This calculator estimates the theoretical fair value of a European-style call option using the Black-Scholes model. A call option gives the buyer the right, but not the obligation, to purchase an underlying asset (like a stock) at a specified price (the strike price) on or before a certain date.
Key Inputs Explained:
Current Stock Price (S): The current market price of the underlying asset.
Strike Price (K): The price at which the option holder can buy the underlying asset.
Time to Expiration (T): The remaining lifespan of the option, expressed in years. For example, 3 months would be 0.25 years.
Volatility (σ): A measure of how much the price of the underlying asset is expected to fluctuate. Higher volatility generally leads to higher option premiums. It's typically expressed as an annual standard deviation.
Risk-Free Interest Rate (r): The theoretical rate of return of an investment with zero risk, often represented by the yield on government bonds. This accounts for the time value of money.
Dividend Yield (q): The expected annual dividend payments from the underlying stock, expressed as a percentage of its price. Dividends reduce the expected future stock price, thus impacting call option value.
The Black-Scholes Model for Call Options
The Black-Scholes model is a foundational mathematical model for option pricing. For a call option, the formula is:
C = S * e^(-qT) * N(d1) - K * e^(-rT) * N(d2)
Where:
C is the theoretical price of the call option.
S is the current stock price.
K is the strike price.
r is the risk-free interest rate.
q is the dividend yield.
T is the time to expiration in years.
e is the base of the natural logarithm (approximately 2.71828).
N(x) is the cumulative standard normal distribution function (often calculated using approximations or lookup tables).
This calculator implements these formulas to provide a theoretical value. The actual market price of an option can deviate due to supply and demand, market sentiment, and other factors.
Use Cases:
Traders: To estimate a fair price for buying or selling call options.
Portfolio Managers: To understand the potential value of option holdings.
Educators and Students: To learn and demonstrate option pricing principles.
Disclaimer: This calculator is for informational and educational purposes only. It does not constitute financial advice. Option trading involves significant risk and is not suitable for all investors. Always consult with a qualified financial advisor before making investment decisions.
// Function to calculate the cumulative standard normal distribution (N(x))
// This is a common approximation for N(x) used in financial calculations.
// For higher precision, a more sophisticated algorithm or library might be needed.
function standardNormalCDF(x) {
var erf = function(z) {
var t = 1.0 / (1.0 + 0.5 * Math.abs(z));
var ans = 1.0 – t * Math.exp(-z * z – 1.26551223 + t * (1.00002368 + t * (0.37409196 + t * (0.09678418 + t * (-0.18628806 + t * (0.27886807 + t * (-1.13520398 + t * (1.48851587 + t * (-0.82215223 + t * (0.17087277))))))))));
if (z >= 0) return ans;
else return 1.0 – ans;
};
return 0.5 * (1 + erf(x / Math.sqrt(2)));
}
function calculateCallOption() {
var stockPrice = parseFloat(document.getElementById("stockPrice").value);
var strikePrice = parseFloat(document.getElementById("strikePrice").value);
var timeToExpiration = parseFloat(document.getElementById("timeToExpiration").value);
var volatility = parseFloat(document.getElementById("volatility").value);
var riskFreeRate = parseFloat(document.getElementById("riskFreeRate").value);
var dividendYield = parseFloat(document.getElementById("dividendYield").value);
var resultElement = document.getElementById("callPriceOutput");
resultElement.textContent = "–";
if (isNaN(stockPrice) || stockPrice <= 0 ||
isNaN(strikePrice) || strikePrice <= 0 ||
isNaN(timeToExpiration) || timeToExpiration <= 0 ||
isNaN(volatility) || volatility <= 0 ||
isNaN(riskFreeRate) ||
isNaN(dividendYield)) {
alert("Please enter valid positive numbers for all fields. Time to expiration and volatility must be greater than zero.");
return;
}
var sqrtT = Math.sqrt(timeToExpiration);
var sigma_sqrtT = volatility * sqrtT;
var d1 = (Math.log(stockPrice / strikePrice) + (riskFreeRate – dividendYield + 0.5 * volatility * volatility) * timeToExpiration) / sigma_sqrtT;
var d2 = d1 – sigma_sqrtT;
var callPrice = stockPrice * Math.exp(-dividendYield * timeToExpiration) * standardNormalCDF(d1) – strikePrice * Math.exp(-riskFreeRate * timeToExpiration) * standardNormalCDF(d2);
// Display the result, rounded to 2 decimal places for currency
resultElement.textContent = "$" + callPrice.toFixed(4); // Displaying with 4 decimal places for more precision in options pricing
}