Index of Pricing and Hedging of Derivative Securities. By Lars Tyge Nielsen. Textbook in continuous-time finance theory. Oxford University Press, 1999.
absolutely continuous measure, 359
adapted process, 15
adjusted local convergence in mean, 232
admissible trading strategy, 146
almost everywhere, 339
almost everywhere identical processes, 3
almost everywhere measurable, 339
almost simple trading strategy, 126
almost surely, 339
arbitrage strategy, 145
Arnold, L., 36-38, 42, 43, 49-51, 89
Arrow, K. J., 297
asset-or-nothing call option, 211
asset-or-nothing put option, 214
associativity of the product measure, 327
augmented sigma-algebra, 327
generated
by a class of subsets, 327
autoregressive process, 111
Ball, C. A., 96, 115, 139, 144, 159
Ball-Torous model, 96, 139, 144
Baxter, M., 90
Bensoussan, A., 160, 162, 194
Billingsley, P., 6, 50, 317, 326, 329, 331, 332, 338, 339, 341, 343, 346, 348, 350, 353, 355, 359-361, 365, 366, 368
Black, F., 163, 194, 197, 247, 248
Black-Scholes formula, 122, 215
put option, 225
Black-Scholes model, 13, 197
admissible trading strategies, 147, 154
cash-or-nothing call option, 177, 188, 208
discounted stock price process, 70
doubling strategy, 148
price of risk, 143
pricing and replication, 122
risk-adjusted probabilities, 79
securities price processes, 13, 55, 74, 118
self-financing trading strategies, 120
stock price process, 22
Black-Scholes PDE
solution, 205
terminal data
adjusted local convergence in mean, 232
convergence almost everywhere, 235
double convergence, 234
bond price
extended Vasicek model, 292
Ho-Lee model, 306
Merton model, 281
simplified Hull-White model, 303
Vasicek model, 256
Borel sigma-algebra, 329, 336
Brown, J. V., 248
Brownian bridge process, 97
Brownian motion, 5, 10
arithmetic, 10
correlated, 12
generalized, 10
geometric, 12
standard, 5
budget constraint, 130
instantaneous, 119
intertemporal, 119
buy-and-hold strategy, 120
calibration
extended Vasicek model, 298, 300
Ho-Lee model, 307, 308
simplified Hull-White model, 304, 305
Cameron, R. H., 90
Campbell, J. Y., 316
Cantor’s set, 333
Carverhill, A., 316
cash-or-nothing call option, 178, 188, 200, 201, 208
cash-or-nothing put option, 210
Cauchy-Schwartz inequality, 368
Chan, K. C., 257, 316
Cheng, S. T., 116, 140, 142
Chung, K. L., 31, 46, 49-51, 54
Churchill, R. V., 247, 248
complement of a set, 318
complete markets theorem, 165, 168, 172, 173, 192
complete measure space, 327
conditional Bayes’ rule, 366
conditional covariance function, 92
conditional expectation, 365
conditional Fubini theorem, 366
conditional integrability, 238
conditional mean function, 92
conditionally Gaussian process, 92
conditional covariance function, 92
conditional mean function, 92
one-factor, 103
conductivity, 370
Constantinides, G. M., 163, 315
contingent claim, 166
marketed, 166
path-independent, 201
continuous process, 2
convergence
adjusted local, in mean, 232
almost everywhere, 235
double, 234, 377
in Lp, 369
in p-norm, 369
in probability, 340
convergence almost everywhere, 235, 378
countable set, 319
counting measure, 324
covariance function, 91
Cox, J. C., 162
cumulative distribution function, 341, 342
cumulative dividend process, 121
cumulative gains process, 119
cumulative net withdrawal process, 121
de Pinto, I., 247
Debreu, G., 163
decreasing function, 338
delivery price, 220, 250
delta, 177
asset-or-nothing call option, 213
asset-or-nothing put option, 214
cash-or-nothing call option, 181
cash-or-nothing put option, 210
path-independent claim, 206
