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No Arbitrage Criteria for Exponential Lévy Models A.V. Selivanov Moscow State University
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Financial Mathematics 3 ”columns” (Z. Bodie, R.C. Merton ”Finance”): Pricing Resource Allocation Utility maximization Risk Management Equilibrium Arbitrage Pricing Theory
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Concept of No Arbitrage Practical : Mathematical : A – set of incomes No Free Lunch (NFL) (J.M. Harrison, D.M. Kreps 1979)
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No Arbitrage condition in discrete time Fundamental Theorem of Asset Pricing: – any sequence, NFL J.M. Harrison, S.R. Pliska 1981 – finite R.C. Dalang, A. Morton, W. Willinger 1990 – general case
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No Arbitrage conditions in continuous time No Free Lunch with Vanishing Risk (NFLVR) F. Delbaen, W. Schachermayer 1994 No Generalized Arbitrage (NGA) A.S. Cherny 2004
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Definition of sigma-martingales A semimartingale M is a sigma-martingale if there exist predictable sets such that or is a uniformly integrable martingale for any n The definition is given by T. Goll and J. Kallsen
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Sigma-martingales and local martingales local martingales positive sigma-martingales sigma-martingales
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Classes of Martingale Measures S – any process, integrable martingale}
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Fundamental Theorem of Asset Pricing no arbitrage condition finite time horizon infinite time horizon F. Delbaen, W. Schachermayer NFLVR S – semimartingale A.S. Cherny NGA S – any positive process
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absence of arbitrage existence of certain martingale measure completeness of the model uniqueness of the measure
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Models under consideration exponential Lévy model: time-changed exponential Lévy model: L–nonzero Lévy process –independent increasing non-constant process
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Black-Scholes and Merton models Black-Scholes model B – Brownian motion, Merton model – Poisson process,
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Theorem for models with finite time horizon statement condition for model (1) condition for model (2) process S is not monotone Let Then Black-Scholes or Merton model Black-Scholes or Merton model; is deterministic and continuous NFLVR NGA
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Theorem for model (1) with infinite time horizon statementcondition for model (1) either the process S is a P-martingale, or and the jumps of S are not bounded from above Let Then Black-Scholes or Merton model NFLVR always GA
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Theorem for model (2) with infinite time horizon Suppose that P-a.s. Then always GA.
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An example: NFLVR and GA NFLVR is satisfied; NGA is not satisfied Strategy:
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Conclusions the criteria for the NFLVR and the NGA conditions for models with finite time horizon; for these models the criteria for the NFLVR and the NGA conditions for models without time change and with infinite time horizon; for these models the NGA is never satisfied, while the NFLVR is satisfied in certain cases We have obtained:
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