Derivative Pricing : a Problem-Based Primer

451 Pages·2018·12.02 MB·English

by  Lo

Derivative Pricing
A Problem-Based Primer
CHAPMAN & HALL/CRC
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Derivative Pricing
A Problem-Based Primer
Ambrose Lo
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Derivative Pricing
A Problem-Based Primer
Ambrose Lo
CRC Press
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Contents
List of Figures ix
List of Tables xi
Preface xiii
Symbols xvii
I Conceptual Foundation on Derivatives 1
1 An Introduction to Forwards and Options 3
1.1 Forwards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.1 Call Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.2 Put Options. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 Classification of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2 Forwards and Futures 27
2.1 Alternative Ways to Buy a Stock . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Prepaid Forwards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.1 Nondividend-paying Stocks . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.2 Dividend-paying Stocks . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3 Forwards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.1 Forward Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.2 Cash-and-Carry Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . 39
2.3.3 Digression: Market Frictions . . . . . . . . . . . . . . . . . . . . . . . 44
2.4 Futures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.4.1 Differences between Futures and Forwards . . . . . . . . . . . . . . . 46
2.4.2 Marking to Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3 Option Strategies 57
3.1 Basic Insurance Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.1.1 Insuring a Long Position: Floors . . . . . . . . . . . . . . . . . . . . 57
3.1.2 Insuring a Short Position: Caps . . . . . . . . . . . . . . . . . . . . . 61
3.1.3 Selling Insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.1.4 A Simple but Useful Observation: Parallel Payoffs, Identical Profit . 65
3.2 Put-call Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2.1 Synthetic Forwards . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2.2 The Put-call Parity Equation . . . . . . . . . . . . . . . . . . . . . . 68
3.3 Spreads and Collars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.3.1 Spreads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
v
vi Contents
3.3.2 Collars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.4 Volatility Speculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.4.1 Straddles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.4.2 Strangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.4.3 Butterfly Spreads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
II Pricing and Hedging of Derivatives 113
4 Binomial Option Pricing Models 115
4.1 One-period Binomial Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.1.1 Pricing by Replication . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.1.2 Risk-neutral Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.1.3 Constructing a Binomial Tree . . . . . . . . . . . . . . . . . . . . . . 125
4.2 Multi-period Binomial Trees . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.3 American Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
4.4 Options on Other Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
4.4.1 Case Study 1: Currency Options . . . . . . . . . . . . . . . . . . . . 149
4.4.2 Case Study 2: Options on Futures . . . . . . . . . . . . . . . . . . . 150
4.5 Epilogue: Pricing by Real Probabilities of Stock Price Movements . . . . . 152
4.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
5 Mathematical Foundations of the Black-Scholes Framework 171
5.1 A Lognormal Model of Stock Prices . . . . . . . . . . . . . . . . . . . . . . 171
5.2 Lognormal-Based Probabilistic Quantities . . . . . . . . . . . . . . . . . . . 174
5.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
6 The Black-Scholes Formula 185
6.1 Black-Scholes Formula for Stocks Paying Continuous Proportional Dividends 185
6.2 Applying the Black-Scholes Formula to Other Underlying Assets . . . . . . 191
6.2.1 Case study 1: Stocks paying non-random, discrete dividends. . . . . 192
6.2.2 Case Study 2: Currency options. . . . . . . . . . . . . . . . . . . . . 196
6.2.3 Case Study 3: Futures options. . . . . . . . . . . . . . . . . . . . . . 200
6.3 Option Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
6.3.1 Option Delta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
6.3.2 Option Gamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
6.3.3 Option Greeks of a Portfolio . . . . . . . . . . . . . . . . . . . . . . 211
6.3.4 Option Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
6.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
7 Option Greeks and Risk Management 231
7.1 Delta-hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
7.2 Hedging Multiple Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
7.3 Delta-Gamma-Theta Approximation . . . . . . . . . . . . . . . . . . . . . . 244
7.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
8 Exotic Options 261
8.1 Gap Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
8.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
8.1.2 All-or-Nothing Options . . . . . . . . . . . . . . . . . . . . . . . . . 264
8.1.3 Pricing and Hedging Gap Options . . . . . . . . . . . . . . . . . . . 268
8.2 Exchange Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
Contents vii
8.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
8.2.2 Pricing Exchange Options . . . . . . . . . . . . . . . . . . . . . . . . 274
8.2.3 Pricing Maximum and Minimum Contingent Claims . . . . . . . . . 280
8.3 Compound Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
8.4 Asian Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
8.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
8.4.2 Pricing Asian Options . . . . . . . . . . . . . . . . . . . . . . . . . . 290
8.5 Lookback Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
8.6 Shout Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
8.7 Barrier Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
8.8 Other Exotic Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
8.8.1 Chooser Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
8.8.2 Forward Start Options . . . . . . . . . . . . . . . . . . . . . . . . . . 311
8.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
III Epilogue 335
9 General Properties of Option Prices 337
9.1 Put-Call Parity and Duality . . . . . . . . . . . . . . . . . . . . . . . . . . 337
9.1.1 Generalized Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
9.1.2 Currency Put-call Duality . . . . . . . . . . . . . . . . . . . . . . . . 340
9.2 Upper and Lower Bounds on Option Prices . . . . . . . . . . . . . . . . . . 343
9.3 Comparing Options with Respect to Contract Characteristics . . . . . . . . 347
9.3.1 Strike Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
9.3.2 Maturity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
9.4 Early Exercise Decisions for American Options . . . . . . . . . . . . . . . . 357
9.4.1 Proof 1: A Proof Based on No-arbitrage Bounds . . . . . . . . . . . 357
9.4.2 Proof 2: A Cost-benefit Dissection Proof . . . . . . . . . . . . . . . . 358
9.4.3 Early Exercise Criterion for American Puts . . . . . . . . . . . . . . 360
9.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
Appendix A Standard Normal Distribution Table 367
Appendix B Solutions to Odd-Numbered End-of-Chapter Problems 369
B.1 Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
B.2 Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
B.3 Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
B.4 Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
B.5 Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
B.6 Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
B.7 Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
B.8 Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
B.9 Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422
Bibliography 427
Index 429

