.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;
}
.calc-input-group {
margin-bottom: 18px;
display: flex;
flex-direction: column;
}
.calc-input-group label {
margin-bottom: 8px;
color: #555;
font-size: 1em;
font-weight: 600;
}
.calc-input-group input[type="number"],
.calc-input-group select {
padding: 10px 12px;
border: 1px solid #ccc;
border-radius: 6px;
font-size: 1em;
width: calc(100% – 24px);
box-sizing: border-box;
transition: border-color 0.3s ease;
}
.calc-input-group input[type="number"]:focus,
.calc-input-group select:focus {
border-color: #007bff;
outline: none;
box-shadow: 0 0 0 3px rgba(0, 123, 255, 0.25);
}
.calc-input-group input[type="radio"] {
margin-right: 5px;
margin-left: 15px;
}
.calc-input-group input[type="radio"] + label {
display: inline-block;
margin-bottom: 0;
font-weight: normal;
}
.calc-input-group input[type="radio"]:first-of-type {
margin-left: 0;
}
.calculate-button {
background-color: #007bff;
color: white;
padding: 12px 25px;
border: none;
border-radius: 6px;
cursor: pointer;
font-size: 1.1em;
font-weight: 600;
width: 100%;
transition: background-color 0.3s ease, transform 0.2s ease;
margin-top: 15px;
}
.calculate-button:hover {
background-color: #0056b3;
transform: translateY(-2px);
}
.calculate-button:active {
transform: translateY(0);
}
.calc-results {
background-color: #e9f7ff;
border: 1px solid #cce5ff;
border-radius: 8px;
padding: 20px;
margin-top: 25px;
}
.calc-results h3 {
color: #0056b3;
margin-top: 0;
margin-bottom: 15px;
font-size: 1.4em;
text-align: center;
}
.calc-results p {
margin-bottom: 10px;
font-size: 1.05em;
color: #333;
display: flex;
justify-content: space-between;
}
.calc-results p strong {
color: #000;
flex-basis: 50%;
}
.calc-results p span {
flex-basis: 50%;
text-align: right;
font-weight: 500;
}
// Standard Normal Cumulative Distribution Function (approximation)
function normSCDF(x) {
var a1 = 0.31938153;
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 – (((((a5 * K + a4) * K + a3) * K + a2) * K + a1) * K) * Math.exp(-L * L / 2.0) / Math.sqrt(2 * Math.PI);
if (x < 0) {
w = 1.0 – w;
}
return w;
}
// Standard Normal Probability Density Function
function normPDF(x) {
return (1 / Math.sqrt(2 * Math.PI)) * Math.exp(-x * x / 2);
}
function calculateGreeks() {
var stockPrice = parseFloat(document.getElementById('stockPrice').value);
var strikePrice = parseFloat(document.getElementById('strikePrice').value);
var timeToExpirationDays = parseFloat(document.getElementById('timeToExpiration').value);
var volatility = parseFloat(document.getElementById('volatility').value) / 100; // Convert to decimal
var riskFreeRate = parseFloat(document.getElementById('riskFreeRate').value) / 100; // Convert to decimal
var dividendYield = parseFloat(document.getElementById('dividendYield').value) / 100; // Convert to decimal
var isCall = document.getElementById('callOption').checked;
var errorMessages = [];
if (isNaN(stockPrice) || stockPrice <= 0) errorMessages.push("Underlying Stock Price must be a positive number.");
if (isNaN(strikePrice) || strikePrice <= 0) errorMessages.push("Strike Price must be a positive number.");
if (isNaN(timeToExpirationDays) || timeToExpirationDays <= 0) errorMessages.push("Time to Expiration must be a positive number of days.");
if (isNaN(volatility) || volatility <= 0) errorMessages.push("Volatility must be a positive number.");
if (isNaN(riskFreeRate) || riskFreeRate < 0) errorMessages.push("Risk-Free Rate must be a non-negative number.");
if (isNaN(dividendYield) || dividendYield 0) {
document.getElementById('errorMessages').innerHTML = errorMessages.join(");
document.getElementById('optionPriceResult').innerText = ";
document.getElementById('deltaResult').innerText = ";
document.getElementById('gammaResult').innerText = ";
document.getElementById('thetaResult').innerText = ";
