## Option Pricing in Incomplete Markets: Modeling Based on Geometric Levy Processes and Minimal Entropy Martingale MeasuresThis volume offers the reader practical methods to compute the option prices in the incomplete asset markets. The [GLP & MEMM] pricing models are clearly introduced, and the properties of these models are discussed in great detail. It is shown that the geometric L(r)vy process (GLP) is a typical example of the incomplete market, and that the MEMM (minimal entropy martingale measure) is an extremely powerful pricing measure. This volume also presents the calibration procedure of the [GLP \& MEMM] model that has been widely used in the application of practical problem |

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### Contents

1 Basic Concepts in Mathematical Finance | 1 |

2 Levy Processes and Geometric Levy Process Models | 7 |

3 Equivalent Martingale Measures | 21 |

4 EsscherTransformed Martingale Measures | 29 |

5 Minimax Martingale Measures and Minimal Distance Martingale Measures | 41 |

6 Minimal Distance Martingale Measures for Geometric Levy Processes | 47 |

7 The GLP MEMM Pricing Model | 75 |

8 Calibration and Fitness Analysis of the GLP MEMM Model | 99 |

9 The GSP MEMM Pricing Model | 111 |

10 The MultiDimensional GLP MEMM Pricing Model | 121 |

Appendix A Estimation | 141 |

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181 | |

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### Common terms and phrases

assume CGMY process Chapter characteristic function compound Poisson process corresponding deﬁned Deﬁnition denoted distance function distance martingale measure entropy martingale measure equivalent martingale measure Esscher martingale measure Esscher-transformed martingale measure ESSMM European call options exponential utility ﬁrst following condition following equation following form following theorem geometric Lévy process geometric stable process given GLP & MEMM implied volatility Incomplete Markets inﬁnitely divisible distribution l\/IEMM l\/Iiyahara large number Lévy measure Lévy process models Mathematical Finance measure of Zt MEMM world method of moments minimal distance martingale minimal entropy martingale Miyahara MLqEMM MVEMM Nagoya City University obtain the following option pricing P(ESSMM P(MEMM parameters portfolio pricing model probability measure probability space Problem DP Process & MEMM relative entropy Remark risk process satisﬁes the following Section Set the following solution stable process model Suppose Theorem 6.1 tingale utility function variance gamma process volatility smile VOMM Wiener process