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Lectures on Mathematical Finance
M. Jeanblanc
City University, HONG KONG
June 2001

Contents
1 Pricing and Hedging 3
1.1 Discrete time . . . . 3
1.1.1 Binomial approach . . . . . . . 3
1.1.2 Two dates, several assets and several states of the world . . . . 5
1.1.3 Multiperiod discrete time model . 6
1.2 Continuous time model 8
1.2.1 The Bachelier model  . . . . . 8
1.2.2 Martingales . 8
1.2.3 Black and Scholes model . . . . 8
1.2.4 PDE approach . 9
1.2.5 Martingale approach . . . . . . 10
1.2.6 Discounted processes  . . . . . 10
1.2.7 Girsanov's theorem . . . . . . . 11
1.2.8 European options 12
1.2.9 Derivative products . . . . . . 13
2 Single jump and Default processes 17
2.1 A toy model . . . . . 17
2.1.1 Payment at Maturity . . . . . . 18
2.1.2 Payment at hit . 19
2.1.3 Risk neutral probability measure, martingales . . 20
2.2 Successive default times 21
2.2.1 Two times . . 21
2.2.2 Copulas . . . 22
2.2.3 More than two times . . . . . . 23
2.3 Elementary martingale . 23
2.3.1 Intensity process 24
2.3.2 Representation theorem . . . . 25
2.3.3 Partial information  . . . . . . 26
2.4 Cox Processes and Extensions . . . . . 26
2.4.1 Construction of Cox Processes with a given stochastic intensity 26
2.4.2 Conditional Expectations . . . 27
2.4.3 Conditional Expectation of F1-Measurable Random Variables 29
2.4.4 Defaultable Zero-Coupon Bond 29
2.4.5 Stochastic boundary . . . . . . 30
2.4.6 Representation theorem . . . . 30
2.4.7 Hedging contingent claims . . . 31
2.5 General Case . . . . 32
2.5.1 Conditional expectation . . . . 32
2.5.2 Ordered Random Times . . . . 33
2.6 In_mum and supremum, general case  35
2.7 Correlated default time 36
3 Optimal portfolio 39
3.1 Discrete time . . . . 39
3.1.1 Two dates, 2 assets, complete case 39
3.1.2 Two dates Model, d + 1 assets 41
3.1.3 Incomplete markets . . . . . 42
3.1.4 Complete case . 43
3.1.5 Multiperiod Discrete time model . 43
3.1.6 Markovitz e_cient portfolio . . 44
3.2 Continuous time models. Maximization of terminal wealth in a complete market. . . 45
3.2.1 A continuous time two assets model . . . . . . . 45
3.2.2 Historical probability . . . . . . 45
3.2.3 The Dynamic programming method . . . . . . . 47
3.3 Consumption and terminal wealth . . 47
3.3.1 The martingale method . . . . 47
3.3.2 The Dynamic programming method . . . . . . . 49
3.3.3 Income . . . 51
4 Portfolio Insurance 55
4.1 Introduction . . . . . 55
4.2 Classical insurance strategies . . . . . 56
4.2.1 Strategic allocation and general framework . . . 56
4.2.2 European versus American guarantee
4.2.3 Stop loss strategy 58
4.2.4 CPPI strategy . 58
4.2.5 OBPI Strategy . 59
4.2.6 Comparison of performances . 61
4.3 OBPI Optimality for a European Guarantee . . . . . . . 61
4.3.1 Choice of the strategic allocation and properties 62
4.3.2 Choice of the tactic allocation . 62
4.3.3 Optimality of the tactic allocation 63
4.4 American case in the Black and Scholes framework . . . 63
4.4.1 American Put Based strategy . 64
4.4.2 Properties of the American put price . . . . . . . 64
4.4.3 An adapted self-_nancing strategy 65
4.4.4 Description of the American Put Based Strategy 66
4.5 American case for general complete markets . . . . . . . 68
4.5.1 Price of an American put . . . 68
4.5.2 Self-_nancing strategy . . . . . 70
4.5.3 Optimality . 70
4.6 Optimality results for general utility functions case . . . 72
4.6.1 European guarantee . . . . . . 72
4.6.2 American guarantee . . . . . . 73
5 Incomplete markets 79
5.1 Discrete time. Example 79
5.1.1 Case of a contingent claim . . . 80
5.1.2 Range of price for a European call . . . . . . .. . . . . 83
5.1.3 Two dates, continuum prices . 84
5.1.4 Bid-ask price 85
5.2 Discrete time, general setting : Bid-ask spread . . . . . .. . . 86
5.3 Continuous time . . 86
5.3.1 Superhedging price . . . . . . . 87
5.3.2 Choice of the model . . . . . . 88
5.3.3 Bounds for stochastic volatility 88
5.3.4 Jump di_usion processes . . . . 89
5.3.5 Transaction costs 91
5.3.6 Variance hedging 91
5.3.7 Remaining risk . 92
5.3.8 Reservation price 92
5.3.9 Davis approach . 92
5.3.10 Minimal entropy 93

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