Delta And Gamma Of An Option

By Rekhit Pachanekar

 

The Greeks have been credited for a number of inventions, achievements and the like. In fact, we use the Greek alphabets as an alternative form of symbols and labels all throughout science. And somehow, they have even seeped into the world of finance and investing. It is here that we start our journey to understand the basics of options and how they work.

In this blog, we will be covering:

Before we talk about the Greeks, let us refresh our memory of what ‘call’ or ‘put’ is, when it comes to options. To find Delta, skip to this section.

How to call or put

For a call option, the call option buyer pays a premium to the option writer to buy the stock at a specified price (strike price) at or before the expiry date.

For a put option, the put option buyer pays a premium to sell the stock at the specified strike price at or before the expiry date.

You can read about the different terms in more detail here.

Let’s go through two cases to better understand the call and put options.

For simplicity’s sake, let us assume the following:

Price of Stock when the option is written: $100

Premium: $5

Expiration date: 1 month after the option is bought

Case 1:

The current price of Stock: $110. Strike price: $120

 

Type of Option Owner Moneyness Result
Call Option buyer Out of the money If the option buyer proceeds to buy the stock from the option seller, the stock will be bought at the price: ($120 +$5 = $125)

Since it is better to buy the stock directly, the buyer would not exercise the option.

Call Option writer Out of the money Since the option buyer does not exercise the option, the option writer makes a profit which is equal to the premium.
Put Option buyer In the money Since the option buyer can sell the stock at ($120 – $5) = $115, which is higher than the stock at $110, the option buyer will exercise the put option.
Put Option writer In the money As the put option is exercised, The option writer reports a loss of ($120 -$110 – $5) = $5

Case 2:

The current price of stock: $120. Strike price: $110

 

Type of Option Owner Moneyness Result
Call Option buyer In the money If the option buyer proceeds to buy the stock from the option seller, the stock will be bought at the price: ($110 + $5 = $115)

Since the current price of the stock is $120, the buyer will make a profit of $5 when they sell the stock in the market. Thus the option will be exercised.

Call Option writer In the money Since the option buyer exercises the option, the option writer will technically book a loss of

($120 -$110 – $5 = $5).

Put Option buyer Out of the money Technically, the put option buyer could sell the stock in the market for $120 than sell it to the option writer at the agreed upon strike price of $110.

Thus the option buyer would let the option expire and register a loss of $5 which is equal to the premium paid.

Put Option writer Out of the money As the put option is not exercised, The option writer reports a profit of $5 which is equal to the premium paid.

So how can you evaluate if the option is really worth buying? Let’s find out.

We will now channel the ancient Greeks to help us understand which options should we choose.

Hello Everyone, let us give a warm welcome to Delta

Delta can be expressed as a ratio of change in the price of the option to the change in the price of the underlying asset.

While the formula for calculating delta is on the basis of the Black-Scholes option pricing model, we can write it simply as,

Delta = [Expected change in Premium] / [Change in the price of the underlying stock].

Let’s understand this with an example for a call option:

We will create a table of historical prices to use as sample data. Let’s assume that the option will expire on 5th March and the strike price agreed upon is $140.

 

Date Option Price Change in price Stock price Change in price Delta =

(change in option price)

(change in stock price)

1 March $10 $100
2 March $15 $5 $110 $10 $5/$10 = 0.5
3 March $22 $7 $120 $10 $7/$10 = 0.7
4 March $30 $8 $130 $10 $8/$10 = 0.8

Thus, if we had to calculate the delta for the option on 2nd March, it would be $5/$10 = 0.5.

Here, we should add that since an option derives its value from the underlying stock, the delta option value will be between 0 and 1. Usually, the delta option creeps towards 1 as the option moves towards “in-the-money”.

While the delta for a call option increases as the price increases, it is the inverse for a put option.

Think about it, as the stock price approaches the strike price, the value of the option would decrease. Thus, the delta put option is always ranging between -0 and 1.

How can we use delta in a strategy?

There are a few option trading strategies based on the delta. A brief overview is given here:

Delta neutral

The aim of this strategy is to buy call or put options in such a way that the resultant delta is always 0, ie neutral. Let’s see an example.

Suppose we take 1 call and 1 put option for the same stock with the delta as 0.5 and -0.5 respectively, the total delta is (0.5 – 0.5 = 0) and thus this is a delta-neutral strategy. In this scenario, one should note that when you buy an option, your loss is capped at the amount of premium you pay while the potential profit could be much larger than the premium paid. But the main reason a delta neutral strategy is taken is to make sure that your overall position is not impacted if the market moves either way.

 

Bull call spread

In a bull call spread, we buy more than one option to offset the potential loss if the trade does not go our way.

Let’s try to understand this with the help of an example.

The following is a table of the available options for the same underlying stock and same expiry date:

 

Call option price Strike price
60 150
48 155
38 160
30 170
15 180
10 200

 

Normally, if we have done the analysis and think that the stock can rise to $200, one way would be to buy a call stock option with a strike price of $180 for a premium of $15. Thus, if we are right and the stock reaches $200 on expiry we buy it at the strike price of $180 and pocket a profit of ($20 -$15) = $5 since we paid the premium of $15.

But if we were not right and the stock price reaches $180 or less, we will not exercise the option resulting in a loss of the premium of $15.

