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Box Spread: Four-Leg Construction, Fixed Payoff, and Arbitrage Limits

For educational purposes only; not investment advice.

Direct answer

Section titled “Direct answer”

A long box spread combines two same-expiration verticals at strikes K₁ < K₂:

  1. Buy the K₁ call and sell the K₂ call.
  2. Buy the K₂ put and sell the K₁ put.

If all four options have matched underlying, multiplier, exercise style, and settlement, the expiration payoff per unit is fixed at K₂ − K₁. A short box reverses every leg and owes that amount at expiration.

The box removes terminal price direction in theory. It does not remove execution, early-assignment, settlement, liquidity, broker, tax, or operational risk.

Why the two verticals add to a constant

Section titled “Why the two verticals add to a constant”

The bull call spread pays from zero up to the strike width as the underlying rises. The bear put spread pays the complementary amount as the underlying falls:

Call vertical payoff = max(S_T − K₁, 0) − max(S_T − K₂, 0)

Put vertical payoff = max(K₂ − S_T, 0) − max(K₁ − S_T, 0)

Adding them gives K₂ − K₁ in every terminal price region. This is also a consequence of put-call parity: two opposite synthetic forwards at different strikes cancel the underlying and leave a fixed strike difference.

Before expiration, quoted leg values may not offset cleanly because of wide spreads, stale markets, dividends, early-exercise value, or different settlement assumptions. An apparent price below discounted payoff is not an arbitrage unless the entire combination can be executed and carried under the assumed terms.

95/105 expiration table

Section titled “95/105 expiration table”

Consider one long 95/105 box with a 100-unit multiplier. Its fixed payoff is ($105 − $95) × 100 = $1,000.

Expiration price 95/105 call spread 105/95 put spread Total per unit
$90 $0 $10 $10
$100 $5 $5 $10
$110 $10 $0 $10

If the combination is bought for $9.70 per unit, it costs $970 and has a theoretical $30 gain before fees at expiration. That $30 is compensation over time, not immediate profit. If aggregate entry, exit, and settlement costs exceed $30, the apparent edge disappears.

Structure and execution checklist

Section titled “Structure and execution checklist”
  • Confirm all legs share underlying, expiration, multiplier, exercise style, and settlement.
  • Submit one complex limit order and judge the executable net market, not four displayed midpoints.
  • Recalculate payoff from each leg; one reversed put or quantity changes the structure entirely.
  • Prefer European-style, cash-settled terms when the objective is a clean fixed payoff.
  • For American equity options, model early assignment, dividends, stock positions, borrow, and funding.
  • Check broker margin, buying power, strategy permission, and liquidation treatment.
  • Include every commission, exchange fee, spread, settlement charge, and tax consideration.
  • Plan expiration handling and verify the official settlement value and timing.

Common misconceptions

Section titled “Common misconceptions”
  • “A flat expiration graph means no risk.” It describes contractual payoff, not the full trading process.
  • “Any box below strike width is free money.” The future payoff must be discounted and all costs included.
  • “Leg midpoints lock the price.” Only a filled complex order establishes the position’s cost.
  • “The Greeks are exactly zero at all times.” Market conventions, dividends, skew, and discrete exercise can create residual exposures.
  • “One assigned short leg leaves the same box.” Assignment creates a different stock-and-options position.
  • “Long and short boxes have symmetric account treatment.” Margin, cash, and liquidation rules may differ.
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Authoritative sources

Section titled “Authoritative sources”