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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 vy models

No Arbitrage Criteriafor Exponential Lévy Models

A.V. SelivanovMoscow State University

financial mathematics
Financial Mathematics

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

concept of no arbitrage

air

  • Mathematical:A – set of incomes

No Free Lunch (NFL)

(J.M. Harrison, D.M. Kreps 1979)

Concept of No Arbitrage
  • Practical:
no arbitrage condition in discrete time

– 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
no arbitrage conditions in continuous time
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
definition of sigma martingales
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 and local martingales

sigma-martingales

local martingales

positive sigma-martingales

slide10

existence of certainmartingale measure

absence of arbitrage

completeness of the model

uniqueness of the measure

models under consideration
Models under consideration
  • exponential Lévy model:
  • time-changed exponential Lévy model:

L – nonzero Lévy process – independent increasing non-constant process

black scholes and merton models
Black-Scholes and Merton models
  • Black-Scholes modelB– Brownian motion,
  • Merton model – Poisson process,
theorem for model 2 with infinite time horizon
Theorem for model (2) with infinite time horizon

Suppose that P-a.s.

Then

always GA.

an example nflvr and ga
An example: NFLVR and GA

NFLVR is satisfied;NGA is not satisfied

Strategy:

conclusions
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