# Binary Put Option Delta

## Binary Put Option Delta measures the change in the price of an option owing to a change in the underlying asset price and is the gradient of the slope of the binary put options price profile versus the underlying.

If one takes a look below at the formula for the Binary Put Option Delta one can see that it is the binary call option delta with a minus sign tagged on the front. In effect the binary put option delta is the binary call option delta reflected through the horizontal axis at zero. As such much of the following description and analysis is a reflection of that of the binary call option delta.

Of all the Greeks the 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 put option with a delta of -0.5 means that if the underlying share price goes up 1¢ then the binary put will decrease in value by ½¢. Another interpretation would be a short 400 contract position in S&P500 binary puts with a delta of -0.25 which would be equivalent to being long 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.

### Binary Put Option Delta & 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 0.1 day price profile of a binary put with Figure 2 showing (in black) the same price profile between the underlying prices of 99.78 and 99.99.

Fig.1 – Binary Put Option Fair Value 0.1-Days to Expiry

Fig.2 – Fair Value & Delta Gradients

The blue ’18 tick chord’ in Figure 2 travels between the point on the put profile 9 ticks below the price of 99.90 to 9 ticks above where the two fair values of the binary put option are 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 Put value at S2

P1         =          Binary Put value at S1

SInc     =         Minimum Underlying Price Change

i.e.  Gradient          =          (54.8254-98.9230) / (99.99-99.81) x 0.01          =          -2.4499

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 Put Delta
Asset Price99.8199.8499.8799.9099.9399.9699.99
δS=0.00-2.3225
δS=0.0394.2052-2.344780.1370
δS=0.0697.3520-2.398168.5748
δS=0.0998.9230-2.449954.8254

As the price difference narrows (as reflected by δS = 0.06 and δS = 0.03) the gradient tends to the delta of -2.3225 at 99.90. The delta is therefore the first differential of the binary put 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 approaches the gradient of the tangent (delta) of the price profile.

### Binary Put Option Delta w.r.t. Implied Volatility

Figure 3 illustrates 5-day binary put profiles with Figure 4 providing the associated deltas over a range of implied volatilities (as in the legend) over an underlying asset price range of 440 ticks.

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 between just -0.04 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 put being a losing bet until the underlying rises to close to the strike where the price profile drops sharply to fall through 0.5 before levelling out at 0. The 1% delta in Figure 4 reflects this dramatic change of binary put price with the 1% delta profile showing zero delta around the underlying of 99.40 followed by a sharply decreasing delta as the binary put price collapss dramatically over a small change in the underlying, followed by a sharply increasing delta as the delta reverts to zero in line with the binary put leveling off at zero. Fig.3 – Binary Put Option Fair Value w.r.t. Implied Volatility This feature of the binary option delta when at the money is that of the Dirac delta function, or δ function, where the area within the delta profile and the horizontal axis at zero is 1. This means that the binary put delta when at-the-money and with time to expiry or implied volatility approaching zero can become infinitely high negative number with a total area of one under the spike. This feature obviously renders delta-neutral hedging as impractical when the binary put option is at-the-money with very little time to expiry or extremely low implied volatility. In practice these conditions and a long at-the-money binary put position in Apple Inc would require the delta-neutral trader to bid for the company in order to get ‘flat’! Fig.4 – Binary Put Option Delta w.r.t. Implied Volatility In the above illustration the 1.00% delta plummets to —3.41 but this falls even more sharply as the time to expiry decreases from 5 days. ### Binary Put Option Delta w.r.t. Time to Expiry Figures 3 & 5 illustrate binary put price profiles which always have a negative slope so the binary put option deltas are always negative. Fig.5 – Binary Put 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 shallowest and lowest absolute value delta profile. At-the-money binary put (and call) options with short time to expiry provide the greatest gearing of any financial instrument as illustrated by the extremely steep at-the-money price profile of Figure 5 and its associated delta in Figure 6. The 0.1-day delta troughs at —4.82 which basically offers gearing of 482% compared to the 100% gearing of a short future position. Fig.6 – Binary Put Option Delta w.r.t. Time to Expiry 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. ### Binary Put Option Delta Application Table 2 shows 10 day, 5% volatility binary put option prices with deltas. Table 2 - Binary Put Option Fair Value with associated Delta Asset Price99.8199.8499.8799.9099.9399.9699.99 Fair Value59.248257.832956.407954.975053.535952.092750.6471 Delta-0.4699-0.4735-0.4764-0.4788-0.4805-0.4816-0.4820 At$99.87 the binary put is worth 56.4079 and has a delta of -0.4764. Therefore, if the underlying rises three ticks from $99.87 to$99.90 the binary put will fall in value to:

