## 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

Contents | 1 |

Dynamic Models in Discrete Time 43 | 42 |

The BlackScholes Formula | 81 |

Portfolios Optimizing Wealth and Consumption 127 | 126 |

The Yield Curve | 159 |

Equilibrium of Financial Markets in Discrete Time | 191 |

Equilibrium of Financial Markets in Continuous Time | 216 |

Incomplete Markets | 237 |

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agent arbitrage opportunities asset prices assume assumption barrier options Black–Scholes Brownian motion calculations Chap coeﬁicients conditional expectation constraints contingent claim continuous converges Cvitanic deﬁned Deﬁnition denotes density discounted prices distribution dynamic equal equivalent euro exists financial markets finite ﬁrst follows formula Gaussian hedging Hence incomplete markets initial value interest rate It6 process Karatzas Karoui lemma market is complete martingale measure Mathematical Finance maturity measure Q methods notation obtain Option pricing pair payoff portfolio probability measure problem Proof Proposition Radner equilibrium random variable result risk risk-neutral measure risk-neutral probability riskless asset satisﬁes satisfying scheme self-ﬁnancing sequence simulate solution spot rate stochastic differential equation stochastic integral strategy strictly positive suppose unique utility functions valuation vector volatility wealth zero coupon bond