The Resource Convex duality and financial mathematics, Peter Carr, Qiji Jim Zhu
Convex duality and financial mathematics, Peter Carr, Qiji Jim Zhu
Resource Information
The item Convex duality and financial mathematics, Peter Carr, Qiji Jim Zhu represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in University of Liverpool.This item is available to borrow from 1 library branch.
Resource Information
The item Convex duality and financial mathematics, Peter Carr, Qiji Jim Zhu represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in University of Liverpool.
This item is available to borrow from 1 library branch.
 Summary
 This book provides a concise introduction to convex duality in financial mathematics. Convex duality plays an essential role in dealing with financial problems and involves maximizing concave utility functions and minimizing convex risk measures. Recently, convex and generalized convex dualities have shown to be crucial in the process of the dynamic hedging of contingent claims. Common underlying principles and connections between different perspectives are developed; results are illustrated through graphs and explained heuristically. This book can be used as a reference and is aimed toward graduate students, researchers and practitioners in mathematics, finance, economics, and optimization. Topics include: Markowitz portfolio theory, growth portfolio theory, fundamental theorem of asset pricing emphasizing the duality between utility optimization and pricing by martingale measures, risk measures and its dual representation, hedging and superhedging and its relationship with linear programming duality and the duality relationship in dynamic hedging of contingent claims.
 Language
 eng
 Extent
 1 online resource.
 Contents

 Intro; Preface; Contents; 1 Convex Duality; 1.1 Convex Sets and Functions; 1.1.1 Definitions; 1.1.2 Convex Programming; 1.2 Subdifferential and Lagrange Multiplier; 1.2.1 Definition; 1.2.2 Nonemptiness of Subdifferential; 1.2.3 Calculus; 1.2.4 Role in Convex Programming; 1.3 Fenchel Conjugate; 1.3.1 The Fenchel Conjugate; 1.3.2 The FenchelYoung Inequality; 1.3.3 Graphic Illustration and Generalizations; 1.4 Convex Duality Theory; 1.4.1 Rockafellar Duality; 1.4.2 Fenchel Duality; 1.4.3 Lagrange Duality; 1.4.4 Generalized FenchelYoung Inequality; Multidimensional FenchelYoung Inequality
 1.5 Generalized Convexity, Conjugacy and DualityRockafellar Duality; Lagrange Duality; 2 Financial Models in One Period Economy; 2.1 Portfolio; 2.1.1 Markowitz Portfolio; 2.1.2 Capital Asset Pricing Model; 2.1.3 Sharpe Ratio; 2.2 Utility Functions; 2.2.1 Utility Functions; 2.2.2 Measuring Risk Aversion; 2.2.3 Growth Optimal Portfolio Theory; 2.2.4 Efficiency Index; 2.3 Fundamental Theorem of Asset Pricing; 2.3.1 Fundamental Theorem of Asset Pricing; 2.3.2 Pricing Contingent Claims; 2.3.3 Complete Market; 2.3.4 Use Linear Programming Duality; 2.4 Risk Measures; 2.4.1 Coherent Risk Measure
