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The insider problem in the trinomial model: a discrete-time jump process approach

Hélène Halconruy

Abstract

In an incomplete market underpinned by the trinomial model, we consider two investors : an ordinary agent whose decisions are driven by public information and an insider who possesses from the beginning a surplus of information encoded through a random variable for which he or she knows the outcome. Through the definition of an auxiliary model based on a marked binomial process, we handle the trinomial model as a volatility one, and use the stochastic analysis and Malliavin calculus toolboxes available in that context. In particular, we connect the information drift, the drift to eliminate in order to preserve the martingale property within an initial enlargement of filtration in terms of the Malliavin derivative. We solve explicitly the agent and the insider expected logarithmic utility maximisation problems and provide a hedging formula for replicable claims. We identify the insider expected additional utility with the Shannon entropy of the extra information, and examine then the existence of arbitrage opportunities for the insider.

The insider problem in the trinomial model: a discrete-time jump process approach

Abstract

In an incomplete market underpinned by the trinomial model, we consider two investors : an ordinary agent whose decisions are driven by public information and an insider who possesses from the beginning a surplus of information encoded through a random variable for which he or she knows the outcome. Through the definition of an auxiliary model based on a marked binomial process, we handle the trinomial model as a volatility one, and use the stochastic analysis and Malliavin calculus toolboxes available in that context. In particular, we connect the information drift, the drift to eliminate in order to preserve the martingale property within an initial enlargement of filtration in terms of the Malliavin derivative. We solve explicitly the agent and the insider expected logarithmic utility maximisation problems and provide a hedging formula for replicable claims. We identify the insider expected additional utility with the Shannon entropy of the extra information, and examine then the existence of arbitrage opportunities for the insider.

Paper Structure

This paper contains 15 sections, 8 theorems, 120 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries and theoretical tools
  3. The jump-binomial model: frame and martingale measures
  4. Clark formula for marked binomial processes
  5. Enlargement of filtration in a discrete setting: existing results
  6. Insider vs agent: the rewards of extra information
  7. Utility maximization problems: setting and notation
  8. Agent's maximum expected utility
  9. Insider's maximum expected utility
  10. Insider's advantage and impact of the extra information
  11. Inside insider's mind
  12. A new interpretation of the information drift
  13. Insider's optimal hedging strategy
  14. Conclusion and perspectives
  15. Proofs

Key Result

Proposition 3.1

For $x\in\mathbb{R}_+^*$, $t\in[\![1,T]\!]$ and $u\in\{\mathbf{log},\mathbf{exp},\mathbf{pow}\}$, let $\widehat{\mathrm V}_{t}^{\EuScript{F},u}$ be the optimal portfolio (discounted) value for the problem $\Phi_t^{\EuScript{F},u}(x)$ defined by optTerAg_eq. We get: Logarithmic utility: Exponential utility: Power utility: Let $\widehat{\mathrm L}_t=(d\widehat{\mathbf P}/d\mathbf P)|\EuScript{F}_

Figures (1)

  • Figure 1: Realization of a MBP on $[\![1,6]\!]\times\{-1,1\}$

Theorems & Definitions (21)

  • Remark 2.1
  • Remark 2.2
  • Proposition 3.1: Agent's portfolio optimization
  • Lemma 3.2
  • Theorem 3.3: Insider's utility optimization in the jump-binomial model
  • Theorem 3.4
  • Remark 3.5
  • Corollary 3.6
  • Proposition 4.1
  • Remark 4.2
  • ...and 11 more