Mathematical models are used by the financial industry to determine the theoretical value of an option based on key parameters such as the price and volatility of the underlying security, time to expiration, interest rates, and others. The relationship of an option’s price to each of these parameters can thus be described mathematically and five such relationships have been given the names of Greek letters.
Options Greeks Explained
Mathematical formulas such as the Black-Scholes Option Pricing Model (BSOPM) are used throughout the industry to determine the theoretical value of any option using the following parameters:
- Type of option (put or call)
- Strike price of the option contract
- Duration of the contract
- Price of the underlying security
- Historical volatility of the underlying security
- Dividends payable from the underlying security
- Interest Rate
When trading options, using them to hedge stock positions, or building stock and option investment strategies, it is important to understand how an option’s price may be expected to change as these parameters change. The option type and strike price are fixed characteristics for each contract, but the other parameters can all change, with the price and volatility of the underlying being especially important to the price of an option.
How the price of an option will change when these parameters change is thus important and the following measures, named for Greek letters, describe these relationships:
- Delta – A measure of how much an option’s price will change relative to the price of the underlying security.
- Vega – A measure of how much an option’s price will change relative to a change in the volatility of the underlying security.
- Theta – A measure of how much an option’s price will change relative to time (i.e. as it gets closer to its expiration).
- Rho – A measure of how much an option’s price will change relative to changes in interest rates.
- Gamma – A measure of how much an option’s Delta will change when the price of the underlying security changes.
The Greeks can measure potential opportunity as well as risk and are frequently used to gauge these factors in an individual option position, a combined stock-option strategy, or an entire portfolio. Greeks are generally available from options pricing services, so investors do not need to calculate them.
Tip: Option pricing models and Greeks provide theoretical values only. While theoretical values provide an important guide and can be of significant assistance in developing investment strategies, actual prices are determined in the marketplace and may not necessarily conform to the models.
Greek Option Delta
Delta is defined as the theoretical change in an option’s price for a $1 change in the price of the underlying security. Delta can be helpful to an investor in the following ways:
- Option selection. The Delta of different available options can be used to fine-tune the desired level of directional exposure to the underlying.
- Strategy enhancement. Delta can be used to create or compare different option strategies for given objectives.
- Probability of expiring in the money. Delta can be used to approximate the probability of an option expiring in the money.
- Risk management. The exposure of an entire portfolio containing options to movements in the market can be determined by the Delta of the portfolio as a whole.
Note: Delta is defined to represent the change in an option’s price for a $1 change in the underlying. If an investor is interested, however, in how much the option is expected to move from say a $5 move in the underlying stock, it is not accurate to simply multiply the Delta by five as the relationship is not linear. Each successive $1 move in the underlying will have a different Delta. The difference is defined by the option’s Gamma.
Delta Option Example
Apple (AAPL) recently traded at 140, and the AAPL Aug 130 put had a market price of $3.60 with a Delta of .26. The Delta implies a 26% probability that the 130 put contract will expire in the money or a 74% chance of keeping the premium of $360 if AAPL expires above 130 at expiration.
Delta Option Greek Considerations
- Deltas are calculated for a $1 move in the underlying stock. Depending on which direction the underlying moves, the delta of the next $1 move will be slightly more or less than the current Delta.
- Options are affected by other variables. These other variables (volatility, time, etc,) can either offset a Delta move or add to it.
- The composite Delta for a diversified portfolio (which is determined by adding the Deltas of all positions in the portfolio) can inform the investor as to their exposure to an up or down movement in the market as a whole.
- As a rule of thumb, the Delta of an at-the-money option will generally be near .5 and will be higher for in-the-money strike prices and lower for out-of-the-money strikes with the same expiration.
Gamma measures the change in an option's Delta, given a $1 move in the underlying security. Gamma is helpful for determining whether an option’s Delta will increase or decrease for moves greater than $1 in the underlying and by how much.
- Gamma values are largest for at-the-money options and smallest for in-the-money or out-of-the-money options.
- Gamma can alert investors to the potential for a "Gamma squeeze", which can cause abnormal price movement in the underlying security near an expiration.
Gamma Option Example
With AAPL trading at 140, the Aug 140 call has a price of $8.80, a Delta of .56, and a Gamma of .05. That means if AAPL moves up $1.00, the option should increase by $.56 but an additional $1 move up will increase the option by another $.61 (all else being equal).
