Introduction

Volatility plays a pivotal role in options pricing as it measures the magnitude of price fluctuations in the underlying asset. In this chapter, we will explore how volatility is captured and integrated into the binomial model, a powerful tool for option pricing. Understanding this concept is essential for accurate option valuation, risk assessment, and decision-making.

Capturing Volatility in the Binomial Model

In the binomial model, volatility is incorporated through the specification of price movements in each time step. The model assumes that the price of the underlying asset can move up or down in each period. The magnitude of these price movements is determined by volatility.

Calculating Upward and Downward Movements

The binomial model uses the following formulas to calculate upward and downward price movements:

• Upward Movement (U): $U = e^{\sigma \sqrt{\Delta t}} $, where $\sigma$ is the volatility and $\Delta t$ is the time interval
• Downward Movement (D): $D = \frac{1}{U}$, which is the reciprocal of the upward movement

These formulas are critical in capturing the impact of volatility on option prices. Higher volatility leads to larger price movements (U) and wider price spreads between up and down movements.

Example: Assume a stock is currently trading at USD 100, and we want to estimate its price after one period in a binomial model. If the annualized volatility is 20%, and the time period is one year, we can calculate $U$ and $D$ as follows:

• $U = e^{0.20\times \sqrt{1}} \approx 1.2214$
• $D = \frac{1}{U} ≈ 0.8187$ This means that in one year, the stock price is expected to go up by approximately 22.14% (U) or down by about 18.87% (D) based on its historical volatility.

Visualizing Volatility with Binomial Trees

Binomial trees are constructed to visualize the price path of the underlying asset under different volatility scenarios. Each node in the tree represents a possible price at a specific time step. By iteratively applying the upward and downward movements, the tree shows how prices evolve over time, capturing the influence of volatility.