ARBITRAGE, CREDIT AND INFORMATIONAL RISKS

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PEKING UNIVERSITY SERIES IN MATHEMATICS

Series Editors: Kung-Ching Chang, Pingwen Zhang, Bin Liu, and Jiping Zhang (Peking University, China)

Vol. 1: An Introduction to Finsler Geometry by Xiaohuan Mo (Peking University, China)

Vol. 2: Numerical Methods for Exterior Problems by Lung-An Ying (Peking University & Xiamen University, China)

Vol. 3: Approaches to the Qualitative Theory of Ordinary Differential Equations: Dynamical Systems and Nonlinear Oscillations by Tongren Ding (Peking University, China)

Vol. 4: Elliptic, Hyperbolic and Mixed Complex Equations with Parabolic Degeneracy: Including Tricomi–Bers and Tricomi–Frankl–Rassias Problems by Guo Chun Wen (Peking University, China)

Vol. 5: Arbitrage, Credit and Informational Risks edited by Caroline Hillairet (Ecole Polytechnique, France), Monique Jeanblanc (Université d'Evry, France) and Ying Jiao (Université Lyon I, France)

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N E W J E R S E Y • L O N D O N • S I N G A P O R E • B E I J I N G • S H A N G H A I • H O N G K O N G • TA I P E I • C H E N N A I

World Scientific

Peking Universi ty Ser ies in Mathematics — Vol . 5

ARBITRAGE, CREDIT AND INFORMATIONAL RISKS

Caroline HillairetEcole Polytechnique, France

Monique JeanblancUniversité d’Evry, France

Ying JiaoUniversité Lyon I, France

Editors

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Published by

World Scientific Publishing Co. Pte. Ltd.5 Toh Tuck Link, Singapore 596224USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.

Peking University Series in Mathematics — Vol. 5ARBITRAGE, CREDIT AND INFORMATIONAL RISKS

Copyright © 2014 by World Scientific Publishing Co. Pte. Ltd.

All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

The image on the back cover is from the website of Beijing International Center for Mathematical Research (BICMR), http://www.bicmr.org/

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 978-981-4602-06-8

Printed in Singapore

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February 24, 2014 9:25 9143 - Arbitrage, Credit and Informational Risks book1 page v

Learning without thought is labor lost;

thought without learning is perilous

Confucius

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May 2, 2013 14:6 BC: 8831 - Probability and Statistical Theory PST˙ws

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February 21, 2014 11:56 9143 - Arbitrage, Credit and Informational Risks book1 page vii

Preface

This book includes some contributions presented at the workshop “Ar-

bitrage, Credit and Informational Risks”. This event has been held at

Beijing International Center for Mathematical Research (BICMR) in June

2013 in the framework of the Sino-French Research Program in Mathemat-

ics (SFRPM). It was dedicated to the presentation of research results and

discussions in the field of financial mathematics around the themes of arbi-

trage, credit and asymmetric information risks. The book is split into three

parts.

In the first part, we collected papers in which two different notions of

arbitrages and related concepts are studied. The first one is the well-known

condition of No Free Lunch with Vanishing Risk (NFLVR); it is equivalent

to the existence of an equivalent local martingale measure under which dis-

counted prices are local martingales. The second one is the condition of No

Unbounded Profit with Bounded Risk (NUPBR), which is also known as

No Arbitrage of the First Kind; it is equivalent to the existence of a pos-

itive local martingale deflator such that prices multiplied by this deflator

are local martingales. This part contains the four following papers:

Claudio Fontana studies the stability of NFLVR and NUPBR, as well

as other arbitrage conditions, under an absolutely continuous change of

probability measure.

Johannes Ruf and Wolfgang J. Runggaldier present a systematic con-

struction of market models that satisfy NUPBR, but not NFLVR, by stating

sufficient assumptions on the first hitting time of zero by the inverse of the

martingale deflator.

In the context of jump-diffusion market models, Jacopo Mancin

and Wolfgang J. Runggaldier construct examples of models that satisfy

NUPBR, but not NFLVR.

