Introduction

Delta is a critical concept in option pricing and risk management. It measures how the price of an option changes in response to small changes in the price of the underlying asset, such as a stock. Understanding delta is crucial for option traders and investors as it helps them assess the sensitivity of their options positions to changes in the underlying asset’s price. In this chapter, we will delve into the concept of delta, learn how to calculate it for stock options using the binomial model, and explore its significance in making informed trading decisions.

Delta

Delta is defined as the rate of change in the option’s price concerning a one-unit change in the price of the underlying asset. It quantifies the option’s sensitivity to changes in the stock’s value. Delta values range from -1 to 1 for put options and 0 to 1 for call options.

Call Option

• A delta of 0.5 means that for every USD 1 increase in the stock price, the call option’s price increases by USD 0.50
• A delta of 1 implies that the call option’s price will increase by USD 1 for every USD 1 increase in the stock price
• A delta of 0 suggests that the call option’s price will not change with changes in the stock price.

Put Option

• A delta of -0.5 means that for every USD 1 increase in the stock price, the put option’s price decreases by USD 0.50
• A delta of -1 implies that the put option’s price will decrease by USD 1 for every USD 1 increase in the stock price
• A delta of 0 suggests that the put option’s price will not change with changes in the stock price

Calculating Delta Using the Binomial Model

Delta can be calculated using the binomial model by examining how the option’s price changes over different stock price movements. Here’s the formula to calculate delta for a call option:

$$ \text{Delta(Call)} = \frac{C(U) - C(D)}{S(U) - S(D)} $$

Where:

• $C(U)$ is the call option’s price when the stock price moves up (U)
• $C(D)$ is the call option’s price when the stock price moves down (D)
• $S(U)$ is the stock price when it moves up
• $S(D)$ is the stock price when it moves down

For a put option, the formula is similar:

$$ \text{Delta(Put)} = \frac{P(U) - P(D)}{S(U) - S(D)}$$

Where:

• $P(U)$ is the put option’s price when the stock price moves up (U)
• $P(D)$ is the put option’s price when the stock price moves down (D)

Example: Let’s say you have a call option on Company X’s stock with a strike price of USD 50. The current stock price is USD 48, and you calculate that if the stock price goes up (U), the call option’s price will be USD 3, and if it goes down (D), the call option’s price will be USD 1. The stock price will be USD 55 if it goes up (U) and USD 45 if it goes down (D).

Using the delta formula:

$$ \text{Delta(Call)} = \frac{C(U) - C(D)}{S(U) - S(D)} $$ $$ \text{Delta(Call)}= \frac{3-1}{55-45} = \frac{2}{10} = 0.2$$

So, the delta of this call option is 0.2. This means that for every USD 1 increase in Company X’s stock price, the call option’s price will increase by USD 0.20.