No Arbitrage Criteria for Exponential L évy Models

1 / 17

# No Arbitrage Criteria for Exponential L évy Models - PowerPoint PPT Presentation

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.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## No Arbitrage Criteria for Exponential L évy Models

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### No Arbitrage Criteriafor Exponential Lévy Models

A.V. SelivanovMoscow 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
• Practical:

– 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 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

existence of certainmartingale measure

absence of arbitrage

completeness of the model

uniqueness of the measure

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 modelB– Brownian motion,
• Merton model – Poisson process,
Theorem for model (2) with infinite time horizon

Suppose that P-a.s.

Then

always GA.

An example: NFLVR and GA

NFLVR is satisfied;NGA is not satisfied

Strategy:

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