56.4079 + 3 x -0.4764 = 54.9787

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

53.5359 + (-3) x -0.4805 = 54.9774

At $99.90 the binary put value in Table 2 is 54.9750 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 high when assessing the upward move from $99.87 to$99.90, while similarly the delta of -0.4805 is too low when evaluating the change in binary put 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 = -0.4776 and should this number be used in the first calculation above then the binary put at $99.90 would be estimated as: 56.4079 + 3 x -0.4776 = 54.9751 an error of 0.0001. 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:

53.5359 + 3 x -0.47965 = 54.97485

an error of just 0.00015.

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

#### Hedging with Binary Put Option Deltas

Example: a binary options trader buys 100 contracts of the $100 strike binary put with 10 days to expiry with the future trading at$99.87 at a price of 56.4079, costing a total of:

56.4079 x $10 x 100 contracts =$56,407.90

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 short 47.64 futures at the futures price of $99.87 so the trader buys 48 futures to hedge. 1) the future falls to$99.81 where the option is worth 59.2482 so the position P&L is now:

Binary Put Option profit:

59.2482 – 56.4079 = 2.8403

which equates to a gain of:

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

Future loses:

99.81-99.87 = -0.06

which equates to a loss of:

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

an overall loss of $39.70 2) the future rises to$99.93 where the option is worth 53.5359 so the position P&L is now:

Binary Put Option loses:

53.5359 – 56.4079 = -2.8720

which equates to a loss of:

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

Future gains:

99.93-99.87 = +0.06

which equates to a gain of:

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

an overall profit of $28.00. This profit on the upside and loss on the downside can partly be explained away by the over-hedging of 48 futures as opposed to 47.64 futures. If 47.64 futures were used then the downside loss would be reduced to: -0.06/0.01 x$10 x 47.64 = -$2,858.40 which generates an overall downside loss of -$18.40.

The upside loss would equate to:

$2,858.40 –$2,872.00 = -\$14.60

Therefore a loss is made on the upside and downside even when an exact delta hedge is assumed. This is because now that this binary put option is in-the-money it has negative gamma. Binary Put Option Gamma

### Binary Put Option Delta v Conventional Put Option Deltas

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

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

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

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

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

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

Fig.7f – 0.01-Day Binary & Conventional Put Option Delta

Points of note are:

1)      Whereas the conventional put deltas are constrained to a value of 0.5 when the option is at-the-money, the binary put 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 put option is lower than the conventional put option, but when time to expiry is reduced (Figs.7d-e) the delta of the binary put becomes higher than the maximum value of 1.0 of the conventional put option.

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

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

### Formula

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

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 put option delta provide instant and easily understood information on the behaviour of the price of a binary put in relation to a change in the underlying. Binary puts always have negative deltas so an increase in the underlying causes a decrease in the value of the binary put.

For the same volatility the binary put option delta which is 50 ticks in-the-money is the same as the delta of the binary put 50 ticks out-of-the-money. In other words, the deltas are horizontally symmetric about the underlying when at-the-money.

When a trader takes a position in any binary put 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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