The Complete Guide On Option Delta

Options are derivatives, meaning they derive their value from the underlying asset. The underlying is usually stock but can be currency, futures or commodities. These derivatives are contracts created every day and cease to exist on their expiration. This leads to characteristics that are fundamentally different than trading straight stock. One of those different characteristics is how option prices react to movement in the underlying (delta), time (theta), volatility (vega), and interest rates (rho).

We are going to dig deeper into option delta; what it means, how it reacts to various changes in volatility and time, and how to neutralize it. Understanding and quantifying an option’s sensitivity to a shift in the underlying price can be the difference between a profitable positon and a losing position.

Where To Find An Option’s Delta

Before we can even begin to define or talk about all the cool attributes of delta, you need to know where to find the delta. An option’s delta, along with the other Greeks, come out of the option pricing model.

Due to the fact that Greeks are a byproduct of a calculation means they have a model risk. A model risk means; the outputs are only as good as the inputs. If you plug in the incorrect implied volatility, you can’t expect to receive proper Greeks.

Now, model risk in terms of option pricing and Greeks, is not a serious risk. Out of the 7 factors that affect an option’s price only one of them is unknown, implied volatility. It would be difficult to mess up the model so much that it throws off your Greeks.


It is more important to understand how Greeks are derived and that you can’t always trust the numbers for how they are.

The best place to find the delta of your option is through an option chain. An option chain displays all the calls and puts for a given expiration and underlying. You can usually customize your option chain to display the various Greeks that you are interested in. Most traders will use their option brokerage, but you can also use free tools such as Nasdaq.com to view an option chain.

What Is Option Delta

Simply, option delta measures how much an option's price moves compared to a dollar move in the underlying. Technically, as Tom puts it, “Delta is the first derivative (or slope) of the price curve of an option.” Let's break this down with an example.

Let's say, The Option Prophet (sym: TOP) is trading at $50.00, and the 50 strike call option is trading at 2.00. You can't expect that if TOP moves up $1.00 to $51.00, that the option will also move up a $1.00 to $3.00. That would be a 50% move in the option’s price compared to a 2% move in the underlying price. If that were the case, we would all trade options, and nobody would trade straight stock.

Delta is expressed as a number between -1 and 1. More specifically, a call option delta will range from 0 to 1 and a put option delta will range from -1 to 0.
Using our example from above, if you are long a call on TOP with a delta of 0.50 and the underlying makes the move up from $50.00 to $51.00, your option would now be worth $2.50. The delta told us that since the underlying increased by $1, it would increase the option price $0.50.

(0.50(delta) x 1(change in price) + 2.00(option price)) = $2.50

If TOP makes another strong move and increases from $51.00 to $52.00, then our option is now worth $3.00. Again, the delta told us to move the option price up $0.50 when the underlying increases a $1.00.

(0.50(delta) x 1(change in price) + 2.50(option price)) = $3.00

Similarly, delta also tells us to move the option price down $0.50 when the underlying drops $1.00. If TOP goes from $52.00 down to $49.00 our option price is now worth $1.50.

(3.00(option price) - 0.50(delta) x 3(change in price)) = $1.50

Puts carry a negative delta because their movement is inverse to the underlying. When a stock moves down $1.00, a put option will increase by the delta.
You are now long a put option with a delta of -0.30 on TOP for $4.00. TOP continues its drop from $49.00 to $45.00, a $4 move. Your option price increases from $4.00 to $5.20.

(-0.30(delta) x 4(change in price) + 4.00(option price)) = $5.20

Why Is Delta Important

Delta is important because it gives us a way to realize the movement in our options. Delta also doubles as a primitive probability analysis. We say primitive because this number is a good approximation but it is not perfect. Turning delta into a percentage gives you the probability of that option ending in-the-money. An at-the-money option has a delta of 0.50, so it has a 50% chance of finishing at-the-money at expiration.

This becomes increasingly useful if you are a seller of options and want to keep options out-of-the-money, or if you purchase out-of-the-money options looking for them to go in-the-money. You will also use delta as our hedge ratio, which we will touch on shortly. Using delta as a probability proxy is only an estimate and in practice it is not precise.

How Delta Changes With Stock Price

The deltas of calls and puts both increase as the stock price increases and decrease as the stock price decreases.

TOP is currently trading at $50.00, and the 50 call has a delta of 0.50. The stock increases to $55.00, taking the call from at-the-money to in-the-money. The delta is now 0.55, increasing as the stock price increases.

The confusing part is when we try to make sense of a put’s delta, increasing as a stock’s price increases. TOP is currently trading at $50.00, and the 50 put has a delta of -0.50. The stock increases to $55.00, taking the put from at-the-money to out-of-the-money. The delta of the put is now -0.45, increasing as the stock price increases. This works because put deltas are negative.

How Delta Changes Based On Strike Price

An option's moneyness tells us if it is in-the-money, at-the-money or out-of-the-money. A call option that is in-the-money will have a delta that is between 0.50 and 1. The more in-the-money the option is, the closer to 1 it becomes. An option that is deep in-the-money will move more like stock compared to an option that is at-the-money or out-of-the-money.

A call option that is at-the-money will have a delta around 0.50. When the option moves out-of-the-money, its delta will between 0.50 and 0. The deeper it is out-of-the-money the closer to 0 it becomes.

Put option deltas are similar to call options. An in-the-money put option has a delta between -0.50 and -1. The more in-the-money the put option is the closer to -1 it becomes. Like a call option, an at-the-money put option has a delta close to -0.50. Put options with a delta between 0 and -0.50 are out-of-the-money.

How Delta Changes Based On Time To Expiration

The delta of your options will change as the time to expiration becomes shorter. As each day passes, your option delta will continue on its current path. If your call option is in-the-money, it will start getting closer to 1. If your put option is in-the-money, it will start begin to get closer to -1.

Similarly, your at-the-money options, both calls and puts, will keep their deltas at 0.50 and -0.50, respectively.

Out-of-the-money call and put options will both head towards 0 as time to expiration gets closer.

How Delta Changes As Volatility Changes

As implied volatility increases, your option deltas will get closer to 0.50. In-the-money option deltas will decrease, at-the-money deltas will stay the same, and out-of-the-money deltas will increase.

A change in implied volatility changes the size of one standard deviation.

The Option Prophet (sym: TOP) is trading at $100 and has a standard deviation (implied volatility) of 20%. A 1-standard deviation move in the stock will put the end price at $80.00 or $120.00

100 + (100 x 20%) = $120.00

100 – (100 x 20%) = $80.00

What happens when you increase your implied volatility from 20% to 40%? Our 120 call goes from being at the end of the volatility range, to only half the distance away. In terms of volatility, since our calls have become closer to at-the-money, their deltas begin to move towards 0.50.

As you can imagine, as implied volatility decreases, your option delta will begin to move further away from at-the-money. In-the-money options will move closer to 1 and -1, at-the-money options will remain at 0.50, and out-of-the-money options will move towards 0.

Position Delta vs Portfolio Delta

Delta doesn't stop at a single option position. It is not often you are only buying or selling one contract, so you need to know how to expand delta into an entire position. Figuring out delta for your whole positions is as easy as multiplying the amount of contracts by the option delta.

You are long 4 TOP calls with a 0.65 delta. The delta for your entire position is 2.60.

4(amount of contracts) x 0.65(delta) = 2.60

With this information you know that if the underlying increases $1, your profit on the position will be $260.

Once you have the deltas figured out for each of your positions, you can expand it to cover your entire portfolio. You can calculate your portfolio delta by adding up each of your position deltas.

You have a portfolio that consist of: 4 long TOP call options with a 0.45 delta. You are also holding 6 long ABC put options with a delta of -0.75.

Your position deltas are:

4(amount of calls) x 0.45(delta) = 1.80

6(amount of puts) x -0.75(delta) = -4.50

Your portfolio delta is -2.70.

1.80(call position) – 4.50(put position) = -2.70

When you know the delta of your portfolio, you can be sure that your positions match your bullish, bearish or neutral stance. If you are bullish on the market, you want a positive portfolio delta, and if you are bearish on the market you want a negative portfolio delta. In our example a negative 2.70 delta means your portfolio is net short.

Being Delta Neutral

When you trade options you are not always bullish or bearish, sometimes you will have a neutral outlook. Managing your deltas in an option position, and your portfolio, is essential to make sure you are trading towards your market bias. A delta neutral position has a position delta that is at 0 or approximately at 0.
Managing deltas is an active management style. Deltas are constantly changing and need constant management to keep them in line. When making adjustments to option positions a lot of focus goes on the deltas. Deltas can be reduced or increased by adding to a position, adding a different option to a position, or even adding stock to a position.

Remember, the underlying stock position will also have a delta of 1 if you are long, and a delta of -1 if you are short the position. Having this knowledge, you can easily adjust your positions back to neutral by buying or selling the underlying asset.

Let’s see how this plays out in an example. You are long an at-the-money straddle on TOP. This is essentially a delta neutral strategy. Your position requires a move in the underlying, but an upward move or downward move doesn’t matter. TOP begins to move higher, generating a profit on your position, but also moving you to a positive delta. If your delta has gone from 0 to +45, you need to make an adjustment to get it back to 0. You have a couple of options. You could sell some of your calls, buy some more puts or short the underlying. If you decide you are going to neutralize your delta by shorting the underlying, you would need to short 45 shares. This would bring your delta back to 0, and return your position to delta neutral.

How a position is adjusted depends on the position and what you are trying to accomplish.

Conclusion

Delta tells us exactly how an option will move as the underlying moves. A call's delta will range from 0 to 1, with in-the-money being closer to 1, and out-of-the-money being closer to 0. A put's delta will range from -1 to 0, with in-the-money being closer to -1, and out-of-the-money closer to 0.

You can also use delta as a probability measure to tell if an option will finish in-the-money at expiration. Making sure your portfolio will profit according to your market bias is crucial. You use your portfolio delta to make sure your portfolio and bias are aligned.

Delta is very dynamic so you should know how your delta will change as the underlying price, time to expiration, and implied volatility changes.

Now that you know Delta, learn about the other Greeks:

The Complete Guide To Option Gamma

The Complete Guide To Option Theta

The Complete Guide To Option Vega

The Complete Guide To Option Rho

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