙t❛rt ❖♣t✐♦♥s ✶✹✳✸ ❈❤♦♦s❡r ❖♣t✐♦♥s ✶✼✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✻ ✶✹✳✹ ❈♦♠♣♦✉♥❞ ❖♣t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✳✺ P❛t❤✲❉❡♣❡♥❞❡♥t ❉❡r✐✈❛t✐✈❡s ✶✼✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✽ ✶✹✳✺✳✶ ❇❛rr✐❡r ❖♣t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✾ ✶✹✳✺✳✷ ▲♦♦❦❜❛❝❦ ❖♣t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾✶ ✶✹✳✺✳✸ ❆s✐❛♥ ❖♣t✐♦♥s ✶✹✳✻ ◗✉❛♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾✺ ✶✹✳✼ ❖♣t✐♦♥s ♦♥ ❉✐✈✐❞❡♥❞✲P❛②✐♥❣ ❙t♦❝❦s ✶✾✼ ✶✹✳✼✳✶ ❈♦♥t✐♥✉♦✉s ❉✐✈✐❞❡♥❞ ❙tr❡❛♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾✼ ✶✹✳✼✳✷ ❉✐s❝r❡t❡ ❉✐✈✐❞❡♥❞ ❙tr❡❛♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾✽ ✶✹✳✽ ❆♠❡r✐❝❛♥ ❈❧❛✐♠s ✐♥ t❤❡ ❇❙▼ ▼♦❞❡❧ ✶✹✳✾ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵✸ ❆ ❙❡ts ❛♥❞ ❈♦✉♥t✐♥❣ ✷✵✾ ❇ ❙♦❧✉t✐♦♥ ♦❢ t❤❡ ❇❙▼ P❉❊ ✷✶✺ ❈ ❆♥❛❧②t✐❝❛❧ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❇❙▼ ❈❛❧❧ ❋✉♥❝t✐♦♥ ✷✶✾ ❍✐♥ts ❛♥❞ ❙♦❧✉t✐♦♥s f (x) = x(x − K)+ ✱ ✶✺✳ ❇② ❈♦r♦❧❧❛r② ✼✳✷✳✺ ✇✐t❤ N N S0 uj dN −j (S0 uj dN −j − K)p∗ j q ∗ N −j j V0 = (1 + i)−N j=m = N S02 (1 + i)N j=m − v 1+i = S02 ✇❤❡r❡ q˜ = q ∗ d2 /v N (u2 p∗ )j (d2 q ∗ )N −j j KS0 (1 + i)N N N N N (up∗ )j (dq ∗ )N −j j j=m N j=m ❛♥❞ ✷✸✺ N j N −j N j N −j − KS0 , p˜ q˜ pˆ qˆ j j j=m qˆ = − pˆ✳ ❙✐♥❝❡ u2 p∗ + d2 q ∗ = v ✱ (˜ p, q˜) ✐s ❛ ♣r♦❜❛❜✐❧✐t② ✈❡❝t♦r ❛♥❞ t❤❡ ❞❡s✐r❡❞ ❢♦r♠✉❧❛ ❢♦❧❧♦✇s✳ ✶✼✳ ❯s❡ ❊①❡r❝✐s❡ ✸✳✶✷ ❛♥❞ t❤❡ ❧❛✇ ♦❢ t❤❡ ✉♥❝♦♥s❝✐♦✉s st❛t✐st✐❝✐❛♥✳ ✶✾✳ ❇② ❊①❡r❝✐s❡ ✶✼ ✇✐t❤ f (x, y) = (x ❤❛✈❡ + y) − K (1 + i)N V0 = + ✱ m = 1✱ ❛♥❞ n = N✱ ✇❡ (A0 + A1 ), ✇❤❡r❡ N −1 A0 := k=0 N A1 := k=1 N − ∗ k ∗ N −k p q (S0 d + S0 uk dN −k − 2K)+ k ❛♥❞ N − ∗ k ∗ N −k p q (S0 u + S0 uk dN −k − 2K)+ k−1 S0 d + S0 uk dN −k > 2K ❢♦r k = N − ❤❡♥❝❡ k N −k1 > 2K ✳ t❤❡r❡ ❡①✐sts ❛ s♠❛❧❧❡st ✐♥t❡❣❡r k1 ≥ s✉❝❤ t❤❛t S0 d+S0 u d k+1 N −k−1 N −1 ❙✐♥❝❡ S0 u + S0 u d > S0 d + S0 u d ❢♦r k = N − 1✱ t❤❡r❡ k +1 N −k2 −1 ❡①✐sts ❛ s♠❛❧❧❡st ✐♥t❡❣❡r k2 ≥ s✉❝❤ t❤❛t S0 u + S0 u d > 2K ✳ ❚❤❡ ❤②♣♦t❤❡s✐s ✐♠♣❧✐❡s t❤❛t ❚❤❡r❡❢♦r❡ N −1 A0 = k=k1 N − ∗ k ∗ N −k p q (S0 d + S0 uk dN −k − 2K) k N −1 = (S0 d − 2K)q ∗ k=k1 N − ∗ k ∗ N −1−k p q k N −1 + S0 dq ∗ k=k1 ∗ N −1 (up∗ )k (dq ∗ )N −1−k k = (S0 d − 2K)q Ψ(k1 , N − 1, p∗ ) + (1 + i)N S0 dq ∗ Ψ(k1 , N − 1, pˆ), ❖♣t✐♦♥ ❱❛❧✉❛t✐♦♥✿ ❆ ❋✐rst ❈♦✉rs❡ ✐♥ ❋✐♥❛♥❝✐❛❧ ▼❛t❤❡♠❛t✐❝s ✷✸✻ ❛♥❞ N −1 ∗ A1 = p k=k2 N − ∗ k ∗ N −1−k p q (S0 u + S0 uk+1 dN −1−k − 2K) k N −1 = (S0 u − 2K)p N − ∗ k ∗ N −1−k p q k ∗ k=k2 N −1 + S0 up∗ k=k2 N −1 (up∗ )k (dq ∗ )N −1−k k = (S0 u − 2K)p∗ Ψ(k2 , N − 1, p∗ ) + (1 + i)N S0 up∗ Ψ(k2 , N − 1, pˆ) ❈❤❛♣t❡r ✽ ✶✳ ❋♦r ❛♥② r❡❛❧ ♥✉♠❜❡r a✱ E g(X)I{X=a} = g(x)I{a} (x)pX (x) = g(a)pX (a) ❛♥❞ x E Y I{X=a} = pX,Y (a, y)y y ✸✳ ❙✐♥❝❡ XY = X + X(Y − X)✱ ❝♦♥❞✐t✐♦♥✐♥❣ ♦♥ G ②✐❡❧❞s E(XY ) = E X + XE(Y − X|G) = E X + XE(Y − X) = E X + E(X)E(Y − X) = E X ❚❤❡r❡❢♦r❡✱ E(Y − X)2 = E Y − 2E X + E X = E Y − EX ✳ ✺✳ ❇② t❤❡ ✐t❡r❛t❡❞ ❝♦♥❞✐t✐♦♥✐♥❣ ♣r♦♣❡rt②✱ ✐❢ m>n E(Mm |Fn ) = E[E(Mm |Fm−1 )|Fn ] = E(Mm−1 |Fn ) ✼✳ ❙✐♥❝❡ + 2Xn+1 Yn − σ ✱ Mn+1 − Mn = Xn+1 E Mn+1 − Mn |FnX = E Xn+1 + 2Yn E Xn+1 |FnX − σ = ✾✳ ❙✐♥❝❡ Mn+1 = Mn rXn+1 ✱ q p E(Mn+1 |FnX ) = Mn E rXn+1 = Mn p + q p q = Mn ❍✐♥ts ❛♥❞ ❙♦❧✉t✐♦♥s ✶✶✳ ❊①♣❛♥❞✐♥❣ (Am − An )2 ✱ ✇❡ ❤❛✈❡ ❢♦r ✷✸✼ n≤m E (Am − An )2 |Fn = E A2m |Fn + E A2n |Fn − 2E (Am An |Fn ) = E A2m |Fn + A2n − 2An E (Am |Fn ) = E A2m |Fn + A2n − 2A2n = E (Bm |Fn ) + E (Cm |Fn ) − Bn − Cn = E (Cm − Cn |Fn ) ✶✸✳ ❈♦♥❞✐t✐♦♥ ♦♥ Fk ✳ ❈❤❛♣t❡r ✾ ✶✳ ❋♦r ✭❛✮ ❧❡t Ak = {Sk ≤ (S0 + S1 + · · · + Sk−1 )/k}, Ak ∈ FkS ❚❤❡♥ k = 1, 2, , N − ❛♥❞ {τa = n} = A1 A2 · · · An−1 An ∈ FnS , n = 1, 2, , N − 1, ❛♥❞ S S {τa = N } = A1 A2 · · · AN −1 ∈ FN −1 ⊆ FN ✸✳ Pr✐❝❡✿ ✩✷✵✳✾✶✳ ❖♣t✐♠❛❧ ❡①❡r❝✐s❡ t✐♠❡ s❝❡♥❛r✐♦s✿ uudd d ✭✩✷✼✳✵✵✮✱ ud ✭✩✸✺✳✾✶✮❀ ✭✩✷✽✳✹✸✮✳ ✺✳ ❲❡ s❤♦✇ ❜② ✐♥❞✉❝t✐♦♥ ♦♥ k t❤❛t ✭❸✮ vk (Sk (ω)) = f (Sk (ω)) = ❢♦r ❛❧❧ k ≥ n (= τ0 (ω))✳ ❇② ❞❡✜♥✐t✐♦♥ k ≥ n✳ ❙✐♥❝❡ ♦❢ ✭†✮ ❤♦❧❞s ❢♦r ❛r❜✐tr❛r② τ0 ✱ ✭†✮ ❤♦❧❞s ❢♦r k = n✳ ❙✉♣♣♦s❡ vk (Sk (ω)) = max f (Sk (ω)), avk+1 (Sk (ω)u) + bvk+1 (Sk (ω)d) ❛♥❞ ❛❧❧ t❡r♠s ❝♦♠♣r✐s✐♥❣ t❤❡ ❡①♣r❡ss✐♦♥ ♦♥ t❤❡ r✐❣❤t ♦❢ t❤✐s ❡q✉❛t✐♦♥ ❛r❡ ♥♦♥♥❡❣❛t✐✈❡✱ vk+1 (Sk+1 (ω)) = 0✳ ❙✐♥❝❡ vk+1 (Sk+1 (ω)) = max f (Sk+1 (ω)), avk+2 (Sk+1 (ω)u) + bvk+2 (Sk+1 (ω)d) , f (Sk+1 (ω)) = 0✳ ❚❤❡r❡❢♦r❡✱ ✭†✮ ❤♦❧❞s ❢♦r k + 1✳ ❖♣t✐♦♥ ❱❛❧✉❛t✐♦♥✿ ❆ ❋✐rst ❈♦✉rs❡ ✐♥ ❋✐♥❛♥❝✐❛❧ ▼❛t❤❡♠❛t✐❝s ✷✸✽ ✼✳ ❚❤❡ ♣r♦♦❢ ♦❢ ✭❛✮ ✐s ❛ str❛✐❣❤t❢♦r✇❛r❞ ♠♦❞✐✜❝❛t✐♦♥ ♦❢ t❤❛t ♦❢ ❈♦r♦❧✲ ❧❛r② ✾✳✶✳✸✳ ❚♦ ✜♥❞ C0 t❛❦❡ n = 0✱ m = N ✱ ❛♥❞ ✭❛✮✳ ❚❤❡♥ N N ∗ j ∗ N −j N j N −j b u d S0 − K p q j −N C0 = a j=0 = b a N N K N ∗ j ∗ N −j j N −j u d − N p q a j S0 j=m f (x) = (x − K)+ ✐♥ + N N ∗ j ∗ N −j p q j j=m ❈❤❛♣t❡r ✶✵ x−2 dx = sin t dt ⇒ x−1 = cos t + c ⇒ x = (cos t + c)−1 ❀ x(0) = 1/3 ⇒ c = 2✳ ❚❤❡r❡❢♦r❡✱ x(t) = (cos t + 2)−1 ✱ −∞ < t < ∞✳ ✶✳ ✭❛✮ ✭❜✮ x(0) = ⇒ c = −1/2 ⇒ x(t) = (cos t − 1/2)−1 ✱ −π/3 < t < π/3✳ 2 2x dx √ = (2t + cos t) dt ⇒ x = t + sin t + c✳ x(0) = ⇒ c = ⇒ x(t) = t2 + sin t + 1✱ ✈❛❧✐❞ ❢♦r ❛❧❧ t ✭♣♦s✐t✐✈❡ r♦♦t ❜❡❝❛✉s❡ x(0) > 0✮✳ ✭❝✮ (x+1)−1 dx = cot t dt ⇒ ln |x + 1| = ln | sin t|+c ⇒ x+1 = ±ec sin t❀ x(π/6) = 1/2 ⇒ x+1 = ±3 sin t✳ P♦s✐t✐✈❡ s✐❣♥ ✐s ❝❤♦s❡♥ ❜❡❝❛✉s❡ x(π/6)+ > 0✳ ❚❤❡r❡❢♦r❡✱ x(t) = sin t − ✭❞✮ ✸✳ ❯s❡ t❤❡ ♣❛rt✐t✐♦♥s Pn ❞❡s❝r✐❜❡❞ ✐♥ t❤❡ ❡①❛♠♣❧❡ t♦ ❝♦♥str✉❝t ❘✐❡♠❛♥♥✲ ❙t✐❡❧t❥❡s s✉♠s t❤❛t ❞♦ ♥♦t ❝♦♥✈❡r❣❡✳ ✺✳ ❋♦r t❤❡ ✜rst ❛ss❡rt✐♦♥ ✉s❡ t❤❡ ✐❞❡♥t✐t② W (s) + W (t) = W (t) − W (s) + 