When you start trading or investing with stock options, very quickly you will find that understanding the implications of the stock option’s delta is key. Delta is key to getting the leverage, risk-reward, and protection in the underlying stock or ETF that you want. So you may be wondering how to calculate the delta of options.
Actually, that is pretty easy and we will see that in a moment. But the delta of options is more useful than just for an individual option. We’ll also look at how to use delta for multiple positions and delta-neutral strategies.
How to calculate the delta of options
Firstly then, delta is just the dollar amount by which the price of an option moves when the price of the underlying moves one dollar.
If the price of the option goes up by $0.50 when the price of the underlying goes up by $1, then that option has a delta of plus 0.50.
If the price of the option goes down by $0.50 when the price of the underlying goes up by $1, then that option has a delta of minus 0.50.
Call options go up as the price of the underlying increases, so they have a positive delta.
Put options go down as the price of the underlying increases, so they have a negative delta. And here is what that looks like in diagrammatic form.
Simple stuff really. Well … not really. But before we go any further, if any of this terminology is unknown to you, here is an article that explains the basics of stock options.
Simplicity aside, there is an important feature of options that we really need to grasp in order to trade and use them profitably. And that is that some attributes of an option, and in particular the price, rarely stay static. The delta along with a number of other Greek letters is also one of those attributes that tend not to stay static.
There are some things though that are fixed for a specific option. They are:
- the underlying stock or ETF,
- the strike price,
- and the expiration date.
But the price, the delta, and all or most of the other funky Greek letters associated with the option will change. And they will change as:
- the price of the underlying changes
- the expiration date of the option approaches
- interest rates change (if they do, it happens sometimes)
- the market’s skittishness increases or decreases, this is reflected in the implied volatility
- a dividend event in the underlying approaches
That is an awful lot of variables. To get a sense of how each of these works, let’s see what happens when we vary one of them and keep the others constant.
To do that, in most cases, we will have to consider an out-the-money option, an at-the-money option, and an in-the-money option. Also to make things easier we are only going to consider call options. For the moment at least.
Put options work the same way but in a reverse direction.
How delta changes with the price of the underlying
Because there are so many variables that come into play, we are going to need to pick a starting point and then work from there. The starting point or points I will use will be:
- a call option
- 90 days to expiration
- A stock that typically trades at around $100 a share
- Implied volatility of 20%
- Current interest rate 0.0744%
- No upcoming dividend
We will start with a far out-the-money option where the underlying price is one standard deviation from the strike price, i.e. since the strike price is $100, and implied volatility is 20% we will start with the underlying at $80. We will then move in increments until we are deep-in-the-money also by one standard deviation, so the underlying will be $120.
If this doesn’t make sense to you, here is an article that examines implied volatility.
Here is what happens to both the delta and the price of the option as we vary the price of the underlying.
There are a few things to notice here.
Firstly the steepest part of the delta chart is centered around the $100 strike price. What this tells us is that the most rapid rate of change in delta happens around the strike price.
And now, gamma
We can see this from another Greek attribute of options, the gamma. Gamma is the rate of change of delta.
If we plot the gamma for the same option over the same range of underlying prices, this is what we see.
As regards delta, the option price responds more to the movement of the underlying the deeper in the money the option moves and hence the closer delta approaches 1.
We see this in the charge of the option price vs the underlying price. The further away from the strike price and the deeper in the money the option moves, the closer the graph comes to a straight line, as in effect the delta approaches 1.
How delta changes as the expiration date approaches.
Now let’s see what happens with:
- an out-the-money option with a delta starting at 0.25,
- an at-the-money option, and
- an in-the-money option with a delta starting at 0.75
All starting from 90 days to expiration as we approach the expiration date. This is what we get, again for the delta and for the option price.
First for an out-the-money option
And for an at-the-money option
And for an in-the-money option
Let’s look first at the out-the-money option. Because it is out-the-money, it is all extrinsic value. We can see that the price decays pretty much in a linear manner to zero over the last 90 days. However, the delta of the out-the-money option decays with a gentle curve to zero over that same time.
What this confirms is that for an out-the-money option the chances of it expiring in-the-money diminish for this level of implied volatility as we approach expiration.
Theta is the Greek used to express the rate of decline in the price of an option only as a function of time to expiration, i.e. all the other variables are not changing. Theta is usually shown on a day-to-day basis. In other words, if you look at an options board for a stock or an ETF, each option will be shown with a theta value.