standard call option, 217
standard put option, 226
delta function, 177
path-independent claim, 206
delta hedging, 168
density, 358
normal distribution, 181, 203, 372
density process, 185
deterministic integrand, 93, 94
DiBenedetto, E., 371, 377, 384, 385, 387
diffusion kernel, 373
discount bond, 249
discrete trading dates, 125
dispersion, 53
percentage, 74
relative, 74
distribution function, 341, 342
dominated convergence theorem, 350
Dothan, M. U., 162, 247
double convergence, 234, 377
doubling strategy, 145, 148
drift, 53
percentage, 74
relative, 74
Duffie, D., 161, 163, 164, 195
Dybvig, P. H., 164
dynamic hedging, 168
dynamically complete markets, 172
Dynkin, E., 340
Einstein, A., 50
elastic random walk, 116
elasticity, 207
equivalent measures, 360
essential supremum, 367
eta, 207
standard call option, 221
standard put option, 227
Ethier, S. N., 49, 367
event, 323
exercise price
standard call option, 122, 214
standard put option, 225
expectation, 350
expected value, 350
extended real line, 329
extended real numbers, 329
extended Vasicek model, 291
Fatou’s lemma, 350
Feynman, R. P., 388
Feynman-Kac formula, 388
Feynman-Kac theorem, 388
filtration, 14
augmented, 15
generated by a process, 15
right-continuous, 45
forward contract, 220, 250
forward price, 250
forward rate, 252
extended Vasicek model, 294
Ho-Lee model, 307
Merton model, 284
simplified Hull-White model, 303
Vasicek model, 264
forward rate risk premium, 265
extended Vasicek model, 295
Ho-Lee model, 307
Merton model, 286
simplified Hull-White model, 303
Vasicek model, 265
forward yield, 251
Friedman, A., 49, 50
Fubini’s theorem, 357
fundamental solution, 203, 373, 379
fundamental theorem of calculus, 361
gamma, 206
path-independent claim, 206
standard call option, 218
standard put option, 227
gamma function, 206
Gaussian, 373
Gaussian process, 91, 106
covariance function, 91
mean function, 91
Gaussian random variables, 344
general Hull-White model, 291
Gihman, I. I., 50
Girsanov’s theorem, 78
Girsanov, I. V., 90
Green’s function, 373
growth condition, 203, 236, 373, 378
polynomial, 203, 236
Holder inequality, 368
Hansen, L., 163
Harrison, J. M., 50, 89, 125, 126, 160-164, 194, 248
He, H., 163
heat equation, 370
diffusion kernel, 373
fundamental solution, 203, 373, 379
Gaussian, 373
Green’s function, 373
initial data
convergence almost everywhere, 235, 378
double convergence, 377
local convergence in mean, 231, 377
martingale solution, 372, 376
propagator, 373
solution, 230, 370
source function, 373
Heath, D., 90, 271, 287, 297
hedging strategy, 168
history of the world, 318
Ho, T., 316
Ho-Lee model, 291, 305
Huang, C., 161, 162, 164
Hull, J. C., 316
identification of processes, 26
increasing function, 338
increment correlation coefficient, 12
increment covariance matrix, 11, 21
increment mean, 11, 21
independent processes, 4
indicator function, 334
indistinguishable processes, 3
induced sigma-algebra, 336
infimum, 347
information structure, 14
inherited sigma-algebra, 336
initial data
convergence almost everywhere, 235, 378
double convergence, 377
local convergence in mean, 231, 377
instantaneous budget constraint, 119
instantaneous covariance matrix, 43, 53
instantaneous foreign exchange risk, 153
instantaneous forward rate, 252
integrability, 348
above, 348
below, 348
conditional, 238
local, 362
with respect to p, 238, 383
integrable function, 348
integrable process, 16
integral, 348
integration by parts, 70
interest rate, 130
intertemporal budget constraint, 119
Ito, K., 89
Ito process, 52
