This post is published by Hamish Raw of https://hamishraw.com/

The binary call options vega measures the change in price owing to an incremental change in the implied volatility. In general, as with conventional options, an out-of-the-money call will have a positive vega as the option will increase in value as implied volatility rises. This is because the implied volatility would likely (but by no means always) move in tandem with the volatility of the underlying. Therefore the more volatile the underlying price is (e.g. in the below illustrations the underlying price is the price of oil) the greater the probability of the out-of-the-money option becoming a winning in-the-money option. To that end falling implied volatility and decreasing time to expiry have a similar effect on the out-of-the-money binary call.

Unlike a conventional call option where increasing the implied volatility increases the value of the option irrespective of whether the option is in-the-money or not, an in-the-money binary call option has negative binary call options vega, which means that when the underlying is above the strike, increasing the implied volatility will decrease the value of the binary call option. The logic follows on from above whereby the option is in-the-money and is therefore in a winning position in relation to the strike. Increasing the volatility will increase the likelihood of the underlying price falling below the strike thereby reducing the binary call option value, in turn leading to a negative binary call options vega.

Fig.1 show the vega of the Oil $100 strike binary call option. The at-the-money binary call options vega is zero because irrespective of the implied volatility the option has a 50:50 chance of ending in-the-money and will always be worth 50.

The vega of the lower implied volatility (20%) 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 40% 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.

Fig.2 illustrates the effect of the passing of time on vega.

The 0.1 days to expiry profile has a concertinaed profile as the price profile has premium in only a very narrow range as out-of-the-money binary calls are likely to be worthless at any significant distance from the strike. The same rationale applies to the 0.1 day in-the-money binary calls which will be worth 100 at any significant distance from the strike.

Maximum absolute values for binary call options vega are uniformly approximately ±0.8 irrespective of time to expiry.