Options probability refers to the likelihood of an option expiring in-the-money (ITM) or the underlying asset's price moving within a certain range by the option's expiration date. This calculator helps traders estimate these probabilities using key inputs, providing insights into potential outcomes.
How the Calculator Works
This calculator uses a simplified version of the Black-Scholes model, specifically focusing on the 'd2' term, to estimate the risk-neutral probability of an option expiring in-the-money. It also leverages implied volatility to project expected price movements and the probability of the stock price falling within various standard deviation ranges.
Key Inputs Explained:
Current Stock Price: The current market price of the underlying asset (e.g., a stock or ETF). This is the starting point for all price movement calculations.
Option Strike Price: The predetermined price at which the option contract can be exercised. For call options, ITM means the stock price is above the strike; for put options, ITM means the stock price is below the strike.
Days to Expiration: The number of calendar days remaining until the option contract expires. Time is a critical factor, as options lose value as they approach expiration (time decay).
Implied Volatility (%): A forward-looking measure of the market's expectation of how much the underlying asset's price will fluctuate. Higher implied volatility generally means higher option premiums and a wider expected price range. It's entered as a percentage (e.g., 30 for 30%).
Understanding the Outputs:
Probability of Call Option Expiring ITM: The estimated likelihood, as a percentage, that the underlying stock price will be above the specified strike price at expiration.
Probability of Put Option Expiring ITM: The estimated likelihood, as a percentage, that the underlying stock price will be below the specified strike price at expiration.
Expected Price Move (1-Standard Deviation): This is an estimate of how much the stock price is expected to move (up or down) by expiration, based on the implied volatility and days to expiration. It represents one standard deviation of price movement.
Price Ranges (1-SD, 2-SD, 3-SD): These ranges show the expected price boundaries for the underlying asset at expiration, corresponding to 1, 2, and 3 standard deviations of movement from the current price.
Probabilities for Price Ranges:
Within X-SD Range: The probability that the stock price will close within the calculated X-standard deviation price range.
Outside X-SD Range: The probability that the stock price will close outside the calculated X-standard deviation price range.
Above Upper X-SD: The probability that the stock price will close above the upper bound of the X-standard deviation range.
Below Lower X-SD: The probability that the stock price will close below the lower bound of the X-standard deviation range.
Example Calculation:
Let's use the default values:
Current Stock Price: $150
Option Strike Price: $155
Days to Expiration: 45
Implied Volatility: 30%
Upon calculation, you might see results like:
Probability of Call Option Expiring ITM: ~35.80%
Probability of Put Option Expiring ITM: ~39.40%
Expected Price Move (1-SD): ~$15.80
1-SD Price Range: ~$134.20 to ~$165.80
Probability of Price within 1-SD Range: ~68.23%
Probability of Price above Upper 1-SD: ~18.41%
Probability of Price below Lower 1-SD: ~13.36%
These figures suggest that there's roughly a 36% chance the stock will be above $155 at expiration, and a 39% chance it will be below $145. Furthermore, there's about a 68% chance the stock will end up between $134.20 and $165.80.
Important Considerations:
This calculator provides theoretical probabilities based on mathematical models and implied volatility. Actual market movements can deviate significantly from these predictions due to unforeseen events, changes in market sentiment, or limitations of the model. Always use these probabilities as a tool for analysis, not as a guarantee of future performance.