Working in a progressive enlargement setting, and assuming that

NFLVR and NUPBR hold in the reference filtration, Anna Aksamit, Tahir

vii

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viii Preface

Choulli, Jun Deng and Monique Jeanblanc give conditions so that NUPBR

is satisfied in the enlarged filtration, whereas NFLVR fails to hold.

The second part contains four papers devoted to Credit Risk:

Sebastien Hitier and Ying Zhu study the problem of pricing credit

derivatives in a structural model. In structural models, the default time is

the hitting time of a given barrier by the firm value. Structural models,

although based on sound economic arguments, are difficult to handle for

calibration in practice. The authors demonstrate how a structural model

can be calibrated and used for risk-neutral pricing of credit derivatives.

Stephane Crepey proposes and studies a dynamics model for bilateral

counterparty risk on credit derivatives, which goes beyond the classic im-

mersion setting.

Shiqi Song develops mathematical background for a dynamic model of

a single default, based on the fairly general construction of the conditional

law of the default event from its Azema supermartingale. He establishes

a new formula for the semimartingale decomposition of martingales in the

reference filtration and gives results related to the existence of the condi-

tional density of the default time.

Laurence Carassus and Simone Scotti apply the error calculus method-

ology to the problem of an optimal credit allocation under a hidden regime

switching model.

The third part presents four contributions in the area of Control Prob-

lems and Information Risks:

Ivan Guo and Marek Rutkowski consider a class of recursive multi-

player stopping games in a discrete time setting. Such games have interpre-

tations in economic and financial modelling, for example, as multi-person

game options. They prove the existence of an optimal equilibrium and pro-

vide an explicit algorithm for the computation of the value of the game.

Monique Jeanblanc and Anthony Reveillac present examples of Back-

ward Stochastic Differential Equations (BSDEs) whose driver is not inte-

grable and degenerates when the terminal time approaches. Such BSDEs

may have, depending on the terminal conditions, no solution or an infinite

number of solutions.

Caroline Hillairet studies a portfolio optimization problem in a market

model characterized by the presence of different prices for the same asset,

which arise as a consequence of different information settings.

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February 21, 2014 11:56 9143 - Arbitrage, Credit and Informational Risks book1 page ix

Preface ix

In the presence of ‘shadow costs’ of incomplete information, as well

as stochastic volatility and jumps in the dynamics of the asset, Sana Mah-

foudh and Monique Pontier compare the corresponding cost value process

of two different hedging strategies.

Acknowledgement: The workshop “Arbitrage, Credit and Informa-

tional Risks” took place at Beijing International Center for Mathematical

Research in June 2013 in the framework of Sino-French Research Program

in Mathematics. We are grateful for the financial support from National

Natural Science Foundation of China,

and several Chinese and French research institutes such as BICMR, Fonda-

tion Mathematique Jacques Hadamard, Fondation Sciences Mathematiques

de Paris, Institut Fourier at Grenoble and Laboratoire de Probabilite et

Modeles Aleatoires at Paris.

During the preparation of this volume, Marek Rutkowski provided us

with valuable help and many suggestions, we would like to warmly thank

him. We also benefited a lot from the support of the World Scientific Press,

notably from the editors Lai Fun Kwong, Dena Li, Hongbing Shi and Ji

Zhang.

Last but not least, we thank all the participants of the workshop for

their contributions, making this event a great success:

Anna Aksamit, Universite d’Evry Val d’Essonne,

Stefan Ankirchner, Universitat Bonn,

Giuseppe Benedetti, Universite Paris Dauphine,

Christophette Blanchet-Scalliet, Ecole Centrale de Lyon,

Lijun Bo, Xidian University,

Giorgia Callegaro, Universita degli Studi di Padova,

Luciano Campi, Universite Paris Nord,

Laurence Carassus, Universite de Reims Champagne Ardenne,

Sebastien Choukroun, Universite Paris Diderot - Paris 7,

Stephane Crepey, Universite d’Evry Val d’Essonne,

Albina Danilova, London School of Economics,

Cristina Di Girolami, Universita G. D’Annunzio di Pescara,

Nicole El Karoui, Universite Pierre et Marie Curie - Paris 6,

Laure Elie, Universite Paris Diderot - Paris 7,

Claudio Fontana, Inria Paris - Rocquencourt,

Noufel Frikha, Universite Paris-Diderot -Paris 7,

Zorana Grbac, Universitat Berlin,

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February 21, 2014 11:56 9143 - Arbitrage, Credit and Informational Risks book1 page x

