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Contingent Claim Valuation under Increasing Profit, Strong Arbitrage, and Arbitrage of the First Kind

Yukihiro Tsuzuki

Abstract

We study the upper hedging price for contingent claims in market models with strong types of arbitrage: increasing profit, strong arbitrage, and arbitrage of the first kind. The existence of arbitrage may make the price smaller than if it did not exist. For example, when the asset price process has a reflecting boundary, which introduces increasing profit in the market model, the option prices are reduced to those of the corresponding options that knock-out at the boundary. Furthermore, we demonstrate that corporate stock price processes with increasing profit are obtained as a result of corporate stock issuance and repurchase plans.

Contingent Claim Valuation under Increasing Profit, Strong Arbitrage, and Arbitrage of the First Kind

Abstract

We study the upper hedging price for contingent claims in market models with strong types of arbitrage: increasing profit, strong arbitrage, and arbitrage of the first kind. The existence of arbitrage may make the price smaller than if it did not exist. For example, when the asset price process has a reflecting boundary, which introduces increasing profit in the market model, the option prices are reduced to those of the corresponding options that knock-out at the boundary. Furthermore, we demonstrate that corporate stock price processes with increasing profit are obtained as a result of corporate stock issuance and repurchase plans.

Paper Structure

This paper contains 28 sections, 7 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Financial market model
  4. Notions of arbitrage
  5. Contingent claim valuation
  6. Examples based on the three-dimensional Bessel process
  7. Bachelier's model
  8. Three-dimensional Bessel process
  9. Enlargement with the future infimum process $J$
  10. Enlargement with a future infimum process $J_{0}$
  11. Enlargement with an honest time $\tau_{J_{0}}$
  12. Contingent claim valuation under arbitrage
  13. Contingent claim valuation under increasing profits
  14. Contingent claim valuation under strong arbitrage
  15. Contingent claim valuation under arbitrage of the first kind
  16. ...and 13 more sections

Key Result

Lemma 2.1

Let $\theta_{\varepsilon}^{1} := \inf \{t >0 : \varepsilon > \widehat{L}_{t} \}$ for $\varepsilon \in (0,1)$. Then, the stopped process $\widehat{L}^{\rho^{1}}$ is a nonnegative continuous local martingale that satisfies and the reciprocal $1/\widehat{L}^{\theta_{\varepsilon}^{1}}$ is a continuous semimartingale that is bounded by $1/\varepsilon$ and is expressed with a stochastic integral with r

Theorems & Definitions (34)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.1
  • proof
  • Definition 3.1
  • Remark 3.1
  • Theorem 3.1
  • Proposition 3.1
  • ...and 24 more