Understanding Option Delta

What is option delta?

There are many factors that affect the value of an option. These include the volatility of the underlying product against which the option is written, the time until the option expires and the expected interest rate or yield curve that will prevail during the option’s life. But the most significant component of an option’s value in the majority of instances, is the value of the underlying product. After all, an option contract is a derivative, meaning essentially that it derives its value from elsewhere. Delta

Typically, options are theoretically valued using mathematical models. These will incorporate a selection of variables and generate a single value for any option in question. Now to the derivatives trader, the risk associated with any option, or portfolio of options, is that one or more of the influencing variables changes in value. So, for instance, the underlying product may become more volatile or time itself may whittle away at the option’s value. Delta is the risk to an option’s value associated with a change in the price of the underlying product. Specifically, we can define delta as the the change in option value for a change in the price of the underlying product.

Understanding delta is clearly therefore of crucial importance to an options trader. Although it may be easily hedged in the first instance (simply by trading the underlying product in the appropriate size and direction), comprehending how delta evolves and is itself affected by changing circumstance, is a core competency for any options trader.

What determines and affects option delta?

A call will have a positive delta, whilst a put will have a negative delta. This is trivially true by the definitions of calls and puts; a call gives its owner the right but not the obligation to buy the underlying product. It is clear therefore that if the price of the underlying product rises, then the option becomes more valuable; hence call deltas are positive. And vice versa for puts whose deltas must be negative. In practice, it is not uncommon to hear the ‘negative’ dropped for convenience; the delta of the put will be referred to in absolute terms, with the negative being implicit.

After the sign of the delta (positive for calls, negative for puts) the next most important factor is the price of the underlying product relative to the strike price of the option. A call option whose strike is far below the current underlying product price is referred to as deep in-the-money. In this case, any change in the underlying product price will be reflected almost perfectly by the change in the call option value. The delta in this case is therefore approaching +1 or 100% (both are used interchangeably). So, with the underlying product trading at say $100, the $10 strike call is likely to have a delta of 100% and a value of $90; there is very little optionality in this option and it is merely a substitute for the underlying product itself. If the underlying product increases in value to say $101, then the $10 call must rise to $91; the increase in value is one for one, reflecting the 100% delta. The same holds for puts whose strike is considerably above the underlying price. A put of strike $200, will also have a delta of (-)100%.

When an option is a long way out-of-the-money, its delta will be close to zero. A small change in the price of the underlying is unlikely to affect the value of the option greatly as its chances of expiring in-the-money are barely altered. Hence, delta is very low for these options.

For options whose strikes are closer to the underlying price, things are a little more interesting. The option whose strike is very near to the price of the underlying product will have a delta approaching 50%. This is not merely because the so-called at-the-money option is halfway between the deep in-the-money option (with 100% delta) and the deep out-of-the-money option (with 0% delta) but also because the chances of the option expiring in-the-money are about half. This in fact is an alternative interpretation of delta; the probability of expiring in-the-money.

Option delta is affected by the option’s longevity. Clearly, an out-of-the-money option that has a very long life ahead of it, will have a higher (absolute) delta than that of an option of the same strike due to expire out-of-the-money in the next ten minutes. The longer dated option has time on its side and may yet become valuable. Hence a change in the underlying product price will have a greater impact on the longer dated option’s value than on a shorter dated option of the same strike.

Implied volatility is also a key factor in delta terms. Increased implied volatility often has an effect analogous to increasing the time left to an option’s expiry. The more volatile a product is expected to be over the course of an option’s life, the more chance the option has of expiring in-the-money and the higher therefore its delta will be (in absolute terms).

The importance of delta to option traders
Delta can be interpreted as the equivalent exposure in the underlying product to price changes, derived from the options portfolio. In other words, if my options portfolio on stock ABCD is showing a combined delta of +50, then I am synthetically long 50 shares of ABCD. Now this is easily hedged simply be selling 50 shares of ABCD. The position then becomes what is known as delta neutral.

However, the story does not end there, because in the world of derivatives and options, nothing ever remains neutral for long! Whilst the delta of the shares is unchanging (the delta of a share with respect to itself is always +1), the delta of the options portfolio will vary considerably over time, with changes in implied volatility and with changes in the underlying price itself. Furthermore, because of the very nature of options, these changes are likely to be exponential and nonlinear. Risk is therefore magnified.