This post is publushed by Hamish Raw of https://hamishraw.com/

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

Delta= P/S

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

The practical relevance of the binary put options delta is that it provides a ratio that can convert the binary put options position into an equivalent position in the underlying. So, if the out-of-the-money binary put has a delta of ―0.25 then a long position in that binary put of, say, 100 contracts would be equivalent to:

100 binary puts = ―0.25 x 100 = ―25 futures, or short 25 futures.

Since a future has a straight line P&L profile whereas, in general, options have a non-linear P&L profile, that delta and subsequent equivalent position is only good for that underlying price. In fact not only will a change in the underlying have a bearing on the delta, but other factors such as implied volatility, time to expiry, and possibly interest rates and yield too will have a say. The binary put options delta is a dynamic number which has its own delta, the binary put options gamma.

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

Binary put options delta over a range of implied volatilities is provided in Figure 2. Here, even with implied volatility at 15% and 5-days to expiry the absolute delta is in excess of 1.0, the maximum value of a conventional delta.

When the binary put options delta, or any other delta for that matter, is capable of being so high one would not expect an extremely competitive bid/ask spread from the market maker as the directional risk incurred in taking on the trade can easily offset the profit on the bid/ask

Finite Delta

The 5-day, 25% implied volatility $1700 binary put option price profile of Figure 2 of the Binary Put Options page at an underlying gold price of $1725 shows the put to be worth 31.408697. At the underlying gold prices of 1724.5 and 1725.5 the options are worth 31.761051 and 31.058130 respectively. Using the finite difference method:

**Binary Put Option Delta = (P _{1}‒P_{2})/(S_{1}‒S_{2})**

where:

S_{1} = The lower underlying price

S_{2} = The higher underlying price

P_{1} = Binary Put Option price at the lower underlying price

P_{2} = Binary Put Option price at the higher underlying price

so that the above numbers provide a 5-day binary put options delta of:

Binary Put Options Delta = ‒(31.761051‒31.058130)/(1724.5‒1725.5) = ‒0.702921

If the underlying price increment was reduced from 0.5 to 0.00001 then:

S_{1}=1724.99999

S_{2}=1725.00001

P_{1}=31.408704

P_{2}=31.408690

so that the 5-day delta becomes:

Binary Put Options Delta = ‒(31.408704‒31.408690)/(1724.99999‒1725.00001) = ‒0.702929

so that the narrow of the underlying price increment has made little difference. This is because the high implied volatility and time to expiry have 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

This hedge has created a profit on the upside almost equal to the profit on the downside. The hedge has been almost exact.

N.B. Pricing binary call and put options in the range 0-100 requires careful checking of the actual greeks using examples such as above. This also applies to conventional options and binary options where the underlying tick value may not equal the option tick value. Using a worked example such as above immediately provides a check on the greek.