Projection markovienne de processus stochastiques

tel-00766235, version 1 - 17 Dec 2012
2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.4.1 Semimartingales driven by a Markov process . . . . . . 50
2.4.2 Time changed Lévy processes . . . . . . . . . . . . . . 61
3 Forward equations for option prices 67
3.1 Forward PIDEs for call options . . . . . . . . . . . . . . . . . 69
3.1.1 General formulation of the forward equation . . . . . . 69
3.1.2 Derivation of the forward equation . . . . . . . . . . . 73
3.1.3 Uniqueness of solutions of the forward PIDE . . . . . . 78
3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.2.1 Itô processes . . . . . . . . . . . . . . . . . . . . . . . . 89
3.2.2 Markovian jump-diffusion models . . . . . . . . . . . . 90
3.2.3 Pure jump processes . . . . . . . . . . . . . . . . . . . 92
3.2.4 Time changed Lévy processes . . . . . . . . . . . . . . 94
3.2.5 Index options in a multivariate jump-diffusion model . 96
3.2.6 Forward equations for CDO pricing . . . . . . . . . . . 103
4 Short-time asymptotics for marginals of semimartingales 110
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.2 Short time asymptotics for conditional expectations . . . . . . 113
4.2.1 Main result . . . . . . . . . . . . . . . . . . . . . . . . 113
4.2.2 Some consequences and examples . . . . . . . . . . . . 120
4.3 Short-maturity asymptotics for call options . . . . . . . . . . . 125
4.3.1 Out-of-the money call options . . . . . . . . . . . . . . 126
4.3.2 At-the-money call options . . . . . . . . . . . . . . . . 130
5 Application to index options in a jump-diffusion model 146
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.2 Short maturity asymptotics for index options . . . . . . . . . . 148
5.3 Example : the multivariate Merton model . . . . . . . . . . . 153
5.3.1 The two-dimensional case . . . . . . . . . . . . . . . . 156
5.3.2 The general case . . . . . . . . . . . . . . . . . . . . . 159
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