# Greeks Primer

### Options “Greeks” are a set of risk parameters used by options traders to contextualise market conditions and inform trading strategies.

**Contents**

An Overview of the Greeks

Delta (Δ)

Gamma (Γ)

Theta (Θ)

Vega (v)

Rho (ρ)

Minor Greeks

### Synopsis

The four Greeks most salient to cobbers trading on Zeta are delta, gamma, theta, and vega. Each has its own significance and applicability, but grasping how they interplay allows you to paint a more comprehensive picture of risk in individual positions and across your entire portfolio. The Greeks help traders orchestrate options strategies from position entry to exit.

They are derived from calculus. And yes, your calculus professor was wasting your time not focusing explicitly on them. To be fair, she may have purposefully neglected teaching you about the Greeks because she knew markets are PvP, so the less you knew, the better — she thought. In the same way a compass keeps you oriented when you’re lost in the desert, the Greeks keep you oriented as you navigate markets.

**Delta (Δ)**

Delta is an invaluable hedging parameter that measures the sensitivity of an option's price relative to changes in the price of an underlying asset. You can gauge the delta of individual positions, strategies, or your entire portfolio. It is the first derivative of an option’s price in relation to the price of the underlying. Traders use delta to understand their directional exposure better. It indicates how much an option's price should change in response to a $1 change in the underlying price. Delta is represented as a number between [0;1] for calls and [-1;0] for puts. Positions with a positive delta increase in value if the price of the underlying increases. Positions with a negative delta increase in value if the price of the underlying decreases.

The strike price of an at the money (ATM) option is near the spot price — generally with a delta in the 40-60 range (approximately). Out of the money (OTM) options and ITM options generally lie on the opposite ends of this 40-60 delta range. An ITM option will have a delta > 60, and an OTM option will have a delta < 40 (approximately). A deep ITM option (delta approaching 1) will begin to trade close to dollar-for-dollar with the underlying. Delta is a valuable indicator for determining how many options contracts are needed to hedge long or short positions in your portfolio.

If you anticipate upward price action, you want to construct a portfolio with a positive delta. If you anticipate downward price action, you want to construct a portfolio with a negative delta. If you have no directional bias but expect volatility (or a lack thereof), you would then construct a delta-neutral portfolio. Delta neutral strategies are resistant (not immune) to directional swings — they capitalize on theta decay or overstated IV.

It is important to remember that the delta of an option fluctuates as the underlying moves. If delta neutrality is desired, it must be maintained by recalibrating your positions to account for the changes to their deltas. Delta is a proxy for the likelihood that an option will move ITM. How IV impacts delta depends on the moneyness of an option’s strike (i.e., if it is ATM, ITM, or OTM).

In the case of ATM strikes, delta remains static when IV rises or falls. For ITM strikes, delta decreases in response to increasing IV and increases in response to decreasing IV. For OTM strikes, delta increases in response to increasing IV and decreases in response to decreasing IV. Traders often refer to option deltas in basis points (e.g., an option with a 0.15 delta would be called a 15 delta option).

Examples (assuming all else equal):

The underlying price of a call option (0.35 delta) increases by $1

A $0.35 increase in the option's value would be expected

The underlying price of a put option (-0.67 delta) decreases by $1

A $0.67 increase in the option's value would be expected

A 0.40 delta SOL option trades at a $1.80 premium while SOL trades at $33. The price of SOL then gaps up to $38

The resulting premium would be $1.80 + 0.40 x ($38 - $33) = $3.80

### Gamma (Γ)

Gamma refers to the rate of change of an option's delta in response to changes in the underlying asset's price. Specifically, it represents the expected change in delta given a $1 move in the underlying. Gamma can be thought of as the delta of an option's delta. Gamma is to delta as acceleration is to speed. Acceleration is the rate at which speed changes — gamma is the rate at which delta changes.

Gamma is the second derivative of an option's price relative to the underlying price. It signals the stability of an option's delta. Gamma poses unique risks for option sellers and unique benefits for option buyers. For option buyers, the profit rate accelerates with each favorable move in the underlying price, while losses decelerate with each unfavorable move. For option sellers, the inverse is true. Option buyers are long gamma.

