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Smoothing Binary Options

A binary (or digital) option entitles the holder to gain a fixed amount of money if the underlying is above or below a predetermined level, the strike, on a given date. By adjusting the distance from the strike, and the time to maturity, one can easily reach a desired premium / payoff ratio. If compared to a vanilla option, the binary is often considered simpler, because the trader exactly knows in advance which is the best and the worst case he will face at maturity. Another advantage for the option buyer is indeed a bit more technical, but absolutely valuable: the binary option is the mathematical limit of a call spread with the difference of the strikes going to zero and the amount going to infinite. This cumbersome definition reveals a nice thing indeed, i.e. that the option buyer may even get, under some circumstances, a cheaper price than for a vanilla option, simply because the call spread cuts off part of the time value of the vanilla, which is proportional to volatility.

Having said that, one must recognize that BOs are quite frustrating, both for the buyer and for the seller, when the underlying moves close to the strike price with little time to maturity. One of the two will see the price rapidly vanishing without being able to do nothing for hedging his position, exactly like red betting on the roulette table. The reason is that the BO price function is discontinuous at maturity, consequently the Greeks tend to infinite!

A smart way to solve this situation, is to build a portfolio with many BOs for small amounts each, and different maturities. This technique is called "accrual", and gives the possibility to earn a fixed amount of money for each day the underlying stays above or below the strike price. The accrual structure is attractive for medium term investors, since it can lead to good profits even if the market moves in the wrong way when close to maturity, provided that it stayed in the desired zone until then. This situation would result profitable for a BO buyer, while resulting loser for a vanilla option buyer. However, I’d never recommend an accrual structure for short term trading, since it is much more complicated and expensive to unwind. Last but not least, a positive effect of such a portfolio of binary options is to display smoothed Greeks: this enables the experienced option trader to neutralize the portfolio exposure by using vanilla options and the underlying itself, provided that the hedging is performed on a continuous basis. Another usage of accrual structures is when the trader whishes to bet on the underlying remaining in a range (i.e. between two barriers) most of the time. In this situation he would buy a "daily range accrual", which is built with two sets of binary options: a set of daily long Binary calls at the lower barrier, and a set of daily short Binary calls at the higher barrier.

Another version of the Binary option is the "American" version, i.e. the payoff condition is checked not only at maturity, but during the whole life of the option. There are two versions of American Binary options: the so called "One Touch" and the "No Touch". In the first case, the option pays if the underlying touches the strike level, at any time before maturity, and in the second case the option pays if the option never touches the level during the time to maturity. American BOs are more attractive than their "European" brothers checked at maturity only, but are much more expensive, and even more tricky from a quantitative point of view, since the pricing is strongly affected by the term structure of volatility. This often means higher bid-ask spreads and lesser counterparties available to quote them.

Disclosure: "No positions"