# Delta Hedging

For some reason, I always get tripped up conceptually when it comes to Derivatives…

I get the math\formulas behind Delta hedging, but I just don’t quite understand how the underlying security is subsequently hedged against movements in stock price.

I’ll illustrate with an example - Let’s say John Doe has a \$10 million portfolio of Stock X. Stock X currently trades at \$50\share and there are European call options for the stock with a Delta of .5977.

To neutralize John’s equity position in stock X from changes in the stock price of X, he would need to sell 334,616 of the Eurpoean call options (I only know this because of the formula -> (\$10 mil divided by \$50/share)(1/.5977)).

I just can’t wrap my mind around this from a logical standpoint. My thinking is, if you write a bunch of calls, you’re still going to be susceptible to company X’s stock price decreasing significantlly or going to \$0. Conversely, if you sell so many calls to the point where the premiums bring in more than the portfolio value of stock X, you will be susceptible to significant price increases (because you would consequently be writing some uncovered calls)

If someone could briefly explain what I’m missing (I’m sure I’m missing something), it would be greatly appreciated!

Simply draw pay off diagrams you’ll understand quickly. You will see that when the underlying (price of stock) increases and you hold the stock, your payoff increases (logical right?). When you write a call option, your pay off decreases when the stock price increases. Hence one offsets the other. The delta hedging simply tells you HOW MANY call options you need to sell (write, or short) in order to offset the increase in pay off due to the appreciation of stock price.

In addition to the above, it is important to remember that the option delta changes, hence the number of options needed to have a perfect hedge also changes with stock price movements…

The option delta also changes with time even if the stock price remains unchanged.

That’s a very important remark that I missed, thanks Sir.

My pleasure.

Thank you both for the help. I think my misunderstanding stemmed from the idea that the Delta hedge could be constant. I did not realize that it was an instantaneous hedge only (which leads me to wonder how practical and common it is (I’m no options trader)).

For anyone else who was following this post, I actually found a very helpful explanation:

'A delta-neutral portfolio is perfectly hedged against small price changes in the underlying asset. This is true both for price increases and decreases. That is, the portfolio value will not change significantly if the asset price changes by a small amount. However, large changes in the underlying will cause the hedge to become imperfect. This means that overall portfolio value can change by a significant amount if the price change in the underlying asset is large"

In Delta hedging you are offsetting the change in price of the underlying asset with the change in price of the options. So if you are long the underlying, you need to go short on the option position.

However, option prices do not have a linear relationship with the underlying asset price. Delta is the measure we use to determine how much of a change in the underlying asset price translates into a change in the option price at any specific price point.

Continuing with this non-linear relationship, you can see how Delta will change depending on which price point you pick. For a price point which puts the option way out of the money, a change in the underlying asset price will have an minimal impact on the option price. As prices of the underlying start nearing the strike price, a change in the underlying asset price will start translating into more of a change in the option price. For a price point which puts the option way into the money, a change in the underlying asset prices translates into a 1 for 1 price change in the option.

This change in Delta that is dependent on the change in the underlying asset price, or from one price point to another, is called Gamma. As Delta is the measure of the change in price of the option in relation to the change in price of the underlying asset, Gamma is the measure of the change in Delta of the option in relation to the change in the price of the underlying asset.

As illustrated above with our non-linear relationship, when the option was way in or out of the money, the change in Delta was more or less minimal; being way out of the money resulted in almost no change in price for the option, and being way in the money resulted in always a 1 for 1 trade off for the price of the option and underlying asset. However, when options are at the money, the change in price of the option becomes more volatile because it is difficult to tell if the option will expire in or out of the money.

As such, you can see that Gamma, the change in Delta in relation to the change in the underlying price, is smallest for both way in and out of the money options, and largest for options at the money.

Because Gamma is non-zero, which means our Delta will change when the price of the underlying changes, our Delta hedges will never be perfect. Delta hedges for this reason need to be constantly rebalanced too. If you did all the math, you would see that we would have minor gains / losses on our hedges.