The Theta value says by how much that option would decline over the next day if nothing else changes. By nothing else changing, we mean that the price of the underlying, the implied volatility, the interest rate all stay constant and there is no new dividend announcement.
How delta moves when implied volatility changes
Now let’s take a look at what happens to options when the implied volatility varies from 0% to 100%. On a practical note, volatility of even low volatility stocks usually hovers at least at around or above 10%. For many stocks in a regular sort of market 20% implied volatility is typical.
But when markets get jittery, implied volatility can shoot up, not just for individual stocks but for whole indexes. The most-watched volatility index, the VIX, is implied volatility calculated from a basket of options on the Standard and Poor 500 Index, all with around 30 days to expiration.
Personally, I’ve never seen a stock with an implied volatility of less than 10% but that doesn’t mean it doesn’t happen. Be that as it may, this is what we get, again starting with our same out-the-money, at-the-money, and in-the-money options.
First for an out-the-money option
And for an at-the-money option
And lastly for an in-the-money option.
In all three cases, the price increases in direct and linear proportion to increasing implied volatility. This is not the case for delta though.
For an in-the-money option, the delta at low volatility is higher and closer to 1, but as implied volatility increases the delta drops. This makes sense intuitively.
The more volatile a stock is expected to be the greater are the chances that an in-the-money option could move out-the-money. Whereas since it is already in-the-money, if volatility is expected to be very low it will more likely stay in-the-money.
For an out-the-money option, we see the other side of a similar effect. At very low levels of expected volatility, it is more likely that an ou-the-money option will stay out-the-money, hence the option price is less responsive to movements in the underlying price hence delta is very low.
But, as expected volatility increases, the chances of an out-the-money option becoming in-the-money also increase, so the option price becomes progressively more responsive, and hence its delta increases.
This introduces us to another Greek, the vega. Vega is a measure of how an option price changes with respect to changes in implied volatility while all other factors remain unchanged.
In fact, vega is quoted as the change in the option price for a 1% rise in implied volatility while, the price of the underlying, the days to expiration, and interest rates all remain unchanged and there is no new dividend announcement.
Impact on delta if interest rates change
Interest rates are still close to all-time historical lows at present, though they have been creeping up. At current levels, though the impact of interest rates on options prices is minimal.
Last one, rho
You probably guessed it by now, there is another Greek letter that measures how options prices respond to changes in base interest rates. That Greek is rho.
Because holders of call options gain leverage, holding deep-in-the-money options is like holding stock at a lower cost.
For the sake of completeness, let’s look at what happens to delta and options prices when interest rates move between two ranges. First, we’ll look from 0% to 0.1% since current interest rates are somewhere in the middle of this range. We will then also look from 0% to 20%. just to see what happens.
First an out-the-money option for the current low-interest-rate range.
Then for the same out-the-money option for a wider range of interest rates.
Then for an at-the-money option for the current low-interest-rate range.
And now the same at-the-money option for a wider range of interest rates
Then for an in-the-money option using the current low-interest-rate range.
And to complete the picture, the same in-the-money option using the wider range of interest rates.
There are only a couple of things to note here.
First that the current very low-interest rates are so low that the graph shows kinks. If we added more digits behind the decimal point those kinks would go away, if the option calculator could handle it.
Secondly, and importantly, that if interest rates rise call option prices and option deltas rise. For an explanation, we have to consider the difference between holding stock and holding deep-in-the-money call options.
An investor who holds a deep-in-the-money call gains the same benefit as a holder of 100 shares in the stock. But the option holder is free to invest the difference in cost in risk-free investments at the prevailing risk-free interest rate. Effectively the market balances the two choices by increasing the fair price of options as interest rates rise.
The reverse is the case for put options. As interest rates rise, the fair price of put options decreases.
Approaching dividend event affecting delta
Dividends also affect option prices. The effect is most prevalent around the ex-dividend date. The holders of in-the-money call options can choose to exercise their options early before an ex-dividend date. The question to consider is whether the option extrinsic value is greater than or less than the dividend value.
The market will tend to adjust option prices to reflect dividend events but not always efficiently. This means in practice you can sometimes benefit from early exercise of in-the-money call options trading the extrinsic value when it is low for a dividend.
Putting them all together
It is easy with this kind of analysis to get lost in the weeds and lose sight of the big picture. So let’s remind ourselves that it is really going to be the approaching expiration date and any change in market sentiment and hence implied volatility that can seriously affect options prices.