One workaround is to buy both, a call option at $180 and sell a call option for $200 at $10. Thus, when the stock’s price reaches $200 on expiry, we exercise the call option for a profit of $5 (as seen above) and also pocket a profit of the premium of $10 since it will not be exercised by the owner. Thus, in this way, the total profit is ($5 + $10) = $15.

If the stock price goes above $200 and the put option is exercised by the owner, the increase in the profit from bought call option at $180 will be the same as the loss accumulated from the sold call option at $200 and thus, the profit would always be $15 no matter the increase in the stock price above $200 at expiry date.

Let’s construct a table to understand the various scenarios.

 

Stock price at expiry Impact on call option bought for stock price at $180 Impact on call option sold for stock price at $200 Total profit
Below $180 Expires worthless. Loss of premium of $15 Expires worthless. Profit of $10 -$15 + $10 = -$5
Between $180 and $200 In the money and profit is equal to (Stock price – $180 – $15) Expires worthless.

Profit of $10

(Stock price – $180 – $15) + $10

= (Stock price – $180 – $5)

Above $200 In the money and profit of (Stock price – $180  -$15) In the money. Loss of (Stock price -$200 – $10) (Stock price – $180  -$15) – (Stock price – $200 – $10)

= $15

You can go through this informative blog to understand how to implement it in Python.

Bear put spread

The bull call spread was executed when we thought the stock would be increasing, but what if we analyse and find the stock price would decrease. In that case, we use the bear put spread.

Let’s assume that we are looking at the different strike prices of the same stock with the same expiry date.

 

Put option price Strike price
10 140
15 160
30 180
38 200

 

One way to go about it is to buy the put option for the strike price of 160 at a premium of $15 while selling a put option for the strike price of $140 for the strike price of $10.

Thus, we create a scenario table as follows

 

Stock price at expiry Impact on put option bought for stock price at $160 Impact on put option sold for stock price  at $140 Total profit
Above $160 Expires worthless.

Loss of premium = $15

Profit of premium $10 as put option buyer will not exercise the option. Profit = -$15 + $10 = -$5
Between $140 and $160 Option is exercised. Profit = ($160 – Strike price – $15) Expires worthless. Profit of $10 Profit = ($160 – Strike price – $15) + $10

= ($160 – Strike price – $5)

Below $140 Option is exercised. Profit = ($160 – Strike price – $15) Option is exercised.

Hence, loss = ($140 – Strike price – $10)

Profit = ($160 – Strike price – $15) – ($140 – Strike price – $10)

= $15

 

In this way, knowing delta, we can minimize our losses by simultaneously buying and selling options.

So far we have talked about Delta, ie the ratio of change in the price of the option to the change in the price of the underlying asset. Delta helps us understand if the option is in the money, at the money or out of the money. We also know that depending on the strike price and expiry date of a stock, the delta of the asset changes.

You can go through this informative blog to understand how to implement it in Python.

But what if we want to know how fast the delta of an option will change? Is this possible? It is here where we introduce the term Gamma.

 

Hello Everyone, I am Gamma!

Gamma is defined as the measure of change of a portfolio’s delta with respect to the change in the price of the underlying asset.

Gamma = [Change in an option delta] / [Unit change in price of underlying asset]

Gamma signifies the speed with which an option will go either in-the-money or out-of-the-money due to change in the price of the underlying asset. This helps us in

Let’s see an example of how delta changes with respect to Gamma. Consider a call option of stock at a strike price of $300 for a premium of $15.

Strike price: $300

Initial Stock price: $150

Delta: 0.2

Gamma: 0.005

Premium: $15

New stock price: $180

Change in stock price: $180 – $150 = $30

Now,

Gamma Formula

Thus, Change in Premium = Delta * Change in price of stock = 0.2 * 30 = 6.

Thus, new premium = $15 + $6 = $21

Change in delta = Gamma * Change in stock price = 0.005 * 30 = 0.15

Thus, new delta = 0.2 + 0.15 = 0.35.

Let us take things a step further and assume the stock price increases another 30 points to $210.

Now,

New stock price: $210

Change in stock price: $210 – $180 = $30

Now,

Gamma Formula

Change in Premium

 

 

Change in delta = Gamma * Change in stock price = 0.005 * 30 = 0.15

Thus, new delta = 0.35 + 0.15 = 0.5.

In this way, delta and gamma of an option changes with the change in the stock price.

We should note that Gamma is the highest for a stock call option when delta of an option is at the money. Since a slight change in the underlying stock leads to a dramatic increase in the delta.

Similarly, the gamma is low for options which are either out of the money or in the money as the delta of a stock changes marginally with changes in the stock option.

Can we use Gamma in helping us choose the right option?

Usually, a high gamma indicates high volatility (slight change in stock price has a significant impact on the delta and thus the premium). Thus, with a mix of stock call and put options, investors try to keep the total gamma of the options at a low level and thus seek to keep the volatility at the lowest.

In conclusion, we have understood the importance of Delta and Gamma and how it can be used to analyse different options of the same stock as well as how to hedge the option risk by using different strategies. You can go through the different trading strategies by enrolling for the options trading bundle.

Disclaimer: All investments and trading in the stock market involve risk. Any decisions to place trades in the financial markets, including trading in stock or options or other financial instruments is a personal decision that should only be made after thorough research, including a personal risk and financial assessment and the engagement of professional assistance to the extent you believe necessary. The trading strategies or related information mentioned in this article is for informational purposes only.