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No Arbitrage Criteria for Exponential Lévy Models A.V. Selivanov Moscow State University.

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Presentation on theme: "No Arbitrage Criteria for Exponential Lévy Models A.V. Selivanov Moscow State University."— Presentation transcript:

1 No Arbitrage Criteria for Exponential Lévy Models A.V. Selivanov Moscow State University

2 Financial Mathematics 3 ”columns” (Z. Bodie, R.C. Merton ”Finance”): Pricing Resource Allocation Utility maximization Risk Management Equilibrium Arbitrage Pricing Theory

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

4 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

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 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

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

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

9 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

10 absence of arbitrage existence of certain martingale measure 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 model B – Brownian motion, Merton model  – Poisson process,

13 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

14 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

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 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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