Paper: arXiv 2410.04867

Abstract

We investigate the well-posedness in the Hadamard sense and the absence of price manipulation in the optimal execution problem within the Almgren-Chriss framework, where the temporary and permanent impact parameters vary deterministically over time. We present sufficient conditions for the existence of a unique solution and provide second-order conditions for the problem, with a particular focus on scenarios where impact parameters change monotonically over time. Additionally, we establish conditions to prevent transaction-triggered price manipulation in the optimal solution, i.e. the occurence of buying and selling in the same trading program. Our findings are supported by numerical analyses that explore various regimes in simple parametric settings for the dynamics of impact parameters.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper is mathematically dense, focusing on well-posedness, existence/uniqueness of solutions, and analytical conditions using calculus of variations and differential equations, while empirical validation is limited to simple numerical experiments without real data or detailed backtesting.

Research Flowchart

  flowchart TD
  A["Research Goal:<br/>Well-Posedness & Price Manipulation<br/>in Optimal Execution w/ Varying Liquidity"] --> B["Methodology:<br/>Extension of Almgren-Chriss Model<br/>Deterministic Time-Varying Parameters"]
  B --> C{"Analysis 1:<br/>Hadamard Well-Posedness"}
  B --> D{"Analysis 2:<br/>Price Manipulation Conditions"}
  C --> E["Result:<br/>Existence & Uniqueness<br/>Second-Order Conditions"]
  D --> F["Result:<br/>Absence of Transaction-Triggered<br/>Price Manipulation"]
  E --> G["Outcome:<br/>Validated via Numerical Simulations<br/>for Monotonic & Parametric Regimes"]
  F --> G

Cited References (47)

  1. Price manipulation and quasi-arbitrage (2004)
  2. Order book resilience, price manipulation, and the positive portfolio problem (2012)
  3. Optimal liquidation under stochastic price impact (2019)
  4. How markets slowly digest changes in supply and demand (2009)
  5. Optimal execution of portfolio transactions (2001)
  6. Linear models for the impact of order flow on prices I. Propagators: Transient vs (2018)
  7. Optimal trading strategy and supply/demand dynamics (2013)
  8. Bid, ask and transaction prices in a specialist market with heterogeneously informed traders (1985)
and 39 more
  1. Continuous auctions and insider trading (1985)
  2. No-dynamic-arbitrage and market impact (2010)
  3. Optimal execution with limit and market orders (2015)
  4. Tikhonov regularization and total least squares (1999)
  5. The Financial Mathematics of Market Liquidity: From optimal execution to market making (2016)
  6. Comparative statics and heterogeneity (2019)
  7. A class of P-matrices with applications to the localization of the eigenvalues of a real matrix (2001)
  8. The price impact of order book events (2014)
  9. Optimal trade execution and price manipulation in order books with time-varying liquidity (2014)
  10. Optimal trade execution in order books with stochastic liquidity (2019)
  11. Co-impact: Crowding effects in institutional trading activity (2020)
  12. Beyond the square root: Evidence for logarithmic dependence of market impact on size and participation rate (2015)
  13. Master curve for price-impact function (2003)
  14. Why is equity order flow so persistent? (2015)
  15. Optimal control of execution costs (1998)
  16. Convex Optimization (2004)
  17. On shooting methods for boundary value problems (1969)
  18. Sur les probl `e mes aux d 'e riv 'e es partielles et leur signification physique (1902)
  19. Fluctuations and response in financial markets: the subtle nature ofrandom’price changes (2003)
  20. Optimal trade execution and absence of price manipulations in limit order book models (2010)
  21. Efficient capital markets: A review of theory and empirical work (1970)
  22. Black was right: Price is within a factor 2 of value (2017) — link
  23. Cross-impact and no-dynamic-arbitrage (2019)
  24. Trading lightly: Cross-impact and optimal portfolio execution (2017) — link
  25. The Sturm-Picone Theorem; Pseudo-Oscillatory Solutions (1962)
  26. The new calculus of variations (1906)
  27. Numerical methods for two-point boundary-value problems (2018)
  28. Incorporating order-flow into optimal execution (2016)
  29. Measuring the information content of stock trades (1991)
  30. Measuring price impact and information content of trades in a time-varying setting — → 2212.12687
  31. Generalized autoregressive score models with applications (2013)
  32. A simple approximate long-memory model of realized volatility (2009)
  33. Price discovery in high resolution (2021)
  34. Variational inference with normalizing flows (2015)
  35. Trading with Concave Price Impact and Impact Decay-Theory and Evidence (2023)
  36. Trades, quotes and prices: financial markets under the microscope (2018)
  37. Mathematics for economists (1994)
  38. The flash crash: High-frequency trading in an electronic market (2017)
  39. Information-theoretic optimality of observation-driven time series models for continuous responses (2015)

1 of 47 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.