Digital puts are all-or-nothing options that settle at:
- 1.00, 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
- 0.50 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 digital 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.
Digital Put Options at Expiry
Figure 1 shows the expiry profile of a $100.00 strike digital 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 digital call option reflected through the vertical axis at 0.50. This is a fair assessment since:
Digital Put Option Fair Value = 1.00 ― Digital Call Option Fair Value
when interest rates and the asset’s yield is zero.
Digital Puts v Digital Calls
Selling a digital put is the same as buying the same strike, same expiry digital call option.
If the digital put option is so trivial what might be the rationale of offering a digital put as well as a digital 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 digital put option possibly provides the trader a slightly easier calculation of downside risk than selling a digital call option.
Digital Put Options and Time to Expiry
Figure 2 illustrates how the digital 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 digital 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 digital 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.
|European Digitals||Digital Put Delta||Digital Put Gamma||Digital Put Theta||Digital Put Vega|
Digital Put Options and Volatility
Figure 3 shows the put value as volatility changes.
Here we see the 18% volatility contributing to a lack of gearing with a very shallow price profile. As volatility falls from 18%, when the option is in-the-money, the digital put increases in value. This is because a progressively lower volatility increases the chance of the asset price staying below the strike.
Above the strike the opposite takes place as a higher volatility increases the likelihood of the asset price falling to settle below 100.00.
- The theta of the digital put is positive below the strike and negative above it.
- Theta is zero for at-the-money digital put options.
- The vega of the digital put is negative when the asset price is below the strike and positive when the asset price is above the strike.
- Vega is zero when the digital put is at-the-money.
- As volatility decreases the digital put price profile tends to the vertical hence, along with the digital calls, they become the most highly geared of instruments.