# Binary Call Option Delta

## Binary call option delta measures the change in the price of a binary call option owing to a change in the underlying asset price and is the gradient of the slope of the binary options price profile versus the underlying asset price (the ‘underlying’).

Of all the Greeks, the binary call option delta could probably be considered the most useful in that it can also be interpreted as the equivalent position in the underlying, i.e. the delta translates options, whether individual options or a portfolio of options, into an equivalent position of the underlying.

A binary call option with a delta of 0.5 means that if the underlying share price goes up 1¢ then the binary call will increase in value by ½¢. Another interpretation would be a short 400 contract position in S&P500 binary calls with a delta of 0.25 which would be equivalent to being short 100 S&P500 futures.

It is important to realise that the delta is dynamically changing as a function of many variables, including a change in the underlying price, and that a change in any of those variables will most likely cause a change in the delta. Therefore, if any or all of the variables, including the underlying price, time to expiry and implied volatility, change then the above option will not necessarily have a delta of 0.5 and increase in value by ½¢ or the equivalent S&P position be short 100 S&P500 futures.

This practicality and simplicity of concept contributes to deltas, out of all the Greeks, being the most utilised amongst traders, especially market-makers.

The following provides an analysis of:

• the finite difference method to evaluate deltas,
• examples of using the delta to hedge with,
• comparisons of conventional call options delta with binary call option delta, and finally
• a closed-form formula for the binary call option delta.

### Binary Call Option Delta and Finite Delta

The delta Δ of any option is defined by:

Δ = δP / δS

where:

P          =          price of the option

S          =          price of the underlying

δP        =          a change in the value of P

δS        =          a change in the value of S

Figure 1 shows the 1 day price profile of a binary call with Figure 2 showing (in black) the same price profile between the underlying prices of 99.78 and 99.99.

Fig.1 – Binary Call Option Price Profile

Fig.2 – Fair Value & Delta Gradients

The blue ’18 tick chord’ travels between the point on the call profile 9 ticks below the price of 99.90 to 9 ticks above. The fair value of the binary call option at 99.81 is 3.4592 and at 99.99 is 46.1739 as provided in the bottom row of Table 1.. The gradient of this chord is defined by:

Gradient          =          ( P2 – P1 ) / ( S2 – S1 ) x SInc

where:

S2         =          S + δS

S1         =          S – δS

P2         =          Binary Call value at S2

P1         =          Binary Call value at S1

SInc     =         Minimum Underlying Asset Price Change

i.e.  Gradient       =          (46.1739-3.4592) / (99.99-99.81) x 0.01

=        2.3730

as indicated in the bottom row of the central column of Table 1.

The gradients of the ‘12 tick chord’ and ‘6 tick 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 Delta
99.81 99.84 99.87 99.90 99.93 99.96 99.99
δS = 0.00 2.4149
δS = 0.03 10.6905 2.4117 25.1610
δS = 0.06 6.2990 2.3999 35.0971
δS = 0.09 3.4592 2.3730 46.1739
From Gradient of Chord to Call Delta

As the price difference narrows, i.e. as δS → 0  (as reflected by δS = 0.06 and δS = 0.03) the gradient tends to the delta of 2.4149 at 99.90. The binary call option delta is therefore the first differential of the binary call option fair value with respect to the underlying and can be stated mathematically as:

δS → 0, Δ = dP / dS

which means that as δS falls to zero the gradient of the price profile approaches the gradient of the tangent (delta) at the underlying asset price.

### Binary Call Option Delta and Implied Volatility

Figure 3 illustrates 5-day binary call profiles with Figure 4 providing the associated deltas over a range of implied volatilities as in the legends.

