Black-Scholes Option Pricing Calculator
From spot, strike, time, rate and volatility, compute the theoretical European call and put prices (no dividend).
Input Data
Results
At a glance:Black-Scholes-Merton prices a no-dividend European option from S, K, T, r, sigma. Output: call and put. The normal CDF is approximated (Abramowitz-Stegun). Example: S=K=100, T=1, r=3%, sigma=20% → call ~HK$10.45. The result is a theoretical fair value; the market price usually differs due to the volatility smile/skew, dividends, early exercise (American) and costs. In Hong Kong, options trade on the HKEX. Use this to learn pricing and roughly check quotes — not for trading. WARNING: Buyers can lose the whole premium; uncovered sellers face unlimited risk. Education only, not advice.
Formula
Call = S × N(d1) − K × e^(−rT) × N(d2).
Put = K × e^(−rT) × N(−d2) − S × N(−d1).
d1 = [ln(S/K) + (r + σ²/2)T] / (σ√T), d2 = d1 − σ√T.
Where S = spot, K = strike, T = years, r = risk-free rate, σ = volatility, N = normal CDF.
$$d_1 = \dfrac{\ln(S/K) + (r + \sigma^2/2)\,T}{\sigma\sqrt{T}}, \quad d_2 = d_1 - \sigma\sqrt{T}$$$$C = S\,N(d_1) - K e^{-rT} N(d_2)$$$$P = K e^{-rT} N(-d_2) - S\,N(-d_1)$$How to Use
- Enter the spot and strike prices.
- Enter time to expiry, the risk-free rate and volatility.
- View the call and put prices.
FAQ
Can I use this price directly for trading?
No. It is a theoretical fair value and usually differs from the market quote because of the volatility smile/skew (the model assumes one volatility for all strikes, but reality varies), dividends, early exercise (American options) and transaction costs. Use it to understand pricing and roughly check quotes; trade on live exchange prices and broker Greeks.
Why does volatility matter so much?
Volatility is the main uncertain input. Higher sigma raises both call and put prices (more chance of a profitable move). Because future volatility is unknown, the input sigma drives the result; that is why implied volatility (backed out from market price) is closely watched.
What is the difference between European and American options?
European options can be exercised only at expiry; American can be exercised any time before. This calculator is European. Many HKEX stock options are American, so their value can be slightly higher. The difference is usually small except near dividends.
What is the risk-free rate in Hong Kong?
Often proxied by Exchange Fund Bills / government bond yields. It enters the model as the continuously compounded rate; a higher rate raises call prices and lowers put prices slightly.
How risky are options really?
High. A buyer can lose the entire premium; an uncovered (naked) seller's loss is theoretically unlimited. Options are leveraged and can move fast. Understand the product, use position sizing, and consult the IFEC and HKEX materials before trading. This tool does not constitute advice.
Related Tools
References
Content review: Calculatorism Finance Team. Results are for reference only; please refer to the relevant authorities for the official figures.