Introduction

Understanding the relationship between delta, theta, gamma, and vega is essential for managing options effectively. These sensitivity measures, often referred to as the “Greeks,” play a pivotal role in assessing how options behave under different market conditions. In this chapter, we will explore how these Greeks are interrelated and how changes in one can impact the others. This knowledge is crucial for option traders and investors looking to make informed decisions.

Delta

Delta, denoted by $\mathrm{\Delta }$, measures the rate of change in the option’s price concerning changes in the underlying asset’s price. It indicates how much an option’s price will change for a one-point change in the underlying asset. Delta can be positive or negative, depending on whether the option is a call or put.

Theta

Theta, denoted by $\theta$, represents time decay in the option’s price. It quantifies how much the option’s value erodes as time passes. Theta is negative for both call and put options, indicating that options lose value as time elapses. Theta increases as the option’s expiration date approaches.

Gamma

Gamma, denoted by $\mathrm{\Gamma }$, measures the rate of change of delta concerning changes in the underlying asset’s price. In essence, it quantifies how delta itself changes as the underlying asset moves. Gamma is positive for both call and put options and is highest for at-the-money options.

Vega

Vega, denoted by $\nu$, reflects the option’s sensitivity to changes in implied volatility. It measures the change in the option’s price for a one-percentage-point change in implied volatility. Vega is typically positive for both call and put options, indicating that options become more valuable as volatility increases.

Relationships Between Greeks

Understanding the relationships between these Greeks is crucial for options traders. Here are some key insights:

1) Delta and Gamma: Delta and gamma are closely related. As the underlying asset’s price changes, delta changes, and gamma measures this change in delta. In other words, gamma tells you how fast delta is changing. Relationship between delta and gamma is given below:

$$\mathrm{\Delta }\mathrm{\Delta }S=\mathrm{\Gamma }$$ Where:

• $\mathrm{\Delta }\mathrm{\Delta }S$ = change in delta
• $\mathrm{\Gamma }$ = gamma

2) Delta and Theta: Delta and theta have an inverse relationship. As an option approaches its expiration date, theta increases, which means delta becomes more sensitive to changes in the underlying asset’s price.

3) Delta and Vega: Delta and vega are positively correlated for options. When implied volatility increases (positive vega), the option’s delta can change, making it more or less sensitive to price movements.