2W (s), ✐♥❞❡♣❡♥❞❡♥❝❡✱ ❛♥❞ ❊①❛♠♣❧❡ ✸✳✻✳✷✳ Xt = ✼✳ ❇② ❚❤❡♦r❡♠ ✶✵✳✻✳✸✱ t F (s) dW (s) ❤❛s ♠❡❛♥ ③❡r♦ ❛♥❞ ✈❛r✐❛♥❝❡ t E F (s) ds V Xt = ✭❛✮ E sWs2 = s2 ✭❜✮ ❙✐♥❝❡ ❤❡♥❝❡ ds = t3 /3✳ W (s) ∼ N (0, s)✱ E exp (2Ws2 ) = √ ✇❤❡r❡ t s V Xt = 2πs α = s−1 − 4✳ ■❢ ∞ e2x e−x −∞ s ≥ 1/4✱ t❤❡♥ /2s dx = √ α≤0 2πs ∞ e−αx /2 dx, −∞ ❛♥❞ t❤❡ ✐♥t❡❣r❛❧ ❞✐✈❡r❣❡s✳ ❍✐♥ts ❛♥❞ ❙♦❧✉t✐♦♥s ❚❤❡r❡❢♦r❡✱ V Yt =√ +∞ ❢♦r t ≥ 1/4✳ y = αx✱ ✇❡ ❤❛✈❡ ■❢ s✉❜st✐t✉t✐♦♥ E exp (2Ws2 ) = √ s♦ t❤❛t e−y s ≤ t < 1/4✱ /2 −∞ t❤❡♥✱ ♠❛❦✐♥❣ t❤❡ dy = √ = (1 − 4s)−1/2 sα t V Xt = ✭❝✮ ❋♦r ∞ 2πsα ✷✸✾ (1 − 4s)−1/2 ds = 21 [1 − √ − 4t] s > 0✱ E |Ws | = √ 2πs V Xt = π ∞ xe−x ✭❜✮ ❋r♦♠ ❱❡rs✐♦♥ ✷✱ /2s dx = ❤❡♥❝❡ ✾✳ ✭❛✮ ❯s❡ ❱❡rs✐♦♥ ✶ ✇✐t❤ t √ s ds = 2s π 3/2 t π f (x) = ex ✳ d(tW ) = 2tW dW + (W + t) dt✳ f (t, x, y) = x/y ✳ ❙✐♥❝❡ ft = 0✱ fx = 1/y ✱ fy = −x/y ✱ fxx = 0✱ fxy = −1/y ✱ ❛♥❞ fyy = 2x/y ✱ ✇❡ ❤❛✈❡ ✭❝✮ ❯s❡ ❱❡rs✐♦♥ ✹ ✇✐t❤ d ❋❛❝t♦r✐♥❣ ♦✉t X Y = dX X X − dY + (dY )2 − dX · dY Y Y Y Y X Y ❣✐✈❡s t❤❡ ❞❡s✐r❡❞ r❡s✉❧t✳ ✶✶✳ ❚❛❦✐♥❣ ❡①♣❡❝t❛t✐♦♥s ✐♥ ✭✶✵✳✶✾✮ ❣✐✈❡s E Xt = e−βt E X0 + α βt (e − 1) β ❈❤❛♣t❡r ✶✶ ✶✳ ✭❛✮ ❝❛❧❧✿ ✩✹✳✻✺❀ ♣✉t✿ ✩✵✳✺✵❀ ✭❜✮ ❝❛❧❧✿ ✩✶✳✷✼❀ ♣✉t✿ ✩✷✳✾✽✳ ✸✳ ∂P ∂C = −1 = Φ(d1 )−1 < 0✱ lims→∞ P = 0✱ ❛♥❞ lims→0+ P = Ke−rτ ✳ ∂s ∂s ✺✳ ❚❛❦✐♥❣ f (z) = AI(K,∞) (z) ✐♥ ❚❤❡♦r❡♠ ✶✶✳✸✳✷ ②✐❡❧❞s Vt = e−r(T −t) G(t, St ), ❖♣t✐♦♥ ❱❛❧✉❛t✐♦♥✿ ❆ ❋✐rst ❈♦✉rs❡ ✐♥ ❋✐♥❛♥❝✐❛❧ ▼❛t❤❡♠❛t✐❝s ✷✹✵ ✇❤❡r❡✱ ❛s ✐♥ t❤❡ ♣r♦♦❢ ♦❢ ❈♦r♦❧❧❛r② ✶✶✳✸✳✸✱ ∞ G(t, s) = −∞ √ AI(K,∞) s exp σ T − t y + (r − σ /2)(T − t) ϕ(y) dy = AΦ d1 (T − t, s, K, σ, r) ✼✳ ❙✐♥❝❡ VT = ST I(K1 ,∞) (ST ) − ST I[K2 ,∞) (ST ), V0 ✾✳ ✶✶✳ ✐s t❤❡ ❞✐✛❡r❡♥❝❡ ✐♥ t❤❡ ♣r✐❝❡s ♦❢ t✇♦ ❛ss❡t✲♦r✲♥♦t❤✐♥❣ ♦♣t✐♦♥s✳ Vt = C(T − t, St , F ) + (F − K)e−r(T −t) ❛♥❞✱ ✐♥ ♣❛rt✐❝✉❧❛r✱ V0 = C0 + (F − K)e−rT ✱ ✇❤❡r❡ C0 ✐s t❤❡ ❝♦st ♦❢ ❛ ❝❛❧❧ ♦♣t✐♦♥ ♦♥ t❤❡ st♦❝❦ ✇✐t❤ rT str✐❦❡ ♣r✐❝❡ F ✳ ❚❤❡r❡❢♦r❡✱ K = F + e C0 ✳ ST > K ✐✛ σWT +(µ−σ /2)T > ln (K/S0 ) ❤❡♥❝❡ t❤❡ ❞❡s✐r❡❞ ♣r♦❜❛❜✐❧✐t② ✐s 1−Φ ln (K/S0 ) − (µ − σ /2)T √ σ T ✶✸✳ ❚❤❡ ❡①♣r❡ss✐♦♥ ❢♦r EC =Φ ln (S0 /K) + (µ − σ /2)T √ σ T ❢♦❧❧♦✇s ❢r♦♠ ❚❤❡♦r❡♠ ✶✶✳✹✳✶✭✐✮ ❛♥❞ t❤❡ ❇❧❛❝❦✲ ❙❝❤♦❧❡s ❢♦r♠✉❧❛✳ ❚♦ ✈❡r✐❢② t❤❡ ❧✐♠✐ts✱ ✇r✐t❡ −1 EC =1−α Φ(d2 ) , sΦ(d1 ) ❛♥❞ ♥♦t❡ t❤❛t ✭❛✮ ❢♦❧❧♦✇s ❢r♦♠ α := Ke−rT lims→∞ Φ(d1,2 ) = 1✳ ❋♦r ✭❜✮✱ ❛♣♣❧② ❧✬❍♦s♣✐t❛❧✬s ❘✉❧❡ t♦ ♦❜t❛✐♥ sΦ(d1 ) sϕ(d1 )(βs)−1 + Φ(d1 ) = lim+ Φ(d2 ) ϕ(d2 )(βs)−1 s→0 s→0 √ sϕ(d1 ) Φ(d1 ) = lim 1+β , β := σ T + ϕ(d2 ) ϕ(d1 ) s→0 −1 −1 α(1 − EC ) = lim+ ❙✐♥❝❡ d22 − d21 = (d2 − d1 )(d2 + d1 ) = −β(d2 + d1 ) = 2[ln(K/s) − rT ], s ϕ(d1 ) = s exp [ 12 (d22 − d21 )] = s exp [ln(K/s) − rT ] = α ϕ(d2 ) ❤❡♥❝❡ −1 − EC −1 = α−1 lim+ s→0 sΦ(d1 ) Φ(d1 ) = + β lim+ Φ(d2 ) s→0 ϕ(d1 ) ❍✐♥ts ❛♥❞ ❙♦❧✉t✐♦♥s ✷✹✶ ❇② ❧✬❍♦s♣✐t❛❧✬s ❘✉❧❡✱ lim+ s→0 ❚❤❡r❡❢♦r❡✱ Φ(d1 ) ϕ(d1 )(βs)−1 = − lim+ = lim+ = ϕ(d1 ) s→0 ϕ(d1 )(−d1 )(βs)−1 d s→0 −1 lims→0+ − EC −1 = 1✱ ✇❤✐❝❤ ✐♠♣❧✐❡s ✭❜✮✳ ✶✺✳ ▼❛❦❡ t❤❡ s✉❜st✐t✉t✐♦♥ √ z = s exp σ T − t y + (r − 12 σ )(T − t) ❈❤❛♣t❡r ✶✷ E eλX = peλ + q ✶✳ ✭❛✮ Ee ✭❜✮ λX = pe λ n ✳ − qeλ −1 ✳ ✸✳ ❋♦r ✭❛✮✱ E(Ws Wt ) = E E Ws Wt |FsW = E Ws E Wt |FsW = E(Ws2 ) = s, ❛♥❞ ❢♦r ✭❜✮✱ E(Wt −Ws |Ws ) = E E(Wt − Ws |FsW )|Ws ) = E [E(Wt − Ws )|Ws )] = ✺✳ ▲❡t A = {(u, v) | v ≤ y, u + v ≤ x} ❛♥❞ f (x, y) = s(t − s) √ ϕ ❇② ✐♥❞❡♣❡♥❞❡♥t ✐♥❝r❡♠❡♥ts✱ x t−s ϕ y √ s P(Wt ≤ x, Ws ≤ y) = P (Wt − Ws , Ws ) ∈ A = f (u, v) du dv A y x −∞ −∞ = ✼✳ M ✐s ❛ ♠❛rt✐♥❣❛❧❡ ✐✛ ❢♦r ❛❧❧ f (u − v, v) du dv ≤ s ≤ t✱ E eα[W (t)−W (s)] |Fs = eh(s)−h(t) ✳ ❇② ✐♥❞❡♣❡♥❞❡♥❝❡ ❛♥❞ ❊①❡r❝✐s❡ ✻✳✶✹✱ E eα[W (t)−W (s)] |Fs = E eα[W (t)−W (s)] = eα ❚❤❡r❡❢♦r❡✱ M ✐s ❛ ♠❛rt✐♥❣❛❧❡ ✐✛ (t−s)/2 h(t) − h(s) = α2 (s − t)/2✳ ✷✹✷ ❖♣t✐♦♥ ❱❛❧✉❛t✐♦♥✿ ❆ ❋✐rst ❈♦✉rs❡ ✐♥ ❋✐♥❛♥❝✐❛❧ ▼❛t❤❡♠❛t✐❝s ✾✳ ❇② ❊①❛♠♣❧❡ ✶✷✳✷✳✸ ❛♥❞ ✐t❡r❛t❡❞ ❝♦♥❞✐t✐♦♥✐♥❣✱ E(Wt2 |Ws ) = E[E(Wt2 −t|FsW )|Ws ]+t = E(Ws2 −s|Ws )+t = Ws2 +t−s ❙✐♠✐❧❛r❧②✱ ❜② ❊①❡r❝✐s❡ ✽✱ E(Wt3 − 3tWt |Ws ) = E[E(Wt3 − 3tWt |FsW )|Ws ] = E(Ws3 − 3sWs |Ws ) = Ws3 − 3sWs ❚❤❡r❡❢♦r❡✱ ❜② ❊①❡r❝✐s❡ ✸✱ E(Wt3 |Ws ) = 3tE(Wt |Ws ) + Ws3 − 3sWs = Ws3 + 3(t − s)Ws ✶✶✳ ❋♦r ❛♥② x✱ P∗ (X ≤ x) = E∗ I(−∞,x] (X) T E I(−∞,x] (X)e−αWT T E(I(−∞,x] (X))E e−αWT = e− α = e− α = P(X ≤ x), t❤❡ ❧❛st ❡q✉❛❧✐t② ❢r♦♠ ❊①❡r❝✐s❡ ✻✳✶✹✳ ❈❤❛♣t❡r ✶✸ ✶✳ ❇② ▲❡♠♠❛ ✶✸✳✷✳✷✱ t❤❡ ❝❛❧❧ ✜♥✐s❤❡s ✐♥ t❤❡ ♠♦♥❡② ✐✛ WT∗ > σ −1 ln (K/S0 ) − (r − 21 σ )T ❚❤❡r❡❢♦r❡✱ t❤❡ 1−Φ P∗ ✲♣r♦❜❛❜✐❧✐t② t❤❛t t❤❡ ❝❛❧❧ ✜♥✐s❤❡s ✐♥ t❤❡ ♠♦♥❡② ✐s ln (K/S0 ) − (r − 21 σ )T √ σ T ✸✳ ❚❤✐s ❢♦❧❧♦✇s ❢r♦♠ e−(r+σ )t = Φ d2 (T, S0 , K, σ, r) ∗∗ St = S0 eσWt − 12 σ t ❛♥❞ ❊①❛♠♣❧❡ ✶✷✳✷✳✻✳ ❈❤❛♣t❡r ✶✹ ✶✳ P❛rts ✭❛✮ ❛♥❞ ✭❜✮ ❢♦❧❧♦✇ ❢r♦♠ t❤❡ ■t♦✲❉♦❡❜❧✐♥ ❢♦r♠✉❧❛ ❛♣♣❧✐❡❞ t♦ f (t, x) = exp σx + (rd − re − σ )t ❍✐♥ts ❛♥❞ ❙♦❧✉t✐♦♥s ✷✹✸ ❛♥❞ f (t, x) = exp −σx − (rd − re − σ )t , r❡s♣❡❝t✐✈❡❧②✳ ✸✳ ❯s❡ ❊q✉❛t✐♦♥ ✭✶✹✳✽✮✱ ✐ts ❛♥❛❧♦❣ ❢♦r ❛ ♣✉t✲♦♥✲❝❛❧❧ ♦♣t✐♦♥✱ ❛♥❞ t❤❡ ✐❞❡♥t✐t② K0 − C(s) + − C(s) − K0 + + C(s) = K0 ✺✳ ❆s ❛ ✜rst st❡♣✱ ˆ (ST − K)IA Z −1 V0 = e−rT E T = e−(r+β ✇❤❡r❡ ˆT ˆ eλW E IA ˆT ˆT ˆ eβ W ˆ eγ W IA − K E IA S0 E /2)T , ✐s ❣✐✈❡♥ ❜② ✭✶✹✳✶✾✮✳ ❆♥ ♦❜✈✐♦✉s ♠♦❞✐✜❝❛t✐♦♥ ♦❢ ✭✶✹✳✶✻✮ s❤♦✇s t❤❛t ˆT ˆ eλW E IA = eλx gˆm (x, y) dA, D := {(x, y) | b ≤ y ≤ 0, x ≥ y} D ❙✐♥❝❡ D ❤❛s t❤❡ s❛♠❡ ❢♦r♠ ❛s ✐♥ ❋✐❣✉r❡ ✶✹✳✷✱ t❤❡ ✐♥t❡❣r❛❧ ❡✈❛❧✉❛t❡s t♦ ✭✶✹✳✶✾✮ ❛♥❞ ❤❡♥❝❡✱ ❛s ✐♥ t❤❡ t❡①t✱ ❧❡❛❞s t♦ ✭✶✹✳✾✮ ❛♥❞ ✭✶✹✳✶✵✮ ✇✐t❤ M = c✳ ✼✳ ❋r♦♠ ✭✶✹✳✶✶✮✱ t❤❡ ❝♦st ♦❢ t❤❡ ♦♣t✐♦♥ ✐s C0do = e−rd T E∗ [(ST − K)IB ], ✇❤❡r❡ S r = rd − re ✳ ✐s ❣✐✈❡♥ ❜② ✭✶✹✳✶✷✮ ✇✐t❤ ❚❤❡ ❝❛❧❝✉❧❛t✐♦♥s ❧❡❛❞✐♥❣ t♦ ❊q✉❛t✐♦♥ ✭✶✹✳✾✮ ②✐❡❧❞✱ ❛s ✐♥ t❤❡ t❡①t✱ C0do = e− (2rd +β −γ)T