Quantitative Finance & Derivatives Lab

Black-Scholes Option Pricing Calculator

Model European call and put option prices, Black-Scholes Greeks (Delta, Gamma, Theta, Vega, Rho), and implied volatility matrix.

Pricing Engine Volatility Matrix Black-Scholes Theory FAQ

Option Contract Presets

Load calibrated derivatives models.

Step 1: Underlying Spot, Strike, Expiration Days, Volatility & Rates

Option Parameters & Market Inputs

📊 Underlying Asset & Contract Terms

Calendar days until option expiration date.

⚡ Volatility & Interest Rates

Call Option Price
$6.62
Intrinsic: $0.00 + Extrinsic Time Value: $6.62
Put Option Price
$5.51
Intrinsic: $0.00 + Extrinsic Time Value: $5.51
Option Delta (Δ)
+0.553 Call Δ
Put Δ: -0.447 | Gamma (Γ): 0.0263 per $1 move
Daily Theta Decay (Θ)
-$0.034 / day
Vega (ν): $0.198 / 1% vol | Rho (ρ): $0.119 / 1% rate

Option Matrix: Underlying Stock Price vs. Implied Volatility (σ)

Simulates European Call Price ($), Put Price ($), and Call Delta (Δ) across underlying price moves and volatility spikes.

Underlying Spot Price 15% Volatility 25% Volatility 35% Volatility 45% Volatility 60% Volatility

Derivatives & Quantitative Finance

Understanding Black-Scholes & Greeks

Key quantitative options pricing, risk management, and derivatives principles:

  • The Merton Continuous Dividend Extension: Adjusts spot asset prices by $e^{-qT}$ to account for dividend cash drag before expiration.
  • Delta ($Delta$) as Hedge Ratio: Delta indicates the number of shares needed to create a delta-neutral, risk-free hedge. It also approximates the risk-neutral probability of expiring in-the-money.
  • Gamma ($Gamma$) Risk: Measures Delta convexity. High Gamma near ATM expiration means market makers must aggressively buy and sell underlying shares to stay delta-neutral.
  • Theta ($Theta$) Time Decay: Options are wasting assets. Extrinsic value decays non-linearly, accelerating in the final 30–45 days before expiration.

Evaluate portfolio Value at Risk in the VaR Lab.

Mathematical Formulation

Black-Scholes & Greeks equations

d_1 = rac{ln(S / K) + (r - q + rac{sigma^2}{2})T}{sigma sqrt{T}} quad ext{and} quad d_2 = d_1 - sigma sqrt{T}

ext{Call} = S e^{-qT} N(d_1) - K e^{-rT} N(d_2)

ext{Put} = K e^{-rT} N(-d_2) - S e^{-qT} N(-d_1)

Delta_{ ext{call}} = e^{-qT} N(d_1) quad ext{and} quad Delta_{ ext{put}} = e^{-qT} [N(d_1) - 1]

Gamma = rac{e^{-qT} N'(d_1)}{S sigma sqrt{T}} quad ext{and} quad ext{Vega } ( u) = S e^{-qT} sqrt{T} N'(d_1)

Theta_{ ext{call}} = - rac{S sigma e^{-qT} N'(d_1)}{2sqrt{T}} - r K e^{-rT} N(d_2) + q S e^{-qT} N(d_1)

Analyze real asset expansion options in the Real Options Lab.

FAQ

Black-Scholes option pricing & Greeks questions

What is the Black-Scholes Option Pricing Model?

The Black-Scholes-Merton model is a mathematical formula for pricing European financial derivative options using current stock price, strike price, time to expiration, risk-free interest rate, dividend yield, and implied volatility.

What are the primary Option Greeks?

The Greeks measure risk sensitivities: Delta (price change per $1 stock move), Gamma (rate of Delta change), Theta (daily time decay), Vega (price change per 1% volatility change), and Rho (interest rate sensitivity).

What is the difference between intrinsic and time value?

Intrinsic value is the immediate payoff if exercised today (Max(0, S-K) for calls). Extrinsic time value represents the premium buyers pay for the remaining potential of favorable price movement before expiration.

How does implied volatility (IV) affect call and put option prices?

Higher implied volatility expands the distribution of potential price outcomes, increasing the extrinsic time value and price of both call and put options.

Can I export Black-Scholes option pricing schedules to CSV?

Yes. You can export complete call/put valuations, Greek risk profiles, d1/d2 parameters, and 6x5 volatility sensitivity matrices as a UTF-8 CSV spreadsheet with formula defense.

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