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// Standard Normal CDF approximation
function normalCDF(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_inv_sqrt_2pi = 0.3989422804014327; // 1/sqrt(2*PI)
var t = 1 / (1 + gamma * Math.abs(x));
var b = pi_inv_sqrt_2pi * Math.exp(-x * x / 2);
var prob = b * (((((a5 * t + a4) * t + a3) * t + a2) * t + a1) * t);
if (x >= 0) {
return 1 – prob;
} else {
return prob;
}
}
function calculateOptionsProbability() {
var currentStockPrice = parseFloat(document.getElementById("currentStockPrice").value);
var strikePrice = parseFloat(document.getElementById("strikePrice").value);
var daysToExpiration = parseFloat(document.getElementById("daysToExpiration").value);
var impliedVolatility = parseFloat(document.getElementById("impliedVolatility").value);
var resultDiv = document.getElementById("result");
resultDiv.innerHTML = ""; // Clear previous results
if (isNaN(currentStockPrice) || isNaN(strikePrice) || isNaN(daysToExpiration) || isNaN(impliedVolatility) ||
currentStockPrice <= 0 || strikePrice <= 0 || daysToExpiration <= 0 || impliedVolatility <= 0) {
resultDiv.innerHTML = "Please enter valid positive numbers for all fields.";
return;
}
var volatilityDecimal = impliedVolatility / 100;
var timeToExpirationYears = daysToExpiration / 365;
// Calculate d2 for ITM probabilities (simplified, assuming risk-free rate r=0)
var d2_numerator = Math.log(currentStockPrice / strikePrice) – (0.5 * Math.pow(volatilityDecimal, 2) * timeToExpirationYears);
var d2_denominator = volatilityDecimal * Math.sqrt(timeToExpirationYears);
if (d2_denominator === 0) { // Avoid division by zero if volatility or time is extremely small
resultDiv.innerHTML = "Cannot calculate with zero volatility or expiration time.";
return;
}
var d2 = d2_numerator / d2_denominator;
var probCallITM = normalCDF(d2) * 100;
var probPutITM = normalCDF(-d2) * 100;
// Calculate Expected Move (1-Standard Deviation Price Range)
var expectedMove = currentStockPrice * volatilityDecimal * Math.sqrt(timeToExpirationYears);
// Calculate Price Ranges for 1, 2, and 3 Standard Deviations
var lower1SD_price = currentStockPrice – expectedMove;
var upper1SD_price = currentStockPrice + expectedMove;
var lower2SD_price = currentStockPrice – (2 * expectedMove);
var upper2SD_price = currentStockPrice + (2 * expectedMove);
var lower3SD_price = currentStockPrice – (3 * expectedMove);
var upper3SD_price = currentStockPrice + (3 * expectedMove);
// Function to calculate d_value for a given price bound
function calculate_d_value(priceBound) {
if (priceBound <= 0) {
return -Infinity; // Effectively 0 probability below or at this point
}
var d_numerator = Math.log(priceBound / currentStockPrice) – (0.5 * Math.pow(volatilityDecimal, 2) * timeToExpirationYears);
return d_numerator / d2_denominator;
}
// Probabilities for 1-SD Range
var d_lower1SD = calculate_d_value(lower1SD_price);
var d_upper1SD = calculate_d_value(upper1SD_price);
var probWithin1SD = (normalCDF(d_upper1SD) – normalCDF(d_lower1SD)) * 100;
var probOutside1SD = (100 – probWithin1SD);
var probAboveUpper1SD = (1 – normalCDF(d_upper1SD)) * 100;
var probBelowLower1SD = normalCDF(d_lower1SD) * 100;
// Probabilities for 2-SD Range
var d_lower2SD = calculate_d_value(lower2SD_price);
var d_upper2SD = calculate_d_value(upper2SD_price);
var probWithin2SD = (normalCDF(d_upper2SD) – normalCDF(d_lower2SD)) * 100;
var probOutside2SD = (100 – probWithin2SD);
var probAboveUpper2SD = (1 – normalCDF(d_upper2SD)) * 100;
var probBelowLower2SD = normalCDF(d_lower2SD) * 100;
// Probabilities for 3-SD Range
var d_lower3SD = calculate_d_value(lower3SD_price);
var d_upper3SD = calculate_d_value(upper3SD_price);
var probWithin3SD = (normalCDF(d_upper3SD) – normalCDF(d_lower3SD)) * 100;
var probOutside3SD = (100 – probWithin3SD);
var probAboveUpper3SD = (1 – normalCDF(d_upper3SD)) * 100;
var probBelowLower3SD = normalCDF(d_lower3SD) * 100;
var resultsHtml = "
Probability of Expiring In-The-Money (ITM):
";
resultsHtml += "Probability of Call Option Expiring ITM: " + probCallITM.toFixed(2) + "%";
resultsHtml += "Probability of Put Option Expiring ITM: " + probPutITM.toFixed(2) + "%";
resultsHtml += "