Introduction

In this chapter, we delve into the fascinating world of option sensitivity measures, often referred to as the “Greeks”. These measures play a crucial role in assessing and managing the risks and opportunities associated with options. We will explore four key Greeks: theta, gamma, vega, and rho. Understanding these measures is essential for option traders and portfolio managers, as they provide insights into how options respond to changes in various market factors.

Theta ($\theta$)

Theta, also known as time decay, measures the rate at which the value of an option erodes with the passage of time, all else being equal. It’s an essential concept for option buyers and sellers. Theta is calculated as follows:

$$\Theta=\frac{\partial V}{\partial t}$$ Where:

• $\Theta$ = Theta
• $V$ = Option’s Value
• $t$ = Time

Example: Suppose you hold a call option on a stock with a theta of -0.03 . This means that, all else being equal, your option will lose USD 0.03 in value for each day that passes. Theta is particularly relevant for options with shorter time to expiration, as they experience faster time decay.

Gamma ($\Gamma$)

Gamma measures the rate of change of an option’s delta concerning changes in the underlying asset’s price. It quantifies how an option’s delta changes as the underlying asset’s price fluctuates. Gamma is calculated as follows:

$$\Gamma=\frac{\partial^2 V}{\partial S^2}$$

Where:

• $\Gamma$ = Gamma
• $V$ = Option’s Value
• $S$ = Underlying Asset’s Price

Example: Suppose you own a call option with a gamma of 0.05. If the underlying stock’s price increases by USD 1, your option’s delta might increase by 0.05, indicating a more significant price sensitivity to further price changes.

Vega ($\nu$)

Vega measures an option’s sensitivity to changes in implied volatility. It quantifies how an option’s price changes when implied volatility levels shift. Vega is calculated as follows:

$$\nu=\frac{\partial V}{\partial \sigma}$$

Where:

• $\nu$ = Vega
• $V$ = Option’s Value
• $\sigma$ = Implied Volatility

Example: Suppose you hold a put option with a vega of 0.03. If the implied volatility of the underlying asset increases by 1%, your option’s value might increase by USD 0.03. Vega is particularly important for traders who want to hedge or speculate on changes in volatility.

Rho ($\rho$)

Rho measures an option’s sensitivity to changes in interest rates. It quantifies how an option’s price changes when interest rates fluctuate. Rho is calculated as follows:

$$\rho=\frac{\partial V}{\partial r}$$

Where:

• $\rho$ = Rho
• $V$ = Option’s Value
• $r$ = Risk-Free Interest Rate

Example: Suppose you have a call option with a rho of 0.02. If the risk-free interest rate increases by 1%, your option’s value might increase by USD 0.02. Rho is particularly relevant for options with longer maturities.

Calculating Gamma and Vega for a Portfolio

To calculate the gamma and vega for a portfolio of options, you need to aggregate the individual gammas and vegas of each option in the portfolio. The gamma and vega of a portfolio represent the overall sensitivity to changes in the underlying asset’s price and implied volatility, respectively.