A Digital With A Twist: Buy High and Sell Low
by Rich Tanenbaum

(from Derivatives Week, December 1996)

Buy high and sell low on volatility, that is. Which still sounds like a paradox, given conventional wisdom. As we will see, with certain Digital options, and at certain volatilities, it actually makes perfect sense.

What are Digital Options?

Digital, or Binary options, are very much like standard, plain vanilla puts and calls, with one exception: when a digital option ends up in the money, it is worth $1 no matter how much in the money it is. Therein lies the root of our paradox, the fact that the value of a Digital option stays constant at $1 even as the option gets more into the money. In other words, the option holder's payoff stays flat even as the standard option's intrinsic value increases. Unlike the typical option instrument which rewards the option holder with an upside that is unlimited, the Digital option holder looks forward to an upside that is fixed.

Digital options may be bought and sold outright, but they are often part of a larger, more complex structure. Digital options are, therefore, more commonplace than you might expect. For instance, let's look at the Deferred Payment option, where a buyer pays the premium only at expiration, and only if the underlying asset is in the money. Such a structure actually decomposes into a standard option and some Digitals, where the option holder is long the standard option and short some Digitals. An Accrual Swap (or a Range Note) is effectively a strip of Digital options, each expiring on an accrual date. You will also find Digital options embedded within certain Barrier structures. The rebate portion of a Barrier option with rebate is made up of a Digital option, albeit an American Digital option.

Relationship to Volatility

Let's observe the behavior of Digital options with respect to volatility. In particular, let's examine a European Digital FX call option with one year to expiration, a strike price of 100, an underlying asset price of 99, and a domestic and a foreign interest rate of 7%. We will look at volatilities ranging from 5 to 25%.

As you can see from the above graph, as volatility (on the x-axis) increases, at first the Digital option's value increases, as we would expect. But something happens when volatility is at 15%: the Digital option actually starts to decline in value as volatility increases. In other words, once volatility reaches a certain level, more volatility will actually hurt the option holder. This defies common options logic. Beyond volatilities of 15%, the Digital option's vega (the change in option price with respect to change in volatility) is actually negative, not something we normally witness.

Complicating these problems is the fact that this counter-intuitive volatility effect is present in certain Digital options, but not all of them. Each trade therefore must be analyzed on its own merits. The obvious questions after all this are "Why do some Digital options decline in value as volatility increases?" and

"Which Digital options will exhibit a negative vega?"

We can get some insight into how to answer these questions by examining the pricing formula for Digital options. The valuation of European Digital options is quite straightforward, as the formula for a Digital option's value is a part of the Black-Scholes formula (or Garman-Kohlhagen, in the presence of a yield on the underlying asset). It is:

exp(r*t) * N(d2)

where

d2 = (ln(S/K)-.5(v^2-r*t))/(v*sqrt(t))

N is the cumulative normal distribution function

Valuing American Digital options is somewhat more complex, but a closed form formula is available for that, as well (see Rubinstein and Riner). But the European formula is worth concentrating on here. Although the formula looks complicated, one simple interpretation of it is that the value of a Digital option is simply the present value of the probability the option will end up in the money. For example, if the present value of $1 to be paid in one year is .94, and there is a 50% chance the underlying will end up above the strike, then a Digital call will be worth .47.

Two notable features about this particular Digital option: it was a European option, and it was an option that was about to go into the money.

If a Digital call option is out of the money (on a forward, not spot, basis), then at very low volatility there is very little chance it will rise above the strike and finish in the money. As volatility increases, we also increase the chance it will finish in the money, and thus the value of the option will rise, as well.

But if the option is already in the money, then at a very low volatility there is very little chance it will fall below the strike and finish out of the money. Thus, in the money Digital options are worth the most when volatility is low, but are actually worth less when volatility is high, because high volatilities will increase the odds that the Digital will not finish in the money. And out of the money Digital options are worth the most when volatility is high, which increases the option's chances of finishing in the money (the same logic holds for puts, as well).

The reason standard options do not exhibit this behavior is that the Digital option can at most be worth 1 at expiration. Volatility helps a standard option because it increases the upside potential of the payoff, but this doesn't happen with Digitals. For in the money Digital options, increasing volatility increases the probability of experiencing the downside, but the upside stays the same.

Four important implications arise from this phenomenon:

The first, just touched upon, is that the Digital option buyer is not always buying volatility. A long Digital position can actually be a bad bet if it turns out implied volatility rises during the life of the trade.

Second, standard implied volatility calculations will return one volatility for an option price. But in this graph, we can clearly see that the implied volatility for an option price of .4 is both 7% and 27%. Therefore any function designed to return "the" implied volatility will really only return only one of two implied volatilities in existence.

Third, a trader who wants to bid at a low volatility and offer at a high volatility may end up paying more for a Digital option that he or she sells it for. This may seem unlikely, for a trader to say "I'll pay you 41, and sell at 40," but it is quite possible to bid for an option with one strike and expiration, and offer another option with another strike and expiration, at two prices which will be out of line with each other, but not in an obvious way.

Fourth, any mark to market of a derivatives book, say at year end, should be done using conservative bid/offer vols. For most option books, a conservative approach would mean pricing long positions at the volatility on the bid side, and pricing short positions at the volatility on the offer side. But with Digital options, the opposite could be true. What seems conservative from a volatility standpoint may actually be aggressive from a profit and loss standpoint.

Because of the side effects of lognormal distributions, it turns out the point at which vega becomes negative is for European options is slightly out of the money, as our example shows: for low volatilities, an increase in volatility increases the option value, while for high volatilities it decreases the option value.

For American Digitals like rebates attached to Barrier options, the situation is even more complex, since volatility affects both the Barrier and the Digital in different ways. For this reason, it is important to examine each option position on an individual basis.