European Binary Options | Binary Put Options Delta | Binary Put Options Gamma | Binary Put Options Theta | Binary Put Options Vega |

Binary put options are a fixed odds bet that the price of the underlying asset is going to be below a specific price at some time in the future. The specific price is know as the strike price.

### Binary Put Options at Expiry

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

- 100.00, if in-the-money at expiry, i.e. the underlying asset price is below the strike, or at
- 0 if out-of-the-money, i.e. the asset price is above the strike at zero, or at
- 50 if at-the-money, i.e. the ‘dead heat’ rule applies as the asset price settles exactly on the strike at expiry.

The price of the binary put 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.

Figure 1 shows the expiry profile of a $100.00 strike binary put option. With this example the trader long the put will win if the asset price settles below the strike of $100.00

The profile would appear to be a binary call option reflected through the vertical axis at 50. This is a fair assessment since:

**Binary Put Option Fair Value** = **100 ― Binary Call Option Fair Value**

#### Binary Put v Binary Call

Selling a binary put is the same as buying the same strike, same expiry binary call option.

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

- 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 a deep in-the-money call. Yet selling this deep-in-the-money call is no different to buying the same strike far out-of-the-money put.
- The client’s broker simply does not permit the trader to short options.
- Thirdly, buying a binary put option possibly provides the trader a slightly easier calculation of downside risk than selling a binary call option.

### Binary Puts and Time to Expiry

Figure 2 illustrates how the binary put’s expiry profile of Figure 1 is arrived at over time.

The profiles always slope down to the right generating a delta that is always negative or zero. The 0.1-day profile is the steepest of the profiles close to the strike and therefore has the most negative delta.

The buyer of this binary put is betting that the asset price is below $100. The 0.1-day profile will become almost vertical around the asset price of $100.00 as the time to expiry approaches zero. Yet again we are looking at, along with the binary call, the most highly geared financial asset in existence.

What is also apparent from the profiles over time is that the bet increases in value when the asset price is below the strike price. Above the strike it decreases in value. Therefore the out-of-the-money has a negative theta and the in-the-money has a positive theta. When the option is at-the-money it has a theta of zero. This is unlike conventional put options where theta is always negative irrespective of where the strike is to the asset price.

### Binary Puts and Volatility

Figure 3 shows the put value as volatility changes.

Here we see the 45% volatility contributing to a lack of gearing with a very shallow price profile. As volatility falls from 45%, when the option is in-the-money, the binary put increases in value. This is because a progressively lower volatility increases the chance of the asset price staying above the strike.

Below the strike the opposite takes place as a higher volatility increases the likelihood of the asset price rising to settle above 100.00.

Therefore the vega of the binary put is negative when the asset price is below the strike and positive when the asset price is above the strike.

By: Hamish Raw