Delta Formula

Updated on January 3, 2024
Article byHarsh Katara
Edited byAnkush Jain
Reviewed byDheeraj Vaidya, CFA, FRM

Delta Meaning

Delta in the theoretical sense is the change in price of an underlying security of an option’s value if its underlying asset moved up or down by $1. It is often looked at as a risk metric that allows traders to hedge their risks and eventually become delta neutral. The most common strategy used for a delta-neutral position is a calendar spread.

Delta-Formula

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Source: Delta Formula (wallstreetmojo.com)

Delta formula is a type of ratio that compares the changes in the price of an asset to the corresponding price changes in its underlying. The numerator is the change in the price of the asset, which reflects how the asset changed since its last price. The asset could be any derivative like a call option or put optionCall Option Or Put OptionCall options offer the right but not the obligation to buy the underlying asset at a particular date for a pre-decided strike price. In contrast, put options offer the rights but not the obligation of selling the underlying asset at a particular date for the pre-decided strike price.read more. For accounting most traders use a excel delta formula and document their trades.

Key Takeaways

  • The delta formula is a tool utilized to assess the impact of changes in the underlying asset’s price on an option’s price. 
  • It estimates how much the option’s price is expected to change for a given change in the underlying asset’s price.
  • Delta represents the sensitivity of the option price in relation to fluctuations in the underlying asset’s price. 
  • A delta of 1 indicates that the option price will move in lockstep with the underlying asset. In contrast, a delta of 0 implies no correlation between the option price and the underlying asset’s price.

Delta Formula Explained

The Delta formula is used by traders and investors to understand the change in the price of a security derivative taking effect from the positive or negative change in the underlying security’s prices. The delta for security can be positive or negative. For a call option, it always ranges between 0 to 1, and for a put option, it ranges from -1 to 0.

Option delta formula is used by traders as a part of their options trading strategy where they initially try to establish a delta-neutral position by buying and selling options simultaneously in the proportion to the neutral ratio.

These options have stock as their underlying, and that is the key aspect that affects the prices of these assets. In capital markets,In Capital Markets,A capital market is a place where buyers and sellers interact and trade financial securities such as debentures, stocks, debt instruments, bonds, and derivative instruments such as futures, options, swaps, and exchange-traded funds (ETFs). There are two kinds of markets: primary markets and secondary markets.read more, this Delta is also referred to as the Hedge Ratio.

There are multiple ways in which traders execute their strategies relating to this form of trading. However, a calendar spread is one of the most common strategies where they buy options with different expiration dates to develop a delta-neutral position.

Delta is a vital calculation that is mostly executed through software in modern times, as this is one of the key reasons that the prices of the option move in a particular direction, and this is an indicator as to how to invest. The behavior of the put option and call option delta can be greatly predictable and can be very useful to traders, portfolio managers, individual investors, and hedge fund managers.

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Formula

The option delta formula acts as a basis for trading in the derivatives market and for us to understand the concept in detail. Let us discuss the formula as mentioned below.

Delta = Change in Price of Asset / Change in Price of Underlying.

However, even the Black and Scholes model is used to determine the value of Delta, where there is a variable in it, which is N(d1), which can be calculated using computer software.

Examples

Let us understand the concept with the help of a few examples. These examples include an Excel delta formula and an Excel template for the perusal of the reader.

You can download this Delta Formula Excel Template here – Delta Formula Excel Template

Example #1

Suppose that the change in the price of the asset is 0.6733, and the change in the price of the underlying is 0.7788. You are required to calculate Delta.

Solution:

We are given both the figures that change in the price of the asset, which is 0.6733, and change in the price of the underlying, which is 0.7788. Therefore, we can use the above equation to calculate the Delta.

Use the below given data for the calculation of Delta.

  • Change in Price of Underlying: 0.7788
  • Change in Price of Asset: 0.6733

The calculation of Delta is as follows,

Excel 1.1

Delta =0.6733 / 0.7788

Therefore,

Delta Formula Example 1.2

Delta = 0.8645

Hence, the Delta will be 0.8645

Example #2

ABC stock has been listed for a number of years but has remained quite volatile in nature. The traders and investors have been suffering losses in the stock due to its unnatural price movement. The stock has been listed for five years and is now eligible to enter the derivatives market. John already holds the position of this stock in his portfolio.

The current price of the stock is $88.92, and the call option of strike priceStrike PriceExercise price or strike price refers to the price at which the underlying stock is purchased or sold by the persons trading in the options of calls & puts available in the derivative trading. Thus, the exercise price is a term used in the derivative market.read more $87.95 is trading at $1.35, which has an expiration of 1 month left. John wants to hedge his position, and hence he wants to calculate the Delta for this stock. Next trading day, he notices that the stock price has moved down to $87.98, so the call option price moved down slightly to $1.31.

