Black Scholes Model Calculator

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Black-Scholes Model Calculator

// Function to calculate the cumulative standard normal distribution (N(x)) function N(x) { // Abramowitz and Stegun approximation for N(x) var a1 = 0.319381530; var a2 = -0.356563782; var a3 = 1.781477937; var a4 = -1.821255978; var a5 = 1.330274429; var p = 0.2316419; var c2 = 0.3989422804014327; // 1 / sqrt(2 * PI) var L = Math.abs(x); var K = 1.0 / (1.0 + p * L); var W = c2 * Math.exp(-L * L / 2.0); var Nd = W * (((((a5 * K + a4) * K + a3) * K + a2) * K + a1) * K); if (x < 0) { return 1.0 – Nd; } else { return Nd; } } function calculateBlackScholes() { var S = parseFloat(document.getElementById('stockPrice').value); var K = parseFloat(document.getElementById('strikePrice').value); var T = parseFloat(document.getElementById('timeToExpiration').value); var sigma = parseFloat(document.getElementById('volatility').value); var r = parseFloat(document.getElementById('riskFreeRate').value); var q = parseFloat(document.getElementById('dividendYield').value); var resultDiv = document.getElementById('blackScholesResult'); resultDiv.innerHTML = ''; // Clear previous results if (isNaN(S) || isNaN(K) || isNaN(T) || isNaN(sigma) || isNaN(r) || isNaN(q) || S <= 0 || K <= 0 || T <= 0 || sigma < 0 || r < 0 || q < 0) { resultDiv.innerHTML = 'Please enter valid positive numbers for all fields. Volatility, Risk-Free Rate, and Dividend Yield can be zero or positive.'; return; } var d1_numerator = Math.log(S / K) + (r – q + (sigma * sigma / 2)) * T; var d1_denominator = sigma * Math.sqrt(T); var d1 = d1_numerator / d1_denominator; var d2 = d1 – sigma * Math.sqrt(T); // Calculate Call Option Price var callPrice = (S * Math.exp(-q * T) * N(d1)) – (K * Math.exp(-r * T) * N(d2)); // Calculate Put Option Price var putPrice = (K * Math.exp(-r * T) * N(-d2)) – (S * Math.exp(-q * T) * N(-d1)); resultDiv.innerHTML = '

Calculated Option Prices:

' + 'Call Option Price: $' + callPrice.toFixed(2) + " + 'Put Option Price: $' + putPrice.toFixed(2) + "; } .calculator-container { font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif; background-color: #f9f9f9; padding: 25px; border-radius: 10px; box-shadow: 0 4px 12px rgba(0, 0, 0, 0.1); max-width: 500px; margin: 30px auto; border: 1px solid #e0e0e0; } .calculator-container h2 { text-align: center; color: #333; margin-bottom: 25px; font-size: 1.8em; } .calculator-inputs label { display: block; margin-bottom: 8px; color: #555; font-weight: bold; font-size: 0.95em; } .calculator-inputs input[type="number"] { width: calc(100% – 22px); padding: 12px; margin-bottom: 18px; border: 1px solid #ccc; border-radius: 6px; font-size: 1em; box-sizing: border-box; transition: border-color 0.3s ease; } .calculator-inputs input[type="number"]:focus { border-color: #007bff; outline: none; box-shadow: 0 0 5px rgba(0, 123, 255, 0.3); } .calculator-inputs button { width: 100%; padding: 14px; background-color: #007bff; color: white; border: none; border-radius: 6px; font-size: 1.1em; cursor: pointer; transition: background-color 0.3s ease, transform 0.2s ease; margin-top: 10px; } .calculator-inputs button:hover { background-color: #0056b3; transform: translateY(-2px); } .calculator-inputs button:active { transform: translateY(0); } .calculator-results { margin-top: 25px; padding: 20px; background-color: #e9f7ef; border: 1px solid #d4edda; border-radius: 8px; color: #155724; font-size: 1.1em; line-height: 1.6; } .calculator-results h3 { color: #0f5132; margin-top: 0; margin-bottom: 15px; font-size: 1.4em; text-align: center; } .calculator-results p { margin-bottom: 8px; } .calculator-results p strong { color: #0f5132; }

