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E-grāmata: Model-free Hedging: A Martingale Optimal Transport Viewpoint

(Société Générale, Paris, France)
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Model-free Hedging: A Martingale Optimal Transport ViewpointPalielināt
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Model-free Hedging: A Martingale Optimal Transport Viewpoint focuses on the computation of model-independent bounds for exotic options consistent with market prices of liquid instruments such as Vanilla options. The author gives an overview of Martingale Optimal Transport, highlighting the differences between the optimal transport and its martingale counterpart. This topic is then discussed in the context of mathematical finance.
Preface ix
1 Pricing and hedging without tears
1(24)
1.1 An insurance viewpoint
1(4)
1.1.1 Utility preference
2(1)
1.1.2 Quantile approach
3(2)
1.2 A Trader Viewpoint
5(15)
1.2.1 Super-replication: Linear programming
6(3)
1.2.2 Arbitrage-free prices and bounds
9(2)
1.2.3 A Worked-Out Example: The Binomial Model
11(1)
1.2.4 Replication paradigm
12(1)
1.2.5 Geometry of M1: Extremal points
13(1)
1.2.6 Mean-variance: Quadratic programming
14(2)
1.2.7 Utility function: Convex programming
16(1)
1.2.8 Quantile hedging
16(1)
1.2.9 Utility indifference price
17(1)
1.2.10 A Worked-Out Example: The Trinomial Model
18(2)
1.3 A Cautious Trader Viewpoint
20(5)
2 Martingale optimal transport
25(64)
2.1 Optimal transport in a nutshell
25(19)
2.1.1 Trading T-Vanilla options
25(1)
2.1.2 Super-replication and Monge-Kantorovich duality
26(5)
2.1.3 Formulation in Rd+ and multi-dimensional marginals
31(1)
2.1.4 Frechet-Hoeffding solution
31(3)
2.1.5 Brenier's solution
34(3)
2.1.6 Axiomatic construction of marginals: Stieltjes moment problem
37(2)
2.1.7 Some symmetries
39(2)
2.1.8 Robust quantile hedging
41(1)
2.1.9 Multi-marginals and infinitely-many marginals case
42(1)
2.1.10 Link with Hamilton-Jacobi equation
43(1)
2.2 Martingale optimal transport
44(27)
2.2.1 Dual formulation
47(4)
2.2.2 Link with Hamilton-Jacobi-Bellman equation
51(2)
2.2.3 A Discrete Martingale Frechet-Hoeffding Solution
53(5)
2.2.4 OT versus MOT: A summary
58(1)
2.2.5 Martingale Brenier's solution
58(2)
2.2.6 Symmetries in MOT
60(1)
2.2.7 c-cyclical monotonicity
61(1)
2.2.8 Martingale McCann's interpolation
61(3)
2.2.9 Multi-marginals extension
64(3)
2.2.10 Robust quantile hedging
67(2)
2.2.11 Model-independent arbitrage
69(1)
2.2.12 Market frictions
70(1)
2.3 Other optimal solutions
71(5)
2.4 Numerical experiments
76(1)
2.5 Constrained MOT
76(13)
2.5.1 VIX constraints
78(4)
2.5.2 Entropy penalty
82(4)
2.5.3 American options
86(3)
3 Model-independent options
89(24)
3.1 Probabilistic setup
89(1)
3.2 Exotic options made of Vanillas: Nice martingales
90(8)
3.2.1 Variance swaps
90(2)
3.2.2 Covariance options
92(1)
3.2.3 Lookback/Barrier options
93(3)
3.2.4 Options on spot/variance
96(1)
3.2.5 Options on local time
97(1)
3.3 Timer options
98(7)
3.3.1 Dirichlet options
98(2)
3.3.2 Neumann options
100(2)
3.3.3 Some generalizations
102(1)
3.3.4 Model-dependence
103(2)
3.4 Ocone's martingales
105(8)
3.4.1 Lookback/Barrier options
108(1)
3.4.2 Options on variance
109(1)
3.4.3 Options on local time
110(3)
4 Continuous-time MOT and Skorokhod embedding
113(66)
4.1 Continuous-time MOT and robust hedging
113(3)
4.1.1 Pathwise integration
113(2)
4.1.2 Continuous-time MOT
115(1)
4.2 Matching marginals
116(12)
4.2.1 Bass's construction
117(1)
4.2.2 Local variance Gamma model
118(2)
4.2.3 Local volatility model
120(1)
4.2.4 Local stochastic volatility models and McKean SDEs
121(1)
4.2.5 Local Levy's model
122(1)
4.2.6 Martingale Frechet-Hoeffding solution
123(5)
4.3 Digression: Matching path-dependent options
128(1)
4.4 Link with Skorokhod embedding problem
129(2)
4.5 A (singular) stochastic control approach
131(3)
4.6 Review of solutions to SEP and its interpretation in mathematical finance
134(26)
4.6.1 Azema-Yor solution
134(12)
4.6.2 Root's solution
146(6)
4.6.3 Perkins solution
152(4)
4.6.4 Vallois' solution
156(4)
4.7 Matching marginals through SEP
160(3)
4.7.1 Through Azema-Yor
161(1)
4.7.2 Through Vallois
162(1)
4.7.3 Optimality in Mc((Pt)tε(0,T])
162(1)
4.8 Martingale inequalities
163(5)
4.8.1 Doob's inequality revisited
163(2)
4.8.2 Burkholder-Davis-Gundy inequality
165(2)
4.8.3 Inequalities on local time
167(1)
4.9 Randomized SEP
168(11)
4.9.1 Robust pricing with partial information
169(1)
4.9.2 p-mixed SEP
170(2)
4.9.3 Optimality
172(7)
References 179(10)
Index 189
Pierre Henry-Labordere works in the Global Markets Quantitative Research team at Societe Generale. He holds a Ph.D. in Theoretical Physics from Ecole Normale Superieure (Paris) and a habilitation thesis in Applied Mathematics from University Paris-Dauphine. More importantly, Pierre has a longstanding experience in tek diving, particularly mixed-gas closed-circuit rebreathers. Pierre is also professor (charge de cours) at Ecole Polytechnique and research associate at CMAP (Ecole Polytechnique). He was the recipient of the 2013 "Quant of the Year" award from Risk magazine and the 2014 Institute Louis Bachelier award for his paper on MOT written in collaboration with M. Beiglbock and F. Penkner from University of Vienna.
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