# Binary Call Options

Binary call options are all-or-nothing options that settle at:

• 100.00 if in-the-money at expiry, or
• zero if out-of-the-money.
• If the underlying at expiry is exactly on the strike, ‘at-the-money‘, settlement can be treated in numerous ways:
• the option is treated as in-the-money or out-of-the-money and is 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. This would allow the call and put settlements to 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.

On this website the method of settling the options at 50 is adopted when the asset price and strike price are exactly equal at expiry.

The price of binary call options could be interpreted as the probability of the event happening. This assumes a zero cost-of-carry and yield, i.e. interest rates are zero with no dividends or coupon payments.

### Binary Calls at Expiry

Figure 1 shows the expiry profile of a \$100 strike binary call:

• Below the strike price of 100.00 the binary call has lost and settles at 0.
• Above the strike price of 100.00 the binary call has won and settles at 100.
• In the event of the asset price settling exactly on the strike of 100.00 on this website is adopted the value of 50. This represents a dead heat.

If the graph was flipped upside down around the vertical axis at 50 it becomes the binary put option expiry profile.

### Binary Calls over Time

Figure 2 shows the P&L profile illustrating how the expiry profile was arrived at over time. The price profiles are always flat or rising from left to right indicating that the binary call always has a positive or zero delta.

Below the strike price it is clear that the price profiles become steeper as they approach the strike price. Conversely they become less steep above the strike price. This indicates the unusual (if you are a conventional options trader) feature of the binary call gamma being positive below the strike and negative above it.

The buyer of this binary call is betting that the asset price will be above \$100 at expiry. The 25-day profile is shallow but over time this animal changes its spots. It becomes 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. Conversely it increases in value when in-the-money. Therefore, the out-of-the-money call has a negative binary call option theta while the in-the-money has a positive call theta. Furthermore, the at-the-money has a binary call option theta of zero assuming that the above ‘dead heat’ rule is applied.

### Binary Call Options and Volatility

Implied volatility is a critical input into the pricing of binary options. The level of implied volatility determines whether one is buying the binary option cheaply or too expensively. Figure 3 displays the binary call options price profile over a range of 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 indicates a positive vega. Above the strike an increase in ‘vol’ lowers the option value; This indicates negative vega.

Why does the price increase at 97.00? This is because with a low volatility the probability of the underlying price rising above the strike is low. Over time this will, in turn, lead to a worthless binary call option. On the other hand, as volatility increases and the underlying swings around more there is a greater chance of the asset price moving above the strike.

### Summary

As long as the asset price is above the strike the binary call option is referred to as being ‘in-the-money’.

When the call option is in-the-money the option holder will have a better chance of being a winner if volatility falls. An increase in implied volatility decreases the value of the option. This is because the option has a higher probability of the underlying sliding back down through the strike. Therefore, above the strike the binary call option has a negative vega.

By: Hamish Raw

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