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A general framework for pricing and hedging under local viability

Huy N. Chau, Miklos Rasonyi

Abstract

In this paper, a new approach for solving the problems of pricing and hedging derivatives is introduced in a general frictionless market setting. The method is applicable even in cases where an equivalent local martingale measure fails to exist. Our main results include a new superhedging duality for American options when wealth processes can be negative and trading strategies are subject to a cone constraint. This answers one of the questions raised by Fernholz, Karatzas and Kardaras.

A general framework for pricing and hedging under local viability

Abstract

In this paper, a new approach for solving the problems of pricing and hedging derivatives is introduced in a general frictionless market setting. The method is applicable even in cases where an equivalent local martingale measure fails to exist. Our main results include a new superhedging duality for American options when wealth processes can be negative and trading strategies are subject to a cone constraint. This answers one of the questions raised by Fernholz, Karatzas and Kardaras.

Paper Structure

This paper contains 16 sections, 17 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. The model
  3. Pricing and hedging American options
  4. Pricing and hedging European options
  5. Future extensions
  6. Proofs
  7. Some closedness results
  8. Proof of Theorem \ref{['thm:local_closed_0']}
  9. Proof of Proposition \ref{['pro:1']}
  10. Proof of Corollary \ref{['cor:density_A']}
  11. Proof of Proposition \ref{['pro:reduce']}
  12. Proof of Theorem \ref{['thm:hedge_American']}
  13. Proof of Proposition \ref{['pro:appro_hedging']}
  14. Appendix
  15. Convex compactness and maximal elements in $(L^0_+)^{\mathbb{N}}$
  16. ...and 1 more sections

Key Result

Theorem 2.7

For a locally bounded semimartingale $S$, the condition NFLVR$_{\mathfrak{C}}$ when $\mathfrak{C} = \mathfrak{C}^u$ is equivalent to the existence of a probability $Q\sim P$ such that $S$ is a local $Q$-martingale.

Theorems & Definitions (45)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7
  • Definition 2.8
  • Remark 3.2
  • Remark 3.3
  • ...and 35 more