Binomial Option Pricing Model Calculator

Binomial Option Pricing Model Calculator

Estimated Option Price

function calculateBOPM() { var S = parseFloat(document.getElementById('stockPrice').value); var K = parseFloat(document.getElementById('strikePrice').value); var T = parseFloat(document.getElementById('timeToMaturity').value); var r = parseFloat(document.getElementById('riskFreeRate').value) / 100; var v = parseFloat(document.getElementById('volatility').value) / 100; var n = parseInt(document.getElementById('steps').value); var type = document.getElementById('optionType').value; if (isNaN(S) || isNaN(K) || isNaN(T) || isNaN(r) || isNaN(v) || isNaN(n) || n <= 0) { alert("Please enter valid positive numbers for all fields."); return; } var dt = T / n; var u = Math.exp(v * Math.sqrt(dt)); var d = 1 / u; var a = Math.exp(r * dt); var p = (a – d) / (u – d); var q = 1 – p; // Initialize price tree at maturity var prices = []; for (var i = 0; i = 0; j–) { for (var i = 0; i <= j; i++) { prices[i] = (p * prices[i] + q * prices[i + 1]) / a; } } var resultDiv = document.getElementById('calcResult'); var resultVal = document.getElementById('resultValue'); var metrics = document.getElementById('additionalMetrics'); resultDiv.style.display = "block"; resultVal.innerHTML = prices[0].toFixed(4); metrics.innerHTML = "Up Factor (u): " + u.toFixed(4) + " | Down Factor (d): " + d.toFixed(4) + " | Risk-Neutral Prob (p): " + p.toFixed(4); }

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Understanding the Binomial Option Pricing Model (BOPM)

The Binomial Option Pricing Model is a powerful iterative method used to value options. Developed by Cox, Ross, and Rubinstein, it provides a mathematical framework for estimating the fair value of an option by simulating the path of an underlying asset over discrete time steps.

Key Components of the Model

• S₀ (Current Stock Price): The market price of the asset at the moment of valuation.
• K (Strike Price): The predetermined price at which the option holder can buy (call) or sell (put) the asset.
• Volatility (σ): A measure of how much the stock price is expected to fluctuate. Higher volatility increases option premiums.
• Risk-Free Rate (r): The theoretical rate of return on an investment with zero risk, usually based on government bonds.
• Up and Down Factors (u & d): Calculated from volatility and time, these represent the possible price movements at each step.

How the Calculation Works

The model builds a "binomial tree" of potential future stock prices. At each step, the price can move up by factor u or down by factor d. Once the tree reaches the expiration date (maturity), the final option value is calculated based on the difference between the stock price and the strike price.

The calculator then works backward from the expiration date to the present day, discounting the expected values using risk-neutral probabilities. This process accounts for the time value of money and the probability of various price outcomes.

Example Scenario

Imagine a stock trading at 100. You want to price a 6-month (0.5 year) Call option with a strike of 105. If the risk-free rate is 5% and volatility is 20%, using a 10-step model:

1. The model calculates small price movements every 0.05 years.
2. It determines the probability of the stock being above 105 at the end of 6 months.
3. It discounts those potential profits back to today's dollars.

In this case, the resulting theoretical price represents what a trader should pay for the option given these specific market conditions.

Advantages of the Binomial Model

Unlike the Black-Scholes model, which is a continuous-time formula, the Binomial model is discrete and more intuitive. It is particularly useful for valuing American options (which can be exercised at any time) because it allows for checking the optimal exercise strategy at every node in the tree.