 2.4.2 Equivalent Characterization of Coherent Risk MeasuresDual Representation; Coherent Acceptance Cone; Coherent Preference; Valuation Bounds and Price System; 2.4.3 Good Deal; 2.4.4 Several Commonly Used Risk Measures; Standard Deviation; Drawdown; Value at Risk; Conditional Value at Risk; Estimating CVaR; 3 Finite Period Financial Models; 3.1 The Model; 3.1.1 An Example; 3.1.2 A General Model; 3.2 Arbitrage and Admissible Trading Strategies; 3.3 Fundamental Theorem of Asset Pricing; 3.3.1 Fundamental Theorem of Asset Pricing
 3.3.2 Relationship Between Dual of Portfolio Utility Maximization, Lagrange Multiplier and MartingaleMeasure3.3.3 Pricing Contingent Claims; 3.3.4 Complete Market; 3.4 Hedging and Super Hedging; 3.4.1 Super and Subhedging Bounds; 3.4.2 Towards a Complete Market; 3.4.3 Incomplete Market Arise from Complete Markets; 3.5 Conic Finance; 3.5.1 Modeling Financial Markets with an AskBid Spread; 3.5.2 Characterization of No Arbitrage by Utility Optimization; 3.5.3 Dual Characterization of No Arbitrage; 3.5.4 Pricing and Hedging; 4 Continuous Financial Models; 4.1 Continuous Stochastic Processes
 4.1.1 Brownian Motion and Martingale4.1.2 The Itô Formula; Itô Processes; The Multidimensional Itô Formula; Martingale Representation; Dual Itô Formula; 4.1.3 Girsanov Theorem; 4.2 Bachelier and BlackScholes Formulae; 4.2.1 Pricing Contingent Claims; Bachelier Formula; BlackScholes Formula; 4.2.2 Convexity; 4.2.3 Duality; 4.3 Duality and Delta Hedging; 4.3.1 Delta Hedging; 4.3.2 Duality; 4.3.3 Time Reversal; 4.4 Generalized Duality and Hedging with Contingent Claims; 4.4.1 Preservation of Generalized Convexity in the Value Function of a Contingent Claim; Consistency of Generalized Convexity
 Isbn
 9783319924915
 Label
 Convex duality and financial mathematics
 Title
 Convex duality and financial mathematics
 Statement of responsibility
 Peter Carr, Qiji Jim Zhu
 Language
 eng
 Summary
 This book provides a concise introduction to convex duality in financial mathematics. Convex duality plays an essential role in dealing with financial problems and involves maximizing concave utility functions and minimizing convex risk measures. Recently, convex and generalized convex dualities have shown to be crucial in the process of the dynamic hedging of contingent claims. Common underlying principles and connections between different perspectives are developed; results are illustrated through graphs and explained heuristically. This book can be used as a reference and is aimed toward graduate students, researchers and practitioners in mathematics, finance, economics, and optimization. Topics include: Markowitz portfolio theory, growth portfolio theory, fundamental theorem of asset pricing emphasizing the duality between utility optimization and pricing by martingale measures, risk measures and its dual representation, hedging and superhedging and its relationship with linear programming duality and the duality relationship in dynamic hedging of contingent claims.
 Assigning source
 Provided by publisher
 Cataloging source
 N$T
 http://library.link/vocab/creatorDate
 1958
 http://library.link/vocab/creatorName
 Carr, Peter
 Dewey number
 650.01/51
 Index
 index present
 LC call number
 HF5691
 Literary form
 non fiction
 Nature of contents

