DFEM - A Discussion of Financial Economics in Actuarial...

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A Discussion of Financial Economics in Actuarial Models A Preparation for the Actuarial Exam MFE/3F Marcel B. Finan Arkansas Tech University c All Rights Reserved Preliminary Draft October 10, 2011
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2 To Pallavi and Amin
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Preface This is the third of a series of books intended to help individuals to pass actuarial exams. The present manuscript covers the financial economics segment of Exam M referred to by MFE/3F. The flow of topics in the book follows very closely that of McDonald’s Derivatives Markets . The book covers designated sections from this book as suggested by the 2009 SOA Syllabus. The recommended approach for using this book is to read each section, work on the embedded examples, and then try the problems. Answer keys are provided so that you check your numerical answers against the correct ones. Problems taken from previous SOA/CAS exams will be indicated by the symbol . This manuscript can be used for personal use or class use, but not for commercial purposes. If you find any errors, I would appreciate hearing from you: [email protected] Marcel B. Finan Russellville, Arkansas May 2010 3
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4 PREFACE
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Contents Preface 3 Parity and Other Price Options Properties 9 1 A Review of Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 Put-Call Parity for European Options . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3 Put-Call Parity of Stock Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4 Conversions and Reverse Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5 Parity for Currency Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 6 Parity of European Options on Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 7 Put-Call Parity Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 8 Labeling Options: Currency Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 9 No-Arbitrage Bounds on Option Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 10 General Rules of Early Exercise on American Options . . . . . . . . . . . . . . . . . . 69 11 Effect of Maturity Time Growth on Option Prices . . . . . . . . . . . . . . . . . . . . 78 12 Options with Different Strike Prices but Same Time to Expiration . . . . . . . . . . . 84 13 Convexity Properties of the Option Price Functions . . . . . . . . . . . . . . . . . . . 90 Option Pricing in Binomial Models 99 14 Single-Period Binomial Model Pricing of European Call Options . . . . . . . . . . . . 100 15 Risk-Neutral Option Pricing in the Binomial Model: A First Look . . . . . . . . . . . 109 16 Binomial Trees and Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 17 Multi-Period Binomial Option Pricing Model . . . . . . . . . . . . . . . . . . . . . . . 120 18 Binomial Option Pricing for European Puts . . . . . . . . . . . . . . . . . . . . . . . . 125 19 Binomial Option Pricing for American Options . . . . . . . . . . . . . . . . . . . . . . 131 20 Binomial Option Pricing on Currency Options . . . . . . . . . . . . . . . . . . . . . . 137 21 Binomial Pricing of Futures Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 22 Further Discussion of Early Exercising . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 23 Risk-Neutral Probability Versus Real Probability . . . . . . . . . . . . . . . . . . . . . 155 5
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6 CONTENTS 24 Random Walk and the Binomial Model . . . . . . . . . . . . . . . . . . . . . . . . . . 165 25 Alternative Binomial Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 26 Estimating (Historical) Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 The Black-Scholes Model 183 27 The Black-Scholes Formulas for European Options . . . . . . . . . . . . . . . . . . . . 184 28 Applying the Black-Scholes Formula To Other Assets . . . . . . . . . . . . . . . . . . 190 29 Option Greeks: Delta, Gamma, and Vega . . . . . . . . . . . . . . . . . . . . . . . . . 200 30 Option Greeks: Theta, Rho, and Psi . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 31 Option Elasticity and Option Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . 216 32 The Risk Premium and Sharpe Ratio of an Option . . . . . . . . . . . . . . . . . . . . 223 33 Profit Before Maturity: Calendar Spreads . . . . . . . . . . . . . . . . . . . . . . . . . 231 34 Implied Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 Option Hedging 243 35 Delta-Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 36 Option Price Approximations: Delta and Delta-Gamma Approximations . . . . . . . . 252 37 The Delta-Gamma-Theta Approximation and the Market-Maker’s Profit . . . . . . . . 258 38 The Black-Scholes Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 39 Delta-Gamma Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
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