This calculator uses the binomial option pricing model to estimate the fair value of a European call or put option. Enter the parameters and press "Calculate".
Results
| Metric | Call Option | Put Option |
|---|---|---|
| Option Price |
Binomial Option Pricing Model Explanation
The Binomial Option Pricing Model estimates the price of options by building a binomial tree of potential future stock prices. The model is based on discrete time steps and assumes that, in each step, the stock price can move either up or down by a specific factor.
Key Parameters
- Stock Price (S): The current price of the underlying asset.
- Strike Price (K): The price at which the option can be exercised.
- Risk-free Interest Rate (r): The risk-free rate used to discount future payoffs.
- Volatility (σ): The expected volatility of the stock price.
- Time to Expiration (T): The time remaining until the option expires.
- Number of Steps (N): The number of time steps in the binomial tree.
Binomial Formulae
The option price is calculated by first estimating the possible stock prices at each step in the binomial tree and then working backwards to determine the option price at the initial step.
The stock price movement at each step is modeled as:
\( u = e^{\sigma \sqrt{\Delta t}} \quad \text{(up factor)} \)
\( d = \frac{1}{u} \quad \text{(down factor)} \)
The risk-neutral probability of an upward movement is:
\( p = \frac{e^{r \Delta t} – d}{u – d} \)
The value of the option is calculated by working backwards through the tree, where the value at each node is:
\( C = e^{-r \Delta t} \left[ p C_{\text{up}} + (1 – p) C_{\text{down}} \right] \)
Steps in the Binomial Model
- Build the binomial tree by calculating the possible stock prices at each node.
- Calculate the option payoff at each final node (expiration).
- Work backwards through the tree, applying the binomial formula at each node.
- Arrive at the option price at the root of the tree (current time).
Advantages of the Binomial Model
- Can handle American-style options, which allow early exercise.
- More flexible than the Black-Scholes model for handling various market conditions.
- Provides a more intuitive understanding of option pricing by simulating different stock price paths.
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