 dictionaries
 bibliography
 http://library.link/vocab/relatedWorkOrContributorName
 Zhu, Qiji J.
 Series statement
 SpringerBriefs in mathematics,
 http://library.link/vocab/subjectName

 Business mathematics
 Convex functions
 Duality theory (Mathematics)
 Label
 Convex duality and financial mathematics, Peter Carr, Qiji Jim Zhu
 Antecedent source
 unknown
 Bibliography note
 Includes bibliographical references and index
 Carrier category
 online resource
 Carrier category code

 cr
 Carrier MARC source
 rdacarrier
 Color
 multicolored
 Content category
 text
 Content type code

 txt
 Content type MARC source
 rdacontent
 Contents

 Intro; Preface; Contents; 1 Convex Duality; 1.1 Convex Sets and Functions; 1.1.1 Definitions; 1.1.2 Convex Programming; 1.2 Subdifferential and Lagrange Multiplier; 1.2.1 Definition; 1.2.2 Nonemptiness of Subdifferential; 1.2.3 Calculus; 1.2.4 Role in Convex Programming; 1.3 Fenchel Conjugate; 1.3.1 The Fenchel Conjugate; 1.3.2 The FenchelYoung Inequality; 1.3.3 Graphic Illustration and Generalizations; 1.4 Convex Duality Theory; 1.4.1 Rockafellar Duality; 1.4.2 Fenchel Duality; 1.4.3 Lagrange Duality; 1.4.4 Generalized FenchelYoung Inequality; Multidimensional FenchelYoung Inequality
 1.5 Generalized Convexity, Conjugacy and DualityRockafellar Duality; Lagrange Duality; 2 Financial Models in One Period Economy; 2.1 Portfolio; 2.1.1 Markowitz Portfolio; 2.1.2 Capital Asset Pricing Model; 2.1.3 Sharpe Ratio; 2.2 Utility Functions; 2.2.1 Utility Functions; 2.2.2 Measuring Risk Aversion; 2.2.3 Growth Optimal Portfolio Theory; 2.2.4 Efficiency Index; 2.3 Fundamental Theorem of Asset Pricing; 2.3.1 Fundamental Theorem of Asset Pricing; 2.3.2 Pricing Contingent Claims; 2.3.3 Complete Market; 2.3.4 Use Linear Programming Duality; 2.4 Risk Measures; 2.4.1 Coherent Risk Measure
 2.4.2 Equivalent Characterization of Coherent Risk MeasuresDual Representation; Coherent Acceptance Cone; Coherent Preference; Valuation Bounds and Price System; 2.4.3 Good Deal; 2.4.4 Several Commonly Used Risk Measures; Standard Deviation; Drawdown; Value at Risk; Conditional Value at Risk; Estimating CVaR; 3 Finite Period Financial Models; 3.1 The Model; 3.1.1 An Example; 3.1.2 A General Model; 3.2 Arbitrage and Admissible Trading Strategies; 3.3 Fundamental Theorem of Asset Pricing; 3.3.1 Fundamental Theorem of Asset Pricing
 3.3.2 Relationship Between Dual of Portfolio Utility Maximization, Lagrange Multiplier and MartingaleMeasure3.3.3 Pricing Contingent Claims; 3.3.4 Complete Market; 3.4 Hedging and Super Hedging; 3.4.1 Super and Subhedging Bounds; 3.4.2 Towards a Complete Market; 3.4.3 Incomplete Market Arise from Complete Markets; 3.5 Conic Finance; 3.5.1 Modeling Financial Markets with an AskBid Spread; 3.5.2 Characterization of No Arbitrage by Utility Optimization; 3.5.3 Dual Characterization of No Arbitrage; 3.5.4 Pricing and Hedging; 4 Continuous Financial Models; 4.1 Continuous Stochastic Processes
 4.1.1 Brownian Motion and Martingale4.1.2 The Itô Formula; Itô Processes; The Multidimensional Itô Formula; Martingale Representation; Dual Itô Formula; 4.1.3 Girsanov Theorem; 4.2 Bachelier and BlackScholes Formulae; 4.2.1 Pricing Contingent Claims; Bachelier Formula; BlackScholes Formula; 4.2.2 Convexity; 4.2.3 Duality; 4.3 Duality and Delta Hedging; 4.3.1 Delta Hedging; 4.3.2 Duality; 4.3.3 Time Reversal; 4.4 Generalized Duality and Hedging with Contingent Claims; 4.4.1 Preservation of Generalized Convexity in the Value Function of a Contingent Claim; Consistency of Generalized Convexity
 Dimensions
 unknown
 Extent
 1 online resource.
 File format
 unknown
 Form of item
 online
 Isbn
 9783319924915
 Level of compression
 unknown
 Media category
 computer
 Media MARC source
 rdamedia
 Media type code

 c
 Quality assurance targets
 not applicable
 Reformatting quality
 unknown
 Sound
 unknown sound
 Specific material designation
 remote
 System control number

 on1045796474
 (OCoLC)1045796474
 Label
 Convex duality and financial mathematics, Peter Carr, Qiji Jim Zhu
 Antecedent source
 unknown
 Bibliography note
 Includes bibliographical references and index
 Carrier category
 online resource
 Carrier category code

 cr
 Carrier MARC source
 rdacarrier
 Color
 multicolored
 Content category
 text
 Content type code

