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Analysis of Employee Stock Options and Guaranteed Withdrawal Benefits for Life by Premal Shah B

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Analysis of Employee Stock Options and Guaranteed Withdrawal Benefits for Life by Premal Shah B Analysis of Employee Stock Options and Guaranteed Withdrawal Benefits for Life by Premal Shah B. Tech. & M. Tech., Electrical Engineering (2003), Indian Institute of Technology (IIT) Bombay, Mumbai Submitted to the Sloan School of Management in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Operations Research at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 2008 c Massachusetts Institute of Technology 2008. All rights reserved. Author.............................................................. Sloan School of Management July 03, 2008 Certified by. Dimitris J. Bertsimas Boeing Professor of Operations Research Thesis Supervisor Accepted by . Cynthia Barnhart Professor Co-director, Operations Research Center 2 Analysis of Employee Stock Options and Guaranteed Withdrawal Benefits for Life by Premal Shah B. Tech. & M. Tech., Electrical Engineering (2003), Indian Institute of Technology (IIT) Bombay, Mumbai Submitted to the Sloan School of Management on July 03, 2008, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Operations Research Abstract In this thesis we study three problems related to financial modeling. First, we study the problem of pricing Employee Stock Options (ESOs) from the point of view of the issuing company. Since an employee cannot trade or effectively hedge ESOs, she exercises them to maximize a subjective criterion of value. Modeling this exercise behavior is key to pricing ESOs. We argue that ESO exercises should not be modeled on a one by one basis, as is commonly done, but at a portfolio level because exercises related to different ESOs that an employee holds would be coupled. Using utility based models we also show that such coupled exercise behavior leads to lower average ESO costs for the commonly used utility functions such as power and exponential utilities. Unfortunately, utility based models do not lead to tractable solutions for finding costs associated with ESOs. We propose a new risk management based approach to model exercise behavior based on mean-variance portfolio maximization. The resulting exercise behavior is both intuitive and leads to a computationally tractable model for finding ESO exercises and pricing ESOs as a portfolio. We also study a special variant of this risk-management based exercise model, which leads to a decoupling of the ESO exercises and then obtain analytical bounds on the implied cost of an ESO for the employer in this case. Next, we study Guaranteed Withdrawal Benefits (GWB) for life, a recent and popular product that many insurance companies have offered for retirement plan- ning. The GWB feature promises to the investor increasing withdrawals over her lifetime and is an exotic option that bears financial and mortality related risks for the insurance company. We first analyze a continuous time version of this product in a Black Scholes economy with simplifying assumptions on population mortality and obtain an analytical solution for the product value. This simple analysis reveals the high sensitivity the product bears to several risk factors. We then further in- vestigate the pricing of GWB in a more realistic setting using different asset pricing models, including those that allow the interest rates and the volatility of returns to be 3 stochastic. Our analysis reveals that 1) GWB has insufficient price discrimination and is susceptible to adverse selection and 2) valuations can vary substantially depending on which class of models is used for accounting. We believe that the ambiguity in value and the presence of significant risks, which can be challenging to hedge, should create concerns to the GWB underwriters, their clients as well as the regulators. Finally, many problems in finance are Sequential Decision Problems (SDPs) under uncertainty. We find that SDP formulations using commonly used financial metrics or acceptability criteria can lead to dynamically inconsistent strategies. We study the link between objective functions used in SDPs, dynamic consistency and dynamic programming. We then propose ways to create dynamically consistent formulations. Thesis Supervisor: Dimitris J. Bertsimas Title: Boeing Professor of Operations Research 4 Acknowledgments First, I would like to express my sincere gratitude towards Prof. Dimitris Bertsimas for his guidance and support in the work related to this thesis and beyond. From day one of my association with him, Prof. Bertsimas was never short of interesting research ideas and insights into problems. His emphasis on the practicability of research has kept me true. Without him this work would have been but an exercise in futility. Dimitris has been a great source of inspiration, encouragement and counsel to me outside research and academics as well. His versatility in terms of dealing with a broad