document.getElementById('vegaResult').innerText = ";
document.getElementById('rhoResult').innerText = ";
return;
} else {
document.getElementById('errorMessages').innerHTML = ";
}
var timeToExpirationYears = timeToExpirationDays / 365;
var d1 = (Math.log(stockPrice / strikePrice) + (riskFreeRate – dividendYield + (volatility * volatility) / 2) * timeToExpirationYears) / (volatility * Math.sqrt(timeToExpirationYears));
var d2 = d1 – volatility * Math.sqrt(timeToExpirationYears);
var optionPrice, delta, gamma, theta, vega, rho;
if (isCall) {
optionPrice = (stockPrice * Math.exp(-dividendYield * timeToExpirationYears) * normSCDF(d1)) – (strikePrice * Math.exp(-riskFreeRate * timeToExpirationYears) * normSCDF(d2));
delta = Math.exp(-dividendYield * timeToExpirationYears) * normSCDF(d1);
theta = (- (stockPrice * Math.exp(-dividendYield * timeToExpirationYears) * normPDF(d1) * volatility) / (2 * Math.sqrt(timeToExpirationYears))) – (riskFreeRate * strikePrice * Math.exp(-riskFreeRate * timeToExpirationYears) * normSCDF(d2)) + (dividendYield * stockPrice * Math.exp(-dividendYield * timeToExpirationYears) * normSCDF(d1));
rho = strikePrice * timeToExpirationYears * Math.exp(-riskFreeRate * timeToExpirationYears) * normSCDF(d2);
} else { // Put option
optionPrice = (strikePrice * Math.exp(-riskFreeRate * timeToExpirationYears) * normSCDF(-d2)) – (stockPrice * Math.exp(-dividendYield * timeToExpirationYears) * normSCDF(-d1));
delta = Math.exp(-dividendYield * timeToExpirationYears) * (normSCDF(d1) – 1);
theta = (- (stockPrice * Math.exp(-dividendYield * timeToExpirationYears) * normPDF(d1) * volatility) / (2 * Math.sqrt(timeToExpirationYears))) + (riskFreeRate * strikePrice * Math.exp(-riskFreeRate * timeToExpirationYears) * normSCDF(-d2)) – (dividendYield * stockPrice * Math.exp(-dividendYield * timeToExpirationYears) * normSCDF(-d1));
rho = -strikePrice * timeToExpirationYears * Math.exp(-riskFreeRate * timeToExpirationYears) * normSCDF(-d2);
}
gamma = (Math.exp(-dividendYield * timeToExpirationYears) * normPDF(d1)) / (stockPrice * volatility * Math.sqrt(timeToExpirationYears));
vega = stockPrice * Math.exp(-dividendYield * timeToExpirationYears) * normPDF(d1) * Math.sqrt(timeToExpirationYears);
// Display results
document.getElementById('optionPriceResult').innerText = optionPrice.toFixed(4);
document.getElementById('deltaResult').innerText = delta.toFixed(4);
document.getElementById('gammaResult').innerText = gamma.toFixed(4);
document.getElementById('thetaResult').innerText = (theta / 365).toFixed(4); // Theta is typically quoted per day
document.getElementById('vegaResult').innerText = (vega / 100).toFixed(4); // Vega is typically quoted per 1% change in volatility
document.getElementById('rhoResult').innerText = (rho / 100).toFixed(4); // Rho is typically quoted per 1% change in interest rate
}
// Initial calculation on page load
window.onload = calculateGreeks;
Understanding Option Greeks: Your Guide to Option Sensitivity
Option Greeks are a set of measures that quantify the sensitivity of an option's price to changes in various underlying factors. They are crucial tools for option traders and investors to understand and manage the risks associated with their option positions. By providing insights into how an option's value will react to market movements, time decay, or changes in volatility, Greeks enable more informed decision-making.
The Black-Scholes Model
The calculator above uses the Black-Scholes model, a widely recognized mathematical model for pricing European-style options. Developed by Fischer Black, Myron Scholes, and Robert Merton, it provides a theoretical estimate of an option's price based on several key inputs:
- Underlying Stock Price (S): The current market price of the asset on which the option is based.
