| Foreword |
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xiii | |
Part
1. Facts. Models |
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1 | (380) |
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Chapter I. Main Concepts, Structures, and Instruments. Aims and Problems of Financial Theory and Financial Engineering |
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2 | (78) |
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1. Financial structures and instruments |
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3 | (32) |
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1a. Key objects and structures |
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3 | (3) |
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6 | (14) |
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1c. Market of derivatives, Financial instruments |
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20 | (15) |
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2. Financial markets under uncertainty. Classical theories of the dynamics of financial indexes, their critics and revision. Neoclassical theories |
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35 | (34) |
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2a. Random walk conjecture and concept of efficient market |
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37 | (9) |
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2b. Investment portfolio. Markowitz's diversification |
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46 | (5) |
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2c. CAPM: Capital Asset Pricing Model |
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51 | (5) |
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2d. APT: Arbitrage Pricing Theory |
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56 | (4) |
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2e. Analysis, interpretation, and revision of the classical concepts of efficient market. I |
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60 | (5) |
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2f. Analysis, interpretation, and revision of the classical concepts of efficient market. II |
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65 | (4) |
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3. Aims and problems of financial theory, engineering, and actuarial calculations |
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69 | (11) |
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3a. Role of financial theory and financial engineering. Financial risks |
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69 | (2) |
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3b. Insurance: a social mechanism of compensation for financial losses |
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71 | (6) |
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3c. A classical example of actuarial calculations: the Lundberg Cramer theorem |
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77 | (3) |
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Chapter II. Stochastic Models. Discrete Time |
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80 | (108) |
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1. Necessary probabilistic concepts and several models of the dynamics of market prices |
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81 | (36) |
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1a. Uncertainty and irregularity in the behavior of prices. Their description and representation in probabilistic terms |
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81 | (8) |
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1b. Doob decomposition. Canonical representations |
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89 | (6) |
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1c. Local martingales. Martingale transformations. Generalized martingales |
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95 | (8) |
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1d. Gaussian and conditionally Gaussian models |
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103 | (6) |
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1e. Binomial model of price evolution |
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109 | (3) |
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1f. Models with discrete intervention of chance |
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112 | (5) |
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2. Linear stochastic models |
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117 | (35) |
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2a. Moving average model MA(q) |
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119 | (6) |
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2b. Autoregressive model AR(p) |
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125 | (13) |
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2c. Autoregressive and moving average model ARMA(p, q) and integrated model ARIMA(p, d, q) |
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138 | (4) |
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2d. Prediction in linear models |
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142 | (10) |
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3. Nonlinear stochastic conditionally Gaussian models |
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152 | (24) |
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3a. ARCH and GARCH models |
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153 | (10) |
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3b. EGARCH, TGARCH, HARCH, and other models |
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163 | (5) |
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3c. Stochastic volatility models |
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168 | (8) |
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4. Supplement: dynamical chaos models |
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176 | (12) |
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4a. Nonlinear chaotic models |
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176 | (7) |
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4b. Distinguishing between `chaotic' and `stochastic' sequences |
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183 | (5) |
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Chapter III. Stochastic Models. Continuous Time |
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188 | (126) |
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1. Non-Gaussian models of distributions and processes |
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189 | (32) |
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1a. Stable and infinitely divisible distributions |
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189 | (11) |
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200 | (7) |
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207 | (7) |
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1d. Hyperbolic distributions and processes |
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214 | (7) |
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2. Models with self-similarity. Fractality |
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221 | (15) |
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2a. Hurst's statistical phenomenon of self-similarity |
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221 | (3) |
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2b. A digression on fractal geometry |
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224 | (2) |
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2c. Statistical self-similarity. Fractal Brownian motion |
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226 | (6) |
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2d. Fractional Gaussian noise: a process with strong aftereffect |
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232 | (4) |
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3. Models based on a Brownian motion |
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236 | (42) |
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3a. Brownian motion and its role of a basic process |
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236 | (4) |
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3b. Brownian motion: a compendium of classical results |
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240 | (11) |
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3c. Stochastic integration with respect to a Brownian motion |
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251 | (6) |
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3d. Ito processes and Ito's formula |
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257 | (7) |
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3e. Stochastic differential equations |
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264 | (7) |
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3f. Forward and backward Kolmogorov's equations. Probabilistic representation of solutions |
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271 | (7) |
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4. Diffusion models of the evolution of interest rates, stock and bond prices |
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278 | (16) |
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4a. Stochastic interest rates |
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278 | (6) |
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4b. Standard diffusion model of stock prices (geometric Brownian motion) and its generalizations |
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284 | (5) |
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4c. Diffusion models of the term structure of prices in a family of bonds |
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289 | (5) |
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294 | (20) |
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5a. Semimartingales and stochastic integrals |
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294 | (7) |
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5b. Doob-Meyer decomposition. Compensators. Quadratic variation |
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301 | (6) |
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5c. Ito's formula for semimartingales. Generalizations |
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307 | (7) |
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Chapter IV. Statistical Analysis of Financial Data |
