# Black-Scholes Calculator

Calculate European option prices using the Black-Scholes model. Get call and put prices plus the Greeks (delta, gamma, theta, vega). Free options calculator.

## What this calculates

Price European call and put options using the Black-Scholes model. Enter the stock price, strike price, time to expiry, risk-free rate, and volatility to get theoretical option prices and the key Greeks.

## Inputs

- **Stock Price (S)** ($) — min 0.01 — Current price of the underlying stock.
- **Strike Price (K)** ($) — min 0.01 — The option's strike (exercise) price.
- **Time to Expiry** (years) — min 0.001, max 10 — Time until expiration in years (e.g., 6 months = 0.5).
- **Risk-Free Rate** (%) — min 0, max 25 — Annual risk-free interest rate (e.g., Treasury yield).
- **Volatility (σ)** (%) — min 1, max 200 — Annualized implied volatility of the underlying stock.
- **Dividend Yield** (%) — min 0, max 20 — Annual continuous dividend yield (0 if no dividends).

## Outputs

- **Call Option Price** — formatted as currency — Theoretical price of a European call option.
- **Put Option Price** — formatted as currency — Theoretical price of a European put option.
- **Call Delta** — Rate of change in call price per $1 move in stock.
- **Put Delta** — Rate of change in put price per $1 move in stock.
- **Gamma** — Rate of change in delta per $1 move in stock.
- **Vega** — Change in option price per 1% change in volatility.
- **Call Theta (per day)** — Daily time decay for the call option.
- **Put Theta (per day)** — Daily time decay for the put option.

## Details

The Black-Scholes model (developed by Fischer Black, Myron Scholes, and Robert Merton in 1973) is the foundation of modern options pricing. It calculates the theoretical fair value of European-style options, which can only be exercised at expiration.

The formula uses five inputs: stock price (S), strike price (K), time to expiration (T), risk-free interest rate (r), and volatility (sigma). The model assumes the stock follows a geometric Brownian motion with constant volatility and that markets are frictionless.

## The Greeks

- **Delta** measures how much the option price changes when the stock moves $1. A call with delta 0.60 gains $0.60 when the stock rises $1.
- **Gamma** measures how fast delta changes. High gamma means delta is very sensitive to stock moves.
- **Theta** is time decay, the amount the option loses each day just from the passage of time. It accelerates as expiration approaches.
- **Vega** measures sensitivity to volatility. A vega of 0.15 means the option gains $0.15 per contract for each 1% increase in implied volatility.

The Black-Scholes model has limitations: it assumes constant volatility (real markets have a volatility smile), no dividends in the basic form (this calculator uses the extension for continuous dividends), and European-style exercise only. Despite these simplifications, it remains the industry standard starting point for options pricing.

## Frequently Asked Questions

**Q: What is implied volatility and where do I find it?**

A: Implied volatility (IV) is the market's expectation of how much the stock will move. It is derived from current option prices rather than historical price data. You can find IV on your brokerage platform's option chain, on financial sites like Yahoo Finance or CBOE, or through data providers. Typical IV for large-cap stocks is 15-30%, while volatile stocks or around earnings announcements can be 40-100%+.

**Q: Why does Black-Scholes not work for American options?**

A: The Black-Scholes formula assumes the option can only be exercised at expiration (European style). American options can be exercised at any time before expiration, which gives them extra value, especially for puts and for calls on dividend-paying stocks. American options require numerical methods like the binomial model or finite difference methods to price correctly. For non-dividend stocks, American and European call prices are identical.

**Q: What risk-free rate should I use?**

A: Use the yield on a US Treasury security with a maturity closest to your option's expiration date. For a 6-month option, use the 6-month T-bill yield. For a 1-year option, use the 1-year Treasury yield. You can find current rates on the US Treasury website or the Federal Reserve's H.15 release. As of early 2025, short-term Treasury yields have been around 4-5%.

**Q: What does negative theta mean?**

A: Negative theta means the option loses value each day due to time decay. Almost all long option positions have negative theta. For example, a theta of -0.05 means the option loses $0.05 per day (or $5 per contract of 100 shares). Theta accelerates as expiration approaches, especially for at-the-money options. Sellers of options benefit from theta decay, which is why selling options is sometimes called collecting theta.

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Source: https://vastcalc.com/calculators/finance/black-scholes
Category: Finance
Last updated: 2026-04-08