This page provides the derivation of the binary call option vega formula from first principles, illustrates the binary call option vega with respect to time to expiry and implied volatility, followed by the formula itself. Zero interest rates are assumed as usual. The vega has crucial importance when conducting binary options portfolio risk management or when simply taking a single speculative position.
The trader using binary options to take directional views needs to understand the effect of vega since a purchase of binary calls might well be complemented with a rise in the underlying, but a change in implied volatility could negatively affect the value of the binary call option after the move. Figure 1 shows binary call option price profiles over different implied volatilities. Figure 2 shows how with seven static underlying prices, the binary call options change in value as the implied volatility rises from 1.
What also might be recognised is that the legend is inverted from the same illustration in binary put option vega. This being because at When the underlying price is What this suggests is that as implied volatility rises the option increases in value when out-of-the-money positive vega and decreases in value when in-the-money negative vega. Figure 2 shows how the binary call options change value for a particular underlying price where implied volatility is shown on the horizontal axis.
The gradient of an individual profile for a particular implied volatility will provide the vega for that binary call option. It is evident that below the Fair Value of 50, i. At the same time above the fair value price of 50 the options are falling in value as implied volatility rises, leading to negatively sloping profiles and negative vegas.
As the implied volatility continues to rise to The vega as represented by the above formula Eq 1 measures the gradient of the slopes in Figure 2. Chords have been added centred around Since the price profile is increasing exponentially, the gradient of the chords decrease the longer the length of the chord. As the difference between implied volatilities narrows the gradient tends to the vega of 0.
The vega is therefore the first differential of the binary call fair value with respect to implied volatility and can be stated mathematically as:. Figure 1 illustrates 4-day to expiry binary call profiles with Figure 4 providing the associated vegas for the same implied volatilities. Irrespective of the implied volatility the vega when at-the-money is always zero. When out-of-the-money the binary call option vega is always positive as with out-of-the-money conventional call options but when in-the-money the binary call option vega is negative unlike in-the-money conventional call options.
As the implied volatility falls from The maximum absolute vega in Figure 6 is fairly steady at around 2. Irrespective of time to expiry the binary call option vega travels through zero for the now familiar reason that at-the-money binaries are priced at 50, or very close to it.
This formula is based on binary call option prices that range between 0 and 1. Should a vega be required for binary call option prices that range between 0 and then the vega should be multiplied by Vega is an indispensible metric for the binary options market-maker but can also be used proficiently by the speculator, especially the speculator who is trading one-touch calls and puts and double no-touch strategies.
Assessing the change of vega due to a move in the underlying can be critically important so that when buying and selling options it is sometimes just not good enough to forecast the direction of the underlying, it is also important to forecast what implied volatility will do should your directional forecast prove correct.
You must be logged in to post a comment. Binary Call Option Vega Call option vega measures the change in the price of an option owing to a change in implied volatility and is the gradient of the slope of the binary call options price profile versus implied volatility.
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