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.
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.
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Digital Call Delta and Volatility
Figure 2 shows various digital call deltas with 5 days to expiry where the delta remains manageable.
At 100.00 the Figure 2 5-day profile would sit between the 2 and 8-day profiles for Figure 1. As volatility falls from 10% to 6% and then 2% the delta increases exponentially. This is why market-making (providing a two-sided price) on short term digital calls and puts is so risky. Yet at with volatility of 18% no one is going to get rich or poor trading a 5-day digital call, the gearing is just not available.
Summary
As can be seen from the peaks of the 0.1 day and 2% volatility profiles. Hedging a short at-the-money equity digital call with just seconds to expiry would entail bidding for the company to hedge this one lot short call. This kind of distribution is known in the quant world as a Dirac distribution (after the guy who discovered it) as the peak literally tends to infinity.
Much has been written about the danger of attempting to hedge a short-term at-the-money digital option. The most practical method is to limit downside risk by straddling a conventional call spread around the digital call strike and hedging the digital call using the call spread’s delta. This is not a difficult process to fathom but too long to add to this starter page. More on this later.