Digital call options are all-or-nothing options that settle at:

- 100.00 if in-the-money at expiry, or
- zero if out-of-the-money.
- If the underlying at expiry is exactly on the strike, ‘at-the-money‘, settlement is (1 + 0)/2 = 0.5.

The price of digital call options could be interpreted as the probability of the event happening. This assumes a zero cost-of-carry and yield, i.e. interest rates are zero with no dividends or coupon payments.

### Digital Call Options at Expiry

Figure 1 shows the expiry profile of a $100 strike digital call:

- Below the strike price of 100.00 the digital call has lost and settles at 0.
- Above the strike price of 100.00 the digital call has won and settles at 1.00.
- In the event of the asset price settling exactly on the strike of 100.00 on this website is adopted the value of 0.50. This represents a dead heat.

If the graph was flipped upside down around the vertical axis at 0.50 it becomes the digital put option expiry profile.

### Digital Call Options over Time

Figure 2 shows the P&L profile illustrating how the expiry profile was arrived at over time. The price profiles are always flat or rising from left to right indicating that the digital call always has a positive or zero delta.

Below the strike price it is clear that the price profiles become steeper as they approach the strike price. Conversely they become less steep above the strike price. This indicates the unusual (if you are a conventional options trader) feature of the digital call gamma being positive below the strike and negative above it.

The buyer of this digital call is betting that the asset price will be above $100 at expiry. The 25-day profile is shallow but over time this animal changes its spots. It becomes the most highly geared and dangerous instrument in the world of finance. It is doubtful that any other single instrument can offer a P&L profile that can exceed an angle of 45°. Indeed the angle of an at-the-money moments before expiry tends to the vertical and becomes absolutely unhedgeable.

#### Positive Theta?

What is also apparent from the profiles over time is that the bet decreases in value when out-of-the-money. Conversely it increases in value when in-the-money. Therefore, the out-of-the-money call has a negative digital call option theta while the in-the-money has a positive call theta. Furthermore, the at-the-money has a theta of zero assuming that the above ‘dead heat’ rule is applied.

European Options | Digital Call Delta | Digital Call Gamma | Digital Call Theta | Digital Call Vega |

### Digital Call Options and Volatility

Implied volatility is a critical input into the pricing of digital options. The level of implied volatility determines whether one is buying the digital option cheaply or too expensively. Figure 3 displays the digital call options price profile over a range of implied volatility.

At the underlying price of $98.50, as implied volatility increases, so does the value of the out-of-the-money option. This indicates a positive vega. Above the strike an increase in ‘vol’ lowers the option value; this indicates negative vega.

Why does the price increase at 98.50? This is because with a low volatility the probability of the underlying price rising above the strike is low. Over time this will, in turn, lead to a worthless digital call option. On the other hand, as volatility increases and the underlying swings around more there is a greater chance of the asset price moving above the strike.

At 101.50 this call is in-the-money, the asset price is above the strike price and the ‘long’ (the trader who has bought the option) is in a winning position. Yet if the underlying asset becomes more volatile there would be a higher chance of the asset price falling back through the strike to a losing position. If the asset price ground to a halt at 101.50 and volatility falls then there is now a better chance of the bet remaining a winner and the bet’s value increases.

### Summary

With the very first instrument analysed it is noticeable that there is a big difference between the characteristics of the digital call and those of the conventional call.

When the call option is in-the-money the option holder will have a better chance of being a winner if volatility falls. An increase in implied volatility decreases the value of the option. This is because the option has a higher probability of the underlying sliding back down through the strike. Therefore, above the strike the digital call option has a negative vega.