Options Pricing Calculator
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Options Pricing Calculator
Calculated Options Price
Enter details and click "Calculate Price"
Understanding Options Pricing: The Black-Scholes Model
Options contracts give the buyer the right, but not the obligation, to buy or sell an underlying asset at a specific price (the strike price) on or before a certain date. Determining the fair price of an option is crucial for traders and investors. The most widely used model for pricing European-style options is the Black-Scholes-Merton (BSM) model.
The Black-Scholes Formula
The BSM model provides a theoretical estimate of the price of European-style options. It relies on several key inputs and assumptions:
- Current Price of the Underlying Asset (S): The current market price of the stock or asset.
- 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 contract expires, expressed in years.
- Implied Volatility (σ): The market's expectation of the future volatility of the underlying asset's price. This is a key input and is often derived from the market prices of other options.
- Risk-Free Interest Rate (r): The theoretical rate of return of an investment with zero risk, typically represented by government bond yields.
- Dividend Yield (q): The rate at which the underlying asset is expected to pay dividends.
How it Works (Simplified)
The BSM model calculates the probability of an option finishing "in-the-money" (profitable) at expiration and then discounts that probability back to the present value, considering the risk-free rate and volatility.
For a Call Option, the formula is:
C = S₀ * N(d₁) – K * e^(-rT) * N(d₂)
For a Put Option, the formula is:
P = K * e^(-rT) * N(-d₂) – S₀ * N(-d₁)
Where:
- C is the price of the call option.
- P is the price of the put option.
- S₀ is the current price of the underlying asset.
- K is the strike price.
- r is the annual risk-free interest rate.
- T is the time to expiration in years.
- q is the annual dividend yield.
- σ is the implied volatility.
- N(x) is the cumulative standard normal distribution function.
- d₁ = [ln(S₀/K) + (r – q + σ²/2) * T] / (σ * √T)
- d₂ = d₁ – σ * √T
- e is the base of the natural logarithm (approximately 2.71828).
- ln is the natural logarithm.
The calculator above implements the Black-Scholes model to provide a theoretical price based on your inputs.
Use Cases
- Valuation: Determining a fair price for an option contract.
- Trading Strategies: Assisting in the development and execution of options trading strategies.
- Risk Management: Understanding the potential value changes based on market factors.
- Education: Learning how different variables affect option prices.
Disclaimer: This calculator is for informational and educational purposes only. It uses the Black-Scholes model, which has several assumptions and limitations. Actual market prices may differ. Consult with a qualified financial advisor before making any investment decisions.
function erf(x) {
var a1 = 0.254829592;
var a2 = -0.284496736;
var a3 = 1.421413741;
var a4 = -1.453152027;
var a5 = 1.061405429;
var p = 0.3275911;
var sign = 1;
if (x < 0) {
sign = -1;
}
x = Math.abs(x);
var t = 1.0 / (1.0 + p * x);
var y = 1.0 – (((((a5 * t + a4) * t) + a3) * t + a2) * t + a1) * t * Math.exp(-x * x);
return sign * y;
}
function cndf2(x) {
var PI = Math.PI;
return 0.5 * (1.0 + erf(x / Math.sqrt(2.0)));
}
function calculateOptionsPrice() {
var S = parseFloat(document.getElementById("underlyingPrice").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 optionType = document.getElementById("optionType").value;
var resultElement = document.getElementById("result");
if (isNaN(S) || isNaN(K) || isNaN(T) || isNaN(sigma) || isNaN(r) || isNaN(q)) {
resultElement.innerText = "Please enter valid numerical values for all fields.";
return;
}
if (T <= 0 || sigma <= 0) {
resultElement.innerText = "Time to expiration and volatility must be greater than zero.";
return;
}
var d1 = (Math.log(S / K) + (r – q + 0.5 * sigma * sigma) * T) / (sigma * Math.sqrt(T));
var d2 = d1 – sigma * Math.sqrt(T);
var price = 0;
if (optionType === "call") {
price = S * Math.exp(-q * T) * cndf2(d1) – K * Math.exp(-r * T) * cndf2(d2);
} else { // put
price = K * Math.exp(-r * T) * cndf2(-d2) – S * Math.exp(-q * T) * cndf2(-d1);
}
if (isNaN(price) || !isFinite(price)) {
resultElement.innerText = "Calculation resulted in an invalid number. Check inputs.";
} else {
resultElement.innerText = "$" + price.toFixed(2);
}
}