# Binary Put Options

## Binary Put Options are all-or-nothing options that settle at 100 (if in-the-money at expiry, i.e. the underlying price is below the strike) or at zero (if out-of-the-money, i.e. the underlying price is above the strike).

If at expiry the underlying price is exactly on the strike price then the settlement price of binary puts can be established in different ways, i.e. the two obvious alternatives are that binary puts are treated as in-the-money or out-of-the-money and are settled at 100 or 0 respectively. A possibly more rational method would be to treat the settlement as a ‘dead heat’ and settle the bet at 50. Should there be binary calls and binary puts trading then it is imperative that the rules ensure that the sum of binary calls and binary puts settlement prices comes to 100. To that end it makes eminent sense that the expiry settlement price of at-the-money binary calls and binary puts trading on the same underlying with same expiry should both be 50.

The price of binary puts could be interpreted as the probability of the underlying price being below the strike assuming zero cost-of-carry, i.e. interest rates are zero.

## Binary Put Option’s Greeks

For those looking for a high level overview of the binary put options Greeks then the ‘Descriptive’ page may be suitable, while a more in-depth understanding of the mechanics, plus formulae, are provided in the ‘Analytic’ version:

DescriptiveAnalyticOut-of-the-MoneyIn-the-Money
GammaGamma+ve-ve
ThetaTheta-ve-ve
VegaVega+ve-ve

### Binary Put Option Formula

Binary Put Option Fair Value = $e^{-rt}.\left (1 - N\left ( d_{2} \right )\right)$

where:

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

and:

S = price of the underlying asset

E = strike / exercise price

r = risk free interest rate

D = continuous dividend yield of the underlying asset

t = time in years to expiry

σ = annualised standard deviation of asset returns

### Binary Put Options Price Profiles

Figure 1 shows the expiry profile of a Gold $1700 binary put option. Fig.1 – Gold$1700 Binary Put Options Settlement Values at Expiry

The profile would appear to be a binary call option reflected through the horizontal axis at 50 which would be a fair assessment since:

Binary Put Options Fair Value  =  100 – Binary Call Option Fair Value

so that selling binary call options is the same as buying same strike, same expiry binary put options.

If binary put options are so trivial what might be the rationale of offering binary put options as well as a binary call options to customers?

i.            Firstly, many retail speculators and investors remain uncomfortable with the idea of ‘shorting’ financial instruments. Going ‘short’ is what those reprobates of financial markets, hedge funds, do. Shorting is for professionals. This might mean that a speculator who wishes to bet that a market will fall will be uncomfortable selling deep in-the-money binary calls whereas buying lower premium out-of-the-money binary put options is more palatable.

ii.            Secondly, buying binary put options possibly provide slightly easier calculation of downside risk than selling  binary call options.

Figure 2 illustrates a price profile illustrating how the expiry profile of Figure 1 is arrived at over time. The buyer of this binary put option is betting that the Gold price is below $1,700. The 25-day profile is almost horizontal and one could be excused for thinking that this must surely be the most prosaic, boring financial instrument in existence. But over time this animal changes its spots to become the most highly geared and dangerous instrument in the world of finance. It is doubtful that any other single instrument can offer a P&L profile that can exceed an angle of 45°. Indeed the angle of an at-the-money moments before expiry tends to the vertical and becomes absolutely unhedgeable. What is also apparent from the profiles over time is that the bet decreases in value when out-of-the-money and increases in value when in-the-money, i.e. the out-of-the-money has a negative theta, the in-the-money has a positive theta while the at-the-money has a theta of zero assuming that the above ‘dead heat’ rule is applied. The profiles also slope down to the right with the rising underlying price generating a delta that is always negative or zero. The 0.5-day profile is the steepest of the profiles close to the strike and therefore has the most negative delta. #### Binary Put Options and Implied Volatility Figure 3 presents binary put options over a range of implied volatility. The profiles are all fairly close together since the implied volatility is relatively high so that large incremental changes are required to substantially shift the fair value. Fig.3 – Gold$1700 5-Day Binary Put Options Fair Value w.r.t. Implied Volatility

To underline this point the same profiles are displayed in Figure 4 with just 0.5-days to expiry and the profiles remain tightly spread.

Fig.4 – Gold \$1700 0.5-Day Binary Put Options Fair Value w.r.t. Implied Volatility

The fact that the profiles are in a narrow range means that the vega is low. If one were to draw a vertical line from an underlying price along the bottom axis then the change of 1% measures the binary put option vega.

The attraction of binary put options is the limited risk nature of both buying and selling this instrument. The gearing obtained from the 0.5-day Gold binary put option is phenomenal whatever the implied volatility and such gearing is unavailable with any other limited risk financial instrument.

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