Optimal Execution: Price Caps, Order Flow and Dark Pools | Algorithmic & HFT Ch. 7

The classical optimal execution schedule is a fixed curve: it trades the same way whatever the price does, whatever other traders are doing and wherever the order could be sent. What changes when the trader is allowed to react? How should a buyer behave when she must never pay above a price limit, how should a seller respond to the order flow of other traders, and how does the schedule change when part of the order can rest in a dark pool? This lecture builds each answer from scratch. We re-derive the base problem with one Bellman step, then add the price cap and reduce the dynamic programming equation to a Fisher-type PDE for one function h(t, S), with the trading speed ν* = y h / k. We bound h, solve it numerically, work out the perpetual case in closed form and recreate the chapter's figures from exact simulations. Other traders' orders are modelled as Poisson shot noise and solved with a quadratic ansatz; the dark pool is solved in closed form when every match fills the post, and numerically for partial fills. What you will learn: why the textbook schedule is deterministic and blind to the price, and which three ingredients change that how the ansatz H = yS + y²h(t, S) without permanent impact reduces the problem to a Fisher-type PDE, and why the bounds χ(t) ≤ h ≤ α hold how to solve the PDE with Crank-Nicolson and a Newton iteration, and check it by a verification theorem and Monte Carlo what the cap does to the speed: 3.77 per day at the open against 3.17 without it, and 31.7 % of the paths touching the cap how Poisson order flow tilts the strategy: buy flow while selling slows her down, sell flow speeds her up how a dark pool reshapes the schedule: slower at first, faster later, and a non-convex inventory curve how the three exercises of the chapter are solved and checked: random jump sizes, orders ahead in the dark-pool queue and a drift in the midprice The lecture goes beyond the textbook: the base problem is re-derived, the proofs are filled in and every printed formula is checked, with the slips we found (the statement of Theorem 7.1, the initial guess of the iteration, signs in the dark-pool equations) marked on screen. CHAPTERS 0:00 Why a fixed schedule is not enough 5:00 The classical problem from scratch 17:08 Buying under a price cap: the control problem 25:24 Without permanent impact: the reduced PDE 34:58 Bounds on h and the artificial boundary 44:37 Numerics: Crank-Nicolson, Newton, Monte Carlo check 54:13 The perpetual case 1:02:21 Figure 7.1: the speed map over time and price 1:09:49 Simulating the capped strategy 1:20:03 Order flow as Poisson shot noise 1:28:06 Solving the order-flow equation 1:38:47 The strategy as a forecast of the flow 1:45:59 Simulating the order-flow strategy 1:54:45 Lit market versus dark pool: the model 2:02:54 Solving the dark-pool equation 2:14:00 Figure 7.6: slower first, faster later 2:23:37 Partial fills of random size 2:31:23 The three exercises of the chapter 2:40:25 Take-home: three extensions, one method Based on chapter 7 (Optimal Execution with Continuous Trading II) of "Algorithmic and High-Frequency Trading" by Álvaro Cartea, Sebastian Jaimungal, José Penalva (Cambridge University Press, 2015). This is an independent educational explainer: not affiliated with or endorsed by the authors or the publisher. Not financial advice. Animations are made with manim; all simulations shown were run for this video; narration is a synthetic text-to-speech voice. #OptimalExecution #AlgorithmicTrading #QuantFinance #DarkPools