List of Figures
1.1.1 Payoff diagrams of a long forward (left) and a short forward (right). . . . 6
1.2.1 Payoff and profit diagrams of a long call (left) and a short call (right). . . 10
1.2.2 The profit functions of the three calls in Example 1.2.2. . . . . . . . . . . 13
1.2.3 Payoff and profit diagrams of a long put (left) and a short put (right). . . 16
3.1.1 The payoff diagrams of a long asset (unhedged, dashed) and a long asset
coupled with a long K-strike put (hedged, bold). . . . . . . . . . . . . . . 58
3.1.2 The payoff diagrams of a short asset (unhedged, dashed) and a short asset
coupled with a long K-strike call (hedged, bold). . . . . . . . . . . . . . . 62
3.2.1 The payoff diagram of a long synthetic forward constructed by K-strike
long call and short put options. . . . . . . . . . . . . . . . . . . . . . . . . 67
3.3.1 Payoff diagram of a call K -K bull spread. . . . . . . . . . . . . . . . . . 76
1 2
3.3.2 Payoff diagram of K -K bear spreads constructed by calls (left) and by
1 2
puts (right).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.3.3 Payoff diagram of a long K -K collar. . . . . . . . . . . . . . . . . . . . . 84
1 2
3.3.4 Illustration of the construction of two different zero-cost collars: a K -K
1 2
zero-cost collar and a K(cid:48)-K(cid:48) zero-cost collar. . . . . . . . . . . . . . . . . 90
1 2
3.4.1 The payoff and profit diagrams of a long K-strike straddle. . . . . . . . . 91
3.4.2 Payoff diagram of a long K -K strangle. . . . . . . . . . . . . . . . . . . 93
1 2
3.4.3 The payoff diagrams of the student’s “strange” and a genuine strangle in
Example 3.4.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.4.4 PayoffdiagramofalongK -K -K butterflyspreadconstructedbyashort
1 2 3
K -strike straddle coupled with a long K -K strangle. . . . . . . . . . . 97
2 1 3
3.4.5 Payoff diagram of a long K -K -K call (or put) butterfly spread. . . . . 98
1 2 3
4.1.1 A generic one-period binomial stock price model. The derivative payoffs
are shown in parentheses. . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.2.1 A generic two-period binomial stock price tree. . . . . . . . . . . . . . . . 132
4.3.1 The two-period binomial tree for Example 4.3.1. . . . . . . . . . . . . . . 142
4.3.2 The two-period binomial forward tree for Example 4.3.2. . . . . . . . . . 144
4.3.3 The two-period binomial tree for Example 4.3.3. . . . . . . . . . . . . . . 145
4.3.4 The two-period binomial tree for Example 4.3.4. . . . . . . . . . . . . . . 147
4.4.1 The exchange rate evolution in Example 4.4.1. . . . . . . . . . . . . . . . 151
4.4.2 The binomial futures price tree in Example 4.4.2. . . . . . . . . . . . . . . 153
5.1.1 AstockpricepathintheBlack-ScholesstockpricemodelwithS(0)=100,
α=0.08, and σ =0.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.2.1 The cash flows between different parties in Example 6.2.4. . . . . . . . . . 200
6.3.1 Thevariationofcallandputdeltaswiththecurrentstockpricefordifferent
times to expiration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
ix