Option Price Calculator

Option Price Calculator (Black-Scholes)

// Function to calculate the cumulative standard normal distribution (N(x)) function N(x) { var a1 = 0.319381530; var a2 = -0.356563782; var a3 = 1.781477937; var a4 = -1.821255978; var a5 = 1.330274429; var gamma = 0.2316419; var pi = Math.PI; var k = 1.0 / (1.0 + gamma * Math.abs(x)); var N_x = 1.0 – ( ( ( (a5 * k + a4) * k + a3) * k + a2) * k + a1) * k * Math.exp(-0.5 * x * x) / Math.sqrt(2 * pi); if (x < 0) { N_x = 1.0 – N_x; } return N_x; } function calculateOptionPrice() { var S = parseFloat(document.getElementById('underlyingPrice').value); var K = parseFloat(document.getElementById('strikePrice').value); var T_days = parseFloat(document.getElementById('timeToExpiration').value); var sigma_percent = parseFloat(document.getElementById('volatility').value); var r_percent = parseFloat(document.getElementById('riskFreeRate').value); var q_percent = parseFloat(document.getElementById('dividendYield').value); var optionType = document.querySelector('input[name="optionType"]:checked').value; var resultDiv = document.getElementById('result'); // Input validation if (isNaN(S) || S <= 0) { resultDiv.innerHTML = "Please enter a valid positive Underlying Asset Price."; return; } if (isNaN(K) || K <= 0) { resultDiv.innerHTML = "Please enter a valid positive Strike Price."; return; } if (isNaN(T_days) || T_days <= 0) { resultDiv.innerHTML = "Please enter a valid positive Time to Expiration (in days)."; return; } if (isNaN(sigma_percent) || sigma_percent <= 0) { resultDiv.innerHTML = "Please enter a valid positive Volatility percentage."; return; } if (isNaN(r_percent) || r_percent < 0) { resultDiv.innerHTML = "Please enter a valid non-negative Risk-Free Rate percentage."; return; } if (isNaN(q_percent) || q_percent < 0) { resultDiv.innerHTML = "Please enter a valid non-negative Dividend Yield percentage."; return; } // Convert inputs to required units var T = T_days / 365.0; // Time to expiration in years var sigma = sigma_percent / 100.0; // Volatility as a decimal var r = r_percent / 100.0; // Risk-free rate as a decimal var q = q_percent / 100.0; // Dividend yield as a decimal // Calculate d1 and d2 var d1_numerator = Math.log(S / K) + (r – q + (sigma * sigma) / 2) * T; var d1_denominator = sigma * Math.sqrt(T); var d1 = d1_numerator / d1_denominator; var d2 = d1 – sigma * Math.sqrt(T); var optionPrice; if (optionType === 'call') { // Call option price formula optionPrice = (S * Math.exp(-q * T) * N(d1)) – (K * Math.exp(-r * T) * N(d2)); resultDiv.innerHTML = "Calculated Call Option Price: $" + optionPrice.toFixed(2); } else { // put option // Put option price formula optionPrice = (K * Math.exp(-r * T) * N(-d2)) – (S * Math.exp(-q * T) * N(-d1)); resultDiv.innerHTML = "Calculated Put Option Price: $" + optionPrice.toFixed(2); } }

Understanding the Option Price Calculator

Options are financial derivatives that give the buyer the right, but not the obligation, to buy or sell an underlying asset at a specified price (the strike price) on or before a certain date (the expiration date). This calculator uses a simplified version of the Black-Scholes-Merton model, one of the most widely used models for pricing European-style options.

How Options are Priced: The Black-Scholes Model

The Black-Scholes model, developed by Fischer Black, Myron Scholes, and Robert Merton, provides a theoretical estimate for the price of European call and put options. It relies on several key inputs to determine an option's fair value:

Key Inputs Explained:

• Underlying Asset Price (S): This is the current market price of the asset on which the option is based (e.g., a stock, index, or commodity). A higher underlying price generally increases call option values and decreases put option values.
• Strike Price (K): Also known as the exercise price, this is the fixed price at which the option holder can buy (for a call) or sell (for a put) the underlying asset. The relationship between the underlying price and strike price is crucial for an option's "moneyness."
• Time to Expiration (T): This is the remaining time until the option contract expires, typically measured in days and converted to years for the calculation. Generally, the longer the time to expiration, the higher the option's value, as there's more time for the underlying asset's price to move favorably.
• Volatility (σ): This measures the expected fluctuation of the underlying asset's price over the life of the option. Higher volatility means a greater chance of significant price movements, which increases the value of both call and put options. It's expressed as an annualized percentage.
• Risk-Free Rate (r): This is the theoretical rate of return of an investment with zero risk, often approximated by the yield on short-term government bonds. A higher risk-free rate generally increases call option values and decreases put option values. It's expressed as an annualized percentage.
• Dividend Yield (q): For options on dividend-paying stocks, the dividend yield represents the expected annual dividend payments as a percentage of the stock price. Dividends reduce the underlying stock price, which generally decreases call option values and increases put option values. It's expressed as an annualized percentage.

Call vs. Put Options

• Call Option: Gives the holder the right to buy the underlying asset at the strike price. Investors buy calls when they expect the underlying asset's price to rise.
• Put Option: Gives the holder the right to sell the underlying asset at the strike price. Investors buy puts when they expect the underlying asset's price to fall.

How the Calculator Works (Simplified Black-Scholes)

The calculator takes your inputs and applies the Black-Scholes formulas to estimate the theoretical price of the option. The core of the calculation involves two key components, d1 and d2, which incorporate all the input variables. These values are then used with the cumulative standard normal distribution function (N(x)) to derive the final option price.

Example Calculation:

Let's say you want to price a Call Option with the following parameters:

• Underlying Asset Price: $100
• Strike Price: $100
• Time to Expiration: 90 days
• Volatility: 20%
• Risk-Free Rate: 5%
• Dividend Yield: 0%

Using these inputs in the calculator, you would find the theoretical Call Option Price to be approximately $4.39. If you switch to a Put Option with the same parameters, the price would be approximately $3.25.

Limitations of the Black-Scholes Model

While powerful, the Black-Scholes model has limitations:

• It assumes European-style options (exercisable only at expiration), not American-style (exercisable any time before expiration).
• It assumes constant volatility, which is rarely true in real markets.
• It assumes no dividends or a known, constant dividend yield.
• It assumes a constant risk-free rate.
• It assumes efficient markets with no transaction costs.

Despite these limitations, the Black-Scholes model remains a fundamental tool for understanding and estimating option prices, providing a valuable benchmark for traders and investors.