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Föllmer H., Schied A. — Stochastic finance
Föllmer H., Schied A. — Stochastic finance

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Название: Stochastic finance

Авторы: Föllmer H., Schied A.

Аннотация:

This book is an introduction to financial mathematics for mathematicians. It is intended both for graduate students with a certain background in probability theory as well as for professional mathematicians in industry and academia. In contrast to many textbooks on mathematical finance, only discrete-time stochastic models are considered. This setting has the advantage that the text can concentrate from the beginning on typical problems which are suggested by financial applications. Moreover, certain principles, such as the general incompleteness of realistic market models, become thus more transparent and visible. On the other hand, all models are based on general probability spaces, and so the text captures the interplay between probability theory and functional analysis which is typical for modern mathematical finance.
The first part of the book contains a study of financial investments in a static one-period market model. Here, an investor faces intrinsic risk and uncertainty, which cannot be hedged away. The tools presented to deal with this situation range from the classical theory of expected utility until the more recent development of measures of risk.
In the second part of the book, the idea of dynamic hedging and arbitrage-free pricing of contingent claims is developed in a multi-period framework. Such market models are typically incomplete, and particular focus is given to
methods combining the dynamic hedging of a risky position with the tools of assessing risk and uncertainty as presented in part.
Contents: Mathematical finance in one period: Arbitrage theory. Expected utility. Optimal investments. Measures of risk Dynamic Arbitrage Theory: Dynamic hedging of contingent claims. American contingent claims. Optional decomposition and super-hedging. Efficient hedging in incomplete markets. Minimizing the hedging error. Hedging under constraints References. Index


Язык: en

Рубрика: Экономика и финансы/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2002

