Stochastic methods in finance : lectures given at the C.I.M.E.-E.M.S. Summer School held in Bressanone/Brixen, Italy, July 6-12, 2003

This volume includes the five lecture courses given at the CIME-EMS School on "Stochastic Methods in Finance" held in Bressanone/Brixen, Italy 2003. It deals with innovative methods, mainly from stochastic analysis, that play a fundamental role in the mathematical modelling of finance and insurance: the theory of stochastic processes, optimal and stochastic control, stochastic differential equations, convex analysis and duality theory. Five topics are treated in detail: Utility maximization in incomplete markets; the theory of nonlinear expectations and its relationship with the theory of risk measures in a dynamic setting; credit risk modelling; the interplay between finance and insurance; incomplete information in the context of economic equilibrium and insider trading.

Kerry BackTomasz R. Bielecki and Monique Jeanblanc and Marek RutkowskiChristian HippShige PengWalter Schachermayer

Incomplete and Asymmetric Information in Asset Pricing Theory	p. 1
1 Filtering Theory	p. 1
1.1 Kalman-Bucy Filter	p. 3
1.2 Two-State Markov Chain	p. 4
2 Incomplete Information	p. 5
2.1 Seminal Work	p. 5
2.2 Markov Chain Models of Production Economies	p. 6
2.3 Markov Chain Models of Pure Exchange Economies	p. 7
2.4 Heterogeneous Beliefs	p. 11
3 Asymmetric Information	p. 12
3.1 Anticipative Information	p. 12
3.2 Rational Expectations Models	p. 13
3.3 Kyle Model	p. 16
3.4 Continuous-Time Kyle Model	p. 18
3.5 Multiple Informed Traders in the Kyle Model	p. 20
References	p. 23
Modeling and Valuation of Credit Risk	p. 27
1 Introduction	p. 27
2 Structural Approach	p. 29
2.1 Basic Assumptions	p. 29
Defaultable Claims	p. 29
Risk-Neutral Valuation Formula	p. 31
Defaultable Zero-Coupon Bond	p. 32
2.2 Classic Structural Models	p. 34
Merlon's Model	p. 34
Black and Cox Model	p. 37
2.3 Stochastic Interest Rates	p. 43
2.4 Credit Spreads: A Case Study	p. 45
2.5 Comments on Structural Models	p. 46
3 Intensity-Based Approach	p. 47
3.1 Hazard Function	p. 47
Hazard Function of a Random Time	p. 48
Associated Martingales	p. 49
Change of a Probability Measure	p. 50
Martingale Hazard Function	p. 53
Defaultable Bonds: Deterministic Intensity	p. 53
3.2 Hazard Processes	p. 55
Hazard Process of a Random Time	p. 56
Valuation of Defaultable Claims	p. 57
Alternative Recovery Rules	p. 59
Defaultable Bonds: Stochastic Intensity	p. 63
Martingale Hazard Process	p. 64
Martingale Hypothesis	p. 65
Canonical Construction	p. 67
Kusuoka's Counter-Example	p. 69
Change of a Probability	p. 70
Statistical Probability	p. 72
Change of a Numeraire	p. 74
Preprice ofa Defaultable Claim	p. 77
Credit Default Swaption	p. 79
A Practical Example	p. 82
3.3 Martingale Approach	p. 84
Standing Assumptions	p. 85
Valuation of Defaultable Claims	p. 85
Martingale Approach under (H.1)	p. 87
3.4 Further Developments	p. 88
Default-Adjusted Martingale Measure	p. 88
Hybrid Models	p. 89
Unified Approach	p. 90
3.5 Comments on Intensity-Based Models	p. 90
4 Dependent Defaults and Credit Migrations	p. 91
4.1 Basket Credit Derivatives	p. 92
The i th -to-Default Contingent Claims	p. 92
Case of Two Entities	p. 93
4.2 Conditionally Independent Defaults	p. 94
Canonical Construction	p. 94
Independent Default Times	p. 95
Signed Intensities	p. 96
Valuation of FDC and LDC	p. 96
General Valuation Formula	p. 97
Default Swap of Basket Type	p. 98
4.3 Copula-Based Approaches	p. 99
Direct Application	p. 100
Indirect Application	p. 100
Simplified Version	p. 102
4.4 Jarrow and Yu Model	p. 103
Construction and Properties of the Model	p. 103
Bond Valuation	p. 105
4.5 Extension of the Jarrow and Yu Model	p. 106
Kusuoka's Construction	p. 107
Interpretation of Intensities	p. 108
Bond Valuation	p. 108
4.6 Dependent Intensities of Credit Migrations	p. 109
Extension of Kusuoka's Construction	p. 109
4.7 Dynamics of Dependent Credit Ratings	p. 112
4.8 Defaultable Term Structure	p. 113
Standing Assumptions	p. 113
Credit Migration Process	p. 116
Defaultable Term Structure	p. 117
Premia for Interest Rate and Credit Event Risks	p. 119
Defaultable Coupon Bond	p. 120