In Figure 3 the 9% fair value profile is fairly shallow in comparison to the other four profiles which is reflected in Figure 4 where the 9% delta profile fluctuates just 0.16 from a delta of 0.22 at the wings to 0.38 when at-the-money and is the flattest of the five delta profiles. In Figure 3, with the volatility at 1% and underlying below $100, there is little chance of the binary call being a winning bet until the underlying gets close to the strike where the price profile steepens sharply to travel up through 0.5 before levelling out short of the binary call price of 100. Fig.3 – Binary Call Option Fair Value w.r.t. Volatility The 1% delta in Figure 4 reflects this dramatic change of binary call price with the 1% delta profile showing zero delta followed by a sharply increasing delta as the binary call price changes dramatically over a small change in the underlying, followed by a sharply decreasing delta as the binary call option delta reverts to zero as the binary call levels off at the higher price. For the same volatility the delta of the binary call which is 50 ticks in-the-money is the same as the delta of the binary call 50 ticks out-of-the-money. In other words the deltas are horizontally symmetric about the underlying when at-the-money, i.e. when the underlying is at$100.

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

This feature of the binary call option delta when at the money is that of the Dirac delta function, or δ function, where the area below the profile is 1. This means that the binary call option delta when at-the-money and with time to expiry or implied volatility approaching zero can become infinitely high with a total area of one under the spike. This feature obviously renders delta-neutral hedging as impractical when the binary call option is at-the-money with very little time to expiry or extremely low implied volatility. In practice these conditions and a short at-the-money binary call position in Apple Inc would require the delta-neutral trader to bid for the company in order to get ‘flat’!

### Binary Call Option Delta and Time to Expiry

In the above illustration (Fig.4) the 1.00% delta peaks off the scale at 3.41 but this value increases sharply as the time to expiry decreases from 5 days.

Figures 3 & 5 illustrate binary call price profiles which always have a positive slope so the binary call options delta is always positive.

Fig.5 – Binary Call Option Fair Value w.r.t. Time to Expiry

The 25-day price profile in Figure 5 has the longest time to expiry and subsequently has the lowest gearing which is illustrated in Figure 6 by the lowest value delta profile.

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

Short time to expiry binary call (and put) options provide the greatest gearing of any financial instrument as illustrated by the extremely steep price profile of Figure 5 and its associated delta in Figure 6. The 0.1-day delta peaks at 4.82 which basically offers gearing of 482% compared to the 100% gearing of a long future position.

Decreasing volatility and decreasing time to expiry have a similar impact on the price of a binary option which is borne out by the similar delta profiles of Figures 4 & 6.

Table 2 shows 10 day, 5% volatility binary call option prices with deltas.

Table 2 - Binary Call Option Fair Value with associated Delta
Asset Price$99.81$99.84$99.87$99.90$99.93$99.96$99.99 Fair Value40.751842.167143.592145.025046.464147.907349.3529 Delta0.46990.47350.47640.47880.48050.48160.4820 At$99.87 the binary call is worth 43.5921 and has a delta of 0.4764. Therefore, if the underlying rises three ticks from $99.87 to$99.90 the binary call will rise in value to:

43.5921 + 3 x 0.4764 = 45.0213

If the underlying fell 3 ticks from $99.93 to$99.90 the binary call would be worth:

46.4641 + (-3) x 0.4805 = 45.0226

At $99.90 the binary call value in Table 2 is 45.0250 so there is a slight discrepancy between the values calculated above and true value in the table. This is because the deltas of 0.4764 and 0.4805 are the deltas for just the two underlying levels of$99.87 and $99.93 respectively, i.e. the deltas change with the underlying. At$99.90 the delta is 0.4788 so the value of 0.4764 is too low when assessing the upward move from $99.87 to$99.90, while similarly the delta of 0.4805 is too high when evaluating the change in binary call price when the underlying falls from $99.93 to$99.90. The average of the two deltas at $99.87 and$99.90 is:

( 0.4764 + 0.4788 ) / 2 = 0.4772

and should this number be used in the first calculation above then the binary call at $99.90 would be estimated as: 43.5921 + 3 x 0.4772 = 45.0237 an error of 0.0013. The average delta between$99.90 and $99.93 is: ( 0.4788 + 0.4805 ) / 2 = 0.47965 The second calculation above would now generate a price at$99.90 of:

46.4641 + (-3) x 0.47965 = 45.02515

an error of just 0.00015.

