## Hedging DerivativesValuation and hedging of financial derivatives are intrinsically linked concepts. Choosing appropriate hedging techniques depends on both the type of derivative and assumptions placed on the underlying stochastic process. This volume provides a systematic treatment of hedging in incomplete markets. Mean-variance hedging under the risk-neutral measure is applied in the framework of exponential L(r)vy processes and for derivatives written on defaultable assets. It is discussed how to complete markets based upon stochastic volatility models via trading in both stocks and vanilla options. Exponential utility indifference pricing is explored via a duality with entropy minimization. Backward stochastic differential equations offer an alternative approach and are moreover applied to study markets with trading constraints including basis risk. A range of optimal martingale measures are discussed including the entropy, Esscher and minimal martingale measures. Quasi-symmetry properties of stochastic processes are deployed in the semi-static hedging of barrier options. This book is directed towards both graduate students and researchers in mathematical finance, and will also provide an orientation to applied mathematicians, financial economists and practitioners wishing to explore recent progress in this field." |

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### Contents

1 Introduction | 1 |

2 Stochastic Calculus | 11 |

3 Arbitrage and Completeness | 35 |

4 Asset Price Models | 57 |

5 Static Hedging | 85 |

6 MeanVariance Hedging | 103 |

7 Entropic Valuation and Hedging | 133 |

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### Common terms and phrases

admissible strategy arbitrage arbitrage opportunities asset price assume barrier options Brownian motion BSDE complete markets condition convex defined Definition denoted density process duality dynamics entropy martingale measure entropy measure equation equivalent martingale measure Esscher measure exists exp(X exponential utility finite variation follows function hedging strategy hence KW decomposition Lemma Lévy measure Lévy process local martingale Mathematical Finance mean-variance hedging measure Q minimal entropy martingale Moreover optimal martingale measure option prices payoff Poisson process portfolio price process probability measure Proof Proposition Protter Q-martingale random variable replicating respect result Rheinländer risk risk-neutral measure Schweizer self-dual semi-martingale Shiryaev ſº solution square-integrable stochastic exponential stochastic integral Stochastic Processes stochastic volatility Theorem trading unique utility indifference price value process vanilla options zero