On the basis of the given data, you are required to calculate the Delta, which shall be a basis for the hedge ratioHedge RatioThe hedge ratio is the open position's hedge ratio's comparative value with the position's aggregate size itself. Also, it can be the comparative value of the futures contracts purchased or sold with a value of cash commodity that is being hedged.read more for the trader.

Solution:

Use the below given data for the calculation of Delta.

  • Call Option Price at Beginning: 1.35
  • Call Option Price at End: 1.31
  • Stock Price at Beginning: 88.92
  • Stock Price at End: 87.98

The calculation of Delta is as follows,

Delta Formula Example 2.1

Here, the asset is the call option, and it is underlying it’s the stock. So, first, we will find out the changes in the price of the asset, which is the change in the price of call option which shall be $1.35 less $1.31 that is equal to $0.04, and now the change in underlying price would be $88.92 less $87.98 which shall equal to $0.94.

We can use the above equation to calculate Delta (rough figure, a true figure can be obtained through other complex models like Black and Scholes)

Delta =$0.04 00/ $0.9400

Therefore,

Calculation 2.2

Delta = $0.0426

Hence, the Delta will be $0.0426.

Example #3

JP Morgan is one of the biggest investment banksBiggest Investment BanksTop 10 bulge bracket investment banks are - Blackstone, Goldman Sach & co, Morgan Stanley, J.P Morgan Chase & co, Bank of America Merrill Lynch, Credit Suisse, Citi, Deutsche Bank, HSBC, UBS.read more in the United States of America. They have multiple stock, bond, derivatives positions sitting in their balance sheetTheir Balance SheetA balance sheet is one of the financial statements of a company that presents the shareholders' equity, liabilities, and assets of the company at a specific point in time. It is based on the accounting equation that states that the sum of the total liabilities and the owner's capital equals the total assets of the company.read more. One such position is in WMD stock, which is trading at $52.67. The company has long exposure to this stock. On the next trading day, the stock trades at $51.78. The trader who is acting on behalf of the company has an option that shall hedge the losses.

The strike price of the put option is $54.23 and when it is currently trading at $3.92. The price of the put option closed at $3.75 yesterday. The trader wants to know the rough Delta and asks you to calculate the Delta of the WMD put option.

Solution:

Use the below given data for the calculation of Delta.

  • Put option Price at Beginning: 3.75
  • Put Option Price at End: 3.92
  • Stock Price at Beginning: 52.67
  • Stock Price at End: 51.87

The calculation of Delta is as follows,

Example 3.1

Here, the asset is the put optionPut OptionPut Option is a financial instrument that gives the buyer the right to sell the option anytime before the date of contract expiration at a pre-specified price called strike price. It protects the underlying asset from any downfall of the underlying asset anticipated.read more, and it is underlying it’s the stock. So, first, we will find out the changes in the price of the asset, which is the change in the price of the put option which shall be $3.75 less $3.92 that is equal to $-0.17 and now the change in underlying price would be $52.67 less $51.78 which shall equal to $0.99.

We can use the above equation to calculate Delta (rough figure, the true figure can be obtained through other complex models like Black and Scholes)

Delta = $-0.1700 / $0.8000

Therefore,

Delta Formula Example 3.2

Delta =$-0.2125

Hence, the Delta will be $-0.2125.

Calculator

The calculator to options delta formula is provided below for the perusal of the reader.

Change in Price of Asset
Change in Price of Underlying
Delta
 

Delta =
Change in Price of Asset
=
Change in Price of Underlying
0
= 0
0

Frequently Asked Questions (FAQs)

1. What are the benefits of using the delta formula?

The delta formula is a mathematical tool used in options trading to measure the sensitivity of an option’s price to changes in the price of the underlying asset. The delta formula’s benefits include understanding the potential profit or loss of an options position, assessing the probability of an option expiring in-the-money, and managing risk by adjusting the position based on changes in the delta value.

2. What are the limitations of the delta formula?

The limitations of the delta formula include its assumption of constant factors such as volatility, interest rates, and time until expiration, which may not hold true in real-world scenarios. Additionally, delta alone does not capture all the complexities of options pricing, such as higher-order Greeks like gamma and vega, which measure additional aspects of risk and sensitivity.

3. Who derived the delta formula?

The delta formula was derived by economists Fischer Black and Myron Scholes. Their work on options pricing, which led to the development of the Black-Scholes-Merton model, revolutionized the understanding of options and derivatives. The delta formula is a key component of their groundbreaking work, and it remains a widely used tool in options trading and risk management.

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