Black-Scholes Calculator

Black-Scholes Calculator – Option Pricing, Greeks and Implied Volatility

This Black-Scholes Calculator helps you price European call and put options, compute option Greeks, and estimate implied volatility from a market option price. It provides a practical way to study how option values change with volatility, time and underlying price, without manually applying complex formulas.

The Black-Scholes model is one of the most widely used frameworks in options pricing. It assumes constant volatility, continuous compounding and frictionless markets, and is strictly valid for European options that can only be exercised at expiry. Even with its limitations, it remains a standard reference for traders, analysts and students.

How the Black-Scholes Model Works

The model prices a European call option using the formula:

and a European put option using:

Here:

• S = current underlying price
• K = strike price
• T = time to expiry in years
• r = risk-free interest rate (annual)
• q = dividend yield (annual)
• σ = volatility of the underlying (annualized)
• N(·) = cumulative distribution function of the standard normal distribution

d₁ and d₂ are defined as:

Mode 1: Option Pricing

The Option Price tab calculates Black-Scholes call and put prices for a given set of inputs:

• Underlying price (S)
• Strike price (K)
• Time to expiry in years (T)
• Risk-free interest rate (r)
• Dividend yield (q)
• Volatility (σ)
• Selected option type (call or put)

The calculator returns:

• Theoretical call price
• Theoretical put price
• Intrinsic value for the selected option type
• Time value, defined as option price minus intrinsic value

Intrinsic Value and Time Value

Intrinsic value reflects the amount an option is in the money:

• Call intrinsic value = max(S − K, 0)
• Put intrinsic value = max(K − S, 0)

Time value is the extra premium an investor pays for the possibility of favorable moves before expiry:

Mode 2: Option Greeks

The Greeks tab calculates the main sensitivities for the selected call or put:

• Delta – sensitivity to underlying price
• Gamma – sensitivity of Delta to underlying price
• Vega – sensitivity to volatility (per 1% change)
• Theta – sensitivity to time decay (per day)
• Rho – sensitivity to interest rate changes (per 1% change)

Standard Black-Scholes Greeks

For a call option:

• Delta = e^−qT · N(d₁)
• Gamma = e^−qT · φ(d₁) ÷ (S σ √T)
• Vega ≈ S · e^−qT · φ(d₁) √T
• Theta includes time decay, interest cost, and dividends
• Rho ≈ K T e^−rT · N(d₂)

For a put option, the formulas adjust using N(−d₁) and N(−d₂). Theta is shown per day, while Vega and Rho are reported per 1% change in volatility or interest rate to make interpretation easier.

Mode 3: Implied Volatility

The Implied Volatility tab solves the reverse Black-Scholes problem: given all inputs and a market option price, it finds the volatility that would produce that price in the model. This is done numerically using an iterative approach.

• You enter S, K, T, r, q and market option price
• Select option type (call or put)
• The calculator searches for a volatility level where theoretical price ≈ market price

The output includes:

• Implied volatility in percent
• Model price at the computed implied vol
• Difference between model and market price
• A simple label indicating if the option appears slightly over or under priced relative to the solved price

Interpreting the Results

Theoretical Black-Scholes prices are not guarantees of future value. Instead, they give a consistent baseline under the model assumptions. Traders often compare these values with actual market prices to identify relative richness or cheapness, keeping in mind factors such as transaction costs, early exercise, and market microstructure.

Greeks help you understand how an option will react when the underlying price moves, when volatility changes, or when time passes. For example:

• High Delta means the option behaves more like the underlying asset.
• High Gamma indicates Delta will change quickly with price moves.
• High Vega means option value is very sensitive to changes in volatility.
• Negative Theta reflects the typical time decay of long options positions.

Limitations of the Black-Scholes Model

While powerful, the Black-Scholes model has limitations:

• Assumes constant volatility and interest rates.
• Assumes log-normal distribution of underlying returns.
• Designed for European options with no early exercise.
• Does not explicitly handle jumps, fat tails or volatility smiles.

In practice, traders use implied volatility surfaces and more advanced models, but Black-Scholes remains a useful starting point and a standard benchmark.

How to Use This Tool Effectively

• Use the Option Price tab to compute fair value under Black-Scholes and to separate intrinsic from time value.
• Use the Greeks tab to analyze risk exposure and sensitivity before opening or adjusting a position.
• Use the Implied Volatility tab to back out the market’s implied volatility and compare it with historical volatility or your own expectations.
• Combine results with your risk management rules and not as a sole basis for trading decisions.

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