# 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.

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. For the options market-maker who is conducting dynamic portfolio risk management the vega is in effect what the delta-neutral market-maker is trading, constantly buying and selling ‘vol’ and hedging away the deltas via trading the underlying. So for the market-maker, knowing ones vega is the same as a futures trader knowing how many futures contracts they are long/short.

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.

### Binary Call Option Vega and Finite Vega

The vega V of any option is defined by:

V = δP / δσ

where:

P     =     price of the option

σ     =     implied volatility

δP     =    a change in the value of P

δσ     =    a change in the value of σ

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 1.0% to 45.0%, so in effect a profile from Figure 2 is a vertical cross section at that underlying price in Figure 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 99.75 in the put option example the option is in-the-money, while with the call option version here, the option is out-of-the-money.

When the underlying price is 100.00 the option is at-the-money and the changes in implied volatility has no effect on the price of the binary option as it is always 50. The 18.0% profile of Figure 1 is the highest of profiles when out-of-the-money (where S<100.00) but the lowest of the profiles when the binary call option is in-the-money (S>100.00). 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).

Fig.1 – Binary Call Option Price profiles w.r.t. Implied Volatility

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.e. where the options are out-of-the-money, the value of the option increases as implied volatility rises along the lower axis, meaning positively sloping profiles and hence positive vegas. 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 45.0% all the profiles concertina around 50 and flatten out leading to very low vega at very high implied volatilities.

Fig.2 – Binary Call Option Price profiles with Fixed Underlying Prices

The vega (as represented by the above formula Eq(1) measures the gradient of the slopes in Figure 2.

Figure 3 is the S=99.75 price profile running from 4.0% implied volatility to 16.0% implied volatility, it is a section of the 99.75 profile of Fig.2. Chords have been added centred around 10.0% implied volatility so that, for example, the 6.0% chord stretches from 7.0% ‘vol’ to 13.0% ‘vol’. Since the price profile is increasing exponentially, the gradient of the chords decrease the longer the length of the chord.

The gradient of the chord is defined by:

Gradient     =      ( P2 – P1 ) / ( σ2 – σ1 )

where:

σ2    =    σ + δσ

σ1    =    σ – δσ

P2    =     Binary Call value at σ2

P1    =     Binary Call value at σ1

i.e.        Gradient    =    (42.4366 ― 36.4953) / (13 ‒ 7)     =    0.9902

as indicated in the δt = 6% row of the central column of Table 1.

Fig.3 – Slope of the Vega at \$99.75 plus approximating Vega ‘chords’

The gradients of the ’10.0% chord’ and ‘2.0%  chord’ are calculated in the same manner and are also presented in the central column of Table 1.

Table 1 - From Gradient of Chord to Call Vega
Implied Volatility5%7%9%10%11%13%15%
δσ =0.0%0.9056
δσ =2.0%39.34300.914341.1717
δσ =6.0%36.49530.990242.4366
δσ =10.0%31.53151.182743.3583

As the difference between implied volatilities narrows the gradient tends to the vega of 0.9056 at 10.0% implied volatility, i.e. where δσ = 0.0%. The vega is therefore the first differential of the binary call fair value with respect to implied volatility and can be stated mathematically as:

as δσ → 0,     V = dP / dσ

which means that as δσ falls to zero the gradient approaches the tangent (vega) of the price profile of Figure 2 at 10.0% implied volatility.

### Binary Call Option Vega w.r.t. Implied Volatility

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).

Fig.4 – Binary Call Option Vega w.r.t. Implied Volatility

As the implied volatility falls from 18.0% (where the absolute values of the vega are the lowest of the profiles) the peaks and troughs of the vegas increase absolutely while the peaks and troughs move closer to the strike.

### Binary Call Option Vega w.r.t. Time to Expiry

Figures 5 & 6 provide the binary call options price profiles over time to expiry with the associated binary call option vega.

The maximum absolute vega in Figure 6 is fairly steady at around 2.43 irrespective of the time to expiry, although the time to expiry does determine how close to the strike the peak and trough in vega is.

Fig.5 – Binary Call Option Price profiles w.r.t. Time to Expiry

Fig.6 – Binary Call Option Vega w.r.t. Time to Expiry

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.

Points of note are:

1)    Whereas conventional call option vegas are always positive as an increase in implied volatility always increases the value of the option, the effect of an increase in implied volatility with binary call options can be positive or negative dependent on whether they are in- or out-of-the-money.

2)    Whereas with conventional call options vega is always at its absolute highest when at-the-money, the binary call option vega when at-the-money is always zero.

3)    Out-of-the-money binary call options have positive or zero vega, in-the-money binary call options have zero or negative vega.

### Formula

$\textup{Binary&space;Call&space;Option&space;Vega}=-e^{-rt}{N}'\left&space;(&space;d_{2}&space;\right&space;)\left&space;(&space;\sqrt{t}+\frac{d_{2}}{\sigma&space;}&space;\right&space;)$

where:

$d_{2}=\frac{log\left&space;(&space;\frac{S}{E}&space;\right&space;)+\left&space;(&space;r-D-\frac{\sigma&space;^{2}}{2}&space;\right&space;)t}{\sigma&space;\sqrt{t}}$

${N}'\left&space;(&space;x&space;\right&space;)=\frac{1}{\sqrt{2\pi&space;}}.e^{-0.5x^{2}}$

and:

$\textup{S}=\textup{price&space;of&space;the&space;underlying}$

$\textup{E}=\textup{strike/exercise&space;price}$

$\textup{r}=\textup{risk&space;free&space;rate&space;of&space;interest}$

$\textup{D}=\textup{continuous&space;dividend&space;yield&space;of&space;underlying}$

$\textup{t}=\textup{time&space;in&space;years&space;to&space;expiry}$

$\sigma&space;=\textup{annualised&space;standard&space;deviation&space;of&space;asset&space;returns}$

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 100 then the vega should be multiplied by 100.

### Summary

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.

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