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314 | (67) |
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1. Empirical data. Probabilistic and statistical models of their description. Statistics of `ticks' |
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315 | (12) |
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1a. Structural changes in financial data gathering and analysis |
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315 | (3) |
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1b. Geography-related features of the statistical data on exchange rates |
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318 | (3) |
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1c. Description of financial indexes as stochastic processes with discrete intervention of chance |
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321 | (3) |
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1d. On the statistics of `ticks' |
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324 | (3) |
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2. Statistics of one-dimensional distributions |
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327 | (18) |
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2a. Statistical data discretizing |
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327 | (2) |
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2b. One-dimensional distributions of the logarithms of relative price changes. Deviation from the Gaussian property and leptokurtosis of empirical densities |
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329 | (5) |
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2c. One-dimensional distributions of the logarithms of relative price changes. `Heavy tails' and their statistics |
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334 | (6) |
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2d. One-dimensional distributions of the logarithms of relative price changes. Structure of the central parts of distributions |
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340 | (5) |
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3. Statistics of volatility, correlation dependence, and aftereffect in prices |
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345 | (22) |
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3a. Volatility. Definition and examples |
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345 | (6) |
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3b. Periodicity and fractal structure of volatility in exchange rates |
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351 | (3) |
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3c. Correlation properties |
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354 | (4) |
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3d. `Devolatization'. Operational time |
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358 | (6) |
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3e. `Cluster' phenomenon and aftereffect in prices |
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364 | (3) |
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4. Statistical R/S-analysis |
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367 | (14) |
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4a. Sources and methods of R/S-analysis |
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367 | (9) |
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4b. R/S-analysis of some financial time series |
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376 | (5) |
Part
2. Theory |
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381 | (422) |
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Chapter V. Theory of Arbitrage in Stochastic Financial Models. Discrete Time |
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382 | (120) |
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1. Investment portfolio on a (B, S)-market |
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383 | (27) |
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1a. Strategies satisfying balance conditions |
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383 | (12) |
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1b. Notion of `hedging'. Upper and lower prices. Complete and incomplete markets |
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395 | (4) |
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1c. Upper and lower prices in a single-step model |
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399 | (9) |
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1d. CRR-model: an example of a complete market |
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408 | (2) |
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410 | (23) |
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2a. `Arbitrage' and `absence of arbitrage' |
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410 | (3) |
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2b. Martingale criterion of the absence of arbitrage. First fundamental theorem |
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413 | (4) |
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2c. Martingale criterion of the absence of arbitrage. Proof of sufficiency |
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417 | (1) |
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2d. Martingale criterion of the absence of arbitrage. Proof of necessity (by means of the Esscher conditional transformation) |
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417 | (7) |
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2e. Extended version of the First fundamental theorem |
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424 | (9) |
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3. Construction of martingale measures by means of an absolutely continuous change of measure |
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433 | (48) |
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3a. Main definitions. Density process |
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433 | (6) |
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3b. Discrete version of Girsanov's theorem. Conditionally Gaussian case |
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439 | (7) |
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3c. Martingale property of the prices in the case of a conditionally Gaussian and logarithmically conditionally Gaussian distributions |
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446 | (4) |
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3d. Discrete version of Girsanov's theorem. General case |
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450 | (9) |
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3e. Integer-valued random measures and their compensators. Transformation of compensators under absolutely continuous changes of measures. `Stochastic integrals' |
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459 | (8) |
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3f. `Predictable' criteria of arbitrage-free (B, S)-markets |
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467 | (14) |
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4. Complete and perfect arbitrage-free markets |
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481 | (21) |
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4a. Martingale criterion of a complete market. Statement of the Second fundamental theorem. Proof of necessity |
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481 | (2) |
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4b. Representability of local martingales. `S-representability' |
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483 | (2) |
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4c. Representability of local martingales (`Mu-representability' and `(Mu-Nu)-representability') |
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485 | (3) |
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4d. `S-representability' in the binomial CRR-model |
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488 | (3) |
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4e. Martingale criterion of a complete market. Proof of necessity for d = 1 |
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491 | (6) |
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4f. Extended version of the Second fundamental theorem |
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497 | (5) |
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Chapter VI. Theory of Pricing in Stochastic Financial Models. Discrete Time |
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502 | (130) |
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1. European hedge pricing on arbitrage-free markets |
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503 | (22) |
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1a. Risks and their reduction |
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503 | (2) |
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1b. Main hedge pricing formula. Complete markets |
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505 | (7) |
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1c. Main hedge pricing formula. Incomplete markets |
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512 | (6) |
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1d. Hedge pricing on the basis of the mean square criterion |
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518 | (3) |
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1e. Forward contracts and futures contracts |
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521 | (4) |
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2. American hedge pricing on arbitrage-free markets |
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525 | (28) |
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2a. Optimal stopping problems. Supermartingale characterization |
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525 | (10) |
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2b. Complete and incomplete markets. Supermartingale characterization of hedging prices |
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535 | (3) |
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2c. Complete and incomplete markets. Main formulas for hedging prices |
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538 | (8) |
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2d. Optional decomposition |
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546 | (7) |
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3. Scheme of series of `large' arbitrage-free markets and asymptotic arbitrage |