positive, 73
uniqueness of coefficients, 60
Ito’s formula
heuristic multiplication rules, 65
logarithmic transformation, 57
product, 67
ratio, 69
Taylor expansion interpretation, 67
time-dependent, 59
Ito’s lemma, 54
iterated logarithm, 7
Jagannathan, R., 163
Jamshidian, F., 316
Jarrow, R. A., 61, 90, 160, 161, 271, 287, 297
Jensen’s inequality, 274, 353
Kac, M., 388
kappa, 222
Karatzas, I., 7, 8, 21, 44, 49-51, 78, 160, 162, 164, 195, 247, 377, 387, 388
Karolyi, G. A., 257
Kreps, D. M., 125, 162, 163, 248
Kurtz, T. G., 49, 367
lambda, 207
Langevin equation, 116
law of large numbers for Brownian motion, 8
law of the iterated logarithm, 7
Lebesgue integral, 348
Lebesgue measurable sets, 332
Lebesgue measure, 331
Lee, S., 316
left-continuous process, 2
Lehoczky, J., 162, 164
Liapounov inequality, 368
likelihood process, 185
limit inferior, 349
Liptser, R. S., 21, 31, 36-38, 42, 43, 45, 49, 50, 77
Lo, A. W., 316
local convergence in mean, 231, 377
local integrability, 362
locally integrable function, 362
lognormal distribution, 84
truncated, 84
Longstaff, F. A., 257
lower bound, 347
MacKinlay, A. C., 316
Madan, D., 61, 90, 160, 161
Markovian process, 106
Martin, W. T., 90
martingale, 16
martingale method, 166
martingale representation theorem, 44
martingale solution, 372, 376
martingale value, 166
maturity
forward contract, 250
zero-coupon bond, 249
McKean, H. P., 89
mean, 350
mean function, 91
mean reversion, 108
measurable
mapping, 334, 336
process, 15
set, 319
space, 319
measure, 323
measure space, 323
complete, 327
completion of, 328
sigma-finite, 326
Merton, R. C., 160, 162, 248, 316
Merton model, 280
modification, 3
money market account, 13, 131
monotone convergence theorem, 349
Morton, A., 90, 271, 287, 297
Mueller, S. M., 61, 161
Nielsen, L. T., 247
norm
Lp-norm, 367
Novikov condition, 77
null set, 327
numeraire, 121
open set, 330
optional process, 45
optional sigma-algebra, 45
Ornstein, L. S., 116
Ornstein-Uhlenbeck process, 108
Paley, R. E., 90
partition, 320
Pearson, N. D., 163
percentage dispersion, 74, 137
percentage drift, 74, 137
percentage standard deviation, 74
Pliska, S., 89, 125, 126, 160, 161, 163, 164, 194
Pratt, J. W., 297
predictable process, 45
predictable sigma-algebra, 45
previsible process, 44
price process, 118
normalized, 121
prices of risk, 130
minimal, 134, 136
pricing kernel, 130
probability distribution, 339
probability measure, 323
probability space, 323
process, 2
adapted, 15
autoregressive, 111
conditionally Gaussian, 92
one-factor, 103
continuous, 2
Gaussian, 91, 106
integrable, 16
Ito, 52
left-continuous, 2
Markovian, 106
measurable, 15
optional, 45
Ornstein-Uhlenbeck, 108
predictable, 45
progressive, 45
progressively measurable, 45
right-continuous, 2
simple, 32
square integrable, 20
stationary, 113
stochastic integral, 36
time integral, 31
Wiener, 17
processes
almost everywhere identical, 3, 26
identification, 26
independent, 4
indistinguishable, 3, 26
modifications, 3, 26
stochastically equivalent, 3, 26
versions, 3, 26
product measure, 326
associativity, 327
product sigma-algebra, 321
progressive process, 45
progressive sigma-algebra, 45
progressively measurable process, 45
propagator, 373
pull-to-par, 258, 293
put-call parity
asset-or-nothing options, 214
cash-or-nothing options, 210
standard options, 225
Radon-Nikodym theorem, 360
random variable, 339
random vector, 339
rate of appreciation
instantaneous expectation, 153
rate of return
instantaneous dispersion, 137
instantaneous expectation, 137
rectangle, 328
