Binomial Tree Documentation

Overview

The Binomial Tree model is a popular method for option pricing, especially for American-style options where early exercise is possible. Unlike the Black-Scholes model, which assumes a continuous-time process, the binomial tree model uses discrete time steps to model the underlying asset's price changes. The asset price can either move up or down at each step, creating a tree of possible future prices. At each final node (expiration), the payoff is calculated, and the option value is then worked backward to the present by considering the risk-neutral probabilities.

Key Concepts in the Binomial Tree Model

• Discretization of Time: The binomial tree model divides the time to expiration into discrete intervals (time steps). At each time step, the asset price can either go up or down by a certain factor, creating a "binomial" structure.


• Risk-Neutral Valuation: The model assumes that investors are risk-neutral, meaning that they don't require extra compensation for taking on risk. The option price is computed by discounting the expected payoff under the risk-neutral probability measure.


• American vs. European Options: The binomial tree model can handle both American and European options. American options can be exercised at any time before expiration, while European options can only be exercised at expiration.

Function Parameters

• S: Current price of the underlying asset.
• K: Strike price of the option.
• T: Time to expiration in years.
• r: Risk-free interest rate.
• σ: Volatility of the underlying asset (annualized standard deviation of returns).
• n: Number of time steps in the binomial tree. Increasing the number of steps increases the model's accuracy but also the computational effort.
• Option type: The type of option, either 'call' or 'put'.
• Exercise type: Whether the option is American or European.

Explanation of Key Calculations

1. Up and Down Factors: The binomial tree model uses up and down factors (\(u\) and \(d\)) to represent the possible price changes at each time step. These are calculated as follows:
   • \( u = \exp(\sigma \sqrt{\Delta t}) \)
   • \( d = \frac{1}{u} \)
   where \(\Delta t\) is the length of each time step.
2. Risk-Neutral Probabilities: The probability of an upward price movement (\(p\)) and downward price movement (\(1 - p\)) under the risk-neutral measure are given by:
   • \( p = \frac{\exp(r \Delta t) - d}{u - d} \)
   These probabilities ensure that the model is consistent with no arbitrage and risk-neutral valuation principles.
3. Option Payoff at Expiration: At the final step of the tree, the payoff for each possible asset price is calculated:
   • Call Option: \( \text{payoff} = \max(S_T - K, 0) \)
   • Put Option: \( \text{payoff} = \max(K - S_T, 0) \)
4. Backward Induction: Once the payoffs at expiration are known, the option value is determined by working backward through the tree, from the last time step to the first, using the risk-neutral probabilities:
   • Option Value: \( \text{value} = e^{-r \Delta t} \left(p \cdot \text{value}_{up} + (1 - p) \cdot \text{value}_{down}\right) \)
5. Early Exercise (American Options): For American options, at each node, the option value is compared to the immediate payoff from early exercise, and the higher value is taken:
   • For calls: \( \text{value} = \max(\text{immediate payoff}, \text{continuation value}) \)
   • For puts: \( \text{value} = \max(\text{immediate payoff}, \text{continuation value}) \)