Deeper Understanding, Faster Calc: SOA MFE and CAS Exam 3F Yufeng Guo October 4, 2017 ii Contents 1 Introduction to derivatives 1 1.1 What is a derivative? . 1 1.1.1 De…nition of derivatives . 2 1.1.2 Major types of derivatives . 2 1.1.3 Basic vocabulary . 3 1.1.4 Uses of Derivatives . 4 1.1.5 Perspectives on Derivatives . 5 1.1.6 Financial Engineering and Security Design . 5 1.2 The role of …nancial markets . 6 1.2.1 Financial Markets and Averages . 6 1.2.2 Risk-sharing . 6 1.3 Derivatives in practice . 6 1.3.1 Growth in derivatives trading . 6 1.3.2 How are derivatives used? . 6 1.4 Buying and short-selling …nancial assets . 7 1.4.1 The lease rate of an asset . 7 1.4.2 Risk and scarcity in short-selling . 7 1.5 Chapter summary . 7 1.6 Review questions . 8 2 An Introduction to Forwards and Options 9 2.1 Forward contracts . 9 2.1.1 De…nition . 9 2.1.2 The payo¤ on a forward contract . 10 2.1.3 Graphing the payo¤ on a forward contract . 10 2.1.4 Comparing a forward and outright purchase . 11 2.1.5 Zero-coupon bonds in payo¤ and pro…t diagrams . 11 2.1.6 Cash settlement vs. delivery . 12 2.1.7 Credit risk . 12 2.2 Call options . 12 2.2.1 Option terminology . 12 2.2.2 Payo¤ and pro…t for a purchased call option . 12 2.2.3 Payo¤ and pro…t for a written call option . 12 2.3 Put options . 13 iii iv CONTENTS 2.3.1 Payo¤ and pro…t for a purchased put option . 13 2.3.2 Payo¤ and pro…t for a written put option . 13 2.4 Summary of forward and option positions . 13 2.5 Options are insurance . 14 2.6 Example: equity-linked CD . 15 3 Insurance, Collars, and Other Strategies 17 3.1 Basic insurance strategies . 17 3.1.1 Insuring a long position: ‡oors . 17 3.1.2 Insuring a short position: caps . 18 3.1.3 Selling insurance . 18 3.2 Synthetic forwards . 19 3.2.1 Put-call parity . 19 3.3 Spreads and collars . 20 3.3.1 Bull and bear spreads . 20 3.3.2 Box spreads . 21 3.3.3 Ratio spreads . 21 3.3.4 Collars . 22 3.4 Speculating on volatility . 22 3.4.1 Straddle . 22 3.4.2 Butter‡y spread . 23 3.4.3 Asymmetric butter‡y spreads . 23 5 Financial forwards and futures 25 5.1 Alternative ways to buy a stock . 25 5.2 Tailing . 25 5.3 Pricing a forward contract . 26 5.4 Forward premium . 27 5.5 Creating a synthetic forward contract . 27 5.5.1 Synthetic forwards in market-making and arbitrage . 29 5.5.2 No-arbitrage bounds with transaction costs . 30 5.5.3 Quasi-arbitrage . 31 5.6 Futures contracts . 32 5.6.1 Role of the clearing house . 32 5.6.2 S&P 500 futures contract . 33 5.6.3 Di¤erence between a forward contract and a futures contract 33 5.6.4 Margins and markings to market . 33 5.6.5 Comparing futures and forward prices . 36 5.6.6 Arbitrage in practice: S&P 500 index arbitrage . 37 5.6.7 Appendix 5.B: Equating forwards and futures . 38 9 Parity and other option relationships 45 9.1 Put-call parity . 45 9.1.1 Option on stocks . 45 9.1.2 Options on currencies . 55 9.1.3 Options on bonds . 57 CONTENTS v 9.1.4 Generalized parity and exchange options . 57 9.1.5 Comparing options with respect to style, maturity, and strike . 63 10 Binomial option pricing: basic concepts 79 10.1 One-period binomial model: simple examples . 79 10.2 General one-period binomial model . 80 10.2.1 Estimate stock volatility . 92 10.2.2 Two or more binomial trees . 97 10.2.3 Options on stock index . 112 10.2.4 Options on currency . 115 10.2.5 Options on futures contracts . 119 11 Binomial option pricing: selected topics 127 11.1 Understanding early exercise . 127 11.2 Understanding risk-neutral probability . 128 11.2.1 Pricing an option using real probabilities . 129 11.2.2 Binomial tree and lognormality . 136 11.3 Stocks paying discrete dividends . 139 11.3.1 Problems with discrete dividend tree . 140 11.3.2 Binomial tree using prepaid forward . 142 12 Black-Scholes formula 149 12.1 Introduction to the Black-Scholes formula . 149 12.1.1 Call and put option price . 149 12.1.2 When is the Black-Scholes formula valid? . 151 12.2 Derive the Black-Scholes formula . 151 12.3 Applying the formula to other assets . 165 12.3.1 Black-Scholes formula in terms of prepaid forward price . 165 12.3.2 Options on stocks with discrete dividends . 165 12.3.3 Options on currencies . 166 12.3.4 Options on futures . 167 12.4 Option the Greeks . 168 12.4.1 Delta . 168 12.4.2 Gamma . 169 12.4.3 Vega . 169 12.4.4 Theta . 170 12.4.5 Rho . 170 12.4.6 Psi . 170 12.4.7 Greek measures for a portfolio . 170 12.4.8 Option elasticity and volatility . 171 12.4.9 Option risk premium and Sharpe ratio . 172 12.4.10 Elasticity and risk premium of a portfolio . 173 12.5 Pro…t diagrams before maturity . 173 12.5.1 Holding period pro…t . 173 12.5.2 Calendar spread . 176 vi CONTENTS 12.6 Implied volatility . 177 12.6.1 Calculate the implied volatility . 177 12.6.2 Volatility skew . 178 12.6.3 Using implied volatility . 179 12.7 Perpetual American options . 179 12.7.1 Perpetual calls and puts . 179 12.7.2 Barrier present values . 184 13 Market-making and delta-hedging 187 13.1 Delta hedging . 187 13.2 Examples of Delta hedging . 187 13.3 Textbook Table 13.2 . 196 13.4 Textbook Table 13.3 . 198 13.5 Mathematics of Delta hedging . 199 13.5.1 Delta-Gamma-Theta approximation . 199 13.5.2 Understanding the market maker’spro…t . 200 14 Exotic options: I 203 14.1 Asian option (i.e. average options) . 203 14.1.1 Characteristics . 203 14.1.2 Examples . 204 14.1.3 Geometric average . 204 14.1.4 Payo¤ at maturity T . 204 14.2 Barrier option . 205 14.2.1 Knock-in option . 205 14.2.2 Knock-out option . 205 14.2.3 Rebate option . 205 14.2.4 Barrier parity . 206 14.2.5 Examples . 206 14.3 Compound option . 207 14.4 Gap option . 208 14.4.1 De…nition . 208 14.4.2 Pricing formula . 209 14.4.3 How to memorize the pricing formula . 209 14.5 Exchange option . ..