range of research areas in both theory and applications, his repertoire of interests and activities and his drive are stupendous. I consider myself very fortunate to have been able to tap into the same, especially given the several sources competing for his time! I would also like to acknowledge the members of my thesis committee - Prof. Jiang Wang, for valuable inputs and suggestions to improve this thesis and Prof. David Gamarnik, for his encouragement and interest. Discussions with David about any problem have always refined my understanding significantly. I am also thankful to David, Dimitris and Prof. John Tsitsiklis for giving me the opportunity to serve as a teaching assistant with them. Teaching a course with one of them at charge has been a surprisingly good learning experience as well! I would also like to express my gratitude towards the Operations Research Center administrators - Paulette Mosley, Laura Rose and Andrew Carvalho - their efficiency and co-operation have made things so much easier. The years at ORC were sweet and memorable thanks to the fellow ORCers - Ilan Lobel, Yunqing (Kelly) Ye, Vinh Doan, Ruben Lobel, Doug Fearing, Dan Iancu, Carine Simon, Gareth, Juliane, Dung, Mike, Rajiv, Pranava, Shobhit, Carol, Pavithra, Theo, Andy, Yann, Elodie, Katy, Rags, Pallav, Dmitriy, Dave, Nikos, Margret, Lincoln, Karima, Mallory, Shubham, Thibault, Martin, Shashi, Sun Wei, Kostas, Nelson, ... Thank you for the countless wonderful moments inside and outside the ORC. You all will be sorely missed! Thanks is also due to all the room-mates and their extensions - Sudhir Chiluveru, Saurabh Tejwani, Hoda, Jean-Paul, Alec, Kurt, Raf, Mukul, Riccardo, Anatoly who made leaving in the dorms so much more homely. I am also grateful to MIT for the opportunity to meet and make so many won- derful friends - Rajiv Menjoge, Aditya Undurti, Theta Aye, Vikas Sharma, Kranthi 5 Vistakula, Neha Soni, Mithila Azad, Mira Chokshi. Thanks to them and the rest of the kajra-re and wandering gopis gangs - Jen Fang, Sukant, Preet, Tulika, Silpa, Natasha, Tara(2!), Nick, Silpa, Saba(2!), Kartik, Chief, ... for all the fun times! The time spent here, in the cold of Boston, also helped me to thaw some old friendships - Nandan Bhat, Ajay Raghavan, Suchit Kaul and Prafulla Krishna - it was great to have rediscovered you all! I am also thankful to Sangam and the '05 executive council fellow members - Vivek Jaiswal, Saurabh, Biraja for helping to re-create here Diwali, Holi and all the other things that I missed from home. I would also like to thank the members of the Edgerton Board '07 - Yvonne, Nicholas, Kelly, Biz, Grace, Becky, Ron,... and the Edgerton House Masters - David and Pamela Mindell; working with you all was a great experience. The last several months at MIT have been different and not just because of this thesis. Thanks to Tara Sainath for helping with proof-reading this thesis and a million other things and all the care and affection I could have asked for. Finally and most importantly, I would like to thank my parents, my sister Payal and my relatives in India - their love, support and care have been a source of great strength for me. Nothing would have been possible without this foundation of my life. 6 Contents Contents 7 List of Figures 11 List of Tables 13 1 Introduction 15 1.1 Motivation . 15 1.2 Contributions . 16 1.3 Thesis Overview . 19 2 Pricing Employee Stock Options 21 2.1 Introduction . 21 2.1.1 Motivation . 22 2.1.2 Related Work . 24 2.1.3 Contributions . 26 2.1.4 Chapter Layout . 27 2.2 Model . 28 2.3 Nature of ESO Cost Functions . 30 2.4 The multi-period problem for one type of options and nature of cost function . 34 2.5 Option Exercises with Multiple Option Types . 44 2.6 ESO costs for portfolios with multiple ESO types . 46 2.7 Summary . 53 3 Tractable Models for Pricing Employee Stock Options 55 3.1 Introduction . 55 3.2 Model . 58 3.3 Risk Management Models . 61 7 3.4 Exercise Behavior under Myopic Mean Variance Optimizing Policy . 67 3.5 Myopic Mean-Volatility Based Exercise Model . 73 3.6 Pricing ESOs under the fixed threshold delta-barrier exercise strategy 78 3.7 Conclusions and Future Directions . 86 4 Variable Annuities with Guaranteed Lifetime Payouts 89 4.1 Introduction . 89 4.1.1 Related Work . 92 4.1.2 Goals, findings and Contributions . 94 4.1.3 Chapter Layout . 94 4.2 Product Description . 95 4.3 GWB - A Hypothetical Historical Analysis . 98 4.4 Black Scholes Model with Continuous Step-ups and Exponential Mor- tality - CBSME Model . 100 4.5 Numerical Results . 108 4.6 Summary . 117 5 Guaranteed Lifetime Payouts: Further Analysis 119 5.1 Introduction . 119 5.1.1 Findings and Contributions . 120 5.1.2 Chapter Layout . 121 5.2 Valuation Models . 122 5.2.1 General Framework for pricing GWB products . 123 5.2.2 Pricing Models for the GWB for life . 127 5.3 Valuation under Alternate Withdrawal Strategy . 131 5.4 Numerical Results . 134 5.4.1 Valuation and Impact of Model Selection . 137 5.4.2 Valuation Spreads Across Asset mixes and Cohort Ages .
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