Examples of Credit Derivatives	p. 121
4.9 Concluding Remarks	p. 122
References	p. 123
Stochastic Control with Application in Insurance	p. 127
1 Preface	p. 127
2 Introduction Into Insurance Risk	p. 128
2.1 The Lundberg Risk Model	p. 128
2.2 Alternatives	p. 129
2.3 Ruin Probability	p. 129
2.4 Asymptotic Behavior For Ruin Probabilities	p. 131
3 Possible Control Variables and Stochastic Control	p. 132
3.1 Possible Control Variables	p. 132
Investment, One Risky Asset	p. 132
Investment Two or More Risky Assets	p. 133
Proportional Reinsurance	p. 134
Unlimited XL Reinsurance	p. 134
XL-Reinsurance	p. 135
Premium Control	p. 135
Control of New Business	p. 135
3.2 Stochastic Control	p. 136
Objective Functions	p. 136
Infinitesimal Generators	p. 137
Hamilton-Jacobi-Bellman Equations	p. 139
Verification Argument	p. 141
Steps for Solution	p. 143
4 Optimal Investment for Insurers	p. 143
4.1 HJB and its Handy Form	p. 143
4.2 Existence of a Solution	p. 145
4.3 Exponential Claim Sizes	p. 145
4.4 Two or More Risky Assets	p. 147
5 Optimal Reinsurance and Optimal New Business	p. 148
5.1 Optimal Proportional Reinsurance	p. 150
5.2 Optimal Unlimited XL Reinsurance	p. 151
5.3 Optimal XL Reinsurance	p. 152
5.4 Optimal New Business	p. 153
6 Asymptotic Behavior for Value Function and Strategies	p. 154
6.1 Optimal Investment: Exponential Claims	p. 154
6.2 Optimal Investment: Small Claims	p. 154
6.3 Optimal Investment: Large Claims	p. 155
6.4 Optimal Reinsurance	p. 156
7 A Control Problem with Constraint: Dividends and Ruin	p. 157
7.1 A Simple Insurance Model with Dividend Payments	p. 157
7.2 Modified HJB Equation	p. 158
7.3 Numerical Example and Conjectures	p. 159
7.4 Earlier and Further Work	p. 161
8 Conclusions	p. 162
References	p. 163
Nonlinear Expectations, Nonlinear Evaluations and Risk Measures	p. 165
1 Introduction	p. 165
1.1 Searching the Mechanism of Evaluations of Risky Assets	p. 165
1.2 Axiomatic Assumptions for Evaluations of Derivatives	p. 166
General Situations: {{\cal F}}_t^X -Consistent Nonlinear Evaluations	p. 166
{{\cal F}}_t^X -Consistent Nonlinear Expectations	p. 167
1.3 Organization of the Lecture	p. 168
2 Brownian Filtration Consistent Evaluations and Expectations	p. 169
2.1 Main Notations and Definitions	p. 169
2.2 {{\cal F}}_t -Consistent Nonlinear Expectations	p. 171
2.3 {{\cal F}}_t -Consistent Nonlinear Evaluations	p. 173
3 Backward Stochastic Differential Equations: g-Evaluations and g-Expectations	p. 176
3.1 BSDE: Existence, Uniqueness and Basic Estimates	p. 176
3.2 1-Dimensional BSDE	p. 182
Comparison Theorem	p. 183
Backward Stochastic Monotone Semigroups and g-Evaluations	p. 186
Example: Black-Scholes Evaluations	p. 188
Expectations	p. 189
Upcrossing Inequality of ¿ g -Supermartingales and Optional Sampling Inequality	p. 193
3.3 A Monotonie Limit Theorem of BSDE	p. 199
3.4 g-Martingales and (Nonlinear) g-Supermartingale Decomposition Theorem	p. 201
4 Finding the Mechanism: Is an {{\cal F}} -Expectation a g-Expectation?	p. 204
4.1 ¿ ¿ -Dominated {{\cal F}} -Expectations	p. 204
4.2 {{\cal F}}_t -Consistent Martingales	p. 207
4.3 BSDE under {{\cal F}}_t -Consistent Nonlinear Expectations	p. 210
4.4 Decomposition Theorem for \cal E}} -Supermartingales	p. 213
4.5 Representation Theorem of an {{\cal F}} -Expectation by a g-Expectation	p. 216
4.6 How to Test and Find g?	p. 219
4.7 A General Situation: {{\cal F}}_t -Evaluation Representation Theorem	p. 220
5 Dynamic Risk Measures	p. 221
6 Numerical Solution of BSDEs: Euler's Approximation	p. 222
7 Appendix	p. 224
7.1 Martingale Representation Theorem	p. 224
7.2 A Monotonic Limit Theorem of Itô's Processes	p. 226
7.3 Optional Stopping Theorem for {{\cal E}}^g -Supermartingale	p. 232
References	p. 238
References on BSDE and Nonlinear Expectations	p. 240
Utility Maximisation in Incomplete Markets	p. 255
1 Problem Setting	p. 255
2 Models on Finite Probability Spaces	p. 259
2.1 Utility Maximization	p. 266
The complete Case (Arrow)	p. 266
The Incomplete Case	p. 272
3 The General Case	p. 277
3.1 The Reasonable Asymptotic Elasticity Condition	p. 277
3.2 Existence Theorems	p. 281
References	p. 289

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