We’ve already seen that delta is the option’s price sensitivity based on the price change of underlying asset. As the underlying price changes, option’s price also changes based on its delta value. Throughout the time this gave traders an idea that if you hold appropriate amount of underlying asset in your portfolio among with your options you can theoretically create a neutral position. We will dig this interesting idea deeper in this post which will give you a more advanced understanding of some of the most common expert-level option trading practices.

Delta hedging opens a whole another dimension in the option trading. The reason for that is because with delta hedging you can create a direction-neutral position through buying or selling stocks parallel to your option strategy. What happens when you create a direction-neutral position is that you can start betting on a number of other parameters such as volatility or interest rates.

### Delta Revisited

Delta will be a value between 0 and +1 for call options and it will be a value between 0 and -1 for put options.

So, a call option with a delta of 0.50 means a $1 increase in the underlying price will cause the call option to increase $0.50 in price. A put option with a delta of -0.50 means a $1 increase in the underlying price will cause the put option to decrease $0.50 in price.

So, basically if you’re holding 2 call options (long position) with a delta of 0.50 you can delta hedge your position by short selling 1 underlying asset (2 x 0.50). Short selling is the practice of selling a security you don’t own (through lending options) and replacing it at the due date (by buying the security that’s previously sold to return to the lender).

#### Example 1

What happens in this situation is that initially you are exposed to the underlying price change at a rate of 0.50 x 2 (delta times option amount in portfolio). So, when the underlying asset increases $1 in price your call option will increase 0.50 x 2 = $1 in total. As we’re trying to create a direction-neutral position, when you short-sell underlying asset in the required amount (in this case 1), you will be exposed to -$1 every time underlying security increases by $1 because you’ve already sold it and you will have to buy it back at $1 higher than you’ve already sold it. This is the epitome of delta hedging.

#### Example 2

Let’s look at a different call option example that will make more practical sense. Although options are highly customizable and come in all types and shapes, they often trade with a multiplier of 100. This means a call option with a delta of 0.60 will expose you to 100 shares of the underlying stock (if the underlying is a stock) and you will have an actual delta of 60.

In this case, 1 long call option position (buying 1 option) will require you to short sell 60 shares of the underlying stock to achieve a delta-neutral (direction-neutral) position.

#### Example 3

Similarly, let’s say you’re long 5 put options with a delta of -0.40 (multiplier 100). This will expose you to an actual delta of 5 x 100 x -0.40 = -200. This means a price decrease of $1 in the underlying stock’s price will cause your portfolio to increase $200 in value (Total option price increase). If you buy 200 shares of the underlying stock along with your 5 put options, every $1 increase in the underlying stock price will cause your options to lose a total value of $200, since the shares themselves will gain $200 in total value from the $1 price increase, options and underlying stock exposure will offset each other resulting in a direction-neutral portfolio.

### Partial Delta Hedging

Some investors may choose to only hedge their delta exposure partially and seek the risk/return profile of a limited directional exposure. This can be done by only covering the delta position based on a desired ratio. Let’s see an example:

Example 4:

Let’s say the trader has 50 call options with a delta of 0.15 (multiplier = 100). His or her total exposure will be 50 x 100 x 0.15 = 750 (total delta)

This means every time the underlying asset goes down $1 in price the portfolio will lose $750 and every time underlying asset goes up $1 in price the portfolio will gain $750.

To offset this delta exposure by 50%, trader can short sell 750 x 50% = 375 shares (assuming the underlying is a stock). This will cut the delta exposure by half and as the underlying increases $1 portfolio will increase $350 theoretically.

The exact number will be slightly different as delta also changes at a rate of gamma and the further price moves from the initial delta hedging operation the more delta will have changed and delta hedging will need to be adjusted.

### Combining Delta Exposures

Once you have a solid understanding of the basic delta hedging it’s easier to build on that knowledge. However, keep in mind that delta hedging along with many option trading strategies requires diligent record keeping and spreadsheets and quantitative applications are great power tools to perform a proper delta hedging.

You can also combine delta from different strategies on the same underlying asset. The key point here is that you can combine delta exposures only for deltas concerning the same underlying asset.

Let’s look closer to a portfolio where the trader holds both call and put options on the same underlying stock and how this affects the delta position and delta hedging strategy.

#### Example 5:

Let’s say your portfolio is consisted of 2 call options with a delta of 0.30 and 10 put options with a delta of -0.70 and all options have a multiplier of 100.

To calculate the total delta exposure, we can calculate the total delta for call options and put options separately and then combine them.

Delta exposure from call options: 0.30 x 2 x 100 = 60

Delta exposure from put options: -0.70 x 2 x 100 = -140

Total delta = 60 – 140 = -80

This portfolio has a total delta exposure of -80 meaning every $1 price decrease in the underlying stock price will cause the portfolio to increase $80 in value. It can be made delta-neutral by buying 80 shares of the underlying stock.

### Gamma Consideration

If delta is the option’s price sensitivity to underlying price change, gamma is the option’s delta sensitivity to the underlying price change. This means every $1 increase in the underlying asset’s price will cause the delta to increase 0.05 if the option’s gamma is 0.05. Note that gamma is positive for both call and put options and it is a second order greek. This means gamma shows the sensitivity of a first order greek which is delta.

Gamma phenomenon creates a new discussion that’s important in option trading: delta is not a constant and it’s continuously changing. The fact that delta is a changing parameter and gamma is the rate of this change means that we can’t just perform a delta hedging and then forget about it. In order to achieve a delta-neutral strategy trader would have to continuously, or at least periodically, perform delta hedging adjustments. Because as the underlying price changes delta would change at a rate of gamma and this change would impact the neutral position.

Luckily, in practical world most investors and traders don’t seek an absolute delta neutral position and they may only adjust based on the relatively big price movements depending on their risk appetite. However, in a big enough position delta hedging can have a

big impact on the bottom-line and trader can be in a situation to perform a perfect delta hedging.

Institutional traders and portfolio managers will utilize very useful computer programs developed with spreadsheets, databases and other quantitative solutions making delta hedging an easier adjustment to make than it would be if calculated manually and continuously. These computer programs can be built in-house and/or purchased externally with the ideal purpose of making the management of an option portfolio flawless.

#### Example 6:

Let’s say you have 10 put options with a delta of -0.20 and gamma 0.01 (multiplier = 100). Your total delta exposure will be: 10 x 100 x -0.20 = -200

After $1 price increase your delta will increase at a rate of gamma which is 0.01, so,

New Delta: -0.20 + 0.01 = -0.19

New Total Delta: -0.19 x 10 x 100 = -190

Initial Delta Hedging: You will need to buy 200 underlying shares.

New Delta Adjustment: You will need to sell 10 shares to achieve a total of 190 long shares.

As you can see from this example, delta hedging is a continuous operation and it would need to be constantly adjusted to achieve a delta-neutral position as the underlying asset’s price goes up and down. The frequency and precision of these adjustments totally depends on the investor’s goals and risk appetite.