# Digital Call Delta The digital call delta is the metric that enables the digital call trader to know how many underlying they need to buy or sell to be hedged.

### Digital Call Delta Evaluation

Digital call delta is the first derivative of the option price w.r.t. a change in the underlying asset price. It is described thus: where C is the digital call value and S is the asset price.

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

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

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

The practicality of deltas 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 digital (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 call delta rises while the digital call delta falls.

### Digital Call Delta Over Time

Figure 1 illustrates the 100.00 digital call delta against days to expiry. What may come as a surprise to conventional options traders is that the digital call delta is at its highest when at-the-money. Figure 1 – Digital Call Option Delta w.r.t. Time to Expiry

Although the scale might suggest that the digital 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.

#### An Unhedgeable Delta

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 digital call 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.

### Digital Call Delta and Volatility

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