Binary Put Options Delta

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Binary put options delta is the metric that describes the change in the binary put (fair) value due to a change in the underlying price. It is the first derivative of the binary put option fair value (P) with respect to a change in the underlying price(S). It is depicted as:

\Delta = \frac{dP}{dS}

The binary put options delta is the gradient of the price profiles of Figures 1 & 2 on Binary Put Options.

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

100 binary 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 binary put options delta is a dynamic number which has its own delta, the binary put gamma.

The binary put options delta profiles are the binary call delta reflected through the horizontal axis at zero. Therefore the binary 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 binary put options delta will approach negative infinity.

Binary Put Delta over Time

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

Binary Put Options Delta
Figure 1 – Binary Put Options Delta w.r.t. Time to Expiry

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 binary put mutates into (along with the binary call option) the most dangerous instrument in existence.

Binary Put Delta and Volatility

Binary put options 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%.

Binary Put Options
Figure 2 – Binary Put Options w.r.t. Volatility

The 45% delta profile reflects the gradient of the Figure 2 binary put price profile of binary 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 binary 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 binary put option price profile of Figure 2 of the Binary 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:

Binary Put Option Delta = –(P1‒P2)/(S1‒S2)


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

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

Binary 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:


so that the 2-day delta becomes:

Binary 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 binary put options gamma to almost zero.

A Practical Example: At the underlying gold price of $1725 I buy 100 $1700 binary 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

By: Hamish Raw

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