1 options . 1
1.1 what are options? 1
1.2 why do we study options? 2
1.3 how are options traded? 4
1.4 typical option prices 6
1.5 other financial derivatives 7
1.6 notes and references 7
1.7 program of chapter 1 and walkthrough 8
2 option valuation preliminaries 11
2.1 motivation 11
2.2 interest rates 11
2.3 short selling 12
2.4 arbitrage 13
2.5 put-call parity 13
2.6 upper and lower bounds on option values 14
2.7 notes and references 16
2.8 program of chapter 2 and walkthrough 17
3 random variables 21
3.1 motivation 21
3.2 random variables, probability and mean 21
.3.3 independence 23
3.4 variance 24
3.5 normal distribution 25
3.6 central limit theorem 27
3.7 notes and references 28
3.8 program of chapter 3 and walkthrough 29
4 computer simulation 33
4.1 motivation 33
4.2 pseudo-random numbers 33
4.3 statistical tests 34
4.4 notes and references 40
4.5 program of chapter 4 and walkthrough 41
5 asset price movement 45
5.1 motivation 45
5.2 efficient market hypothesis 45
5.3 asset price data 46
5.4 assumptions 48
5.5 notes and references 49
5.6 program of chapter 5 and walkthrough 50
6 asset price model: part i 53
6.1 motivation 53
6.2 discrete asset model 53
6.3 continuous asset model 55
6.4 lognormal distribution 56
6.5 features of the asset model 57
6.6 notes and references 59
6.7 program of chapter 6 and walkthrough 60
7 asset price model: part ii 63
7.1 computing asset paths 63
7.2 timescale invariance 66
7.3 sum-of-square returns 68
7.4 notes and references 69
7.5 program of chapter 7 and walkthrough 71
8 black-scholes pde and formulas 73
8.1 motivation 73
8.2 sum-of-square increments for asset price 74
8.3 hedging 76
8.4 black-scholes pde 78
8.5 black-scholes formulas 80
8.6 notes and references 82
8.7 program of chapter 8 and walkthrough 83
9 more on hedging 87
9.1 motivation 87
9.2 discrete hedging 87
9.3 delta at expiry 89
9.4 large-scale test 92
9.5 long-term capital management 93
9.6 notes 94
9.7 program of chapter 9 and walkthrough 96
10 the greeks 99
10.1 motivation 99
10.2 the greeks 99
10.3 interpreting the greeks 101
10.4 black-scholes pde solution 101
10.5 notes and references 102
10.6 program of chapter 10 and walkthrough 104
11 more on the black-scholes formulas 105
11.1 motivation 105
11.2 where is μ? 105
11.3 time dependency 106
11.4 the big picture 106
11.5 change of variables 108
11.6 notes and references 111
11.7 program of chapter 11 and walkthrough 111
12 risk neutrality 115
12.1 motivation 115
12.2 expected payoff 115
12.3 risk neutrality 116
12.4 notes and references 118
12.5 program of chapter 12 and walkthrough .. 120
13 solving a nonlinear equation 123
13.1 motivation 123
13.2 general problem 123
13.3 bisection 123
13.4 newton 124
13.5 further practical issues 127
13.6 notes and references 127
13.7 program of chapter 13 and walkthrough 128
14 implied volatility 131
14.1 motivation 131
14.2 implied volatility 131
14.3 option value as a function of volatility 131
14.4 bisection and newton 133
14.5 implied volatility with real data 135
14.6 notes and references 137
14.7 program of chapter 14 and walkthrough 137
15 monte carlo method 141
15.1 motivation 141
15.2 monte carlo 141
15.3 monte carlo for option valuation 144
15.4 monte carlo for greeks 145
15.5 notes and references 148
15.6 program of chapter 15 and walkthrough 149
16 binomial method 151
16.1 motivation 151
16.2 method 151
16.3 deriving the parameters 153
16.4 binomial method in practice 154
16.5 notes and references 156
16.6 program of chapter 16 and walkthrough 159
17 cash-or-nothing options 163
17.1 motivation 163
17.2 cash-or-nothing options 163
17.3 black-scholes for cash-or-nothing options 164
17.4 delta behaviour 166
17.5 risk neutrality for cash-or-nothing options 167
17.6 notes and references 168
17.7 program of chapter 17 and walkthrough 170
18 american options 173
18.1 motivation 173
18.2 american call and put 173
18.3 black-scholes for american options 174
18.4 binomial method for an american put 176
18.5 optimal exercise boundary 177
18.6 monte carlo for an american put 180
18.7 notes and references 182
18.8 program of chapter 18 and walkthrough 183
19 exotic options 187
19.1 motivation 187
19.2 barrier options 187
19.3 lookback options 191
19.4 asian options 192
19.5 bermudan and shout options 193
19.6 monte carlo and binomial for exotics 194
19.7 notes and references 196
19.8 program of chapter 19 and walkthrough 199
20 historical volatility 203
20.1 motivation 203
20.2 monte carlo-type estimates 203
20.3 accuracy of the sample variance estimate 204
20.4 maximum likelihood estimate 206
20.5 other volatility estimates 207
20.6 example with real data 208
20.7 notes and references 209
20.8 program of chapter 20 and walkthrough 210
21 monte carlo part ii: variance reduction by antithetic variates 215
21.1 motivation 215
21.2 the big picture 215
21.3 dependence 216
21.4 antithetic variates: uniform example 217
21.5 analysis of the uniform case 219
21.6 normal case 221
21.7 multivariate case 222
21.8 antithetic variates in option valuation 222
21.9 notes and references 225
21.10 program of chapter 21 and walkthrough 225
22 monte carlo part iii: variance reduction by control variates 229
22.1 motivation 229
22.2 control variates 229
22.3 control variates in option valuation 231
22.4 notes and references 232
22.5 program of chapter 22 and walkthrough 234
23 finite difference methods 237
23.1 motivation 237
23.2 finite difference operators 237
23.3 heat equation 238
23.4 discretization 239
23.5 ftcs and btcs 240
23.6 local accuracy 246
23.7 von neumann stability and convergence 247
23.8 crank-nicolson 249
23.9 notes and references 251
23.10 program of chapter 23 and walkthrough 252
24 finite difference methods for the black-scholes pde 257
24.1 motivation 257
24.2 ftcs, btcs and crank-nicolson for black-scholes 257
24.3 down-and-out call example 260
24.4 binomial method as finite differences 261
24.5 notes and references 262
24.6 program of chapter 24 and walkthrough 265
references 267
index ... 271