The magnitude of an option buyer's directional exposure increases with each favorable move in the underlying price, amplifying their potential profits. Option sellers are short gamma. The magnitude of an option seller's directional risk exposure increases with each unfavorable move in the underlying price, amplifying their potential losses. Gamma increases as the underlying price draws near the strike price. It decreases as the underlying price moves away from the strike price. All else equal, gamma is at a peak at the money (ATM) and a trough deep in the money (ITM) or out of the money (OTM).

Gamma has time dependence characteristics (i.e., it is affected by time passage, even if the price of the underlying is static). Options nearing expiry have the highest gamma sensitivity because the delta of near-dated options is imminently converging on 0 or 1. An ATM option’s gamma is at a maximum when expiry is approaching. The gamma of deep OTM and ITM options is at a maximum when expiry is distant.

How implied volatility (IV) influences gamma depends on an option’s moneyness. IV and gamma are negatively correlated for ATM options and positively correlated for OTM and ITM options. For ATM strikes, gamma increases when IV falls and decreases when IV rises. For ITM strikes, gamma decreases when IV falls and increases when IV rises. For OTM strikes, gamma decreases when IV falls and increases when IV rises.

The ATM and deep ITM/OTM gamma differential is greatest when IV is low. When IV is high, the differential is reduced. This is logical, considering ATM gamma remains high in the absence of high IV, whereas low IV conditions spell deep ITM/OTM gamma values closer to zero. One critique of traditional gamma measurements is that they are too localized (i.e., reference a spot range that is too narrow).

Using so-called shadow gamma is a hedging approach that aims to minimize insufficiently hedged positions in a portfolio. Up-gamma refers to the change in delta in response to an incremental increase in the underlying price. Down-gamma refers to the change in delta in response to an incremental decrease in the underlying price.

Shadow up-gamma is equal to the delta at a higher underlying price and vol, less the delta at the original underlying price and vol. Shadow down-gamma is equal to the delta at the original underlying price and vol, less the delta at a lower underlying price and vol. In essence, shadow gamma attempts to reduce risk exposure across positions by incorporating variable underlying prices and vols when forecasting changes in delta. Gamma is calculated by dividing the delta differential by the difference in the underlying price (i.e., Γ = D1 - D2 / P1 - P2).

Examples (assuming equal IV and DTE):

Call option (0.6 delta, 0.05 gamma)

If the underlying price increases by $1, the expected delta would be 0.65

Put option (-0.35 delta, 0.04 gamma)

If the underlying price increases by $1, the expected delta would be -0.31

A $32 SOL call option (0.6 delta) trades at $32 and then rises to $36, moving the option’s delta to 0.75

The option’s resulting gamma would be 0.037

**Theta **(Θ)

Theta measures the sensitivity of an option’s price to the passing of time. Theta is negative for options buyers and positive for options sellers. While theta is useful, it is essential to remember that it assumes constant price movement and IV. Options sellers benefit from the daily decay of an option’s price (i.e., theta decay). For options buyers, conversely, theta decay is a bitter adversary.

Options buyers need directionally favorable price action or IV expansion to outpace theta decay. The effect theta decay has on long positions can be minimized by selling options to collect theta while maintaining net long volatility exposure. Long volatility options spreads (e.g., debit, calendar, and diagonal spreads) are effective to this end. Theta helps inform traders how much extrinsic value an option will lose every day until it expires.

The probability that an option will become profitable declines as it matures. Accordingly, theta decay accelerates as expiry looms. Theta is at a peak for at the money (ATM) options. Theta is lowest for deep in the money (ITM) or deep out of the money (OTM) options. As expiry approaches, theta increases for at (or near) the money options.

Options sellers harvest theta, whereas buyers are subject to its predations. Traders utilise theta to assess how much the value of an underlying asset needs to move to offset the premium destruction resulting from time decay. An option's premium is comprised of intrinsic and extrinsic value. All premium value that is not intrinsic is extrinsic. Theta erodes extrinsic value and does not affect intrinsic value. Intrinsic value is the value of an option if it were to expire today. It is the difference between an option’s strike price and mark price. Extrinsic value refers to the expectation value component of an option’s premium.

