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Position-building in competition with real-world constraints

Neil A. Chriss

TL;DR

The optimal-trading framework developed in arXiv:2409.03586v1 is extended to compute optimal strategies with real-world constraints and two-trader equilibria may be calculated as the end-points of a dynamic process of traders forming repeated adjustments to each other's strategy.

Abstract

This paper extends the optimal-trading framework developed in arXiv:2409.03586v1 to compute optimal strategies with real-world constraints. The aim of the current paper, as with the previous, is to study trading in the context of multi-player non-cooperative games. While the former paper relies on methods from the calculus of variations and optimal strategies arise as the solution of partial differential equations, the current paper demonstrates that the entire framework may be re-framed as a quadratic programming problem and cast in this light constraints are readily incorporated into the calculation of optimal strategies. An added benefit is that two-trader equilibria may be calculated as the end-points of a dynamic process of traders forming repeated adjustments to each other's strategy.

Position-building in competition with real-world constraints

TL;DR

The optimal-trading framework developed in arXiv:2409.03586v1 is extended to compute optimal strategies with real-world constraints and two-trader equilibria may be calculated as the end-points of a dynamic process of traders forming repeated adjustments to each other's strategy.

Abstract

This paper extends the optimal-trading framework developed in arXiv:2409.03586v1 to compute optimal strategies with real-world constraints. The aim of the current paper, as with the previous, is to study trading in the context of multi-player non-cooperative games. While the former paper relies on methods from the calculus of variations and optimal strategies arise as the solution of partial differential equations, the current paper demonstrates that the entire framework may be re-framed as a quadratic programming problem and cast in this light constraints are readily incorporated into the calculation of optimal strategies. An added benefit is that two-trader equilibria may be calculated as the end-points of a dynamic process of traders forming repeated adjustments to each other's strategy.
Paper Structure (31 sections, 3 theorems, 75 equations, 17 figures, 1 algorithm)

This paper contains 31 sections, 3 theorems, 75 equations, 17 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Definitions and terminology
  3. Mathematical overview of trading strategies
  4. Convex combinations of trading strategies and the cost function
  5. Fourier series and trading strategies
  6. Approximating the cost function with Fourier series
  7. Properties of the approximate cost function
  8. Some useful facts about Fourier series of trading strategies
  9. Constraints
  10. How to constrain best-response and equilibrium strategies
  11. Overbuying constraint
  12. Channel constraints
  13. End-strategy constraints
  14. Short-selling constraints
  15. No-sell constraints
  16. ...and 16 more sections

Key Result

Proposition 2.1

The total cost function $\text{Cost}(a, b_\lambda) = \int_0^1 (\dot a + \lambda\dot b_\lambda)\dot a + \kappa (a + \lambda b_\lambda) \dot a \,\texttt{dt}$ may be approximated by the a function $\text{Cost}_{\mathcal{F},a}(b_\lambda)$:

Figures (17)

  • Figure 1: An example of a over-buying strategy (blue line) exceeds the target quantity of one and therefore violates the overbuy constraint.
  • Figure 2: The trajectory channel bounded by $\sinh(4t)/\sinh(4)$ below and $t$ above. We may seek a constrained optimization that restricts to strategies that are within this channel and see that the risk averse strategy $y=t^2$ is always within the channel while the eager strategy is never within the channel.
  • Figure 3: A visual depiction of a end-strategy constraints with $t^*=75\%$ and $c=80\%$. The constraint requires that at time $t=0.75$ the trading strategy is at least 80% completed and does not exceed 100% of the target quantity. The eager strategy (blue line) nor the risk-neutral strategy (red-line) satisfy the constraint. See figure \ref{['fig:best-response-eager-End-strategy_sigma3']} for examples of best-response strategies with end-strategy constraints.
  • Figure 4: Best-response strategies $a$ to a risk-neutral adversary trading $b_\lambda$ with an overbuying constraint. The plots depict two different overbuying buying constraints (500% and 200% of the target quantity) in dashed red and the unconstrained in solid red.
  • Figure 5: Best-response strategies $a$ to a risk-averse competitor trading $b_\lambda$ with risk-aversion parameter $\sigma$ as in \ref{['eq:best-response-strategy-ra-neat']} for $\sigma=3$. The results are shown for both unconstrained (solid red) and with the constrained for various values of $\rho$.
  • ...and 12 more figures

Theorems & Definitions (3)

  • Proposition 2.1
  • Proposition 2.2: Approximate cost function for $b$ with respect to $a$
  • Proposition 2.3: The approximate cost-function converges to the cost-function