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Optimal Investment with Random Endowment in a Black–Scholes Market

Article arXiv papers · Author: Michael Donisch et al.
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Summary

The document studies how an investor with constant relative risk aversion should trade in a Black–Scholes market when they also hold a random endowment. It uses a duality approach to derive an explicit optimal trading strategy.

The strategy is presented as the sum of the optimal strategy for an investor without an endowment and an additional adjustment. The adjustment depends linearly on the endowment relative to wealth and exponentially on the time remaining until maturity. This decomposition helps isolate how the endowment changes the investment decision. The text provides no numerical examples, empirical tests, or detailed derivation, so it does not establish how the result performs in practice or how it extends beyond the stated market and preference assumptions.

Key ideas

  • The setting is a Black–Scholes market with an investor who has constant relative risk aversion.
  • The investor’s random endowment is incorporated into an explicit optimal trading rule.
  • The rule separates into a baseline strategy and an additive endowment adjustment.
  • The adjustment varies linearly with the endowment-to-wealth ratio and exponentially with time to maturity.
  • The result is limited to the assumptions stated; no empirical evaluation is described.

Tags

StatisticsPortfolio constructionRisk management
Full text
# An Explicit Solution for the Problem of Optimal Investment with Random Endowment


# An Explicit Solution for the Problem of Optimal Investment with Random Endowment









We consider the problem of optimal investment with random endowment in a Black--Scholes market for an agent with constant relative risk aversion. Using duality arguments, we derive an explicit expression for the optimal trading strategy, which can be decomposed into the optimal strategy in the absence of a random endowment and an additive shift term whose magnitude depends linearly on the endowment-to-wealth ratio and exponentially on time to maturity.

Shown in full with attribution under the source's licence. Licence: abstract CC0

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.