# Box spread

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, abull spread constructed from calls ("e.g." long a 50 call, short a 60 call) combined with abear 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 thepresent 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 thepresent 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 let 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 $K\_1$ < $K\_2$):-

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.**Prevalence**To what extent are the various instruments introduced above traded on exchanges? Chaput and Ederington, surveyed

Chicago Mercantile Exchange 's market for options onEurodollar futures. For the year 1999-2000 they found that some 25% of the trading volume was in raw options, 25% in straddles and vertical spreads (call-spreads and put-spreads), and about 5% in strangles. Guts constituted only about 0.1%, and box-spreads even less (about 0.01%). [Ratio spreads took more than 15 %, and about a dozen other instruments took the remaining 30 %.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.]**The box spread in futures**A box spread in futures contracts is a spread from two consecutive butterfly spreads, summming 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 they occasionally lead to amusing losses, such as that of sigma derivatives in early August 2008 when December 2008

Short Sterling sold off massively due to banks hedging end of year funding needs.**References*** Ben-Zion, U., S. Danan and J. Yagil, “Box Spread Strategies and Arbitrage Opportunities”, The Journal of Derivatives, Spring 2005, 47-62.

* Bharadwaj, Anu and James B. Wiggins, Box spread and put-call parity tests for the S&P 500 index LEAPS market, "Journal of Derivatives", 8(4) (2001): 62-71. The box-spread reveals an arbitrage profit insufficient to cover transaction costs.

* Billingsley, R.S. and Don M. Chance, Options market efficiency and the box spread strategy, "Financial Review", 20 (1987): 287-301.

* Chance, Don M, "An Introduction to Derivatives", 5th edition, Thomson, 2001.

* Chaput, J. Scott and Louis H. Ederington, Option spread and combination trading, [*http://www.commerce.otago.ac.nz/finc/research/pdf/02/practice_Jan2002.pdf*] , 2002.

* Hemler, Michael L.and Thomas W. Miller, Jr. Box spread arbitrage profits following the 1987 market crash: real or illusory? , "Journal of Financial and Quantitative Analysis", 32(1)(1997): 71-90. Post-market simulations with box-spreads on the S&P 500 Index show that market ineffiency increased after the 1987 crash.

* Hull, John C., "Fundamentals of Futures and Options Markets", 4th edition, Prentice-Hall, 2002.

* Ronn, Edud and Aimee Gerbarg Ronn, The Box spread arbitrage conditions: theory, tests, and investment strategies, "Review of Financial Studies", 2(1) (1989): 91-108. The box-spread is used to test for arbitrage opportunities on Chicago Board Options Exchange data.

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