Calculator Black Scholes

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Black-Scholes Option Pricing Calculator

Calculate theoretical Call and Put option prices using the Black-Scholes-Merton model.

Call Option Price

$0.00

Put Option Price

$0.00

Option Greeks (Call)

Delta
0.00
Gamma
0.00
Vega
0.00
Theta (Day)
0.00
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 = 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 + a2 * Math.pow(K, 2) + a3 * Math.pow(K, 3) + a4 * Math.pow(K, 4) + a5 * Math.pow(K, 5)); if (x < 0) { return 1.0 – w; } return w; } function calculateBlackScholes() { var S = parseFloat(document.getElementById('bs_stock_price').value); var K = parseFloat(document.getElementById('bs_strike_price').value); var days = parseFloat(document.getElementById('bs_days_to_expiry').value); var T = days / 365.0; var r = parseFloat(document.getElementById('bs_risk_free').value) / 100.0; var v = parseFloat(document.getElementById('bs_volatility').value) / 100.0; var q = parseFloat(document.getElementById('bs_dividend').value) / 100.0; if (isNaN(S) || isNaN(K) || isNaN(T) || isNaN(r) || isNaN(v) || T <= 0) { alert("Please enter valid positive numbers. Time to expiry must be greater than 0."); return; } var d1 = (Math.log(S / K) + (r – q + (v * v) / 2) * T) / (v * Math.sqrt(T)); var d2 = d1 – v * Math.sqrt(T); var callPrice = S * Math.exp(-q * T) * cnd(d1) – K * Math.exp(-r * T) * cnd(d2); var putPrice = K * Math.exp(-r * T) * cnd(-d2) – S * Math.exp(-q * T) * cnd(-d1); // Greeks var nd1 = (1 / Math.sqrt(2 * Math.PI)) * Math.exp(-0.5 * d1 * d1); var deltaCall = Math.exp(-q * T) * cnd(d1); var gamma = (nd1 * Math.exp(-q * T)) / (S * v * Math.sqrt(T)); var vega = (S * Math.exp(-q * T) * Math.sqrt(T) * nd1) / 100; var thetaCall = (-(S * v * Math.exp(-q * T) * nd1) / (2 * Math.sqrt(T)) – r * K * Math.exp(-r * T) * cnd(d2) + q * S * Math.exp(-q * T) * cnd(d1)) / 365; document.getElementById('call_result').innerHTML = "$" + callPrice.toFixed(2); document.getElementById('put_result').innerHTML = "$" + putPrice.toFixed(2); document.getElementById('call_delta').innerHTML = deltaCall.toFixed(4); document.getElementById('call_gamma').innerHTML = gamma.toFixed(4); document.getElementById('call_vega').innerHTML = vega.toFixed(4); document.getElementById('call_theta').innerHTML = thetaCall.toFixed(4); document.getElementById('bs_results').style.display = 'block'; }

Understanding the Black-Scholes Model

The Black-Scholes Calculator is a sophisticated financial tool used to estimate the fair market value of European-style stock options. Developed by economists Fischer Black and Myron Scholes in 1973, it revolutionized the derivatives market by providing a mathematical framework for pricing contracts based on the underlying asset's volatility and time to maturity.

Key Input Variables

  • Current Stock Price (S): The current market price of the underlying asset.
  • Strike Price (K): The pre-determined price at which the option holder can buy (Call) or sell (Put) the asset.
  • Time to Expiration: Measured in days, this represents how long the contract remains valid. The model converts this into annual terms (T).
  • Volatility (σ): A measure of how much the stock price is expected to fluctuate. This is often the most critical and subjective input.
  • Risk-Free Rate (r): The theoretical rate of return on an investment with zero risk, typically represented by Treasury Bill rates.
  • Dividend Yield (q): The annual dividend rate of the underlying stock, which reduces the call price and increases the put price.

The "Greeks" Explained

Professional traders use "The Greeks" to manage risk and understand how their portfolio reacts to market changes:

Greek What it Measures
Delta Sensitivity to the stock price. A Delta of 0.50 means the option price moves $0.50 for every $1 move in the stock.
Gamma The rate of change in Delta. High Gamma means Delta changes rapidly.
Vega Sensitivity to volatility. Measures how much the option price changes for a 1% change in σ.
Theta Time decay. Represents how much the option loses value daily as it nears expiration.

Example Calculation

Suppose a stock is trading at $150. You are looking at a Call option with a Strike Price of $155 expiring in 45 days. If the volatility is 30% and the risk-free rate is 4%, the Black-Scholes model might calculate a theoretical price of roughly $3.45. This price helps traders determine if the market is currently overpricing or underpricing the contract.

Limitations

While powerful, the model assumes constant volatility and interest rates, and it only applies to European options (options that cannot be exercised before the expiration date). For American options, other models like the Binomial Options Pricing Model are often preferred.

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