# Binary Call Options Vega

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

#### $V = \frac{dP}{dσ}$

where P is the binary call value and sigma[\latex] 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 binary option.

An in-the-money binary call option has a negative binary call vega. This means that when the underlying is above the strike, increasing the implied volatility will decrease the value of the binary 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 binary call has a positive vega.

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

### Binary Call Vega Over Time

Fig.1 illustrates the effect of the passing of time on vega.

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 binary 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 binary call which will be worth 100 at any significant distance from the strike.

Maximum absolute values for the vega are uniformly approximately ±2.35 irrespective of time to expiry. The shorter the time to expiry the nearer the peaks and troughs are.

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

### Binary Call Vega and Volatility

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

The at-the-money binary 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 vega of the lower implied volatility (2%) profile peaks and troughs at higher absolute values than the higher implied volatility profiles. Furthermore the peak and trough close in on the strike as implied volatility falls so that the 18% profile has a fairly shallow profile.

At the extremes of underlying price the vega is zero since the binary call will be worth 0 or 100 irrespective of incidental changes in implied volatility.

Ultimately, falling implied volatility and decreasing time to expiry have a similar effect on the out-of-the-money binary call.

By: Hamish Raw

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