relative dispersion, 74, 137
relative drift, 74, 137
relative standard deviation, 74
replicating trading strategy, 166, 175
replication, 166, 175
rho
standard call option, 224
standard put option, 228
right-continuous process, 2
risk-adjusted dynamics as primitives, 261
risk-adjusted probability measure, 185
risk-adjusted Wiener process, 187
Rogers, L. C. G., 44, 116, 316
Royden, H. L., 332
sample matrix, 91
sample path, 2
sample vector, 91
Samuelson, P. A., 50
Sanders, A., 257
Scholes, M., 163, 194, 197, 247, 248
self-financing arbitrage strategy, 145
self-financing trading strategy, 119
semi-martingale, 44
sequence, 319
Shiryayev, A. N., 21, 31, 36-38, 42, 43, 45, 49, 50, 77
Shreve, S. E., 7, 8, 21, 44, 49-51, 78, 162, 164, 247, 377, 387, 388
sigma-algebra, 317
augmented, 327
generated
by a class of subsets, 319
by a mapping, 335
by a set of mappings, 335
induced, 336
inherited, 336
optional, 45
predictable, 45
progressive, 45
sigma-field, 317
simple function, 347
simple process, 32
simple trading strategy, 126
simplified Hull-White model, 291, 301
Skorohod, A. V., 50
solution, 230, 370
of the Black-Scholes PDE, 205
conditionally integrable, 239
non-negative, 239, 241
of the heat equation, 230, 370
integrable with respect to p, 383
non-negative, 383
source function, 373
speed of adjustment, 108
square integrable process, 20
squared volatility, 74
stable class of sets, 359
standard call option, 122, 214
standard put option, 225
state of the world, 318
state price process, 130, 182
as a primitive, 182
minimal, 134, 136
stationary process, 113
stochastic exponential, 73
stochastic independence
of events, 342
of random vectors, 343
stochastic integral, 36, 61
deterministic integrand, 93, 94
simple process, 34
stochastic process, 2
stochastically equivalent processes, 3
Strauss, W. A., 373
subsequence, 340
suicide strategy, 164
supermartingale, 76
supremum, 347
symmetric difference, 327
Taylor, M. E., 388
term structure of interest rates, 250
terminal data
adjusted local convergence in mean, 232
convergence almost everywhere, 235
double convergence, 234
theta, 206
path-independent claim, 206
standard call option, 223
standard put option, 228
theta function, 206
time integral process, 31
Tonelli’s theorem, 354
Torous, W. N., 96, 115, 139, 144, 159
trading strategy, 119
admissible, 146
almost simple, 126
arbitrage, 145
buy-and-hold, 120
hedging, 168
replicating, 166, 175
self-financing, 119
simple, 126
trigger price
asset-or-nothing call option, 211
asset-or-nothing put option, 214
cash-or-nothing call option, 178
cash-or-nothing put option, 210
Tychonoff uniqueness theorem, 387, 388
Tychonoff, A. N., 387
Uhlenbeck, G. E., 116
unit of account, 152
upper bound, 347
value function, 204
value process, 119, 166
path-independent, 201
Vasicek, O., 116, 142, 315
Vasicek model, 253
vega, 221
standard call option, 221
standard put option, 227
version, 3
volatility, 56, 74
White, A., 316
Widder uniqueness theorem, 387
Widder, D. V., 377, 387, 388
Wiener process, 17
correlated, 22
generalized, 21
standard, 21
Wiener, N., 50, 90
Williams, D., 44
Williams, R. J., 31, 46, 49-51, 54
with probability one, 339
yield, 250
extended Vasicek model, 298
Ho-Lee model, 308
Merton model, 287
simplified Hull-White model, 304
Vasicek model, 272
yield curve, 250
yield risk premium, 273
extended Vasicek model, 299
Ho-Lee model, 308
Merton model, 289
simplified Hull-White model, 304
Vasicek model, 273
Yushkevich, A., 340
Zame, W., 161
zero-coupon bond, 96, 249
yield, 250
Zygmund, A., 90