Calculate option prices using binomial and trinomial tree models with interactive visualization
Start by entering the fundamental parameters: Current Asset Price (the current price of the underlying asset), Strike Price (the exercise price of the option), and Time to Expiry (in years). These form the foundation of your option pricing calculation.
Configure the market environment: Risk-Free Rate (annual interest rate), Volatility (annualized volatility percentage), and Option Type (Call or Put). These parameters determine the pricing dynamics and risk factors.
Select your preferred tree model (Binomial or Trinomial) and set the Number of Steps. More steps provide higher accuracy but increase complexity. For most cases, 4-6 steps offer a good balance between accuracy and simplicity.
Choose between European (exercise only at expiry) or American (early exercise allowed) options. American options are typically worth more due to early exercise flexibility, especially for put options.
The calculated fair value of the option based on the tree model. This represents the theoretical price at which the option should trade in an efficient market. If market price > calculated price, the option may be overpriced; if market price < calculated price, it may be underpriced.
Determines how much the asset price increases in each time step. Calculated as u = e^(蟽鈭毼攖), where 蟽 is volatility and 螖t is time step. A higher volatility results in larger up movements, creating more price dispersion in the tree.
Determines how much the asset price decreases in each time step. Calculated as d = 1/u to maintain the tree's recombining property. Together with the up factor, these determine the asset price range at expiration.
The probability of an upward movement in the risk-neutral world, calculated as p = (e^(r螖t) - d)/(u - d). This probability is used to weight the expected payoffs when calculating option values, ensuring the tree prices options correctly.
Measures the option's sensitivity to underlying asset price changes. Delta = (Option Value Up - Option Value Down) / (Asset Price Up - Asset Price Down). For calls: 0 鈮?螖 鈮?1; for puts: -1 鈮?螖 鈮?0. Delta indicates how many shares to buy/sell for hedging.
Measures how Delta changes with asset price movements. Gamma = Rate of change of Delta. High Gamma means Delta changes rapidly, requiring frequent hedge adjustments. Gamma is highest for at-the-money options near expiration.
The interactive tree shows asset prices (green nodes) and option values (red nodes) at each time step. Green nodes represent possible asset prices, red nodes show corresponding option values. Hover over nodes to see exact values and understand the pricing evolution.
The binomial tree model assumes that the underlying asset price can move up or down by a specific factor in each time step. This creates a tree structure where each node represents a possible asset price at a given time.
In risk-neutral pricing, we assume that investors are indifferent to risk. The expected return on the underlying asset equals the risk-free rate, allowing us to calculate option prices using risk-neutral probabilities.
European options can only be exercised at expiration, while American options can be exercised at any time before expiration. The binomial tree is particularly useful for pricing American options due to its ability to handle early exercise decisions.
As the number of steps in the tree increases, the binomial tree model converges to the Black-Scholes formula. More steps generally provide more accurate results but increase computational complexity.