布萊克–舒爾斯期權定價計算器

分類： Uncategorized
- August 31, 2026
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Calculate theoretical values and Greeks for European-style options using the Black-Scholes model

期權參數

期權類型
$
現時正股價格
$
期權行使價
Time until option expiration
%
年度無風險利率
%
年度股息率
%
Annual volatility of underlying
Convert Trading Days to Years
Time to Expiry (Years): 1.00

Option Pricing Results

期權價格
$10.45
認購期權
內在價值
$0.00
Amount option is in-the-money
時間值
$10.45
Premium above intrinsic value
Moneyness
At-the-Money
Ratio: 1.00

期權希臘值

Δ
Delta
0.528
Rate of change with respect to underlying price
Γ
Gamma
0.017
Rate of change of Delta with respect to underlying price
Θ
Theta
-0.048
Rate of change with respect to time (per day)
V
Vega
0.391
Rate of change with respect to volatility (per 1% change)
ρ
Rho
0.486
Rate of change with respect to interest rate (per 1% change)

Price Visualization

About the Black-Scholes Model

The Black-Scholes model, developed by Fischer Black, Myron Scholes, and Robert Merton in 1973, is a mathematical model for pricing European-style options. It assumes that the price of the underlying asset follows a geometric Brownian motion with constant volatility.

Black-Scholes Formula

Call Price:
C = S₀e⁻ᵈᵗN(d₁) - Xe⁻ʳᵗN(d₂)
Put Price:
P = Xe⁻ʳᵗN(-d₂) - S₀e⁻ᵈᵗN(-d₁)
Where:
d₁ = [ln(S₀/X) + (r - d + σ²/2)t] / (σ√t)
d₂ = d₁ - σ√t
S₀ Current stock price
X Strike price
t Time to expiration (in years)
r Risk-free interest rate
d Dividend yield
σ Volatility of the underlying
N(x) Cumulative normal distribution function

Key Assumptions

  • European-style options (no early exercise)
  • Efficient markets with no transaction costs or taxes
  • Stock prices follow a lognormal distribution
  • Volatility and risk-free rate are constant over the option's life
  • No arbitrage opportunities exist

Note: The Black-Scholes model has limitations and may not accurately price options in all market conditions, particularly during periods of high volatility or market stress.

Understanding the Greeks

Δ Delta

Measures the rate of change of the option price with respect to changes in the underlying asset's price. Delta ranges from 0 to 1 for calls and -1 to 0 for puts. It represents the equivalent exposure in the underlying asset.

Γ Gamma

Measures the rate of change of Delta with respect to changes in the underlying price. High Gamma means the Delta is highly sensitive to price changes in the underlying asset.

Θ Theta

Measures the rate of change of the option price with respect to time (time decay). Theta is typically negative for both calls and puts, as options lose value as time passes.

V Vega

Measures the rate of change of the option price with respect to changes in volatility. Higher implied volatility generally increases option prices.

ρ Rho

Measures the rate of change of the option price with respect to changes in the risk-free interest rate. Call options generally increase in value when interest rates rise, while put options decrease.

本工具計算什麼

計算歐式認購或認沽期權的理論價格。

公式
C = S·N(d₁) − K·e^(−rT)·N(d₂)，其中 d₁ =［ln(S/K) +（r + σ²/2）T］÷（σ√T），d₂ = d₁ − σ√T
計算示例
S = 100、K = 100、r = 5%、σ = 20%、T = 1 年，認購期權價格約為 10.45。
何時使用
為期權估值，並觀察價格如何隨波幅與時間變化。
常見錯誤
模型假設波幅固定且不可提前行使，實際期權兩者皆不符，這正是引伸波幅因行使價而異的原因。