- Strike Price (K): The price at which the option holder can buy (call) or sell (put) the underlying asset.
- Time to Expiration (T): The remaining time until the option contract expires, typically expressed in years.
- Volatility (σ): A measure of the expected fluctuation in the underlying asset's price over a period. Higher volatility generally leads to higher option prices.
- Risk-Free Rate (r): The theoretical rate of return of an investment with zero risk, often approximated by the yield on government bonds.
- Dividend Yield (q): The annual dividend rate paid by the underlying asset, expressed as a percentage of its price.
While powerful, the Black-Scholes model operates under certain assumptions, such as constant volatility and risk-free rates, and continuous trading, which may not always hold true in real markets. Despite these limitations, it remains a cornerstone of option pricing.
The Key Option Greeks Explained:
Delta (Δ)
Delta measures the sensitivity of an option's price to a $1 change in the underlying asset's price. It ranges from 0 to 1 for call options and -1 to 0 for put options.
- Call Options: A Delta of 0.50 means the option price will increase by $0.50 for every $1 increase in the underlying stock price.
- Put Options: A Delta of -0.50 means the option price will increase by $0.50 for every $1 decrease in the underlying stock price.
Delta also approximates the probability that an option will expire in-the-money.
Gamma (Γ)
Gamma measures the rate of change of an option's Delta with respect to a $1 change in the underlying asset's price. It indicates how much Delta itself will move. Gamma is highest for at-the-money options and decreases as options move further in or out of the money.
- A high Gamma means Delta will change rapidly with small movements in the underlying price, making the option's price more sensitive to these movements.
- Gamma is crucial for understanding the stability of a Delta-hedged position.
Theta (Θ)
Theta measures the sensitivity of an option's price to the passage of time, often referred to as "time decay." It represents the amount an option's price is expected to decrease each day, all else being equal.
- Theta is typically a negative number for long option positions, meaning options lose value as time passes.
- Time decay accelerates as an option approaches its expiration date, especially for at-the-money options.
Vega (ν)
Vega measures the sensitivity of an option's price to a 1% change in the underlying asset's implied volatility. Higher volatility generally increases option prices because there's a greater chance the option will expire in-the-money.
- A positive Vega means the option price will increase if volatility rises and decrease if volatility falls.
- Vega is highest for at-the-money options with longer times to expiration.
Rho (Ρ)
Rho measures the sensitivity of an option's price to a 1% change in the risk-free interest rate. While often less significant than other Greeks, it can be important for long-term options.
- Call Options: Generally have positive Rho; their value increases with higher interest rates.
- Put Options: Generally have negative Rho; their value decreases with higher interest rates.
How to Use This Calculator
Input the current market data for the underlying stock, the option's strike price, time to expiration, estimated volatility, risk-free interest rate, and dividend yield. Select whether you are analyzing a Call or Put option. The calculator will then provide the theoretical option price and the values for Delta, Gamma, Theta, Vega, and Rho, helping you assess the option's characteristics and potential risks.
Example Calculation:
Let's consider a Call option with the following parameters:
- Underlying Stock Price (S): 100
- Strike Price (K): 100
- Time to Expiration (Days): 90
- Volatility (σ): 20%
- Risk-Free Rate (r): 5%
- Dividend Yield (q): 0%
- Option Type: Call
Using these inputs in the calculator, you would get results similar to:
- Option Price: ~4.60
- Delta: ~0.57
- Gamma: ~0.03
- Theta: ~-0.03 (per day)
- Vega: ~0.15 (per 1% vol change)
- Rho: ~0.13 (per 1% rate change)
This means for this specific call option:
- If the stock price increases by $1, the option price will increase by approximately $0.57.
- If the stock price increases by $1, the Delta will increase by approximately 0.03.
- The option will lose approximately $0.03 in value each day due to time decay.
- If volatility increases by 1%, the option price will increase by approximately $0.15.
- If the risk-free rate increases by 1%, the option price will increase by approximately $0.13.
By understanding and utilizing these Greeks, traders can better manage their option portfolios, implement hedging strategies, and make more informed decisions in the dynamic options market.