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Free lesson · Arbitrage

Size, impact and the research decision

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Start with the idea

Later units may cost more to trade. Positive small-size edge does not imply unlimited profitable capacity.

Symbols, units & horizon
  • q: matched units
  • e: edge after flat variable costs USD/unit
  • λ: total-impact coefficient USD/unit², positive
  • F: fixed USD cost
  • q*: unconstrained continuous optimum for e>0
  • Π: completion-scenario USD profit

When and why to use this

Write a defensible acceptance/rejection memo with size, cost and failed-leg sensitivity.

Later units may cost more to trade. Positive small-size edge does not imply unlimited profitable capacity.

Assume linear edge revenue and quadratic total impact to study how size changes a scenario. This simple functional form must be estimated and challenged; it is not a universal law.

Inspect capital limits, lot sizes, failed-leg probabilities and unwind costs. Keep a worst-case scenario distinct from expected profit. A rejected trade with a clear reason is a valid research output.

Π(q)=eq−λq2−F,q∗=e2λ
Quadratic impact assumption and calculus result

Size, impact and the research decision

  1. Write edge revenue eq minus total impact λq² and fixed F.
  2. Differentiate: e−2λq=0 gives q*=e/(2λ).
  3. Second derivative −2λ is negative; enforce feasible size and compare with not trading.
Work it by hand

e=.20, λ=.001, F=2: optimal q=100. Profit=.20×100−.001×100²−2=$8; at 50 units profit is $5.50.

Apply it in a strategy

  • Write a defensible acceptance/rejection memo with size, cost and failed-leg sensitivity.
  • Record the input timestamp, executable quantity, currency and horizon. Reconcile the result with a cash-flow or state table.
  • Stress this failure condition: Quiet-period impact fits can understate stressed costs, and a continuous optimum ignores queues and discrete lots.

Research deliverable

Build and explain a size, impact and the research decision worksheet. Write a defensible acceptance/rejection memo with size, cost and failed-leg sensitivity.

Evidence boundary: Synthetic arithmetic and scenarios illustrate mechanics. They are not historical returns, a paper replication, or evidence of an executable edge. Research sources and their access limitations are recorded at the end of this module.

Primary research and operational references · reviewed 12 September 2026

Further reading: Schmeling, Schrimpf & Todorov — Crypto carry ↗

BIS Working Paper 1087; initially April 2023 · Review: Primary publisher summary and abstract checked 2026-09-12; full paper not reviewed in this authoring pass. Markets/data: Bitcoin and Ether spot/futures; exact sample endpoints not verified in this pass. Interpretation: Motivates separating quoted carry from capital and margin constraints. Limits: Summary-level access; no inference that a quoted basis is a realizable return. No independent replication performed.

Further reading: Aldasoro, Beltrán & Grinberg — Stablecoin flows and spillovers to FX markets ↗

BIS Working Paper 1340, 27 March 2026 · Review: Primary publisher summary and abstract checked 2026-09-12. Markets/data: Four USD stablecoins, 27 fiat currencies, 64 exchanges, 2021–2025. Interpretation: Motivates studying stablecoin conversion as a segmented FX route with balance-sheet constraints. Limits: The identification and quantitative estimates require full-paper review; a parity gap is not a frictionless trading profit. No independent replication performed.

Research sources, review dates and limitations

Connect the ideas: Constraints and survival

Retrieve: A desired position must fit available capital and explicit limits.

Check the change: Portfolio weights, venue collateral, working orders and redemption obligations impose different constraints.

Risk → Portfolio construction → Optimization → Crypto derivatives → Execution & microstructure → Fund operations & capstone

Explain it yourself: Can an offsetting terminal payoff remove a margin problem today?

Self-assessed. Write your explanation before opening this comparison.

No. Cash may be required before the hedge pays, or in another account. Check the path, collateral location and feasible transfer times.

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

# Python 3.10+; standard library unless NumPy is imported below.
# Inputs and outputs use the units defined in this lesson. Synthetic teaching example.
def capacity(edge,impact,fixed,limit):
    if impact<=0 or fixed<0 or limit<0: raise ValueError("Invalid impact or limit")
    q=min(limit,max(0,edge/(2*impact)))
    profit=edge*q-impact*q*q-fixed
    return (q,profit) if profit>0 else (0.,0.)

print(capacity(.20,.001,2,1000))

Continue learning

Arbitrage: Payoffs, Financing and Execution — all lessons
  1. Start with every possible payoff
  2. Bid, ask and the gross-to-net waterfall
  3. Put–call parity from expiration states
  4. Dated cash-and-carry
  5. Triangular currency conversion
  6. ETF baskets and creation access
  7. Haircuts and survival capital
  8. Size, impact and the research decision

Quantitative finance and development glossary · Python resources and libraries · Research sources and limitations