This article introduces the concept of Delta and its practical applications, such as calculating asset exposure and building delta-neutral portfolios, providing valuable insights for options traders.//@Tiger_Academy:

Hello Welcome to Tiger Academy - 「Options Greeks Column」episode 1.

In our previous "「Options Academy Column」" series, we've covered the fundamentals of options. In this new series, we'll guide you through a range of articles on how to construct relative quantitative options portfolios using the Greek values of options. First, let's delve into the first and most important Greek value of options: Delta.

## 1. What is the Delta Value of an Option

We know that for buyers, the greatest value of an option lies in the potential for substantial gains while limiting losses. How do we measure the extent of this leverage? This is where the delta value comes into play. Let's take a call option as an example. If the underlying stock's price increases by \$1, causing the call option's price to also increase by \$1, then the delta of that call option is 1. Conversely, if the underlying stock's price rises by \$1, resulting in a \$1 decrease in the put option's price, the delta of that put option is -1. In simple terms, delta measures how much the option price changes for a \$1 change in the underlying stock price.

Now that we understand the basic principle, let's look at some conclusions. The delta of an option can range from -1 to 1. For call options, the delta value always falls between 0 and 1.00, and for put options, it always ranges from 0 to -1.00. Additionally, at-the-money call options have a delta of 0.5, while in-the-money call options have a delta closer to 1, and out-of-the-money call options have a delta closer to 0.

Why isn't the delta value of an option a constant 1 or -1, but instead influenced by whether it's in or out of the money? This is because being in or out of the money to some extent represents the probability of the option being profitable upon exercise. In-the-money options have a much higher probability of being profitable at expiration compared to out-of-the-money options.

Take a call option as an example. The reason why an increase in the stock price leads to an increase in the option price is primarily because the likelihood of the buyer making a profit upon exercise increases. For deep in-the-money options, consider this: if a call option has a strike price of \$10, a premium of \$5, and the current stock price is \$20, the option is highly likely to be profitable upon exercise since the strike price is much lower than the current stock price. If the stock price then rises to \$21, the call option's price should increase by \$1, making the premium \$6, hence a delta of 1.

However, it's a different story for out-of-the-money options. If a call option has a strike price of \$10, a premium of \$5, and the current stock price is \$8, even if the stock price increases to \$9, the option remains out of the money, and the probability of profit upon exercise remains low. In this case, if the delta of the option is 0.3, the option's price would only increase by \$0.3 with a \$1 increase in the stock price.

So, delta not only quantifies the relationship between stock price and option price but also represents the probability of profit upon exercise. Besides these aspects, delta values serve various practical purposes in trading.

## 2. The Functions of Delta Values

1.Calculating Asset Exposure

Let's say you have \$100,000, and the price of Stock A is \$10 per share. You're very bullish on Stock A. If you were to buy the stock directly, you could purchase 10,000 shares. If the stock price rises to \$11 in the future, you would profit \$10,000 [(11 - 10) * 10,000 shares].

But if you want to profit indirectly through buying call options, how many call options should you purchase to achieve the same profit? This is where delta values come into play. Delta can help you calculate how many shares of the underlying asset you effectively hold through options. If you buy at-the-money call options with a strike price of \$10, their delta coefficient is 0.5. If the stock price increases by \$1, the option price increases by \$0.5. So, buying 20,000 options is equivalent to the same profit as holding 10,000 shares [(20,000 * \$0.5) = (10,000 * \$1)]. If the option premium is \$2, you would only need to invest \$40,000 in options, which is much less than the \$100,000 required to buy the stock directly.

Similarly, if you invest \$100,000 in these options, how many shares do you effectively hold? With \$100,000, you can buy 50,000 options, which is equivalent to holding 25,000 shares of the stock (50,000 * 0.5).

2.Building Delta-Neutral Portfolios

In previous articles, we've learned about straddle options strategies. When you expect significant future price volatility and want to profit from it, you can simultaneously buy call and put options with the same expiration date and strike price. However, there's a potential issue with this approach: if the stock price goes up, the profit from the call option and the loss from the put option may not offset each other.

For example, let's consider Apple's stock, currently trading at \$177.5. If you anticipate significant future price volatility, with a range above \$182.5 or below \$175, you might decide to employ a straddle options strategy by simultaneously buying call and put options with a strike price of \$175. In this case, the delta value of the call option is 0.801, and the delta value of the put option is -0.194. If the stock price falls to \$172.5, the call option's price would decrease by \$4.05 [(5 * 0.801)], while the put option's price would only increase by \$0.97 [(5 * 0.194)]. If you were to close your positions, your net loss would be \$3.08. So, if our purpose in establishing a straddle strategy is to profit when the stock price moves beyond a predefined range, there is a possibility of incurring losses in the scenario mentioned. Therefore, the best approach is to build a delta-neutral position. This means adjusting the number of contracts so that the delta values of the call and put options are the same. For example, in the previous example, if you buy one call option with a delta value of 0.801, you would purchase approximately 4.12 put options (0.801/0.194) proportionally. This would make the overall portfolio's delta value equal to 0 (0.801 + 4.12 * -0.194), achieving delta neutrality.

You might wonder at this point: if achieving delta neutrality means that the portfolio's returns won't be affected by small stock price movements, doesn't that mean there won't be any profits? So, what's the purpose of doing this?

The answer is quite simple. While delta-neutral positions aren't affected by small movements in the underlying, they can profit from substantial price swings in the underlying asset. This is primarily influenced by another Greek value - gamma, which we will explain in the next installment.

Additionally, for those of you interested in options investing, there is a free introductory options course available for your learning.

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