Digital Call Vega | Digital Call Options

The digital call vega measures the change in price due to an incremental change in the implied volatility. The vega is represented by:

$V = \frac{dC}{d\sigma}$

where C is the digital call value and $\sigma$ is volatility.

In general, as with conventional options, an out-of-the-money call will have a positive vega. This means the option price will increase in value as implied volatility rises. The implied volatility would likely move in tandem with the actual volatility of the underlying asset price. (This implied/historic volatility relationship is not carved in stone.)

Increasing the implied volatility increases the value of the conventional option irrespective of whether the option is in-the-money or not. This is not the case with a digital option.

An in-the-money digital call option has a negative digital call vega. This means that when the underlying is above the strike, increasing the implied volatility will decrease the value of the digital call option. Increased volatility leads to a higher probability the asset price can fall below the strike. In other words, the option, when in a winning position, now has an increased probability of falling into a losing position.

It follows that if the asset price is below the strike an increase in volatility increases the probability of the asset price being above the strike. Hence the out-of-the-money digital call has a positive vega.

When the asset price is the same as the strike price then vega is zero since, like digital call theta, the digital call will always have a 50:50 chance of being above or below the strike.

Digital Call Vega Over Time

The 0.1-days to expiry profile has a concertina’d profile as the price profile has premium in only a very narrow range. This is because the far out-of-the-money digital calls are likely to be worthless owing to lack of time to expiry. The same rationale applies to the 0.1-day in-the-money digital call which will be worth 100 at any significant distance from the strike.

Maximum absolute values for the vega are uniformly approximately ±0.235 irrespective of time to expiry. The shorter the time to expiry the nearer the peaks and troughs are to the strike price.

At-the-money digital call vega is always zero.

Digital Call Vega and Volatility

Figure 2 shows the vega of the 100 strike digital call over a range of volatilities.

Digital Call Theta w.r.t Volatility — Figure 2 – Digital Call Vega w.r.t. Volatility

The at-the-money digital call option has zero vega. This is because, irrespective of the implied volatility, the option has a 50:50 chance of ending in-the-money. An analogy might be thus: you toss a coin. It does not matter how high you toss it, it will always come down with a 50% chance of being a head or a tail.

The peaks and troughs of the lowest volatility (2%) vega have higher absolute values than the higher volatility profiles. Furthermore the peak and trough close in on the strike as implied volatility falls. Hence the 18% profile has a fairly shallow profile with the peak and trough at the furthest distance from the strike.

At the extremes of underlying asset price the vega is zero since the digital call will be worth 0.0 or 1.0 irrespective of incidental changes in implied volatility.

Summary

Falling implied volatility and decreasing time to expiry have a similar effect on the digital call vega:

Both variables drive the peak and trough of the vega provile closer to the strike price.
Both variables increase the height of the peak and depth of the trough as:
- volatility decreases
- time to expiry decays

The vega always passes through zero as the asset price equals the strike price.