# Digital Put Delta

The digital put delta is the gradient of the price profiles of Figures 1 & 2 on Digital Put Options.

The practical relevance of the digital put delta is it provides a ratio that converts the digital put position into an equivalent position in the underlying. So, 100 out-of-the-money digital puts with a delta of –0.25 has a short underlying position equivalent to:

100 digital puts = –0.25 x 100 = –25 futures, or short 25 futures

A future has a straight line P&L profile whereas, in general, options have a non-linear P&L profile. The delta changes with the asset price so the equivalent position is only good for the current underlying asset price. Not only will a change in the underlying have a bearing on the delta, but other factors such as implied volatility and time to expiry also will have a say. The digital put option delta is a dynamic number which has its own delta, the digital put gamma.

The digital put option delta profiles are the digital call delta reflected through the horizontal axis at zero. Therefore the digital put options delta is always zero or negative and is at its most negative when at-the-money. As time to expiry approaches zero the digital put options delta will approach negative infinity.

### Digital Put Delta over Time

Digital put delta is displayed against time to expiry in Figure 1. As time to expiry decreases the delta profile becomes increasingly narrow around the strike.

When there are 25-days to expiry and implied volatility is at 25% the absolute value of the delta is low. Yet in the last hours of its life the digital put mutates into (along with the digital call option) the most dangerous instrument in existence.

### Digital Put Delta and Volatility

Digital put delta over a range of implied volatilities is provided in Figure 2. This chart illustrates the increasing influence on the delta as the volatility falls from 45% to 15%.

The 45% delta profile reflects the gradient of the Figure 2 digital put price profile of digital put options. At 97.80 the delta is -0.07, at 100.00 the delta is -0.076 and at 102.20  -0.069, almost flat.

The digital put option with either a high number of days and/or a high volatility is not an immediate directional play. When there are 25 days to expiry or 25% volatility it is difficult to see why the option would have any immediate interest to anyone.

#### Finite Delta

The 2-day, 25% implied volatility $100 digital put option price profile of Figure 2 of the Digital Put Options page at an underlying price of$101.00 shows the put to be worth 29.8599. At the underlying prices of 100.80 and 101.20 the options are worth 33.6760 and 26.2606 respectively. Using the finite difference method:

Digital Put Delta = –(P1‒P2)/(S1‒S2)

where:

S1 = The lower underlying price
S2 = The higher underlying price
P1 = Digital Put Option price at the lower underlying price
P2 = Digital Put Option price at the higher underlying price

so that the above numbers provide a 2-day digital put options delta of:

Digital Put Delta = ‒(33.6760‒26.2606)/(100.80‒101.20) = ‒0.1854

If the underlying price increment was reduced from 0.01 to 0.00001 then:

S1=100.99999
S2=101.00001
P1=29.8601
P2=29.8597

so that the 2-day delta becomes:

Digital Put Delta = ‒(29.8601-29.8597)/(100.99999‒101.00001) = ‒0.185628

so that the narrowing of the underlying price increment has made little difference. This is because the high implied volatility has reduced the digital put options gamma to almost zero.

A Practical Example: At the underlying gold price of $1725 I buy 100$1700 digital put options contracts at a price of 31.408697 with a delta of -0.702929 so that I also buy 100 x ―0.702929 = 70.2929 futures at 1725. If the underlying rises to $1730 the option is worth 27.987386 while if it falls to 1720 it has gained value and is worth 35.008393. How does the P&L look at these two new underlying prices? At$1730 the options P&L:

100 contracts x (27.997386-31.408697) = ―342.1311 ticks

70.21 contracts x (1730-1725) = +351.0503 ticks

Profit = 351.0503-342.1311 = 8.9192

At \$1720 the options P&L:

100 contracts x (35.008393-31.408697) = +359.9696 ticks

70.21 contracts x (1720-1725) = ―351.0503 ticks

Profit = 359.9696-351.0503 = 8.9193

### Summary

• The digital put is 1.0 less the value of the same strike, same expiry digital call.
• Therefore the value of the digital call plus the digital put must equal 1.00.
• If the asset price finishes exactly on the strike both call and put settle at 0.5.