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No arbitrage and multiplicative special semimartingales

Eckhard Platen, Stefan Tappe

Abstract

Consider a financial market with nonnegative semimartingales which does not need to have a numéraire. We are interested in the absence of arbitrage in the sense that no self-financing portfolio gives rise to arbitrage opportunities, where we are allowed to add a savings account to the market. We will prove that in this sense the market is free of arbitrage if and only if there exists an equivalent local martingale deflator which is a multiplicative special semimartingale. In this case, the additional savings account relates to the finite variation part of the multiplicative decomposition of the deflator.

No arbitrage and multiplicative special semimartingales

Abstract

Consider a financial market with nonnegative semimartingales which does not need to have a numéraire. We are interested in the absence of arbitrage in the sense that no self-financing portfolio gives rise to arbitrage opportunities, where we are allowed to add a savings account to the market. We will prove that in this sense the market is free of arbitrage if and only if there exists an equivalent local martingale deflator which is a multiplicative special semimartingale. In this case, the additional savings account relates to the finite variation part of the multiplicative decomposition of the deflator.

Paper Structure

This paper contains 16 sections, 48 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Basic notation and no-arbitrage concepts
  3. The fundamental theorems of asset pricing revisited
  4. The main results and their consequences
  5. Semimartingale term structure models
  6. Equivalent local martingale deflators
  7. Jump-diffusion models with fixed times of discontinuities
  8. Further examples
  9. Dynamic trading strategies
  10. The semimartingale property of the primary security accounts
  11. Filtration enlargements
  12. Financial models in discrete time
  13. Vector stochastic integration
  14. Market transformations
  15. Equivalent local martingale deflators and related concepts
  16. ...and 1 more sections

Key Result

Proposition 2.3

Let $X \in \mathscr{S}^d$ and $Y \in \mathscr{M}_{\sigma}$ be such that $X^i Y \in \mathscr{M}_{\sigma}$ for each $i = 1,\ldots,d$. Then for every $H \in L(X)$ we have $(H \bullet X) Y \in \mathscr{M}_{\sigma}$.

Theorems & Definitions (113)

  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Theorem 3.1
  • proof : Proof of Theorem \ref{['thm-Takaoka']}
  • Theorem 3.2
  • proof : Proof of Theorem \ref{['thm-FTAP-DS']}
  • ...and 103 more