Binomial Option Pricing Model

Core concepts of the binomial model

The binomial model assumes that an asset price can only move up or down by specific factors during each time step. This simplified approach creates a powerful framework for understanding option pricing and replication through dynamic hedging.

Key parameters include:

• Up factor (u): The multiplicative factor for upward price movements
• Down factor (d): The multiplicative factor for downward price movements
• Risk-free rate (r): The interest rate used for discounting
• Probability (p): Risk-neutral probability of an upward movement

The model's mathematical foundation relies on the risk-neutral pricing framework:

$p = \frac{(1 + r) - d}{u - d}$

Where the up and down factors are typically calculated as:

$u = e^{\sigma\sqrt{\Delta t}}$ $d = \frac{1}{u}$

Building the price tree

The binomial tree is constructed by starting with the current asset price (S₀) and creating nodes representing possible future prices through successive up and down movements:

Each level of the tree represents a time step, with the number of steps determining the model's granularity. More steps generally provide better accuracy but increase computational complexity.

Option valuation process

The valuation process works backward through the tree, starting at expiration and discounting expected payoffs:

1. Calculate terminal node payoffs using the option's payoff function
2. Work backwards through the tree using risk-neutral probabilities
3. Discount intermediate values at the risk-free rate

For a European option, the value at each node is:

$V_t = \frac{pV_{t+1}^u + (1-p)V_{t+1}^d}{1 + r}$

Where:

• $V_t$ is the option value at time t
• $V_{t+1}^u$ is the option value after an up movement
• $V_{t+1}^d$ is the option value after a down movement

Early exercise and American options

The binomial model naturally handles American options by comparing the holding value with the immediate exercise value at each node:

$V_t = \max(\text{holding value}, \text{exercise value})$

This flexibility in handling early exercise decisions is one of the model's key advantages over the Black-Scholes Model.

Applications in risk management

The binomial model provides insights into option Greeks and risk management:

• Delta (Δ) can be estimated by comparing option values across adjacent nodes
• Gamma (Γ) is calculated from changes in delta
• Theta (Θ) is measured through time step analysis

This makes the model valuable for both pricing and risk analysis in options trading and portfolio management.

Model limitations and considerations

While powerful, the binomial model has important limitations:

1. Assumes constant volatility and interest rates
2. May require many steps for accurate pricing
3. Computational complexity increases exponentially with steps
4. Simplified assumption of only two possible price movements

These limitations should be considered when applying the model in practice, particularly for complex exotic options or in markets with varying volatility.

Relationship to continuous-time models

As the number of time steps increases, the binomial model converges to the Black-Scholes Model:

$\lim_{n \to \infty} \text{Binomial Price} = \text{Black-Scholes Price}$

This convergence provides a theoretical link between discrete and continuous-time option pricing frameworks, making the binomial model both a practical tool and a pedagogical bridge to more advanced pricing models.