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Digital Options

Binary options are a special category of exotic options which have the particularity of a
discontinuous payoff, i.e. a predetermined payoff which corresponds to a payback (or
rebate) depending on the price of the underlying to be crossed. If this target price is not
reached the final payoff is zero. If the target price is reached or exceeded, the payoff will be equal to the rebate (100% of the rebate) minus the premium of the option, which varies
from 15 to 30%. Digital options are European configuration options with payment at maturity. This implies that during the life of the option, even if the target price is hit several
times if at maturity it is lower than the target price, the payoff will be zero, equal to zero.
For example, if an investor wishes to buy a digital call on the CAC 40 with a strike of 4800
points and a current price of 4500 points and a maturity of 3 months. If during these 3
months the 4800 points are affected but at maturity, the price is lower than 4800 points, the payoff will be zero. However, if at maturity the price is higher than 4800 points, the investor will receive a redemption (the 100% rebate), which corresponds to a predetermined amount of the nominal (expressed as a %).

The graph below illustrates the payoffs of a digital call & a digital put:

Payoffs are instantaneous at maturity if the price is higher than the strike price (both for call and put options). This is the big difference with One-Touch, Instant One Touch, and No Touch options that we will study later. A digital option is a European option with a European-style payoff, i.e. at maturity. For the Greeks of digital options, the delta increases sharply as the expiry date approaches and the price of the underlying moves closer to the strike price. The delta is almost infinite for an option with a very short maturity that is close to the strike. It is not uncommon to find deltas greater than 100 in digital options. Gamma and vega behave in the same way. The closer you get to the expiry and the strike, the more they tend towards zero and even become negative after crossing it. Theta is negative and then becomes positive when the strike is hit or crossed, and the maturity date is approaching. Digital options can be used for speculative purposes, which is why they are also called « bet options ».

On the other hand, here we will look at their advantages for setting up elaborate hedging strategies, combining vanilla and exotic options (such as digital, One Touch, Double One Touch, No Touch, etc.), while remaining premium neutral. These strategies are widely used on FX derivatives markets, given the liquidity available. The equity index derivatives market is a little less liquid, but enough to implement these strategies properly.

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