Gamma Risks & Considerations
- Directional risk. Gamma in an option position can help an investor determine the extent to which either risk or opportunity will increase or decrease as an underlying security continues to move in a given direction.
- Price only. As a derivative of Delta, Gamma strictly addresses sensitivity to price change in an underlying security. Gamma does not consider other factors, such as time or volatility.
- Expiration. A positive Gamma is best for long options but risky for short positions. In particular, Gamma becomes more aggressive as expiration approaches, making things challenging for investors with short options who are not hedged with the underlying stock.
Theta Greek Options
Theta measures the change in an option’s theoretical price relative to one day of time. Since long options decay in value with time, Theta values for long option positions are negative. Theta is often managed at both the position and the portfolio level.
Options Theta Example
With AAPL trading at 140, the Aug 140 call has a price of $8.80 and a Theta of -.07. That means that if one day passes and all other factors remain the same, the price of the Aug 140 call should drop to $8.73.
Considerations When Using Theta
- Holding too long. Investors holding long options tend to underestimate the effects of Theta and hold long options positions too long.
- Rate of decay. Theta has a non-linear relationship with price and accelerates as options approach expiration.
Vega represents the change in an option’s theoretical price for a 1% move in the volatility of the underlying security.
Options Vega Example
With AAPL trading at 140, the Aug 140 call has a price of $8.80 and a Vega of .22. If volatility expands by 1% and all else remains the same, the price of the Aug 140 call should increase by .22 to $9.02.
- Vega can be tricky to interpret because option prices in the marketplace tend to be based more on the perception of future volatility (called implied volatility) than on the historic volatility that models use to calculate theoretical value. Therefore, the market price of an option can sometimes depart considerably from its theoretical value. This might occur when the market is anticipating news, an earnings announcement, or other upcoming events that could materially affect a stock’s future price. In such cases, Vega may not provide useful information as it relates to historical volatility while the market is focused on implied volatility.
- Long positions benefit from volatility expansion because their Vega is positive. Short positions benefit from volatility contraction because their Vega is negative.
- Research shows that volatility expansion, though infrequent, occurs quickly and forcefully and that more than 80% of the time, market volatility is in a contracting mode.
Rho represents the change in an option’s theoretical price given a 1% move in interest rates. Interest rates may not materially change for months or years, but when they do, option prices will reflect that change rather quickly.
Rho can be useful in evaluating the effect of a change in interest rates on different option strategies.
Option Greeks can be very helpful in option selection and in the creation of option strategies. But it is important to remember that the Greeks are theoretical and that options may or may not behave in accordance with them. In particular, the Greeks can be expected to be less reliable when an option’s market price is considerably higher than its theoretical value. Implied volatility is what causes this.
The volatility of its underlying security is a major determinant of an option’s price. It is also a characteristic that varies over time for almost all stocks. One constraint of option models such as Black-Scholes is that they assume historical volatility will remain at the same level going forward.
Market prices for options are predicated on how investors expect the underlying stock’s price will move in the future. Thus, when investors expect a stock to be more or less volatile in the future than in the past, they will price the options accordingly and that price may diverge considerably from the theoretical option price determined mathematically.
As a general rule, the further an option’s market price is from its theoretical price, the less effective the Greeks will be in predicting how an option will behave.
The Greeks mentioned above are the most popular, but there are others and they do not always have Greek letters as names. Examples of others include:
- Charm. This is a measure of how Delta changes with time.
- Color. This is a measure of how Gamma changes over time.
- Vomma. This is a measure of how Vega changes with changes in Implied Volatility.
Greek Options Trading Strategies
Investors can use the Greeks to create or enhance option strategies in a number of ways.
- The Greeks can be used to help select specific options for a given strategy.
- Option strategies can be created with the help of the Greeks.
- The Greeks can provide insights on managing a strategy once implemented, such as substituting one option for another in a complex strategy.
- Investors can use the Greeks to help determine optimal times to close or roll options.
- Greeks on a portfolio of different option strategies can help quantify and manage portfolio risk.
Option Greeks allow investors to measure the risk/return in individual option contracts, complex option strategies, or entire portfolios. They offer a useful mathematical toolkit for selecting options, evaluating risks and opportunities, and optimizing strategies. The Greeks can provide useful insights on options that are not readily apparent from price alone.
This article was written by
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