Optimization

Overview

Constructing a trade, or a portfolio, is an optimization problem: trading expected benefit against risk and cost under constraints, and being explicit about which objective is actually being optimized.

Core questions

• How should the trade-off between execution speed and market impact be set for a given risk tolerance?
• How much of an optimized portfolio's apparent benefit survives estimation error in its inputs?

Mathematical formulation

Almgren–Chriss optimal execution trajectory

Given a position X to liquidate by time T, this gives the remaining position xⱼ to hold at each time tⱼ that minimizes a combination of expected execution cost and the variance of that cost, where κ balances risk aversion λ, volatility σ and the market-impact parameter η̃. As κ → 0, the trajectory becomes linear (uniform liquidation).

Methods we use

• Optimal execution under a cost–risk trade-off

  Almgren, R., & Chriss, N. (2001). Optimal execution of portfolio transactions. Journal of Risk, 3(2), 5–39.

• Robust portfolio construction under estimation noise

  López de Prado, M. (2016). Building diversified portfolios that outperform out of sample. Journal of Portfolio Management, 42(4), 59–69.

• Growth-optimal position sizing

  Kelly, J. L. (1956). A new interpretation of information rate. Bell System Technical Journal, 35(4), 917–926.

Open problems

• How does the optimal execution trajectory change when the impact function itself is uncertain, not a known constant?
• What allocation rule is robust to estimation error in both expected returns and the covariance matrix at the same time?