The Mathematics of Financial Models: Solving Real–World Problems with Quantitative Methods

Praise for The Mathematics of Financial Models Dr. Kannoo Ravindran does a great job in using applied quantitative methods to solve financial problems encountered in practice while discussing the practical nuances associated with the problem. The book explains the concept intuitively so it is very easy for readers to get it. This book is a must–read for anyone new to mathematical modeling in finance and serves as a great complement to any good book on finance and derivatives. Dr. Pin Chung, FRM, ASA, MAAA, Chief Financial Officer and Chief Investment Officer, R+V International Business Services Limited, Dublin, Ireland This is a unique work that provides easy–to–follow practical solutions in addition to the underlying theory for solving problems using quantitative methods. A must read for the new practitioner and a great refresher for the experienced practitioner. Kirk Evans, FSA, MAAA, CFA, FRM, Vice President, Product Development Pricing and Risk Management, Sammons Retirement Solutions The actuary or financial quant entering variable annuity risk management faces a bewildering array of new terminology, concepts, and practices. Dr. Ravindrans book provides a comprehensive but compact introduction for new practitionersincluding clear spreadsheet models for building intuition and practical bench marking. Daniel D. Heyer, FCAS, CQF, Vice President, Quantitative Risk Management, Nationwide Financial This book and the accompanying Excel worksheets are a valuable resource for quants and would–be quants. John Hull, Maple Financial Chair in Derivatives and Risk Management, Joseph L. Rotman School of Management, University of Toronto Dr. Kannoo Ravindrans book is a welcomed addition to a students or practitioners library alike, given the range of topics it covers, the accompanying spreadsheet examples, and all the solid references that can be found at the end of each chapter. The author has done a great job of covering the diverse subject matter, like the chapter on financial guarantees embedded in life insurance products, and the one on hedge strategy effectiveness, and leaving readers with a set of building blocks to help them tackle real risk management problems they would face in the field. Peter M. Phillips, Managing Director, Aon Benfield Securities, Inc. Dr. Ravindran has written a valuable book that bridges the all–too–wide gap between theory and practice in mathematical finance. It is useful and should be required reading for students in quantitative finance programs, and yet is immediately accessible to many who work in the field, from front–office users to risk managers, modelers, programmers, and operations staff. Paul Staneski, Ph.D., Principal, Derivatives Solutions, LLC

Preface ix Acknowledgments xi CHAPTER 1 Setting the Stage 1 Why Is This Book Different? 2 Road Map of the Book 3 References 5 CHAPTER 2 Building Zero Curves 7 Market Instruments 8 Linear Interpolation 16 Cubic Splining 25 Appendix: Finding Swap Rates Using a Floating Coupon Bond Approach 41 References 43 CHAPTER 3 Valuing Vanilla Options 45 Black–Scholes Formulae 47 Adaptations of the Black–Scholes Formulae 53 Limitations of the Black–Scholes Formulae 70 Application in Currency Risk Management 74 Appendix 78 References 80 CHAPTER 4 Simulations 81 Uniform Number Generation 82 Non–Uniform Number Generation 86 Applications of Simulations 93 Variance Reduction Techniques 100 References 104 CHAPTER 5 Valuing Exotic Options 107 Valuing Path–Independent, European–Style Options on a Single Variable 108 Valuing Path–Dependent, European–Style Options on a Single Variable 114 Valuing Path–Independent, European–Style Options on Two Variables 135 Valuing Path–Dependent, European–Style Options on Multiple Variables 152 References 157 CHAPTER 6 Estimating Model Parameters 159 Calibration of Parameters in the Black–Scholes Model 161 Using Implied Black–Scholes Volatility Surface and Zero Rate Term Structure to Value Options 169 Using Volatility Surface 178 Calibration of Interest Rate Option Model Parameters 190 Statistical Estimation 196 References 203 CHAPTER 7 The Effectiveness of Hedging Strategies 205 Delta Hedging 206 Assumptions Underlying Delta Hedging 216 Beyond Delta Hedging 223 Testing Hedging Strategies 230 Analysis Associated with the Hedging of a European–Style Vanilla Put Option 235 References 244 CHAPTER 8 Valuing Variable Annuity Guarantees 245 Basic GMDB 246 Death Benefit Riders 261 Other Details Associated with GMDB Products 269 Improving Modeling Assumptions 273 Living Benefit Riders 276 References 279 CHAPTER 9 Real Options 281  Surrendering a GMAB Rider 282 Adding Servers in a Queue 300 References 314 CHAPTER 10 Parting Thoughts 315 About the Author 317 About the Website 319 Index 321

D R. KANNOO RAVINDRAN consults with corporations on investments, derivatives trading, modeling, and risk management. He also lectures around the world on these topics and runs a private equity fund. Dr. Ravindran pioneered the use of derivatives to manage risks embedded in variable annuity products.