Understanding the Black-Scholes Model for Option Pricing

The Black-Scholes Model is a fundamental mathematical model used for pricing European-style options. Developed by Fischer Black, Myron Scholes, and Robert Merton in 1973, it revolutionized the financial industry by providing a theoretical framework for determining the fair value of an option contract. This model is widely used by traders, investors, and financial institutions to make informed decisions about buying and selling options.

Key Assumptions of the Black-Scholes Model

The model relies on several key assumptions, which are important to understand when using the calculator:

  • European-style options: The option can only be exercised at expiration.
  • No dividends: The underlying stock does not pay dividends during the option's life (though extensions exist for dividend-paying stocks).
  • Efficient markets: Market movements cannot be predicted.
  • No transaction costs: Buying or selling options incurs no fees or commissions.
  • Constant risk-free rate and volatility: The risk-free interest rate and the volatility of the underlying asset are constant over the option's life.
  • Lognormal distribution of stock prices: Stock prices follow a continuous random walk with a constant drift and volatility.

Inputs for the Black-Scholes Calculator

To use the calculator effectively, you need to provide the following inputs:

  • Current Stock Price (S): This is the current market price of the underlying asset (e.g., a stock).
  • Strike Price (K): Also known as the exercise price, this is the price at which the option holder can buy (for a call) or sell (for a put) the underlying asset.
  • Time to Expiration (T): The remaining time until the option contract expires, expressed in years. For example, 6 months would be 0.5 years.
  • Volatility (σ): This measures the standard deviation of the underlying asset's returns. It represents how much the stock price is expected to fluctuate. It's typically expressed as an annualized decimal (e.g., 0.20 for 20%).
  • Risk-Free Rate (r): The annual rate of return on a risk-free investment, such as a government bond, expressed as a decimal (e.g., 0.05 for 5%).
  • Dividend Yield (q): The annual dividend yield of the underlying asset, expressed as a decimal. If the stock does not pay dividends, this value is 0.

How to Use the Calculator

  1. Enter the Current Stock Price: Input the current market price of the stock.
  2. Enter the Strike Price: Input the strike price of the option contract.
  3. Enter Time to Expiration: Input the time remaining until expiration in years (e.g., 0.25 for 3 months, 1.0 for 1 year).
  4. Enter Volatility: Input the annualized volatility of the stock as a decimal (e.g., 0.30 for 30%).
  5. Enter Risk-Free Rate: Input the current risk-free interest rate as a decimal (e.g., 0.04 for 4%).
  6. Enter Dividend Yield: Input the annual dividend yield as a decimal (e.g., 0.01 for 1%). If no dividends, enter 0.
  7. Click "Calculate Option Prices": The calculator will then display the theoretical fair value for both a European Call Option and a European Put Option.

Example Calculation

Let's consider an example:

  • Current Stock Price (S): $100
  • Strike Price (K): $100
  • Time to Expiration (T): 1 year
  • Volatility (σ): 20% (0.20)
  • Risk-Free Rate (r): 5% (0.05)
  • Dividend Yield (q): 0% (0)

Using these inputs in the calculator, you would find the theoretical prices for the call and put options based on the Black-Scholes model.

Limitations of the Model

While powerful, the Black-Scholes model has limitations:

  • It assumes constant volatility, which is rarely true in real markets (volatility often changes).
  • It assumes no dividends or a constant dividend yield, which might not hold.
  • It's designed for European options and may not accurately price American options (which can be exercised any time before expiration).
  • It doesn't account for extreme market events or "fat tails" in return distributions.

Despite these limitations, the Black-Scholes model remains a cornerstone of financial theory and a valuable tool for understanding option pricing dynamics.

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