Black-Scholes Option Price Calculator
The Black-Scholes model is a fundamental concept in financial mathematics, used to estimate the theoretical price of European-style options. Developed by Fischer Black, Myron Scholes, and Robert Merton, it provides a framework for understanding how various factors influence option values.
Understanding the Black-Scholes Model
The Black-Scholes model is a mathematical model for the dynamics of a financial market containing derivative investment instruments. From the Black-Scholes equation, one can deduce the Black-Scholes formula, which gives a theoretical estimate of the price of European-style options.
Key Assumptions of the Black-Scholes Model:
- The option is European and can only be exercised at expiration.
- No dividends are paid out during the option's life.
- Market movements cannot be predicted (efficient markets).
- There are no transaction costs in buying or selling the option.
- The risk-free rate and volatility are constant.
- Returns are log-normally distributed.
Inputs Explained:
- Current Stock Price (S): The current market price of the underlying asset. This is a direct input and reflects the present value of the stock.
- Strike Price (K): 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 expires, expressed in years. For example, 6 months would be 0.5 years.
- Risk-Free Rate (r): The theoretical rate of return of an investment with zero risk, typically represented by the yield on government bonds (e.g., U.S. Treasury bills). It's expressed as an annual percentage.
- Volatility (σ): A measure of the expected fluctuation in the underlying asset's price over a period. It's the annualized standard deviation of the stock's returns, expressed as an annual percentage. Higher volatility generally leads to higher option prices.
Outputs Explained:
- Call Option Price: The theoretical fair value of a call option, which gives the holder the right to buy the underlying asset at the strike price.
- Put Option Price: The theoretical fair value of a put option, which gives the holder the right to sell the underlying asset at the strike price.
Example Calculation:
Let's consider an example with realistic values:
- Current Stock Price (S): $150
- Strike Price (K): $145
- Time to Expiration (T): 0.75 years (9 months)
- Risk-Free Rate (r): 3% (0.03)
- Volatility (σ): 25% (0.25)
Using these inputs in the Black-Scholes formula, the calculator would yield:
- Call Option Price: Approximately $15.78
- Put Option Price: Approximately $7.02
This means that, according to the model, a call option with these parameters would theoretically be worth $15.78, and a put option would be worth $7.02.
Limitations:
While widely used, the Black-Scholes model has limitations. Its assumptions, such as constant volatility and risk-free rates, and no dividends, often do not hold true in real-world markets. Despite these limitations, it remains a powerful tool for option pricing and risk management, especially when adjusted for real-world conditions.
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// Function to calculate N(x) – Cumulative Standard Normal Distribution
// Using Abramowitz and Stegun approximation
function N(x) {
var a1 = 0.319381530;
var a2 = -0.356563782;
var a3 = 1.781477937;
var a4 = -1.821255978;
var a5 = 1.330274429;
var L = Math.abs(x);
var K = 1.0 / (1.0 + 0.2316419 * L);
var W = 1.0 – Math.exp(-L * L / 2.0) / Math.sqrt(2 * Math.PI) * (a1 * K + a2 * K * K + a3 * K * K * K + a4 * K * K * K * K + a5 * K * K * K * K * K);
if (x < 0) {
return 1.0 – W;
} else {
return W;
}
}
function calculateBlackScholes() {
var stockPrice = parseFloat(document.getElementById("stockPrice").value);
var strikePrice = parseFloat(document.getElementById("strikePrice").value);
var timeToExp = parseFloat(document.getElementById("timeToExp").value);
var riskFreeRate = parseFloat(document.getElementById("riskFreeRate").value) / 100; // Convert percentage to decimal
var volatility = parseFloat(document.getElementById("volatility").value) / 100; // Convert percentage to decimal
var resultDiv = document.getElementById("blackScholesResult");
resultDiv.innerHTML = ""; // Clear previous results
// Input validation
if (isNaN(stockPrice) || isNaN(strikePrice) || isNaN(timeToExp) || isNaN(riskFreeRate) || isNaN(volatility) ||
stockPrice <= 0 || strikePrice <= 0 || timeToExp <= 0 || riskFreeRate < 0 || volatility <= 0) {
resultDiv.innerHTML = "Please enter valid positive numbers for all fields. Risk-Free Rate can be zero or positive.";
return;
}
// Black-Scholes Formula Calculation
var d1_numerator = Math.log(stockPrice / strikePrice) + (riskFreeRate + (volatility * volatility) / 2) * timeToExp;
var d1_denominator = volatility * Math.sqrt(timeToExp);
var d1 = d1_numerator / d1_denominator;
var d2 = d1 – volatility * Math.sqrt(timeToExp);
// Call Option Price
var callPrice = (stockPrice * N(d1)) – (strikePrice * Math.exp(-riskFreeRate * timeToExp) * N(d2));
// Put Option Price
var putPrice = (strikePrice * Math.exp(-riskFreeRate * timeToExp) * N(-d2)) – (stockPrice * N(-d1));
resultDiv.innerHTML =
"
Calculated Option Prices:" +
"Call Option Price: $" + callPrice.toFixed(2) + "" +
"Put Option Price: $" + putPrice.toFixed(2) + "";
}