What is a box spread, and how does it create a risk-free arbitrage opportunity?

Question

AcadiFi · Accepted Answer

A box spread is one of the cleanest arbitrage constructions in options theory. It combines a bull call spread with a bear put spread at the same strikes, creating a position with a **guaranteed payoff** regardless of where the stock finishes.

## Construction

**Box Spread = Bull Call Spread + Bear Put Spread**

- Bull Call: Long call at K1, Short call at K2 (K1 < K2)
- Bear Put: Long put at K2, Short put at K1

## Why the Payoff Is Fixed

At expiration, the box spread always pays **(K2 - K1)**, no matter what the stock price is.

**Proof by cases for Waverly Corp options with K1 = $40, K2 = $50:**

| If S_T | Bull Call Value | Bear Put Value | Box Payoff |
|---|---|---|---|
| S_T < $40 | $0 - $0 = $0 | ($50-S_T) - ($40-S_T) = $10 | **$10** |
| $40 < S_T < $50 | (S_T-$40) - $0 | ($50-S_T) - $0 | **$10** |
| S_T > $50 | (S_T-$40) - (S_T-$50) = $10 | $0 - $0 = $0 | **$10** |

Every case yields $10 = K2 - K1.

## Arbitrage Condition

Since the box always pays (K2 - K1) at expiration, its fair price today should be:

**Fair Price = PV(K2 - K1) = (K2 - K1) / (1 + Rf)^T**

If the market price of the box differs from this fair value, there is an arbitrage:

- **Box is cheap:** Buy the box (costs less than PV). You pay less today for a guaranteed future payoff. Profit = PV - cost.
- **Box is expensive:** Sell the box (receive more than PV). You receive more today than the present value of what you owe later.

## Numerical Example

Waverly options, K1 = $40, K2 = $50, T = 6 months, Rf = 5%.

Fair box price = 10 / (1.05)^0.5 = 10 / 1.0247 = **$9.76**

Market prices:
- Long 40 call: $8.20, Short 50 call: -$3.50 → Bull call spread cost: $4.70
- Long 50 put: $5.80, Short 40 put: -$1.30 → Bear put spread cost: $4.50
- **Total box cost: $4.70 + $4.50 = $9.20**

Since $9.20 < $9.76 (fair value), the box is underpriced. Buy the box for $9.20, receive $10.00 in 6 months. Risk-free profit = **$0.56** per share.

**Exam tip:** Box spread questions are rare but high-value. Remember: the box always pays K2 - K1, and its fair value is the present value of that amount.

For more options strategies, check our CFA derivatives course.

What is a box spread, and how does it create a risk-free arbitrage opportunity?

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