Imagine SOL is trading at $34, and the aforementioned call option ($2.72 premium) has a $33 strike. The option would consist of $1.00 of intrinsic value and $1.72 of extrinsic value. $2.72 (premium) - $1.00 (intrinsic value) = $1.72 (extrinsic value). Theta melts away an option’s premium over time (in a non-linear fashion) up until expiry, at which point only intrinsic value remains.

The value of deep ITM options is almost purely intrinsic. The value of OTM options is strictly extrinsic. Extrinsic value is highly influenced by IV. This is logical, considering extrinsic value is the expectation value of an option, and IV is a proxy for market sentiment. Regardless of the moneyness of an option’s strike, IV and theta are positively correlated. For ITM, OTM, and ATM strikes, theta decreases when IV decreases and increases when IV increases — increasing IV results in higher premiums, which means more value for theta to feast on. As IV declines, so too does extrinsic value, leaving less value for theta to erode.

Examples (assuming all else equal):

A trader is holding a $28 SOL put (-0.096 theta) with a $0.78 premium while SOL is trading at $33.60

The option's theoretical value after one day would be $0.684

A call option has a $2.72 premium and a theta value of -0.120, so the option’s premium would lose 0.120 of value with the passing of each day

The option’s theoretical value after one day would be $2.60

### Vega (ν)

Vega is a hedging parameter that measures the rate of change of an option’s price in response to changes in the underlying asset's implied volatility (IV). You can gauge the vega of individual positions, strategies, or your entire portfolio. Vega measures the responsiveness of an option’s price to a one-point (i.e., 1%) move in the IV of the underlying asset. Vega is not a Greek letter; however, it is expressed by the Greek letter nu (v).

It is the first derivative of an option’s price with respect to IV. Factors influencing an option’s vega include days to expiration (DTE), strike price, and IV dynamics (i.e., whether IV is expanding or contracting). Generally, vega is positively correlated with the price of an option. I.e., An increase in vega increases an option’s value. A decrease in vega decreases an option’s value. Vega is positive for long options. Vega is negative for short options.

It is highest for at the money (ATM) options and declines as expiry approaches. Options with more DTE have higher vega. Vega influences an option's extrinsic value. It does not affect the intrinsic value component of an option’s premium. Traders will often juxtapose an option’s vega against its bid-ask spread to assess the competitiveness of the spread. If an option’s vega is greater than its bid-ask spread, the spread is considered competitive — and vice versa.

Net short options strategies (e.g., credit spreads) are adversely affected by increasing vega. Net long options strategies (e.g., debit spreads) benefit from increasing vega. Expanding IV benefits long portfolios. Contracting IV benefits short portfolios. Like theta and extrinsic value, vega follows a bell curve, with moneyness on the x-axis and vega on the y-axis. Vega is at a maximum at the highest point of the arc (ATM) and a minimum at the tails (ITM/OTM).

Vega and extrinsic value share a similar relationship with time. Vega is higher for options with more distant maturities. As expiry approaches, vega falls. How IV impacts vega depends on the moneyness of an option’s strike. For ITM and OTM strikes, vega decreases when IV falls and increases when IV rises. For ATM strikes, vega remains static regardless of whether IV falls or rises.

Traders often refer to IV without the %. An IV of 26% would be described as an IV of 26. Vega values denote how much (in dollar terms) the price of an option is expected to increase in response to IV increasing by 1%. Vega x (New IV - Old IV) = Change in Option Premium. Highly positive/negative vega indicates high sensitivity to changes in IV. When the vega of an option is close to zero, changes in IV have a marginal impact on the value of a position. Like gamma, vega cannot be adjusted by taking a position in the underlying asset.

Gamma hedging protects against large swings in the price of the underlying asset. Vega-hedging protects against changes in underlying IV. Your entire portfolio's vega risk is determined by the potential variability of the vol surface (IV across strikes and expiries). As a result of different options in a portfolio having different IVs, vega-neutrality and gamma-neutrality do not generally co-occur in a portfolio. You would likely need to add two or more options to your portfolio to hedge both vega and gamma.

One critique of simple portfolio vega calculations is that summing up the vegas of all positions does not adequately account for the term structure of IV (i.e., how the IV of options of different maturities will change in the future). For example, the IV of an option with 30 DTE would generally be more sensitive than an option with 55 DTE. Using modified vega is an alternative method that weights vegas to account for varying IV sensitivities across maturities.

