Table Of Contents:
Preface v

1 Introduction 1(10)

1.1 Hedging in complete markets 2(2)

1.1.1 Black & Scholes analysis and its limitations 2(2)

1.1.2 Complete markets 4(1)

1.2 Hedging in incomplete markets 4(5)

1.2.1 Sources of incompleteness 4(1)

1.2.2 Calibration 5(1)

1.2.3 Mean-variance hedging 6(1)

1.2.4 Utility indifference pricing and hedging 7(1)

1.2.5 Exotic options 8(1)

1.2.6 Optimal martingale measures 9(1)

1.3 Notes and further reading 9(2)

2 Stochastic Calculus 11(24)

2.1 Filtrations and martingales 11(3)

2.2 Semi-martingales and stochastic integrals 14(6)

2.3 Kunita-Watanabe decomposition 20(5)

2.4 Change of measure 25(5)

2.5 Stochastic exponentials 30(4)

2.6 Notes and further reading 34(1)

3 Arbitrage and Completeness 35(22)

3.1 Strategies and arbitrage 35(4)

3.2 Complete markets 39(2)

3.3 Hidden arbitrage and local times 41(5)

3.4 Immediate arbitrage 46(1)

3.5 Super-hedging and the optional decomposition theorem 47(4)

3.6 Arbitrage via a non-equivalent measure change 51(4)

3.7 Notes and further reading 55(2)

4 Asset Price Models 57(28)

4.1 Exponential Levy processes 57(13)

4.1.1 A Levy process primer 57(6)

4.1.2 Examples of Levy processes 63(2)

4.1.3 Construction of Levy processes by subordination 65(2)

4.1.4 Risk-neutral Levy modelling 67(2)

4.1.5 Weak representation property and measure changes 69(1)

4.2 Stochastic volatility models 70(13)

4.2.1 Examples 71(1)

4.2.2 Stochastic differential equations and time change 72(1)

4.2.3 Construction of a solution via coupling 73(2)

4.2.4 Convexity of option prices 75(1)

4.2.5 Market completion by trading in options 76(2)

4.2.6 Bubbles and strict local martingales 78(4)

4.2.7 Stochastic exponentials 82(1)

4.3 Notes and further reading 83(2)

5 Static Hedging 85(18)

5.1 Static hedging of European claims 85(2)

5.2 Duality principle in option pricing 87(5)

5.2.1 Dynamics of the dual process 87(2)

5.2.2 Duality relations 89(3)

5.3 Symmetry and self-dual processes 92(9)

5.3.1 Definitions and general properties 92(2)

5.3.2 Semi-static hedging of barrier options 94(1)

5.3.3 Self-dual exponential Levy processes 95(2)

5.3.4 Self-dual stochastic volatility models 97(4)

5.4 Notes and further reading 101(2)

6 Mean-Variance Hedging 103(30)

6.1 Concept of mean-variance hedging 103(2)

6.2 Valuation and hedging by the Laplace method 105(10)

6.2.1 Bilateral Laplace transforms 105(1)

6.2.2 Valuation and hedging using Laplace transforms 106(9)

6.3 Valuation and hedging via integro-differential equations 115(3)

6.3.1 Feynman-Kac formula for the value function 115(2)

6.3.2 Computation of the optimal hedging strategy 117(1)

6.4 Mean-variance hedging of defaultable assets 118(10)

6.4.1 Intensity-based approach 118(2)

6.4.2 Martingale representation 120(2)

6.4.3 Hedging of insurance claims with longevity bonds 122(6)

6.5 Quadratic risk-minimisation for payment streams 128(2)

6.6 Notes and further reading 130(3)

7 Entropic Valuation and Hedging 133(28)

7.1 Exponential utility indiffence pricing 133(2)

7.2 The minimal entropy martingale measure 135(7)

7.3 Duality results 142(7)

7.4 Properties of the utility indifference price 149(5)

7.5 Entropic hedging 154(5)

7.6 Notes and further reading 159(2)

8 Hedging Constraints 161(34)

8.1 Framework and preliminaries 161(3)

8.2 Dynamic utility indifference pricing 164(1)

8.3 Martingale optimality principle 165(4)

8.4 Utility indifference hedging and pricing using BSDEs 169(7)

8.4.1 Backward stochastic differential equations 169(2)

8.4.2 Maximising utility from terminal wealth under trading constraints 171(5)

8.5 Examples in Brownian markets 176(8)

8.5.1 Complete markets 177(3)

8.5.2 Basis risk 180(4)

8.6 Connection to the minimal entropy measure in the unconstrained case 184(8)

8.7 Notes and further reading 192(3)

9 Optimal Martingale Measures 195(24)

9.1 Esscher measure 195(5)

9.1.1 Geometric case 195(2)

9.1.2 Linear case 197(3)

9.2 Minimal entropy martingale measure 200(10)

9.2.1 Optimal martingale measure equation 200(4)

9.2.2 Exponential Levy case 204(1)

9.2.3 Orthogonal volatility case 205(2)

9.2.4 Continuous SV models 207(3)

9.3 Variance-optimal martingale measure 210(2)

9.4 q-optimal martingale measure 212(1)

9.5 Minimal martingale measure 213(5)

9.6 Notes and further reading 218(1)
Appendix A Notation and Conventions 219(2)
Bibliography 221(10)
Index 231