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

TL;DR

This paper models an incomplete financial market with insiders using a trinomial jump process, deriving explicit utility maximization solutions, hedging formulas, and analyzing arbitrage opportunities through advanced stochastic calculus techniques.

Contribution

It introduces a novel approach to the trinomial model via a marked binomial process, connecting information drift with Malliavin calculus, and explicitly solves utility maximization for insiders.

Findings

Insider's additional utility equals the Shannon entropy of extra information.

Explicit solutions for utility maximization and hedging formulas are derived.

Conditions for arbitrage opportunities for insiders are identified.

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.