Trading Dev AcademyFree quant education

Free lesson · Prediction strategies

Subset relations and executable bounds

Open interactive lessonPractice calculationsExplore labs

Start with the idea

If event A can happen only when event B happens, a B claim pays at least as much as an A claim in every binary state.

Symbols, units & horizon
  • Y_B,s and Y_A,s: USD payouts of $1 claims in state s
  • b_A: executable A sale bid USD/share
  • a_B: B purchase ask USD/share
  • c: all costs USD per paired trade
  • Π_s: state profit USD per pair
  • A subset of B assumed

When and why to use this

Search related-event constraints by explicit state implication and attainable order direction.

If event A can happen only when event B happens, a B claim pays at least as much as an A claim in every binary state.

Use a state implication, not textual similarity: A implies B. For example, an exact compatible contract for winning a tournament implies reaching its final, provided cancellation and settlement rules match.

In a frictionless model A cannot be more valuable than B. To translate a violation into a trade, buy B at its ask and sell A at its bid, then verify the short sale or collateralized synthetic equivalent exists.

Πs=YB,s−YA,s+bA−aB−c≥bA−aB−c
Statewise payoff inequality under a subset relation

Subset relations and executable bounds

  1. Enumerate neither, B-only and A-and-B states; B minus A pays 0, 1 and 0.
  2. Add initial sale proceeds b_A and subtract purchase ask a_B.
  3. Subtract costs; the minimum terminal surplus is b_A−a_B−c if all other states are excluded by the contract.
Work it by hand

A bid .61, B ask .58, cost .01: cash surplus=.02. State payoffs add either 0 or 1, so modeled minimum is $.02 per pair.

Apply it in a strategy

  • Search related-event constraints by explicit state implication and attainable order direction.
  • Record the input timestamp, executable quantity, currency and horizon. Reconcile the result with a cash-flow or state table.
  • Stress this failure condition: A sale may require unavailable borrowing or collateral; inconsistent cancellation states can break the subset proof.

Research deliverable

Build and explain a subset relations and executable bounds worksheet. Search related-event constraints by explicit state implication and attainable order direction.

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.

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 subset_floor(a_bid,b_ask,cost):
    if not 0<=a_bid<=1 or not 0<=b_ask<=1 or cost<0: raise ValueError("Invalid quote")
    return min(b-a+a_bid-b_ask-cost for a,b in [(0,0),(0,1),(1,1)])

print(subset_floor(.61,.58,.01))

Continue learning

Prediction Strategies: Logic, Sizing and Market Making — all lessons
  1. Complete-set purchases and redemption
  2. Subset relations and executable bounds
  3. Bounds for joint and union events
  4. Binary Kelly sizing and estimation error
  5. Decision buffers for probability uncertainty
  6. Quoting revenue and adverse selection
  7. Event overlap and portfolio variance
  8. Evaluate the decision process, including failed fills

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