## Financial Markets in Continuous TimeIn modern financial practice, asset prices are modelled by means of stochastic processes, and continuous-time stochastic calculus thus plays a central role in financial modelling. This approach has its roots in the foundational work of the Nobel laureates Black, Scholes and Merton. Asset prices are further assumed to be rationalizable, that is, determined by equality of demand and supply on some market. This approach has its roots in the foundational work on General Equilibrium of the Nobel laureates Arrow and Debreu and in the work of McKenzie. This book has four parts. The first brings together a number of results from discrete-time models. The second develops stochastic continuous-time models for the valuation of financial assets (the Black-Scholes formula and its extensions), for optimal portfolio and consumption choice, and for obtaining the yield curve and pricing interest rate products. The third part recalls some concepts and results of general equilibrium theory, and applies this in financial markets. The last part is more advanced and tackles market incompleteness and the valuation of exotic options in a complete market. |

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### Contents

I | 1 |

IV | 2 |

V | 4 |

VI | 5 |

VII | 6 |

VIII | 8 |

IX | 11 |

X | 12 |

LXXXIII | 172 |

LXXXIV | 176 |

LXXXV | 178 |

LXXXVII | 180 |

LXXXVIII | 183 |

LXXXIX | 184 |

XC | 189 |

XCI | 190 |

XI | 18 |

XIII | 19 |

XIV | 22 |

XV | 23 |

XVI | 28 |

XVII | 29 |

XVIII | 31 |

XIX | 35 |

XX | 41 |

XXI | 42 |

XXII | 43 |

XXIII | 44 |

XXIV | 47 |

XXV | 51 |

XXVI | 52 |

XXVII | 53 |

XXVIII | 54 |

XXIX | 55 |

XXX | 57 |

XXXI | 58 |

XXXII | 62 |

XXXIII | 66 |

XXXIV | 71 |

XXXV | 79 |

XXXVII | 80 |

XXXVIII | 82 |

XXXIX | 83 |

XL | 85 |

XLI | 86 |

XLII | 87 |

XLIII | 88 |

XLIV | 90 |

XLV | 92 |

XLVI | 93 |

XLVII | 94 |

XLVIII | 98 |

XLIX | 99 |

L | 101 |

LI | 105 |

LIII | 106 |

LIV | 107 |

LV | 109 |

LVI | 110 |

LVII | 111 |

LVIII | 112 |

LIX | 125 |

LXI | 128 |

LXIII | 129 |

LXIV | 134 |

LXV | 135 |

LXVI | 139 |

LXVII | 140 |

LXVIII | 142 |

LXIX | 143 |

LXX | 145 |

LXXI | 146 |

LXXII | 147 |

LXXIII | 148 |

LXXIV | 149 |

LXXV | 157 |

LXXVII | 162 |

LXXIX | 164 |

LXXX | 169 |

LXXXII | 170 |

XCII | 192 |

XCIII | 194 |

XCV | 195 |

XCVI | 198 |

XCVII | 199 |

XCVIII | 201 |

XCIX | 203 |

C | 205 |

CI | 208 |

CII | 217 |

CIV | 218 |

CV | 219 |

CVI | 220 |

CVII | 222 |

CVIII | 224 |

CX | 226 |

CXI | 228 |

CXIII | 230 |

CXIV | 231 |

CXV | 232 |

CXVI | 237 |

CXIX | 239 |

CXX | 240 |

CXXI | 242 |

CXXII | 243 |

CXXIV | 245 |

CXXVII | 248 |

CXXVIII | 249 |

CXXIX | 250 |

CXXXI | 251 |

CXXXII | 252 |

CXXXV | 254 |

CXXXVI | 255 |

CXXXVII | 256 |

CXL | 257 |

CXLII | 260 |

CXLIV | 262 |

CXLV | 264 |

CXLVI | 267 |

CXLVII | 268 |

CXLVIII | 270 |

CL | 271 |

CLI | 273 |

CLII | 274 |

CLIV | 277 |

CLV | 278 |

CLVI | 281 |

CLVII | 283 |

CLVIII | 285 |

CLIX | 286 |

CLX | 287 |

CLXI | 288 |

CLXII | 289 |

CLXIII | 290 |

CLXVI | 291 |

CLXVII | 292 |

CLXIX | 293 |

CLXXI | 294 |

CLXXII | 295 |

CLXXIV | 297 |

CLXXV | 299 |

321 | |

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### Common terms and phrases

agent arbitrage opportunities Arrow-Debreu asset prices assume Black-Scholes formula Brownian motion calculations Chap coefficients conditional expectation constraints contingent claim continuous deduce defined Definition denotes density deterministic discounted prices distribution dynamics Economic equal equivalent euro exists final wealth financial markets finite follows Gaussian Girsanov's theorem given hedging Hence HJB equation incomplete markets initial value interest rate investor Ito process Ito's lemma Karatzas Karoui market is complete martingale measure Mathematical Finance maturity measure Q method notation numeraire obtain optimal consumption optimal solution option payoff portfolio probability measure problem Proof Proposition Q-martingale Radner equilibrium random variable respect risk risk-neutral measure risk-neutral probability riskless asset satisfies Sect self-financing strategy spot rate stochastic differential equation stochastic integral strictly positive suppose theory unique utility function valuation variance vector volatility zero coupon bond