Profit diagram of a box spread. It is a combination of positions with a riskless payoff.

In options trading, a box spread is a combination of positions that has a certain (i.e. riskless) payoff, considered to be simply "delta neutral interest rate position". For example, a bull spread constructed from calls (e.g. long a 50 call, short a 60 call) combined with a bear spread constructed from puts (e.g. long a 60 put, short a 50 put), has a constant payoff of the difference in exercise prices (e.g. 10). Under the no-arbitrage assumption the net premium paid out to acquire this position should be equal to the present value of the payoff.

They are often called "alligator spreads" because the commissions eat up all your profit due to the large number of trades required for most box spreads.

The box-spread usually combines two pairs of options; and its name derives from the fact that the prices for these options form a rectangular box in two columns of a quotation.

Note that box spreads also form a strategy in futures trading - see below.

## Background

An arbitrage operation may be represented as a sequence which begins with zero balance in an account, initiates transactions at time t = 0, and unwinds transactions at time t = T so that all that remains at the end is a balance whose value B will be known for certain at the beginning of the sequence. If there were no transaction costs then a non-zero value for B would allow an arbitrageur to profit by following the sequence either as it stands if the present value of B is positive, or with all transactions reversed if the present value of B is negative. However, market forces tend to close any arbitrage windows which might open; hence the present value of B is usually insufficiently different from zero for transaction costs to be covered. This is considered typically to be a "Market Maker/ Floor trader" strategy only, due to extreme commission costs of the multiple-leg spread. If the box is for example 20 dollars as per lower example getting short the box anything under 20 is profit and long anything over, has hedged all risk .

A present value of zero for B leads to a parity relation. Two well-known parity relations are:-

• Spot futures parity. The current price of a stock equals the current price of a futures contract discounted by the time remaining until settlement:

$S = F e^{-rT}$

• Put call parity. A long European call c together with a short European put p at the same strike price K is equivalent to borrowing $K e^{-rT}$ and buying the stock at price S. In other words, we can combine options with cash to construct a synthetic stock:

$c - p = S - K e^{-rT}$

Note that directly exploiting deviations from either of these two parity relations involves purchasing or selling the underlying stock.

## The Box Spread

Now consider the put/call parity equation at two different strike prices $K_1$ and $K_2$. The stock price S will disappear if we subtract one equation from the other, thus enabling one to exploit a violation of put/call parity without the need to invest in the underlying stock. The subtraction done one way corresponds to a long-box spread; done the other way it yields a short box-spread. The pay-off for the long box-spread will be the difference between the two strike prices; and the profit will be the amount by which the discounted payoff exceeds the net premium. For parity, the profit should be zero. Otherwise, there is a certain profit to be had by creating either a long box-spread if the profit is positive or a short box-spread if the profit is negative. [Normally, the discounted payoff would differ little from the net premium, and any nominal profit would be consumed by transaction costs.]

The long box-spread comprises four options, on the same underlying asset with the same terminal date. They can be paired in two ways as shown in the following table (assume strike-prices $K_1$ < $K_2$):-

Long bull call-spread Long bear put-spread Buy call at $K_1$ Sell put at $K_1$ Sell call at $K_2$ Buy put at $K_2$

Reading the table horizontally and vertically we obtain two views of a long box-spread.

• A long box-spread can be viewed as a long synthetic stock at a price $K_1$ plus a short synthetic stock at a higher price $K_2$.
• A long box-spread can be viewed as a long bull call-spread at one pair of strike prices, $K_1$ and $K_2$, plus a long bear put-spread at the same pair of strike prices.

We can obtain a third view of the long box-option by reading the table diagonally. In order to interpret the diagonals we need to introduce the straddle, which is a combination of a long call and a long put both at a strike price equal to the current stock price (at-the-money). This combination is direction neutral, its terminal payoff being dependent not on the direction of movement of the stock price but only on the magnitude of the movement. The band between the break-even points can be extended by separating the strike prices of the two options symmetrically with respect to the current stock price:-

• If both options are in-the-money the combination is called a long gut.
• If both options out-of-the-money the combination is called a long strangle.

Returning to the long box-spread, we see that the leading diagonal is a long gut combination, and the other diagonal is a short strangle combination. Hence a long box-spread may be created as a coupling of a long gut with a short strangle.

The short box-spread can be treated similarly.

## An Example

As an example, consider a three-month option on a stock whose current price is \$100. If the interest rate is 8% pa and the volatility is 30% pa then the prices for the options might be:

Call Put \$13.10 \$ 1.65 \$ 3.05 \$ 10.90

The initial investment for a long box-spread would be \$19.30. The following table displays the payoffs of the 4 options for the three ranges of values for the terminal stock price $S_T$:

$S_T < K_1$ $K_1 < S_T < K_2$ $K_2 < S_T$
$0$ $S_T - 90$ $S_T - 90$ $0$ $S_T - 90$ $0$
$0$ $110 - S_T$ $0$ $110 - S_T$ $110 - S_T$ $0$

The terminal payoff has a value of \$ 20 independent of the terminal value of the share price. The discounted value of the payoff is \$ 19.60. Hence there is a nominal profit of 30 cents to be had by investing in the long box-spread.

## The box spread in futures

A box spread in futures contracts is a spread from two consecutive butterfly spreads, summing to +1 -3 +3 -1 in consecutive, or at least equally spaced, contracts. Often presumed not to move much (as in theory they are practically non directional) and therefore trade in a range.