Tim Olson
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35 lines
1.7 KiB
Markdown
35 lines
1.7 KiB
Markdown
---
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description: "An arbitrage/volatility strategy combining a long synthetic forward and a short synthetic forward (or bull call spread and bear put spread) at two strikes, locking in a fixed payoff of K1 - K2."
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tags: [options, arbitrage, neutral, box]
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---
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# Long Box
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**Section**: 2.52 | **Asset Class**: Options | **Type**: Arbitrage
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## Overview
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The long box strategy can be viewed as a combination of a long synthetic forward and a short synthetic forward, or as a combination of a bull call spread and a bear put spread. It consists of: a long ITM put at K1, a short OTM put at K2 (lower), a long ITM call at K2, and a short OTM call at K1. The trader's outlook is neutral. This is a capital gain strategy. We assume K1 >= K2 + D.
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## Construction
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- Buy 1 ITM put option at strike K1 (higher)
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- Sell 1 OTM put option at strike K2 (lower, K2 < K1)
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- Buy 1 ITM call option at strike K2 (same as short put strike)
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- Sell 1 OTM call option at strike K1 (same as long put strike)
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- All same expiry
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Net debit: D (assumed K1 >= K2 + D)
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## Payoff Profile
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f_T = (K1 - S_T)+ - (K2 - S_T)+ + (S_T - K2)+ - (S_T - K1)+ - D
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= K1 - K2 - D (constant, regardless of S_T)
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- P_max = (K1 - K2) - D (fixed payoff at all stock prices)
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## Key Conditions / Signals
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- Used primarily as an arbitrage strategy when the market price of the box (D) is less than the theoretical value (K1 - K2)
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- Also used as a tax strategy in some jurisdictions (see footnote 31 in the source)
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- No directional risk: the payoff is fixed regardless of stock price at expiry
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## Notes
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The long box has a deterministic payoff of K1 - K2 - D at expiry. If D < K1 - K2 (mispricing), this is a risk-free profit. In practice, transaction costs and bid-ask spreads must be considered. Can also be used as a synthetic loan.
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