Introduction ................................................... ix
1 Elements of Finance ........................................... 1
1.1 Price .................................................... 3
1.2 Arbitrage ............................................... 11
1.3 Time Value of Assets, Arbitrage and No-Arbitrage
Assets .................................................. 14
1.4 Money Market, Bonds, and Discounting .................... 17
1.5 Dividends ............................................... 20
1.6 Portfolio ............................................... 21
1.7 Evolution of a Self-Financing Portfolio ................. 23
1.8 Fundamental Theorems of Asset Pricing ................... 28
1.9 Change of Measure via Radon-Nikodym Derivative .......... 44
1.10 Leverage: Forwards and Futures .......................... 48
2 Binomial Models .............................................. 59
2.1 Binomial Model for No-Arbitrage Assets .................. 60
2.1.1 One-Step Model ................................... 61
2.1.2 Hedging in the Binomial Model .................... 65
2.1.3 Multiperiod Binomial Model ....................... 66
2.1.4 Numerical Example ................................ 67
2.1.5 Probability Measures for Exotic No-Arbitrage
Assets ........................................... 73
2.2 Binomial Model with an Arbitrage Asset .................. 75
2.2.1 American Option Pricing in the Binomial Model .... 78
2.2.2 Hedging .......................................... 79
2.2.3 Numerical Example ................................ 81
3 Diffusion Models ............................................. 91
3.1 Geometric Brownian Motion ............................... 93
3.2 General European Contracts .............................. 99
3.3 Price as an Expectation ................................ 109
3.4 Connections with Partial Differential Equations ........ 111
3.5 Money as a Reference Asset ............................. 114
3.6 Hedging ................................................ 117
3.7 Properties of European Call and Put Options ............ 122
3.8 Stochastic Volatility Models ........................... 127
3.9 Foreign Exchange Market ................................ 130
3.9.1 Forwards ........................................ 131
3.9.2 Options ......................................... 133
4 Interest Rate Contracts ..................................... 137
4.1 Forward LIBOR .......................................... 138
4.1.1 Backset LIBOR ................................... 139
4.1.2 Caplet .......................................... 140
4.2 Swaps and Swaptions .................................... 141
4.3 Term Structure Models .................................. 143
5 Barrier Options ............................................. 149
5.1 Types of Barrier Options ............................... 150
5.2 Barrier Option Pricing via Power Options ............... 152
5.2.1 Constant Barrier ................................ 152
5.2.2 Exponential Barrier ............................. 157
5.3 Price of a Down-and-In Call Option ..................... 160
5.4 Connections with the Partial Differential Equations .... 165
6 Lookback Options ............................................ 171
6.1 Connections of Lookbacks with Barrier Options .......... 171
6.1.1 Case α = 1 ...................................... 173
6.1.2 Case α < 1 ...................................... 174
6.1.3 Hedging ......................................... 178
6.2 Partial Differential Equation Approach for Lookbacks ... 180
6.3 Maximum Drawdown ....................................... 187
7 American Options ............................................ 191
7.1 American Options on No-Arbitrage Assets ................ 192
7.2 American Call and Puts on Arbitrage Assets ............. 194
7.3 Perpetual American Put ................................. 195
7.4 Partial Differential Equation Approach ................. 199
8 Contracts on Three or More Assets: Quantos, Rainbows
and "Friends" ............................................... 207
8.1 Pricing in the Geometric Brownian Motion Model ......... 209
8.2 Hedging ................................................ 213
9 Asian Options ............................................... 219
9.1 Pricing in the Geometric Brownian Motion Model ......... 226
9.2 Hedging of Asian Options ............................... 230
9.3 Reduction of the Pricing Equations ..................... 233
10 Jump Models ................................................. 239
10.1 Poisson Process ........................................ 240
10.2 Geometric Poisson Process .............................. 243
10.3 Pricing Equations ...................................... 248
10.4 European Call Option in Geometric Poisson Model ........ 251
10.5 Levy Models with Multiple Jump Sizes ................... 256
A Elements of Probability Theory .............................. 267
A.l Probability, Random Variables .......................... 267
A.2 Conditional Expectation ................................ 271
A.2.1 Some Properties of Conditional Expectation ...... 274
A.3 Martingales ............................................ 274
A.4 Brownian Motion ........................................ 279
A.5 Stochastic Integration ................................. 283
A.6 Stochastic Calculus .................................... 285
A.7 Connections with Partial Differential Equations ........ 287
Solutions to Selected Exercises ................................ 293
References ..................................................... 313
Index .......................................................... 323
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