No Arbitrage Criteria for Exponential Lvy Models A

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

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

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

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

No Arbitrage conditions in continuous time n n No Free Lunch with Vanishing Risk (NFLVR) F. Delbaen, W. Schachermayer 1994 No Generalized Arbitrage (NGA) A. S. Cherny 2004

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

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

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

Fundamental Theorem of Asset Pricing no arbitrage condition F. Delbaen, W. Schachermayer NFLVR S – semimartingale A. S. Cherny NGA S – any positive process finite time horizon infinite time horizon

absence of arbitrage existence of certain martingale measure completeness of the model uniqueness of the measure

Models under consideration n exponential Lévy model: n time-changed exponential Lévy model: L – nonzero Lévy process – independent increasing non-constant process

Black-Scholes and Merton models n Black-Scholes model n B – Brownian motion, Merton model – Poisson process,

Theorem for models with finite time horizon statement NFLVR NGA Let condition for model (1) condition for model (2) process S is not monotone Then Black-Scholes or Merton model; Merton model is deterministic and continuous

Theorem for model (1) with infinite time horizon statement condition for model (1) either the process S is a P-martingale, or and the jumps of S are not bounded from above NFLVR always GA Let Then Black-Scholes or Merton model

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

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

Conclusions We have obtained: n the criteria for the NFLVR and the NGA conditions for models with finite time horizon; for these models n 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