S0 Φ(d1 ) − e2bγ Φ(δ1 ) − Ke−rd T Φ(d2 ) − e2bβ Φ(δ2 ) ❙✐♥❝❡ ✾✳ ❙✐♥❝❡ 2rd + β − γ = 2re ✱ ˆ t❤❡ ❞❡s✐r❡❞ ❝♦♥❝❧✉s✐♦♥ ❢♦❧❧♦✇s✳ ˆ M W = −mU ✱ ˆ ˆ W ˆ U ˆ T ≤ x, M W ˆT ≥ −x, mUˆ ≥ −y) P( ≤ y) = P( ∞ ∞ = −x ❚❤❡r❡❢♦r❡✱ ✐❢ −y < fˆM (x, y) = ❛♥❞ ❛♥❞ ∂2 ∂x∂y fˆm (u, v) dv du −y −y < −x✱ ∞ −x ∞ −y fˆm (u, v) dv du = gˆm (−x, −y), fˆM (x, y) = ♦t❤❡r✇✐s❡✳ ❙✐♥❝❡ gˆm (−x, −y) = −ˆ gm (x, y)✱ t❤❡ ❢♦r♠✉❧❛ ❢♦❧❧♦✇s✳ ❖♣t✐♦♥ ❱❛❧✉❛t✐♦♥✿ ❆ ❋✐rst ❈♦✉rs❡ ✐♥ ❋✐♥❛♥❝✐❛❧ ▼❛t❤❡♠❛t✐❝s ✷✹✹ ✶✶✳ ❙✐♥❝❡ ❝❛❧❧✳ {ST ≥ K, M S ≥ c} = {ST ≥ K}✱ ✶✸✳ ❚❤❡ ✜rst ❛ss❡rt✐♦♥ ❢♦❧❧♦✇s ❢r♦♠ limc→0+ δ1,2 = −∞✱ ✶✺✳ ■❢ t❤❡ ♣r✐❝❡ ✐s t❤❛t ♦❢ ❛ st❛♥❞❛r❞ limc→S − δ1,2 = d1,2 ❛♥❞ t❤❡ s❡❝♦♥❞ ❢r♦♠ limc→0+ Φ(δ1,2 ) = 0✳ t❤❡ ❧❛tt❡r ✐♠♣❧②✐♥❣ t❤❛t mT /(n + 1) ≤ t < (m + 1)T /(n + 1)✱ m = t(n + 1)/T ✳ t❤❡♥ ❤❡♥❝❡ m ≤ t(n + 1)/T < m + 1✱ ✶✼✳ ❇② ✭✶✹✳✶✶✮ ❛♥❞ ✭✶✹✳✸✺✮✱ C0do = e−rT E∗ (ST − K)IB , ✇❤❡r❡ σ r−δ − σ ∗ St = S0 eσ(Wt +βt) , β := ❲✐t❤ t❤✐s ❝❤❛♥❣❡ ✐♥ β  C0do = e−δT S0 Φ(d1 ) − c S0  2(r−δ) +1 σ2 Φ(δ1 )  − Ke−rT Φ(d2 ) −  2(r−δ) −1 σ2 c S0 Φ(δ2 ) , ✇❤❡r❡ ln (S0 /M ) + (r − δ ± σ )T /2 √ , ❛♥❞ σ T ln c2 /(S0 M ) + (r − δ ± σ )T /2 √ = σ T d1,2 = δ1,2 ✶✾✳ ❚❤❡ ❞❡s✐r❡❞ ♣r♦❜❛❜✐❧✐t② ✐s − P(C)✱ ✇❤❡r❡ C := {mS ≥ c} = {mW ≥ b}, ❚♦ ✜♥❞ P(C)✱ r❡❝❛❧❧ t❤❛t t❤❡ ♠❡❛s✉r❡s b := σ −1 ln (c/S0 ) ˆ P∗ ✱ P ❛♥❞ t❤❡ ♣r♦❝❡ss❡s ❛r❡ ❞❡✜♥❡❞ ❜② dP∗ = e−αWT − α ˆ=e dP T −βWT∗ − 12 β T WT∗ = WT + αT, dP, dP∗ , α= µ−r , σ ❛♥❞ ˆ T = W ∗ + βT = WT + (α + β)T, W T β= σ r − σ ˆ W ∗✱ W ❍✐♥ts ❛♥❞ ❙♦❧✉t✐♦♥s ■t ❢♦❧❧♦✇s t❤❛t ˆ✱ dP = U dP ˆ ✷✹✺ ✇❤❡r❡ U := eλWT − λ T , λ := α + β = µ σ − σ ❚❤❡r❡❢♦r❡✱ ˆ C U ) = e−λ P(C) = E(I T /2 eλx gˆm (x, y) dA, D ✇❤❡r❡ D = {(x, y) | b ≤ y ≤ 0, x ≥ y}, ✭s❡❡ ✭✶✹✳✶✻✮✮✳ ❚❤✐s ✐s t❤❡ r❡❣✐♦♥ ♦❢ ✐♥t❡❣r❛t✐♦♥ ❞❡s❝r✐❜❡❞ ✐♥ ❋✐❣✉r❡ ✶✹✳✷✱ s♦ ❜② ✭✶✹✳✶✾✮✱ P(C) = ❙✐♥❝❡ −b + λT √ T Φ − e2bλ Φ b + λT √ T ±b + λT ± ln (c/S0 ) + (µ − σ /2)T √ √ = T σ T ❛♥❞ 2bλ = 2µ − ln (c/S0 ), σ2 t❤❡ ❞❡s✐r❡❞ ♣r♦❜❛❜✐❧✐t② ✐s 1− Φ(d1 ) − ✇❤❡r❡ d1,2 = c S0 2µ −1 σ2 Φ(d2 ) , ± ln (S0 /c) + (µ − σ /2)T √ σ T ✷✶✳ ❚❤❡ t♦t❛❧ ♣❛②♦✛ ✐s t❤❛t ♦❢ ❛ ♣♦rt❢♦❧✐♦ ❝♦♥s✐st✐♥❣ ♦❢ ❛ ❝❛❧❧ ♦♣t✐♦♥ ♠❛t✉r✐♥❣ ❛t t✐♠❡ T0 ❛♥❞ n ❢♦r✇❛r❞ st❛rt ♦♣t✐♦♥s ♠❛t✉r✐♥❣ ❛t t✐♠❡s T1 , T2 , , Tn ✳ ❚❤❡ ❝♦st ♦❢ t❤❡ ❝❧✐q✉❡t ✐s t❤❡♥ t❤❡ s✉♠ ♦❢ t❤❡ ❝♦sts ♦❢ t❤❡s❡ ♦♣t✐♦♥s✱ ✇❤✐❝❤ ♠❛② ❜❡ ♦❜t❛✐♥❡❞ ❜② ✉s✐♥❣ t❤❡ r❡s✉❧ts ♦❢ ❙❡❝t✐♦♥ ✶✹✳✷✳ This page intentionally left blank ❇✐❜❧✐♦❣r❛♣❤② ❬✶❪ ❇❡❧❧❛❧❛❤✱ ▼✳✱ ✷✵✵✾✱ ❊①♦t✐❝ ❉❡r✐✈❛t✐✈❡s✱ ❲♦r❧❞ ❙❝✐❡♥t✐✜❝✱ ▲♦♥❞♦♥✳ ❬✷❪ ❇✐♥❣❤❛♠✱ ◆✳ ❍✳ ❛♥❞ ❘✳ ❑✐❡s❡❧✱ ✷✵✵✹✱ ❘✐s❦✲◆❡✉tr❛❧ ❱❛❧✉❛t✐♦♥✱ ❙♣r✐♥❣❡r✱ ◆❡✇ ❨♦r❦✳ ❬✸❪ ❊t❤❡r✐❞❣❡✱ ❆✳✱ ✷✵✵✷✱ ❆ ❈♦✉rs❡ ✐♥ ❋✐♥❛♥❝✐❛❧ ❈❛❧❝✉❧✉s✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡r✲ s✐t② Pr❡ss✱ ❈❛♠❜r✐❞❣❡✳ ❬✹❪ ❊❧❧✐♦t✱ ❘✳ ❏✳ ❛♥❞ P✳ ❊✳ ❑♦♣♣✱ ✷✵✵✺✱ ▼❛t❤❡♠❛t✐❝s ♦❢ ❋✐♥❛♥❝✐❛❧ ▼❛r❦❡ts✱ ❙♣r✐♥❣❡r✱ ◆❡✇ ❨♦r❦✳ ❬✺❪ ●r✐♠♠❡tt✱ ●✳ ❘✳ ❛♥❞ ❉✳ ❘✳ ❙t✐r③❛❦❡r✱ ✶✾✾✷✱ Pr♦❝❡ss❡s✱ ❖①❢♦r❞ ❙❝✐❡♥❝❡ P✉❜❧✐❝❛t✐♦♥s✱ ❖①❢♦r❞ ❬✻❪ ❍✐❞❛✱ ❚✳✱ ✶✾✽✵✱ Pr♦❜❛❜✐❧✐t② ❛♥❞ ❘❛♥❞♦♠ ❇r♦✇♥✐❛♥ ▼♦t✐♦♥✱ ❙♣r✐♥❣❡r✱ ◆❡✇ ❨♦r❦✳ ❬✼❪ ❍✉❧❧✱ ❏✳ ❈✳✱ ✷✵✵✵✱ ❖♣t✐♦♥s✱ ❋✉t✉r❡s✱ ❛♥❞ ❖t❤❡r ❉❡r✐✈❛t✐✈❡s✱ Pr❡♥t✐❝❡✲❍❛❧❧✱ ❊♥❣❧❡✇♦♦❞ ❈❧✐✛s✱ ◆✳❏✳ ❬✽❪ ❑❛r❛t③❛s✱ ■✳ ❛♥❞ ❙✳ ❙❤r❡✈❡✱ ✶✾✾✽✱ ▼❡t❤♦❞s ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❋✐♥❛♥❝❡✱ ❙♣r✐♥❣❡r✱ ◆❡✇ ❨♦r❦✳ ❬✾❪ ❑✉♦✱ ❍✳✱ ✷✵✵✻✱ ■♥tr♦❞✉❝t✐♦♥ t♦ ❙t♦❝❤❛st✐❝ ■♥t❡❣r❛t✐♦♥✱ ❙♣r✐♥❣❡r✱ ◆❡✇ ❨♦r❦✳ ❬✶✵❪ ❑✇♦❦✱ ❨✳✱ ✷✵✵✽✱ ▼❛t❤❡♠❛t✐❝❛❧ ▼♦❞❡❧s ♦❢ ❋✐♥❛♥❝✐❛❧ ❉❡r✐✈❛t✐✈❡s✱ ❙♣r✐♥❣❡r✱ ◆❡✇ ❨♦r❦✳ ❬✶✶❪ ▲❡✇✐s✱ ▼✳✱ ✷✵✶✵✱ ❚❤❡ ❇✐❣ ❙❤♦rt✱ ❲✳ ❲✳ ◆♦rt♦♥✱ ◆❡✇ ❨♦r❦✳ ❬✶✷❪ ▼✉s✐❡❧❛✱ ▼✳ ❛♥❞ ▼✳ ❘✉t♦✇s❦✐✱ ✶✾✾✼✱ ▼♦❞❡❧❧✐♥❣✱ ❙♣r✐♥❣❡r✱ ◆❡✇ ❨♦r❦✳ ▼❛t❤❡♠❛t✐❝❛❧ ▼♦❞❡❧s ✐♥ ❋✐♥❛♥❝✐❛❧ ❬✶✸❪ ▼②♥❡♥✐✱ ❘✳✱ ✶✾✾✼✱ ❚❤❡ ♣r✐❝✐♥❣ ♦❢ t❤❡ ❆♠❡r✐❝❛♥ ♦♣t✐♦♥✱ ❆♥♥✳ ❆♣♣❧✳ Pr♦❜✳ ✷✱ ✶✕✷✸✳ ❬✶✹❪ ❘♦ss✱ ❙✳ ▼✳✱ ✷✵✶✶✱ ❆♥ ❊❧❡♠❡♥t❛r② ■♥tr♦❞✉❝t✐♦♥ t♦ ▼❛t❤❡♠❛t✐❝❛❧ ❋✐♥❛♥❝❡✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ❈❛♠❜r✐❞❣❡✳ ❬✶✺❪ ❘✉❞✐♥✱ ❲✳✱ ✶✾✼✻✱ Pr✐♥❝✐♣❧❡s ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s✱ ▼❝●r❛✇✲❍✐❧❧✱ ◆❡✇ ❨♦r❦✳ ✷✹✼ ✷✹✽ ❖♣t✐♦♥ ❱❛❧✉❛t✐♦♥✿ ❆ ❋✐rst ❈♦✉rs❡ ✐♥ ❋✐♥❛♥❝✐❛❧ ▼❛t❤❡♠❛t✐❝s ❬✶✻❪ ❙❤r❡✈❡✱ ❙✳ ❊✳✱ ✷✵✵✹✱ ❙t♦❝❤❛st✐❝ ❈❛❧❝✉❧✉s ❢♦r ❋✐♥❛♥❝❡ ■✱ ❙♣r✐♥❣❡r✱ ◆❡✇ ❙t♦❝❤❛st✐❝ ❈❛❧❝✉❧✉s ❢♦r ❋✐♥❛♥❝❡ ■■✱ ❙♣r✐♥❣❡r✱ ◆❡✇ ❨♦r❦✳ ❬✶✼❪ ❙❤r❡✈❡✱ ❙✳ ❊✳✱ ✷✵✵✹✱ ❨♦r❦✳ ❬✶✽❪ ❙t❡❡❧❡✱ ❏✳ ▼✳✱ ✷✵✵✶✱ ❙t♦❝❤❛st✐❝ ❈❛❧❝✉❧✉s ❛♥❞ ❋✐♥❛♥❝✐❛❧ ❆♣♣❧✐❝❛t✐♦♥s✱ ❙♣r✐♥❣❡r✱ ◆❡✇ ❨♦r❦✳ ❬✶✾❪ ❨❡❤✱ ❏✳✱ ✶✾✼✸✱ ❙t♦❝❤❛st✐❝ Pr♦❝❡ss❡s ❛♥❞ t❤❡ ❲✐❡♥❡r ■♥t❡❣r❛❧✱ ❉❡❦❦❡r✱ ◆❡✇ ❨♦r❦✳ ▼❛r❝❡❧ Finance/Mathematics A First Course in Financial Mathematics Option Valuation: A First Course in Financial Mathematics provides a straightforward introduction to the mathematics and models used in the valuation of financial derivatives It examines the principles of option pricing in detail via standard binomial and stochastic calculus models Developing the requisite mathematical background as needed, the text introduces probability theory and stochastic calculus at an undergraduate level Hugo D Junghenn Option Valuation A First Course in Financial Mathematics Junghenn Largely self-contained, this classroom-tested text offers a sound introduction to applied probability through a mathematical finance perspective Numerous examples and exercises help readers gain expertise with financial calculus methods and increase their general mathematical sophistication The exercises range from routine applications to spreadsheet projects to the pricing of a variety of complex financial instruments Hints and solutions to odd-numbered problems are given in an appendix A First Course in Financial Mathematics The first nine chapters of the book describe option valuation techniques in discrete time, focusing on the binomial model The author shows how the binomial model offers a practical method for pricing options using relatively elementary mathematical tools The binomial model also enables a clear, concrete exposition of fundamental principles of finance, such as arbitrage and hedging, without the distraction of complex mathematical constructs The remaining chapters illustrate the theory in continuous time, with an emphasis on the more mathematically sophisticated Black– Scholes–Merton model Option Valuation Option Valuation K14090 K14090_Cover.indd 10/7/11 11:23 AM [...]... Pr♦❜❛❜✐❧✐t✐❡s✮✳ ❙✉♣♣♦s❡ t❤❛t A1 , A2 , , An ❛r❡ ❡✈❡♥ts ✇✐t❤ P (A1 A2 · · · An−1 ) > 0✳ ❚❤❡♥ P (A1 A2 · · · An ) = P (A1 )P (A2 |A1 )P (A3 |A1 A2 ) · · · P(An |A1 A2 · · · An−1 ) ✭✷✳✷✮ Pr♦❜❛❜✐❧✐t② ❙♣❛❝❡s Pr♦♦❢✳ ✷✶ n✳✮ ❚❤❡ ❝♦♥❞✐t✐♦♥ P (A1 A2 · · · An−1 ) > 0 ❡♥s✉r❡s t❤❛t n = 2✱ ✭✷✳✷✮ ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t②✳ ❙✉♣♣♦s❡ ✭✷✳✷✮ ❤♦❧❞s ❢♦r n = k ≥ 2✳ ■❢ A = A1 A2 · · · Ak ✱ t❤❡♥✱ ❜② t❤❡ ❝❛s❡... ✭✐✮ P (A ∪ B) = P (A) + P(B) − P(AB)❀ ❖♣t✐♦♥ ❱❛❧✉❛t✐♦♥✿ ❆ ❋✐rst ❈♦✉rs❡ ✐♥ ❋✐♥❛♥❝✐❛❧ ▼❛t❤❡♠❛t✐❝s ✶✽ ✭✐✐✮ ✭✐✐✐✮ ✐❢ B ⊆ A t❤❡♥ P (A − B) = P (A) − P(B)❀ ✐♥ ♣❛rt✐❝✉❧❛r P(B) ≤ P (A) ❀ P (A ) = 1 − P (A) ✳ Pr♦♦❢✳ ❋♦r ✭✐✮✱ ♥♦t❡ t❤❛t AB ✱ AB ✱ ❛♥❞ BA ✳ A B ✐s t❤❡ ✉♥✐♦♥ ♦❢ t❤❡ ♣❛✐r✇✐s❡ ❞✐s❥♦✐♥t ❡✈❡♥ts ❚❤❡r❡❢♦r❡✱ ❜② ❛❞❞✐t✐✈✐t②✱ P (A ∪ B) = P(AB ) + P(AB) + P(BA ) ❙✐♠✐❧❛r❧②✱ P (A) = P(AB ) + P(AB) P(B) = P(BA ) + P(AB) ❛♥❞... ❡♥❞ ♦❢ ♣❡r✐♦❞ nt❤ ♣❛②♠❡♥t✱ ✐s An−1 ♣❧✉s t❤❡ ✐♥t❡r❡st ✇✐t❤❞r❛✇❛❧ r❡❞✉❝❡s t❤❛t ✈❛❧✉❡ ❜② P n✱ iAn−1 A0 ❜❡ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♦❢ t❤❡ ❥✉st ❜❡❢♦r❡ ✇✐t❤❞r❛✇❛❧ ♦❢ t❤❡ ♦✈❡r t❤❛t ♣❡r✐♦❞✳ ▼❛❦✐♥❣ t❤❡ s♦ An = aAn−1 − P, a := 1 + i ■t❡r❛t✐♥❣✱ ✇❡ ♦❜t❛✐♥ An = a2 An−2 − (1 + a) P = · · · = an A0 − (1 + a + a2 + · · · + an−1 )P ❚❤✉s✱ 1 − (1 + i)n i (1 + i)n (iA0 − P ) + P = i An = (1 + i)n A0 + P ✭✶✳✺✮ ◆♦✇ ❛ss✉♠❡ t❤❛t... ❤♦❧❞s ❢♦r n = k ≥ 2✳ ■❢ A = A1 A2 · · · Ak ✱ t❤❡♥✱ ❜② t❤❡ ❝❛s❡ n = 2✱ ✭❇② ✐♥❞✉❝t✐♦♥ ♦♥ t❤❡ r✐❣❤t s✐❞❡ ♦❢ ✭✷✳✷✮ ✐s ❞❡✜♥❡❞✳ ❋♦r P (A1 A2 · · · Ak+1 ) = P(AAk+1 ) = P (A) P(Ak+1 |A) , ❛♥❞✱ ❜② t❤❡ ✐♥❞✉❝t✐♦♥ ❤②♣♦t❤❡s✐s✱ P (A) = P (A1 )P (A2 |A1 )P (A3 |A1 A2 ) · · · P(Ak |A1 A2 · · · Ak−1 ) ❈♦♠❜✐♥✐♥❣ t❤❡s❡ r❡s✉❧ts ②✐❡❧❞s ✭✷✳✷✮ ❢♦r ❊①❛♠♣❧❡ ✷✳✹✳✺✳ n = k + 1✳ ❆♥ ❥❛r ❝♦♥t❛✐♥s ✺ r❡❞ ❛♥❞ ✻ ❣r❡❡♥ ♠❛r❜❧❡s✳ ❲❡ r❛♥❞♦♠❧②... ❞❡♣♦s✐t ✐s A0 ❛♥❞ t❤❡ ✐♥t❡r❡st r❛t❡ ♣❡r ♣❡r✐♦❞ ✐s i✳ ■❢ ✐♥t❡r❡st ✐s ❝♦♠♣♦✉♥❞❡❞ ✱ t❤❡♥✱ ❛❢t❡r t❤❡ ✜rst ♣❡r✐♦❞✱ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❛❝❝♦✉♥t ✐s A1 = A0 + iA0 = A0 (1 + i)✱ ❛❢t❡r t❤❡ s❡❝♦♥❞ ♣❡r✐♦❞ t❤❡ ✈❛❧✉❡ ✐s A2 = A1 + iA1 = A1 (1 + i) = A0 (1 + i)2 ✱ ❛♥❞ s♦ ♦♥✳ ■♥ ❣❡♥❡r❛❧✱ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❛❝❝♦✉♥t ❛t t✐♠❡ n ✐s ❈♦♥s✐❞❡r ✜rst ❛♥ ❛❝❝♦✉♥t t❤❛t ♣❛②s ✐♥t❡r❡st ❛t t❤❡ ❞✐s❝r❡t❡ t✐♠❡s An = A0 (1 + i)n , A0 ✐s ❝❛❧❧❡❞... ❢♦r ❡①❛♠♣❧❡✱ t❤❛t P = {A1 , A2 , A3 , A4 }✳ ❚❤❡ ❝♦♠♣❧❡♠❡♥t ♦❢ A1 ∪ A3 ✐s t❤❡♥ A2 ∪ A4 ✳ ▲❡t Ω ✶✼ ❜❡ ❛ ✜♥✐t❡✱ ♥♦♥❡♠♣t② s❡t ❛♥❞ ❛ t❤❛t ✐s✱ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♣❛✐r✇✐s❡ ❞✐s❥♦✐♥t✱ ♥♦♥❡♠♣t② s❡ts ✇✐t❤ ✉♥✐♦♥ ❊①❛♠♣❧❡ ✷✳✸✳✸✳ A ❜❡ ❛♥② ❝♦❧❧❡❝t✐♦♥ ♦❢ s✉❜s❡ts ♦❢ Ω ❛♥❞ ❧❡t {Fλ : λ ∈ Λ} σ ✲✜❡❧❞s ❝♦♥t❛✐♥✐♥❣ A ❚❤❡ ✐♥t❡rs❡❝t✐♦♥ FA ♦❢ t❤❡ σ ✲ ✜❡❧❞s Fλ ✐s ❛❣❛✐♥ ❛ σ ✲✜❡❧❞✱ ❝❛❧❧❡❞ t❤❡ σ ✲✜❡❧❞ ❣❡♥❡r❛t❡❞ ❜② A ■t ✐s t❤❡ s♠❛❧❧❡st... ❢♦r ❡✈❡r② n✱ t❤❡♥✱ ❢♦r ❛♥② ❡✈❡♥t A P (A) = P (A| Bn )P(Bn ) n Pr♦♦❢✳ ❚❤❡ ❡✈❡♥ts AB1 , AB2 , P (A) = ❛r❡ ♠✉t✉❛❧❧② ❡①❝❧✉s✐✈❡ ✇✐t❤ ✉♥✐♦♥ P(ABn ) = n ❊①❛♠♣❧❡ ✷✳✹✳✼✳ A ❤❡♥❝❡ P (A| Bn )P(Bn ) n ✭■♥✈❡st♦r✬s ❘✉✐♥✮ ❙✉♣♣♦s❡ ②♦✉ ♦✇♥ ❛ st♦❝❦ t❤❛t ❡❛❝❤ ❞❛② q = 1 − p✳ ❆ss✉♠❡ x ❛♥❞ t❤❛t ②♦✉ ✐♥t❡♥❞ t♦ s❡❧❧ t❤❡ st♦❝❦ ❛s s♦♦♥ ❛s ✐ts ✈❛❧✉❡ ✐s ❡✐t❤❡r a ♦r b✱ ✇❤✐❝❤❡✈❡r ❝♦♠❡s ✜rst✱ ✇❤❡r❡ 0 < a ≤ x ≤ b✳ ❲❤❛t ❣♦❡s ✉♣ ✩✶ ✇✐t❤... ❊①❛♠♣❧❡ ✷✳✸✳✸✮✳ A t❤❛t t❤❡ s❡❧❡❝t❡❞ ♥✉♠❜❡r x d1 d2 d3 ✇✐t❤ ♥♦ ❞✐❣✐t dj ❡q✉❛❧ t♦ ✸✳ ❙❡t A0 = [0, 1]✳ ❙✐♥❝❡ d1 = 3✱ A ♠✉st ❜❡ ❝♦♥t❛✐♥❡❞ ✐♥ t❤❡ s❡t A1 ♦❜t❛✐♥❡❞ ❜② r❡♠♦✈✐♥❣ ❢r♦♠ A0 t❤❡ ✐♥t❡r✈❛❧ [.3, 4)✳ ❙✐♠✐❧❛r❧②✱ s✐♥❝❡ d2 = 3✱ A ✐s ❝♦♥t❛✐♥❡❞ ✐♥ t❤❡ s❡t A2 ♦❜t❛✐♥❡❞ ❜② r❡♠♦✈✐♥❣ ❢r♦♠ A1 t❤❡ ♥✐♥❡ ✐♥t❡r✈❛❧s ♦❢ t❤❡ ❢♦r♠ [.d1 3, d1 4)✱ d1 = 3✳ ❍❛✈✐♥❣ ♦❜t❛✐♥❡❞ An−1 ✐♥ t❤✐s ✇❛②✱ ✇❡ s❡❡ t❤❛t A ♠✉st ❜❡ ❝♦♥t❛✐♥❡❞... ❡✈❡♥t A ∈ F ❛ ♥✉♠❜❡r P (A) ✱ ❝❛❧❧❡❞ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ A s✉❝❤ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s ❤♦❧❞✿ ✭❛✮ 0 ≤ P (A) ≤ 1❀ ✭❜✮ P(Ω) = 1 ✭❝✮ ❛♥❞ P(∅) = 0❀ ✐❢ A1 , A2 , ✐s ❛ ✜♥✐t❡ ♦r ✐♥✜♥✐t❡ s❡q✉❡♥❝❡ ♦❢ ♣❛✐r✇✐s❡ ❞✐s❥♦✐♥t ❡✈❡♥ts✱ t❤❡♥ An P = n P(An ) n ❚❤❡ tr✐♣❧❡ (Ω, F, P) ✐s t❤❡♥ ❝❛❧❧❡❞ ❛ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡✳ ❆ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❡✈❡♥ts ✐s s❛✐❞ t♦ ❜❡ ❡❛❝❤ ♣❛✐r ♦❢ ❞✐st✐♥❝t ♠❡♠❜❡rs A ❛♥❞ ♠✉t✉❛❧❧② ❡①❝❧✉s✐✈❡ B ✐❢ P(AB)... t❤❛t ✐♥ t❤❡ ♣r❡❝❡❞✐♥❣ ❡①❛♠♣❧❡ P (A| B) = |AB| |AB|/|Ω| P(AB) = = |B| |B|/|Ω| P(B) ❚❤✐s s✉❣❣❡sts t❤❡ ❢♦❧❧♦✇✐♥❣ ❣❡♥❡r❛❧ ❞❡✜♥✐t✐♦♥ ♦❢ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t②✳ ❉❡✜♥✐t✐♦♥ ✷✳✹✳✷✳ ▲❡t (Ω, F, P) ❜❡ ❛ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡✳ ■❢ A ❛♥❞ B ❛r❡ ❡✈❡♥ts ✇✐t❤ P(B) > 0✱ t❤❡♥ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t② ♦❢ A ❣✐✈❡♥ B ✐s P (A| B) = P (A| B) ✐s ✉♥❞❡✜♥❡❞ ✐❢ P(B) = 0✳ ❊①❛♠♣❧❡ ✷✳✹✳✸✳ ♣r♦❜❛❜✐❧✐t② ♦❢ ▲❡t B P(AB) P(B) ❜❡ t❤❡ ■♥ t❤❡ ❞❛rt❜♦❛r❞