# Binary Call Options

## At-the-Money Settlement Rules

If the underlying at expiry is exactly on the strike (at-the-money) settlement can be treated in numerous ways: the two obvious candidates are that the binary call options 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. This approach has a particular advantage if binary call options and puts with the same strike are being offered since the call and put settlements would sum to 100, otherwise with the first two alternatives the aggregate settlement would be 200 or zero. Another approach sometimes used with the underlying settling on the strike is to simply void all bets.

### Binary Call Option’s Greeks

For those looking for a high level overview of the binary call 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 Call Option Formula

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

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 Call Options Price Profiles

The price of binary call options could be interpreted as the probability of the event happening if there is a zero cost-of-carry, i.e. interest rates are zero. What are referred to as prediction markets are sprouting up using binary call options and are now widely seen as a more accurate assessment of the probability of an event happening than analyst’s forecasts.

### Binary Call Options Over Time

The first graph shows the expiry profile of  Oil $100 binary call options while the graph below shows the P&L profile illustrating how the expiry profile was arrived at over time. Zero interest rates are assumed as usual. Fig.1 – Expiration Value of$100 Binary Call Option

Fig.2 – Oil $100 Binary Call Options w.r.t. Time to Expiry The buyer of binary call options is betting that Oil will be above$100 at expiry. The 8-day profile is shallow 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 binary call options theta, the in-the-money has a positive binary call options theta while the at-the-money has a binary call options theta of zero assuming that the above ‘dead heat’ rule is applied.

### Binary Call Options and Implied Volatility

Implied volatility is a critical input into the pricing of binary options and the level of implied volatility determines whether one is buying the binary option cheaply or too expensively. Figure 3 displays the oil binary call price profile over a range of implied volatilities.

Fig.3 – Oil $100 Binary Call Options w.r.t. Implied Volatility At the underlying price of$97.00, as implied volatility increases, so does the value of the out-of-the-money option. This is because with a low volatility the probability of the underlying price rising above the strike is low, which in turn will lead to worthless binary call options. As volatility increases and the underlying swings around more there is a greater chance of the binary option moving in-the-money, which in turn means the option will have a better chance of being a winner. So, if an increase in implied volatility increases the value of the option the option has positive vega.

Alternatively, when the underlying is above the strike the 20% implied volatility profile is worth more than the other volatilities. This is because it is in-the-money so that if the underlying remains static the option will ultimately be worth 100. Increasing the volatility increases the probability that the underlying could slide under the strike thereby ultimately generating an option with a zero final settlement price. When an increase in implied volatility leads to a decrease in the value of the option the option is said to have negative vega.

### Summary

The binary call option is at the root of all financial instruments. Any other instrument invented can be constructed from a portfolio of binary call options. This simplistic instrument is the key to all financial engineering: as software code can ultimately be reduced to a series of 0’s and 1’s, so can the world of financial markets.

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