# 4 ways to understand option delta

By , CEO of Volcube.

## Delta as the change in option value for a change in the underlying product price

The most basic definition of delta is as the change in an option’s value for a change in the price of the underlying product. If you have a call option struck on some cheese, then the delta of your call option tells you how much its value will alter when the price of  the cheese changes. If the call has a delta of 20%, then for every \$1 the price of cheese rises, the option will increase by 20 cents. Put options have a negative delta, so a cheese put with a delta of (-)20% will fall in value by 20 cents for every \$1 the cheese price rises.

### How is this useful?

This interpretation of delta is useful because it indicates the sensitivity of the option’s value to price changes in the underlying. If we have an understanding of how volatile the underlying product price is, then we have a handle on how exposed our option position is to these price changes. If the price of cheese very rarely moves more than 50 cents per day, then we might expect our 20% delta call option to rarely make or lose 10 cents per day. This gives us a useful indication of the basic price exposure we face.

## Delta as the option hedge ratio

Knowing how much the option value will change when the underlying product price changes, allows us to hedge appropriately. So if our put option has a delta of (-)20% then for every 100 options we trade, we need to hedge with 20 lots of the underlying (assuming a contract multiplier on the options of 1). This is easiest to see by working through a simple example. If the price of cheese is \$100 and we own 100 lots of the 20% delta puts on cheese, then a \$1 increase in the price of cheese will result in a 20 cent price fall in our put options. So we lose 20 cents * 100 lots = \$20. To hedge this, we need to buy some cheese. The delta tells us how much cheese to buy as a hedge. We need to buy 20 lots of cheese, so that when its price rises by \$1 we will make a \$20 profit to counter the loss on the puts.

### How is delta useful as a hedge ratio?

It should be obvious. The delta tells us exactly how to hedge options to prevent losses due to changes in the price of the underlying.

## Delta as the likelihood of expiring in-the-money

This one probably gets more weight than it should, but can be useful nevertheless. Basically, delta can under certain assumptions be seen as the probability or likelihood that an option will expire in-the-money. So a 20% delta call could be thought to have a 20% chance of expiring in-the-money. This is used by some traders in order to select which options to trade. For example a trader might look for a put that has only a 10% chance of expiring in-the-money; he might want to take his chances on this put expiring worthless and short (write) the put option. Using the delta, the trader can find out which put (i.e.the put with which strike) this is. If cheese is trading at \$100, the (-)20% delta put option with 3 months to expiry might have a strike of \$92. The trader might therefore interpret the delta of the put to mean that this put has a 20% chance of expiring in-the-money. The same idea works for combinations of options (option strategies) to find the probabilities of a range of prices where the underlying might be when the options expire. For example if the cheese put with a 10% delta has a strike of \$90, we might suggest that there is a 20% chance of the options expiring with the price of cheese below \$92, a 10% chance of it expiring between \$90 and \$92 and a 10% chance of it expiring below \$90.

### Note of caution about option delta and probability of expiring in-the-money

This probability is highly theoretical. It is not a FACT about the options that will always be true. All it means is that if every assumption in the pricing model that has been used to formulate the delta turns out to be true, then the delta can be interpreted as the probability of expiring in-the-money, in some cases. This is very unlikely to be the case consistently or even frequently. Volatility can be higher or lower than expected. Interest rates can move. Indeed, for some options where cost of carry or dividends are relevant, this interpretation of delta is even more precarious. Nevertheless, as a rule of thumb, option delta as the probability of expiring in-the-money is undoubtedly useful to know.

## Delta as the equivalent position in the underlying product

Probably the main use of delta in the markets. If we own 100 call options on cheese with a 20% delta, then this is equivalent to owning 20 lots of cheese from a risk perspective. We could neutralise this risk by selling 20 lots of cheese (the exact same idea as delta viewed as the hedge ratio) or we could trade options to achieve the same effect. For example if we buy 100 lots of the (-)20% delta puts on cheese, this will cancel out our +20% delta call delta. Our equivalent position in the underlying product becomes zero. When traders refer to being long or short deltas they mean long or short an equivalent amount of the underlying, whether this is coming from an option position or a straight position in the underlying.

All these interpretations come from the same definition of option delta. Indeed, they are identities; exactly the same thing but viewed from a different perspective. All have their uses and every option trader must know how and when each is applicable. And of course, the best way to learn this is by trading options on Volcube!

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