Black-Scholes Option Pricing Calculator
From spot, strike, time, rate and volatility, compute the theoretical European call and put prices (no dividend).
輸入資料
計算結果
重點速覽：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.
計算公式
d1 = [ln(S ÷ K) + (r + σ² ÷ 2) × T] ÷ (σ × √T)。
d2 = d1 − σ × √T。
認購 Call = S × N(d1) − K × e^(−rT) × N(d2)。
認沽 Put = K × e^(−rT) × N(−d2) − S × N(−d1)。
$$d_1 = \dfrac{\ln(S/K) + (r + \sigma^2/2)\,T}{\sigma\sqrt{T}}, \quad d_2 = d_1 - \sigma\sqrt{T}$$$$C = S\,N(d_1) - K e^{-rT} N(d_2)$$$$P = K e^{-rT} N(-d_2) - S\,N(-d_1)$$使用說明
- Enter the spot and strike prices.
- Enter time to expiry, the risk-free rate and volatility.
- View the call and put prices.
S = K = HK$100、到期 1 年、無風險利率 5% 下，不同年化波動率對應的期權理論價
| 年化波動率 σ | 認購 Call (HK$) | 認沽 Put (HK$) | 解讀 |
|---|---|---|---|
| 10% | 6.80 | 1.93 | 波動低，期權時間價值小 |
| 20% | 10.45 | 5.57 | 中等波動 (基準情境) |
| 30% | 14.23 | 9.35 | 波動升高，期權價值上升 |
| 40% | 18.02 | 13.15 | 高波動，期權更值錢 |
| 50% | 21.79 | 16.92 | 極高波動，期權價值最高 |
S=100, K=100, T=1, r=5%。可見波動率越高，認購與認沽期權價值都越高 —— 因為波動越大，標的價格『大漲或大跌』突破行使價的機會越大，而期權的下檔損失有限 (最多損失權利金)、上檔潛力大。波動率是期權定價最關鍵的變數。
理財情境案例
案例一：平價期權的理論定價
假設某港股現價 HK$100，投資者考慮一張行使價 HK$100 (平價, at-the-money)、到期 1 年的期權,市場無風險利率 5%、該股年化波動率估計 20%。
計算:d1 = [ln(100/100) + (0.05 + 0.20²/2)×1] ÷ (0.20×√1) = 0.07 ÷ 0.20 = 0.35;d2 = 0.35 − 0.20 = 0.15。N(0.35) ≈ 0.6368、N(0.15) ≈ 0.5596。Call = 100×0.6368 − 100×e^(−0.05)×0.5596 ≈ HK$10.45;Put = 100×e^(−0.05)×N(−0.15) − 100×N(−0.35) ≈ HK$5.57。
結果顯示:即使行使價等於現價 (內在價值為 0),認購期權仍值約 HK$10.45、認沽期權約 HK$5.57 —— 這全是『時間價值』,反映在到期前一年內，股價有機會朝有利方向波動。認購比認沽貴，部分因為無風險利率為正 (持有現金有利息，抬高看漲期權)。這說明期權價格不只看『現在是否賺錢』,更看『未來波動的可能性』。
案例二：價內認購期權 (半年期)
另一情境:標的現價 HK$120、行使價 HK$100 (價內, in-the-money 的認購)、到期 0.5 年、無風險利率 4%、波動率 25%。
計算得 d1 ≈ 1.2329、d2 ≈ 1.0561,Call ≈ HK$23.18、Put ≈ HK$1.20。
認購期權值 HK$23.18，其中約 HK$20 是『內在價值』(現價 120 − 行使價 100),其餘約 HK$3.18 是時間價值。對應的認沽期權因為深度價外 (股價遠高於行使價、不太可能跌破),只值約 HK$1.20，幾乎全是時間價值。這個例子說明:價內認購期權的價格會貼近『現價與行使價之差』,而愈價外的期權價格愈低、但槓桿愈高。投資者要理解，付出的權利金裡有多少是內在價值、多少是會隨時間流逝的時間價值 (time decay)。
常見問題
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.
相關工具
參考資料
內容審核：香港計算器財經團隊。採用經典無股息歐式 Black-Scholes-Merton 公式，標準常態分佈以 Abramowitz-Stegun 近似計算;衍生工具風險資訊參考香港交易所 (HKEX) 及投資者及理財教育委員會 (IFEC)。模型價與市場報價因波動率微笑、股息、提前行使等因素而有差異,結果僅供教學與理論估算，不構成投資建議，期權交易涉及高風險。