Examples (assuming all else equal):

A $36 SOL call option is trading at a $2.82 premium with a vega of 0.024 while IV is at 117. IV moves from 117 to 132 (15pt increase)

The premium would increase by 15 x $0.024 = $0.36. The resulting premium would be $3.18

Instead of increasing, the IV of the option ($2.82 premium, 0.024 vega) in the above example instead drops from 117 to 107 (10pt decrease)

The premium would decrease by 10 x $0.024 = $0.24. The resulting premium would be $2.58

**Rho (**ρ)

A measurement of how sensitive an option is relative to interest rates. It signals the expected change an options contract will incur in response to interest rates fluctuating by one percentage point. If there were ever a Greek to leave by the wayside, it would be Rho, considering its lack of usefulness relative to its counterparts. Despite generally not being an essential factor for most trades, it does offer utility to market makers regularly borrowing funds, and it is worth paying attention to when buying Long-Term Equity Anticipation Securities (LEAPS). This is because far-dated options contracts are far more sensitive to changes in interest rates than near-dated options.

**Minor Greeks**

**Vanna**

Vanna is a second-order Greek that measures the impact a minute (1%) fluctuation in IV has on the delta of an option (i.e., how vega changes in responses to small movements in the price of the underlying asset). It is at a peak for ATM options. Given its utility in gauging the effects of underlying price and volatility, vanna can be a helpful maintenance tool for delta or vega-hedged portfolios.

##### Charm

Charm is a second-order Greek that measures the sensitivity of delta to changing days to expiration (DTE); it is commonly referred to as

*delta bleed (or delta decay)*, as the deltas of options tend to wane as expiry approaches. Like vanna, it peaks for ATM options. Charm is especially useful for traders managing a delta-neutral book.

##### Lambda

Lambda is a first-order Greek that measures the elasticity of an option; specifically, an option’s sensitivity to a 1% move in the price of the underlying asset. It denotes the leverage factor an option confers. Lamda is equal to delta x (underlying price / option premium). When choosing from a basket of options, lambda values can shed light on how useful a given option will be for hedging.

**Zomma**

Zomma is a third-order Greek that measures gamma's sensitivity to volatility changes. It can help traders with the maintenance of a gamma-hedged portfolio. High zomma signals that minute IV fluctuations will result in sizable changes in gamma. The risk profile of a portfolio of options is non-linear and ever in flux; zomma can be a valuable tool for managing this risk.

##### Vomma

Vomma is a second-order Greek that measures vega’s sensitivity to changes in volatility. Vomma is at a peak for OTM options. Looking at vomma and vega together gives you a more precise forecast of how an option’s value will fluctuate in response to changes in volatility. Like vega, vomma is positive for long options and negative for short options. Increasing vomma is generally desirable for long option holders, while short option holders like to see decreasing vomma.

**Color**

Color is a third-order Greek that measures Gamma’s sensitivity to the passing of time. It can be thought of as

*gamma decay*. Color can be a useful tool for the maintenance of a gamma-hedged portfolio. Color becomes less useful when options approach expiry, as it is less stable and, thus, less informative. Color tells us the degree to which gamma is expected to change. Color values denote the expected daily move in gamma values.

##### Speed

Speed is a third-order Greek that measures gamma’s sensitivity to the price action of the underlying asset. Speed is a useful tool for maintaining both delta and gamma hedged portfolios. It informs the extent to which traders may have to recalibrate their hedges in response to fluctuations in the underlying asset price.

**Ultima**

Ultima is a third-order Greek that measures the sensitivity of vomma to changes in volatility. It is a useful indicator of expected vomma fluctuations. Positive ultima signals a positive correlation between vomma and volatility.

##### Veta

Veta is a second-order Greek that measures vega’s rate of change over time (i.e., vega decay, or how the sensitivity of theta in response to minute changes in volatility).

##### Epsilon

Epsilon is a first-order Greek that measures the sensitivity of an option’s premium to fluctuations in the dividend yield of an underlying asset (only applicable in the context of equity options).

##### Vera

Vera is a second-order Greek that measures rho’s sensitivity to volatility. It is sometimes used to hedge rho with more precision.

## Greeks Primer

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