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: