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Conditional Non-Lattice Integration, Pricing and Superhedging

Christian Bender, Sebastian E. Ferrando, Alfredo L. Gonzalez

Abstract

Closely motivated by financial considerations, we develop an integration theory which is not classical i.e. it is not necessarily associated to a measure. The base space, denoted by $\mathcal{S}$ and called a trajectory space, substitutes the set $Ω$ in probability theory and provides a fundamental structure via conditional subsets $\mathcal{S}_{(S,j)}$ that allows the definition of conditional integrals. The setting is a natural by-product of no arbitrage assumptions that are used to model financial markets and games of chance (in a discrete infinite time framework). The constructed conditional integrals can be interpreted as required investments, at the conditioning node, for hedging an integrable function, the latter characterized a.e. and in the limit as we increase the number of portfolios used. The integral is not classical due to the fact that the original vector space of portfolio payoffs is not a vector lattice. In contrast to a classical stochastic setting, where price processes are associated to conditional expectations (with respect to risk neutral measures), we uncover a theory where prices are naturally given by conditional non-lattice integrals. One could then study analogues of classical probabilistic notions in such non-classical setting, the paper stops after defining trajectorial martingales the study of which is deferred to future work.

Conditional Non-Lattice Integration, Pricing and Superhedging

Abstract

Closely motivated by financial considerations, we develop an integration theory which is not classical i.e. it is not necessarily associated to a measure. The base space, denoted by and called a trajectory space, substitutes the set in probability theory and provides a fundamental structure via conditional subsets that allows the definition of conditional integrals. The setting is a natural by-product of no arbitrage assumptions that are used to model financial markets and games of chance (in a discrete infinite time framework). The constructed conditional integrals can be interpreted as required investments, at the conditioning node, for hedging an integrable function, the latter characterized a.e. and in the limit as we increase the number of portfolios used. The integral is not classical due to the fact that the original vector space of portfolio payoffs is not a vector lattice. In contrast to a classical stochastic setting, where price processes are associated to conditional expectations (with respect to risk neutral measures), we uncover a theory where prices are naturally given by conditional non-lattice integrals. One could then study analogues of classical probabilistic notions in such non-classical setting, the paper stops after defining trajectorial martingales the study of which is deferred to future work.

Paper Structure

This paper contains 18 sections, 40 theorems, 172 equations.

Table of Contents

  1. Introduction
  2. Elementary Conditional Integral
  3. Trajectorial Setting
  4. Elementary Vector Spaces $\mathcal{E}_j$ and Elementary Conditional Integrals $I_j$
  5. Almost Everywhere Notions
  6. Conditional Integrals
  7. Conditions for validity of property $(L_j)$
  8. Integrable Functions and Their Characterization
  9. Characterization of Conditional Integrability
  10. Basic Properties of Conditional Integrals
  11. Trajectorial Martingales
  12. Conclusions
  13. Some Proofs for Sections \ref{['integralConstruction']}-\ref{['IntegrabilityCharacterization']}
  14. Properties $(L_j)$, $(K_j)$ and Examples
  15. Establishing Property $(L_j)$
  16. ...and 3 more sections

Key Result

Lemma 1

Assume $\mathcal{S}$ is locally $0$-neutral, $n_0 \geq 0$, and let $F=\{F_i\}_{i\ge n_0}$ be a sequence of non-anticipative functions and $\epsilon>0$. Then, for any $S \in\mathcal{S}$ there exists a sequence of trajectories $\{S^n\}_{n \ge n_0}$ with $S^{n_0}=S$ such that for every $n > n_0$, $S^{n and so,

Theorems & Definitions (103)

  • Definition 1
  • Definition 2
  • Definition 3: $0$-Neutral nodes
  • Definition 4: Up-Down nodes
  • Definition 5: Flat nodes
  • Lemma 1
  • Remark 1
  • Corollary 1: $I_j$ is Well Defined
  • proof
  • Remark 2
  • ...and 93 more