# Binary Call Options Delta

Binary call options delta is the metric that enables the binary call trader to know how many underlying they need to buy or sell to be hedged.

Example: A binary call option on a 10 Year Note future has a delta of 0.30. A long 100 binary call option position is equivalent to:

Binary Call Option = 100 x 0.30

= 30 10 Year Note futures

This ‘greek’ is critical in the hedging of an options portfolio against adverse movements in the underlying asset price.

A binary call option with a delta of 0.5 means that if the asset price goes up 1¢ then the binary call will go up by ½¢. Another interpretation would be a short 400 contract position in S&P500 binary calls with a delta of 0.25. This would be equivalent to being short 100 S&P500 futures.

The practicality of deltas, out of all the Greeks, lend themselves to being the most utilised of Greeks amongst traders.

Deltas move with the underlying. If the option is out-of-the-money and the underlying falls then the binary (and conventional) call delta falls. The caveat here is that a huge rise in volatility can mathematically generate a higher delta.

On the other hand, if the underlying is above the strike and rising then the conventional delta rises while the binary delta falls.

Binary Call Delta Evaluation

Binary call delta is the first derivative of the option price w.r.t. a change in the underlying asset price. It is described thus:

$\Delta = \frac{dP}{dS}$

where P is the binary call value and S is the asset price.

In effect the binary call delta is the gradient of the price profile of the binary call option.

Binary Call Delta Over Time

Figure 1 illustrates the 100.00 binary call delta against days to expiry. What may come as a surprise to conventional options traders is that the binary options call delta is at its highest when at-the-money.

Although the scale might suggest that the binary call delta remains fairly low this would be a grave mistake. The delta of the at-the-money tends to infinity as time to expiry approaches zero.

The 8-day profile remains at a very low level with a maximum value of just 0.27 when at-the-money. What starts off as a placid instrument turns into an unmanageable monster over the last few hours of its life. The at-the-money delta becomes so high that the option becomes unhedgeable. When there is just 0.001 days to expiry (1.44mins) the at-the-money binary call options delta has risen to 24.01. The at-the-money delta will approach ±∞ as the time to expiry approaches 0. This implies the gamma would also approach ±∞.

These deltas would increase hugely should the contract be less volatile than this asset and its 10% volatility. If this were a government bond with 5.0% implied volatility and 0.001 days to expiry the delta would be 48.2 and rising.

Binary Call Delta and Volatility

Figure 2 shows various binary call options deltas with 5 days to expiry where the delta remains manageable.

The greater premium of the 5-day (as opposed to 0.1-day and 0.5-day of Figure 1) binary call means that the gradient of the price profile remains shallow. In effect no one is going to get rich or poor trading a 5 day binary call with implied volatility at 18%, the gearing is just not available.

By: Hamish Raw

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