x Preface

Caroline Hillairet, Ecole Polytechnique,

Sebastien Hitier, BNP Paribas Hong Kong,

Chau Ngoc Huy, Universite Paris Diderot - Paris 7,

Monique Jeanblanc, Universite d’Evry Val d’Essonne,

Ying Jiao, Universite Paris Diderot - Paris 7,

Wanmo Kang, Korea Advanced Institute of Science and Technol-

ogy,

Celine Labart, Universite de Savoie,

Sophie Laruelle, Ecole Centrale Paris,

Jerome Lelong, Ecole Nationale Superieure d’Informatique et

Mathematiques Appliquees de Grenoble,

Marta Leniec, Uppsala Universitet,

Libo Li, Ritsumeikan University,

Shanqiu Li, Universite Paris Diderot - Paris 7,

Chunhua Ma, Nankai University,

Gilles Pages, Universite Pierre et Marie Curie - Paris 6,

Xianhua Peng, Hong Kong University of Science and Technology,

Monique Pontier, Universite Paul Sabatier - Toulouse 3,

Johannes Ruf, University of Oxford,

Wolfgang Runggaldier, Universita degli Studi di Padova,

Marek Rutkowski, University of Sydney,

Abass Sagna, Ecole Nationale Superieure d’Informatique pour

l’Industrie et l’Entreprise,

Simone Scotti, Universite Paris Diderot - Paris 7,

Shiqi Song, Universite d’Evry Val d’Essonne,

Laura Vinckenbosch, INRIA Nancy,

Duo Wang, Peking University,

Lan Wu, Peking University,

Jianming Xia, Chinese Academy of Science,

Dewen Xiong, Shanghai Jiao Tong University,

Jingping Yang, Peking University,

Kai Zhang, Jilin University.

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February 21, 2014 11:56 9143 - Arbitrage, Credit and Informational Risks book1 page xi

Contents

Preface vii

Arbitrage 1

No-arbitrage Conditions and Absolutely Continuous Changes

of Measure 3

Claudio Fontana

A Systematic Approach to Constructing Market Models with

Arbitrage 19

Johannes Ruf, Wolfgang J. Runggaldier

On the Existence of Martingale Measures in Jump Diffusion

Market Models 29

Jacopo Mancin, Wolfgang J. Runggaldier

Arbitrages in a Progressive Enlargement Setting 53

Anna Aksamit, Tahir Choulli, Jun Deng, Monique Jeanblanc

Credit Risk 87

Pricing Credit Derivatives with a Structural Default Model 89

Sebastien Hitier, Ying Zhu

xi

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February 13, 2014 15:23 9143 - Arbitrage, Credit and Informational Risks book1 page xii

xii Contents

Reduced-Form Modeling of Counterparty Risk on

Credit Derivatives 103

Stephane Crepey

Dynamic One-default Model 119

Shiqi Song

Stochastic Sensitivity Study for Optimal Credit Allocation 147

Laurence Carassus, Simone Scotti

Control Problem and Information Risks 169

Discrete-Time Multi-Player Stopping and Quitting

Games with Redistribution of Payoffs 171

Ivan Guo, Marek Rutkowski

A Note on BSDEs with Singular Driver Coefficients 207

Monique Jeanblanc, Anthony Reveillac

A Portfolio Optimization Problem with Two Prices

Generated by Two Information Flows 225

Caroline Hillairet

Option Pricing under Stochastic Volatility, Jumps and Cost

of Information 241

Sana Mahfoudh, Monique Pontier Arb

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