 txt
 Content type MARC source
 rdacontent
 Contents

 Intro; Preface; Contents; 1 Convex Duality; 1.1 Convex Sets and Functions; 1.1.1 Definitions; 1.1.2 Convex Programming; 1.2 Subdifferential and Lagrange Multiplier; 1.2.1 Definition; 1.2.2 Nonemptiness of Subdifferential; 1.2.3 Calculus; 1.2.4 Role in Convex Programming; 1.3 Fenchel Conjugate; 1.3.1 The Fenchel Conjugate; 1.3.2 The FenchelYoung Inequality; 1.3.3 Graphic Illustration and Generalizations; 1.4 Convex Duality Theory; 1.4.1 Rockafellar Duality; 1.4.2 Fenchel Duality; 1.4.3 Lagrange Duality; 1.4.4 Generalized FenchelYoung Inequality; Multidimensional FenchelYoung Inequality
 1.5 Generalized Convexity, Conjugacy and DualityRockafellar Duality; Lagrange Duality; 2 Financial Models in One Period Economy; 2.1 Portfolio; 2.1.1 Markowitz Portfolio; 2.1.2 Capital Asset Pricing Model; 2.1.3 Sharpe Ratio; 2.2 Utility Functions; 2.2.1 Utility Functions; 2.2.2 Measuring Risk Aversion; 2.2.3 Growth Optimal Portfolio Theory; 2.2.4 Efficiency Index; 2.3 Fundamental Theorem of Asset Pricing; 2.3.1 Fundamental Theorem of Asset Pricing; 2.3.2 Pricing Contingent Claims; 2.3.3 Complete Market; 2.3.4 Use Linear Programming Duality; 2.4 Risk Measures; 2.4.1 Coherent Risk Measure
 2.4.2 Equivalent Characterization of Coherent Risk MeasuresDual Representation; Coherent Acceptance Cone; Coherent Preference; Valuation Bounds and Price System; 2.4.3 Good Deal; 2.4.4 Several Commonly Used Risk Measures; Standard Deviation; Drawdown; Value at Risk; Conditional Value at Risk; Estimating CVaR; 3 Finite Period Financial Models; 3.1 The Model; 3.1.1 An Example; 3.1.2 A General Model; 3.2 Arbitrage and Admissible Trading Strategies; 3.3 Fundamental Theorem of Asset Pricing; 3.3.1 Fundamental Theorem of Asset Pricing
 3.3.2 Relationship Between Dual of Portfolio Utility Maximization, Lagrange Multiplier and MartingaleMeasure3.3.3 Pricing Contingent Claims; 3.3.4 Complete Market; 3.4 Hedging and Super Hedging; 3.4.1 Super and Subhedging Bounds; 3.4.2 Towards a Complete Market; 3.4.3 Incomplete Market Arise from Complete Markets; 3.5 Conic Finance; 3.5.1 Modeling Financial Markets with an AskBid Spread; 3.5.2 Characterization of No Arbitrage by Utility Optimization; 3.5.3 Dual Characterization of No Arbitrage; 3.5.4 Pricing and Hedging; 4 Continuous Financial Models; 4.1 Continuous Stochastic Processes
 4.1.1 Brownian Motion and Martingale4.1.2 The Itô Formula; Itô Processes; The Multidimensional Itô Formula; Martingale Representation; Dual Itô Formula; 4.1.3 Girsanov Theorem; 4.2 Bachelier and BlackScholes Formulae; 4.2.1 Pricing Contingent Claims; Bachelier Formula; BlackScholes Formula; 4.2.2 Convexity; 4.2.3 Duality; 4.3 Duality and Delta Hedging; 4.3.1 Delta Hedging; 4.3.2 Duality; 4.3.3 Time Reversal; 4.4 Generalized Duality and Hedging with Contingent Claims; 4.4.1 Preservation of Generalized Convexity in the Value Function of a Contingent Claim; Consistency of Generalized Convexity
 Dimensions
 unknown
 Extent
 1 online resource.
 File format
 unknown
 Form of item
 online
 Isbn
 9783319924915
 Level of compression
 unknown
 Media category
 computer
 Media MARC source
 rdamedia
 Media type code

 c
 Quality assurance targets
 not applicable
 Reformatting quality
 unknown
 Sound
 unknown sound
 Specific material designation
 remote
 System control number

 on1045796474
 (OCoLC)1045796474
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