---
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."
tags: [options, arbitrage, neutral, box]
---

# Long Box

**Section**: 2.52 | **Asset Class**: Options | **Type**: Arbitrage

## Overview
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.

## Construction
- Buy 1 ITM put option at strike K1 (higher)
- Sell 1 OTM put option at strike K2 (lower, K2 < K1)
- Buy 1 ITM call option at strike K2 (same as short put strike)
- Sell 1 OTM call option at strike K1 (same as long put strike)
- All same expiry

Net debit: D  (assumed K1 >= K2 + D)

## Payoff Profile
f_T = (K1 - S_T)+ - (K2 - S_T)+ + (S_T - K2)+ - (S_T - K1)+ - D
    = K1 - K2 - D  (constant, regardless of S_T)

- P_max = (K1 - K2) - D  (fixed payoff at all stock prices)

## Key Conditions / Signals
- Used primarily as an arbitrage strategy when the market price of the box (D) is less than the theoretical value (K1 - K2)
- Also used as a tax strategy in some jurisdictions (see footnote 31 in the source)
- No directional risk: the payoff is fixed regardless of stock price at expiry

## Notes
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.