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Log-optimality with small liability stream

Michail Anthropelos, Constantinos Kardaras, Constantinos Stefanakis

Abstract

In an incomplete financial market with general continuous semimartingale dynamics; we model an investor with log-utility preferences who, in addition to an initial capital, receives units of a non-traded endowment process. Using duality techniques, we derive the fourth-order expansion of the primal value function with respect to the units $ε$, held in the non-traded endowment. In turn, this lays the foundation for expanding the optimal wealth process, in this context, up to second order w.r.t. $ε$. The key processes underpinning the aforementioned results are given in terms of Kunita-Watanabe projections, mirroring the case of lower order expansions of similar nature. Both the case of finite and infinite horizons are treated in a unified manner.

Log-optimality with small liability stream

Abstract

In an incomplete financial market with general continuous semimartingale dynamics; we model an investor with log-utility preferences who, in addition to an initial capital, receives units of a non-traded endowment process. Using duality techniques, we derive the fourth-order expansion of the primal value function with respect to the units , held in the non-traded endowment. In turn, this lays the foundation for expanding the optimal wealth process, in this context, up to second order w.r.t. . The key processes underpinning the aforementioned results are given in terms of Kunita-Watanabe projections, mirroring the case of lower order expansions of similar nature. Both the case of finite and infinite horizons are treated in a unified manner.
Paper Structure (9 sections, 3 theorems, 117 equations)

This paper contains 9 sections, 3 theorems, 117 equations.

Table of Contents

  1. Setup
  2. The financial market
  3. Investment opportunities
  4. Illiquid asset
  5. Utility maximization problem
  6. Fourth order expansion of value function w.r.t. $\epsilon$
  7. Second order expansion of optimal wealth w.r.t. $\epsilon$
  8. Discussion and an example
  9. Utility-based pricing

Key Result

Theorem 2.1

Assume eq:ass_arbitrage, eq:ass_illiquid_stronger and eq:ass_fourth; then we have: as $\epsilon\rightarrow 0+$.

Theorems & Definitions (10)

  • Remark 1.1
  • Remark 1.2
  • Theorem 2.1
  • proof
  • Theorem 3.1
  • proof
  • Example 4.1: Linear payoff on a factor model and an infinite horizon
  • Corollary 5.1
  • proof
  • Remark 5.1