The section on binary call option gamma will provide the answers as to why this discrepancy still exists.

### Hedging with Binary Call Option Delta

If the numbers in Table 2 related to a bond future then it might not be unreasonable to offer a binary option on that future with a settlement value of $1000 equating to$10 per point.

Example: a binary options trader buys 100 contracts of the $100 strike binary with 10 days to expiry with the future trading at$99.87 at a price of 43.5921, costing a total of:

43.5921 x $10 x 100 contracts =$43,592.10

How does the trader hedge away the immediate directional exposure?

100 contracts of the option with delta of 0.4764 equates to a position of 47.64 futures at the futures price of $99.87 so the trader sells 48 futures to hedge (just not possible to sell 0.64 of a future…….the option price of 43.5921 was arrived at by ‘averaging in’!) 1) the future falls to$99.81 where the option is worth 40.7518 so the position P&L is now:

Binary Call Option loses:

40.7518 – 43.5921 = -2.8403

which equates to a loss of:

-2.8403 x $10 x 100 contracts = -$2,840.3

Future loses:

99.81-99.87 = -0.06

which equates to a profit of:

-0.06/0.01 x $10 x -48 = +$2,880

an overall profit of $39.70 2) the future rises to$99.93 where the option is worth 46.4641 so the position P&L is now:

Binary Call Option gains:

46.4641 – 43.5921 = 2.8720

which equates to a profit of:

2.8720 x $10 x 100 contracts = +$2,872.00

Future gains:

99.93-99.87 = +0.06

which equates to a loss of:

0.06/0.01 x $10 x -48 = -$2,880

an overall loss of $8.00. This loss on the upside can be explained away by the over-hedging of 48 futures as opposed to 47.64 futures. If 47.64 futures were used (a spreadbet maybe?) then the overall downside profit would be reduced to +$18.10 while the upside loss of $8.00 would turn into a profit of$13.60.

The constant use of deltas for hedging in this manner is vital for an options market-maker. That using a hedge of 47.64 produces a profit on both the upside and downside is the impact of the gamma, in this case positive gamma.

### Binary Call Option Delta v Conventional Call Option Delta

Figures 7a-e illustrate the difference over time to expiry between the binary call option deltas and their conventional cousins for those already familiar with conventionals.

Fig.7a – 25-Day Binary & Conventional Call Delta

Fig.7b – 10-Day Binary & Conventional Call Option Delta

Fig.7c – 4-Day Binary & Conventional Call Option Delta

Fig.7d – 1-Day Binary & Conventional Call Option Delta

Fig.7e – 0.1 Day Binary & Conventional Call Option Delta

Points of note are:

1)      Whereas the conventional call deltas are constrained to a value of 0.5 when the option is at-the-money, the binary call is at its highest when at-the-money and has no constraint being able to approach infinity as time to expiry approaches 0.

2)      When time to expiry is greater than 1 day (Figs.7a-c) the gearing of the binary call option is lower than the conventional call option, but when time to expiry is reduced (Figs.7d-e) the delta of the binary call becomes higher than the maximum value of 1.0 of the conventional call option.

3)      The conventional call option delta profile resembles the price of the binary call.

4)      Substituting a range of implied volatilities instead of the times to expiry would provide a similar set of illustrations to Figs.7a-e.

### Formula

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

where:

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

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

and:

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

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

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

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

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

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

### Summary

Binary call option delta provides instant and easily understood information on the behaviour of the price of a binary call in relation to a change in the underlying. Binary calls always have positive deltas so an increase in the underlying causes an increase in the value of the binary call.When a trader takes a position in any binary call they are immediately exposed to possible adverse movements in time, volatility and the underlying. The risk of the latter can be immediately negated by taking an opposite position in the underlying equivalent to the delta of the position.

For book-runners and market-makers hedging against an adverse movement in the underlying is of prime importance and hence the delta is the most widely used of the greeks.

Nevertheless, as expiry approaches the delta can reach ludicrously high numbers so one should always observe the tenet: “Beware Greeks bearing silly analysis numbers…”.

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