That is, other than movements in the price of the underlying, which is the big driver and what we mostly focus on anyway.
A dividend event can cause a minor kink which may be in our favor. Interest rates are only going to have any impact if rates soar through the roof from where they are right now.
Calculating position delta
One of the features of options deltas is you can add the deltas of options in the same underlying to get the combined effect. This will work for any multi-leg strategy in the same underlying.
A simple bull call debit spread will illustrate how this works. This article explains bull call spreads.
The delta of a bull call debit spread
Let’s imagine that we have a bull call debit spread in the Standard and Poor’s 500 Index tracking fund, symbol SPY. We hold a SPY May 405/415 call spread. So our position has 43 days to expiration.
SPY closed at $400.61
Our long leg is priced at $2.59 and has a Delta of plus 0.3527.
Our short leg is priced at $0.54 and has a Delta of minus 0.1033.
So simple math tells us that our spread is now worth $2.05 ($2.59 – $0.54) and has a Delta of 0.2406 (0.3527 – 0.1033).
Getting the delta you want
Here is another simple application.
Let’s say we wanted to purchase two options each with a delta of 0.8 so we would have a position that is the equivalent of 160 shares in a particular underlying stock. But because of the available strike prices, we are only able to buy call options with a delta of 0.7 or 0.9.
We would just buy one option with 0.7 delta, and one option with 0.9 delta. The two options combined would have a delta of 1.6 and therefore it would be like owning 160 shares in the underlying.
Uses of delta-neutral strategies
A delta-neutral strategy combines a mix of the underlying stock or ETF with call and/or put options to create a position or a portfolio that has a zero delta. In other words, the positive deltas and negative deltas in the same underlying stock or ETF cancel each other out.
There are two uses for this. One is to gain income from falling implied volatility and the other is as a hedge.
Income from falling implied volatility.
When markets are jittery, options are expensive, and implied volatility is high. You can create a position using a combination of short call and put options in the same underlying that will have a net delta of zero within a specific price range of the underlying.
As long as the price of the underlying stays within your range, as implied volatility drops you can close the position and collect the premiums.
Alternatively, you can let the options expire and the positions will also cancel each other out.
Delta-neutral as a hedge
Another use of a delta-neural strategy is to hedge, for example, if you are concerned about a possible market correction.
Let’s imagine that you have 100 shares of SPY that tracks the Standard and Poor’s 500 Index and you think that a market correction is on the cards. If you buy two at-the-money put options each one will have a Delta of – 0.5 making a total Delta of – 1.0 multiplied by 100 and thus exactly balancing out your 100 shares.
Just note that everyone else is likely to be nervous about a market decline, so options are likely to be costly so setting up this kind of hedge is going to cost you.
Is there a conclusion?
All this may seem like an academic waste of time and you can probably forget about the other Greeks. But, making the effort to become familiar with how the delta of options functions can really make a difference to your options trading.
It isn’t a magic bullet though. And don’t blind yourself with science.
Always be ready to put all the complicated stuff to one side and just see what price and trading volume are doing. Then based on what that tells you about what is going on in the market right now and trade and invest responsibly.
But for those interested, here is an article that looks at options and other derivatives Greeks in more detail.
Questions and answers
Q. What is the delta of an option?
A. The delta of a stock option is a measure of how much an option price moves when the price of the underlying stock moves by $1.
Q. How do you calculate the Greek delta?
A. You calculate the delta of an option from the total dollar amount the option’s price changes when the price of the underlying changes by $1. If the option goes up by $0.60 when the stock price goes up by $1 then the option has a delta of plus 0.60. If the price of the option goes down by $0.40 when the stock price goes up by $1, then that option has a delta of minus 0.40.
Q. How do you calculate the Greek Delta, Gamma, and Theta?
A. Delta and Gamma of a stock option are calculated for a $1 move in the price of the underlying. If delta goes up by $0.75 when the price of the underlying goes up by $1 then, the delta is 0.75.
If prior to that $1 move in the underlying the delta was 0.70 then gamma is approximately the difference between 0.70 and 0.75 i.e. 0.05. Actually, the math is a little more complex which is why this is an approximation.
Theta is the rate at which an option position decays in value for 1 more day as expiration approaches. For long option positions, theta is negative as the extrinsic value declines. For short option positions, theta is positive because we have sold the extrinsic value in the form of a premium.
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