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

Decompose a box spread into call and put verticals, verify its fixed payoff across price states, and understand why apparent arbitrage can fail.

Canonical: https://wiki.fcontext.com/options/box-spread/
Fact checked: 2026-07-22

> For educational purposes only; not investment advice.

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## 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.

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## 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.

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## 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.

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## 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.

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## 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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## Related topics

- [Box Spread Financing](/options/box-spread-financing/)
- [Vertical Spreads](/options/vertical-spread/)
- [Assignment Risk](/options/assignment-risk/)

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## Authoritative sources

- [Characteristics and Risks of Standardized Options](https://www.theocc.com/company-information/documents-and-archives/options-disclosure-document) - Options Clearing Corporation
- [SPX Options Product Specifications](https://www.cboe.com/tradable_products/sp_500/spx_options/specifications/) - Cboe
- [Margin Manual](https://cdn.cboe.com/resources/membership/Margin_Manual.pdf) - Cboe