Количество страниц: 422

Добавлена в каталог: 17.06.2006

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Предметный указатель
$\lambda$-efficient allocation      149
$\lambda$-quantile      179
$\mathcal{L}^2$-admissible strategy      350
$\mathcal{L}^2$-admissible trading strategy      350
$\mathcal{Q}$-martingale      285
$\mathcal{Q}$-submartingale      285
$\mathcal{Q}$-supermartingale      285
$\mathcal{Q}_{\mathfrak{s}}$-Snell envelope upper      345
$\psi$-weak topology      58 104
$\psi$-weak topology for measures      390
Absolutely continuous measure      379
Absolutely continuous probability measure      379
Acceptance set      160 176 191 205 296 316 347
Acceptance set of convex measure of risk      160
Adapted process      210
Adapted stochastic process      210
Admissible strategy      191 310
Affine hull of a convex set      28
Affine numerical representation      50
Allais paradox      59 91
American call option      259 269
American claim hedging strategy      276
American Contingent claim      258
American option      257
American put option      270
Antisymmetric relation      74
Arbitrage opportunity      5 31 193 213 299
Arbitrage-free price European contingent claim      225
Arbitrage-free price of a European claim      225
Arbitrage-free price of American claim      268 273
Arbitrage-free price of Contingent claim      16
Arbitrage-free price, universal bounds      18 303
Archimedean axiom      51 95
Arrow—Debreu equilibrium      142
Arrow—Debreu equilibrium and interest rate      154
Arrow—Pratt coefficient      66
Asian call option      220
Asian option      220
Asian put option      220
Asymmetric relation      44
Asymmetry of preference relation      44
Atom of a probability space      22 187
Atomless probability space      187
Attainable american claim      276
Attainable Contingent claim      20
Attainable European claim      222
Attainable European contingent claim      222 228
Attainable payoff      10
Average price option      220
Average strike option      220
Average Value at Risk      182
Balayage order      81 297 301
Banach—Alaoglu theorem      397
Barrier option      306
Barycenter of a measure      26
Basket call option      137
Basket option      14
Bermuda call option      259
Bermuda option      259
Bernoulli, Daniel      43 63
Bernoulli, Nicholas      63
Binomial model      235 248 270
Black—Scholes formula      249
Black—Scholes p.d.e.      254
Bolzano—Weierstrass theorem, randomized version      37
Boundedly supported measures      57
Brownian motion      251
Call American option      259
Call option      13 137
Call option with barrier      220 306
Call-put parity      14
CARA utility      67 72 114 120 133 136
CARA utility function      67 72 114 120 133 136
Catastrophe bond      15
Central limit theorem      387
Certainty equivalent      62 96
Certainty equivalent as reservation price      141
Certainty independence      94
Change of numeraire      213 218
Characterization of upper $\mathcal{Q}_{\mathfrak{s}}$-Snell envelope      345
Characterization of upper Snell envelope      292
Choquet boundary      297
Coherent measure of risk      159 296
Coherent measure of risk and superhedging      296
Coherent risk      159
Comonotonicity      94
Complete market      22 231
Complete relation      44
Compound lottery      51
Concave stochastic order      81 109
Conditional expectation      380
Conditional Value at Risk      182
Conjugate function      200 378
Connected topological space      48
Constant absolute risk aversion      67
Contingent claim      15
Continuation region      272
Continuity axiom      51 95
Continuity from above      103 168 176
Continuity from below      95 103 169 199
Continuous from above convex measure of risk      168 176
Continuous from above preference relation      103
Continuous from below convex measure of risk      169 199
Continuous from below preference relation      103
Continuous preference relation      47
Convex function proper      376
Convex hull      376
Convex hull of the support of a measure      27
Convex measure of risk      159
Convex measure of risk in financial market      191 316 346
Convex risk      159
Convex stochastic order      81 297 301
Cost of superhedging      290
Cost process      349
Countably convex set      35
Cox—Ross—Rubinstein model      235 248 270
CRR model      235 248 270
Decreasingly risk averse utility function      72
Delta hedging      239 253
Delta of a call option      253
Density function      379
Derivative securities      15 219
Digital option      303
Dini's lemma      172
Dirac measure      389
Discounted American claim      259
Discounted claim      221
Discounting      9 155
Distribution invariant convex measure of risk      187
Diversification      159
Doleans—Dade exponential      359
Donsker's invariance principle      252
Doob decomposition      257 286 360
Doob decomposition uniform      286 336
Doob's stopping theorem      263 277
Down-and-out option      221
Dual space      396
Duality superhedging      18 291 296 345
Dunford—Pettis theorem      398
Early exercise premium      270
Eberlein—Smulian theorem      397
Efficient hedging of European call option      321
Efficient market hypothesis      216
Ellsberg paradox      92
Entropy of a measure      125
Entropy of measure      125
Entropy-minimizing risk-neutral measure      126 137
Equivalence relation      44
Equivalent martingale measure      6 215
Equivalent measure      379
Equivalent probability measure      379
Esscher transform      126
Essential infimum      385
Essential supremum      385
European call option      219 231 321
European Contingent claim      219
European put option      220 249
Exercise strategy for American claim      258
Expected shortfall      182 205 323
Expiration date of a contingent claim      219
Exponential family      122
Exponential utility function      67 72 114 120 133 136
Extreme point of a convex set      232
Fair premium      60
Fair price      60
Favorable bet      68
Feasible allocation      142
Fenchel—Legendre transform      126 129 184 200 378
Filtered probability space      210
Filtration      210
Financial position      158
Finitely additive set function      95 164 393
First order stochastic dominance      84 110
Forward contract      13 299
Functional central limit theorem      252
Fundamental theorem of asset pricing for multi-period market      217
Fundamental theorem of asset pricing in one period      6
Fundamental theorem of asset pricing with contingent initial data      32
Fundamental theorem of asset pricing without a priori measure      302
Gains process      212 349
Gamma of a call option      253
Gauge function      58 103 389
Generalized likelihood quotient test      384
Generalized Neyman—Pearson      383
Generalized trading strategy      348
Geometric Brownian motion      251
Geometric form fundamental theorem of asset pricing      28
Girsanov formula      361
Global quadratic risk      368
Greeks      255
Hahn—Banach theorem      396
Halmos—Savage theorem      36
HARA utility      67 72 114 137
HARA utility function      114 137
Hardy—Littlewood inequality      87 140
Hausdorff space      47
Hausdorff topological space      47
Hedging strategy for American claim      276
Hyperbolic absolute risk aversion      67
Independence axiom      50 91 94 96
Indifference relation      44
Inequality Hardy-Littlewood      87 140
Insurance      15 65 71
Interest rate      155
Intrinsic value      227
Intrinsic value of European call option      226
Inverse function right-continuous      85
James' theorem      398
Knock-in option      220 307
Knock-out option      220 311
Kolmogorov's law of large numbers      42 71 73 188
Krein—Smulian theorem      397
Kreps—Yan theorem      36 331
Kunita—Watanabe decomposition      356
Law of Large Numbers      71 73
Lebesgue decomposition      381
Leverage effect      25
Lexicographical order      46 47
Lexicographical preference relation      46 47
Likelihood quotient test      383
Liquid option      298
Local martingale      328
Local quadratic risk      350
Local risk process      350
Local submartingale      328
Local supermartingale      328
Locally convex space      395
Locally convex topological vector space      395
Locally risk-minimizing strategy      350
Locally riskless bond      211
Log-normal distribution      82 246
Lookback option      221 244
Loss function      199 316
Lottery      43 50
Lower Snell envelope      279 293
Market clearing condition      142
Market portfolio      141
Markov property      302
Martingale      32 215
Martingale convergence theorem      74
Martingale measure      6 31 215
Martingale measure as extension of a pricing rule      302
Martingale representation property      234
Maturity of a contingent claim      219
Mean self-financing generalized trading strategy      350
Mean self-financing strategy      350
Mean-preserving spread      80 109
Mean-variance trade-off process      353 370
Measure of risk in financial market      191 316 346
Measure of risk, continuous from above      168 176
Measure of risk, continuous from below      169
Metric space      386
Minimal martingale measure      358
Mixture space      55
Model uncertainty      96
Moment generating function      121
Monetary measure of risk      158
Monetary risk      158
Monotone measure of risk      158
Monotone preference relation      90
Monotone preference relation on assets      90
Monotone preference relation on distributions      60
Monotone risk      158
Monotone stochastic order      84 110
Multiplicative central limit theorem      246
Negative transitive relation      44
Negative transitivity of preference relation      44
Neyman—Pearson      312 382 383
Non-redundancy condition      114 327
Normal distribution      78 82 122 125
Normalized measure of risk      158
Normalized risk      158
Numeraire      9 31 211
Numerical representation      45 48
Numerical representation of preference relation      45 48 50 56 58 61
Numerical representation of robust Savage form      283
Numerical representation of Savage form      90 265
Numerical representation of von Neumann-Morgenstern form      50 56 58 61
Optimal continuation region      272
Optimal stopping problem      264
Optimal stopping region      272
Optional decomposition      289
Optional decomposition of upper Snell envelope      290
Optional decomposition theorem      287
Optional decomposition theorem under constraints      336
Optional sampling theorem      263
Order dense set      45
Orthogonal decomposition of a contingent claim      355
Partial order      74
Penalty function      164
Penalty function for convex measure of risk      164 167 176 200 205
Penalty function for measure of risk      164
Penalty function for risk      164
Poisson distribution      18 122
Polish space      386
Portfolio      4
Portfolio insurance      14
Portmanteau theorem      388
Positive affine transformations      51
Positive homogeneous measure of risk      159
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