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No Arbitrage Criteria for Exponential L évy Models

No Arbitrage Criteria for Exponential L évy Models. A.V. Selivanov Moscow State University. Financial Mathematics. 3 ”columns” (Z. Bodie, R.C. Merton ”Finance”) :. air. Mathematical : A – set of incomes. No Free Lunch (NFL) (J.M. Harrison, D.M. Kreps 1979). Concept of No Arbitrage.

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No Arbitrage Criteria for Exponential L évy Models

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  1. No Arbitrage Criteriafor Exponential Lévy Models A.V. SelivanovMoscow State University

  2. Financial Mathematics 3 ”columns” (Z. Bodie, R.C. Merton ”Finance”):

  3. air • Mathematical:A – set of incomes No Free Lunch (NFL) (J.M. Harrison, D.M. Kreps 1979) Concept of No Arbitrage • Practical:

  4. – any sequence, NFL No Arbitrage condition in discrete time Fundamental Theorem of Asset Pricing: • J.M. Harrison, S.R. Pliska 1981 – finite W • R.C. Dalang, A. Morton, W. Willinger 1990 – general case

  5. 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

  6. Definition of sigma-martingales The definition is given by T. Goll and J. Kallsen • A semimartingale M is a sigma-martingale if there exist predictable sets such that • or • is a uniformly integrable martingale for any n

  7. Sigma-martingales and local martingales sigma-martingales local martingales positive sigma-martingales

  8. S– any process, integrable martingale} Classes of Martingale Measures

  9. Fundamental Theorem of Asset Pricing

  10. existence of certainmartingale measure absence of arbitrage completeness of the model uniqueness of the measure

  11. Models under consideration • exponential Lévy model: • time-changed exponential Lévy model: L – nonzero Lévy process – independent increasing non-constant process

  12. Black-Scholes and Merton models • Black-Scholes modelB– Brownian motion, • Merton model – Poisson process,

  13. Theorem for models with finite time horizon NFLVR NGA

  14. Theorem for model (1) with infinite time horizon NFLVR always GA

  15. Theorem for model (2) with infinite time horizon Suppose that P-a.s. Then always GA.

  16. An example: NFLVR and GA NFLVR is satisfied;NGA is not satisfied Strategy:

  17. Conclusions We have obtained: • 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

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