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 Fenchel-Young 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 Fenchel-Young Inequality; Multidimensional Fenchel-Young 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 Sub-hedging 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 Ask-Bid 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 Black-Scholes Formulae; 4.2.1 Pricing Contingent Claims; Bachelier Formula; Black-Scholes 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