Understanding Option Pricing and the Black-Scholes Model
The Option Price Calculator uses the Black-Scholes-Merton model, a mathematical formula widely used in financial markets to estimate the theoretical value of European-style options. By inputting key variables such as the current stock price, strike price, and time to expiration, traders can determine if an option is fairly valued, overvalued, or undervalued.
Key Inputs for the Calculator
Current Stock Price: The current market value of the underlying asset.
Strike Price: The predetermined price at which the option holder can buy (Call) or sell (Put) the underlying asset.
Days to Expiration: The remaining lifespan of the option contract. Time decay (Theta) reduces the value as this number approaches zero.
Implied Volatility (IV): A measure of how much the market expects the stock price to fluctuate. Higher volatility increases the price of both Calls and Puts.
Risk-Free Interest Rate: Usually based on the yield of government bonds (like the U.S. Treasury 10-year note).
Example Calculation
Suppose a stock is currently trading at $100. You are looking at a Call option with a $105 strike price that expires in 30 days. If the implied volatility is 20% and the risk-free rate is 5%, the Black-Scholes model might calculate the Call price at approximately $0.45 and the Put price at $5.02. These values help traders assess risk and potential return on investment.
Why is Implied Volatility Important?
Implied Volatility is often considered the most critical "unknown" variable. While price and time are certain, IV reflects market sentiment and uncertainty. When a company is about to release earnings, IV typically rises, making options more expensive even if the stock price remains unchanged.
function calculateOptionPrice() {
var S = parseFloat(document.getElementById("underlyingPrice").value);
var K = parseFloat(document.getElementById("strikePrice").value);
var T_days = parseFloat(document.getElementById("daysToExpiry").value);
var v = parseFloat(document.getElementById("volatility").value) / 100;
var r = parseFloat(document.getElementById("riskFreeRate").value) / 100;
if (isNaN(S) || isNaN(K) || isNaN(T_days) || isNaN(v) || isNaN(r) || S <= 0 || K <= 0 || T_days <= 0) {
alert("Please enter valid positive numerical values for all fields.");
return;
}
var T = T_days / 365;
// Black-Scholes Formula Components
var d1 = (Math.log(S / K) + (r + (v * v) / 2) * T) / (v * Math.sqrt(T));
var d2 = d1 – v * Math.sqrt(T);
// Cumulative Normal Distribution Function
function CND(x) {
var a1 = 0.319381530;
var a2 = -0.356563782;
var a3 = 1.781477937;
var a4 = -1.821255978;
var a5 = 1.330274429;
var L = Math.abs(x);
var K_val = 1.0 / (1.0 + 0.2316419 * L);
var w = 1.0 – 1.0 / Math.sqrt(2.0 * Math.PI) * Math.exp(-L * L / 2.0) * (a1 * K_val + a2 * Math.pow(K_val, 2) + a3 * Math.pow(K_val, 3) + a4 * Math.pow(K_val, 4) + a5 * Math.pow(K_val, 5));
if (x < 0) {
return 1.0 – w;
} else {
return w;
}
}
var callPrice = S * CND(d1) – K * Math.exp(-r * T) * CND(d2);
var putPrice = K * Math.exp(-r * T) * CND(-d2) – S * CND(-d1);
// Safety checks for negative values due to floating point or extreme inputs
callPrice = Math.max(0, callPrice);
putPrice = Math.max(0, putPrice);
document.getElementById("callPriceDisplay").innerText = "$" + callPrice.toFixed(2);
document.getElementById("putPriceDisplay").innerText = "$" + putPrice.toFixed(2);
document.getElementById("opcResults").style.display = "block";
}