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553 | (35) |
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3a. One model of `large' financial markets |
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553 | (2) |
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3b. Criteria of the absence of asymptotic arbitrage |
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555 | (4) |
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3c. Asymptotic arbitrage and contiguity |
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559 | (16) |
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3d. Some issues of approximation and convergence in the scheme of series of arbitrage-free markets |
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575 | (13) |
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4. European options on a binomial (B, S)-market |
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588 | (20) |
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4a. Problems of option pricing |
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588 | (2) |
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4b. Rational pricing and hedging strategies. Pay-off function of the general form |
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590 | (5) |
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4c. Rational pricing and hedging strategies. Markovian pay-off functions |
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595 | (3) |
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4d. Standard call and put options |
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598 | (6) |
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4e. Option-based strategies (combinations and spreads) |
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604 | (4) |
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5. American options on a binomial (B, S)-market |
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608 | (24) |
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5a. American option pricing |
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608 | (3) |
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5b. Standard call option pricing |
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611 | (10) |
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5c. Standard put option pricing |
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621 | (4) |
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5d. Options with aftereffect. `Russian option' pricing |
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625 | (7) |
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Chapter VII. Theory of Arbitrage in Stochastic Financial Models. Continuous Time |
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632 | (102) |
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1. Investment portfolio in semimartingale models |
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633 | (16) |
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1a. Admissible strategies. Self-financing. Stochastic vector integral |
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633 | (10) |
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1b. Discounting processes |
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643 | (3) |
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1c. Admissible strategies. Some special classes |
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646 | (3) |
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2. Semimartingale models without opportunities for arbitrage. Completeness |
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649 | (13) |
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2a. Concept of absence of arbitrage and its modifications |
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649 | (2) |
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2b. Martingale criteria of the absence of arbitrage. Sufficient conditions |
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651 | (4) |
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2c. Martingale criteria of the absence of arbitrage. Necessary and sufficient conditions (a list of results) |
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655 | (5) |
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2d. Completeness in semimartingale models |
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660 | (2) |
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3. Semimartingale and martingale measures |
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662 | (42) |
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3a. Canonical representation of semimartingales. Random measures. Triplets of predictable characteristics |
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662 | (10) |
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3b. Construction of marginal measures in diffusion models. Girsanov's theorem |
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672 | (11) |
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3c. Construction of martingale measures for Levy processes. Esscher transformation |
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683 | (8) |
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3d. Predictable criteria of the martingale property of prices. I |
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691 | (3) |
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3e. Predictable criteria of the martingale property of prices. II |
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694 | (4) |
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3f. Representability of local martingales (`(H^(c), Mu-Nu)-representability') |
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698 | (3) |
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3g. Girsanov's theorem for semimartingales. Structure of the densities of probabilistic measures |
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701 | (3) |
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4. Arbitrage, completeness, and hedge pricing in diffusion models of stock |
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704 | (13) |
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4a. Arbitrage and conditions of its absence. Completeness |
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704 | (5) |
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4b. Price of hedging in complete markets |
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709 | (3) |
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4c. Fundamental partial differential equation of hedge pricing |
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712 | (5) |
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5. Arbitrage, completeness, and hedge pricing in diffusion models of bonds |
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717 | (17) |
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5a. Models without opportunities for arbitrage |
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717 | (11) |
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728 | (2) |
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5c. Fundamental partial differentai equation of the term structure of bonds |
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730 | (4) |
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Chapter VIII. Theory of Pricing in Stochastic Financial Models. Continuous Time |
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734 | (69) |
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1. European options in diffusion (B, S)-stockmarkets |
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735 | (16) |
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735 | (4) |
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1b. Black-Scholes formula. Martingale inference |
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739 | (6) |
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1c. Black-Scholes formula. Inference based on the solution of the fundamental equation |
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745 | (3) |
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1d. Black-Scholes formula. Case with dividends |
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748 | (3) |
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2. American options in diffusion (B, S)-stockmarkets. Case of an infinite time horizon |
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751 | (27) |
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751 | (12) |
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763 | (2) |
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2c. Combinations of put and call options |
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765 | (2) |
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767 | (11) |
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3. American options in diffusion (B, S)-stockmarkets. Finite time horizons |
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778 | (14) |
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3a. Special features of calculations on finite time intervals |
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778 | (4) |
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3b. Optimal stopping problems and Stephan problems |
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782 | (2) |
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3c. Stephan problem for standard call and put options |
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784 | (4) |
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3d. Relations between the prices of European and American options |
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788 | (4) |
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4. European and American options in a diffusion (B, P)-bondmarket |
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792 | (11) |
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4a. Option pricing in a bondmarket |
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792 | (3) |
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4b. European option pricing in single-factor Gaussian models |
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795 | (4) |
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4c. American option pricing in single-factor Gaussian models |
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799 | (4) |
| Bibliography |
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803 | (22) |
| Index |
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825 | (8) |
| Index of symbols |
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833 | |
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