Black-Scholes Model
Black-Scholes answers a simple question: "What should this option cost?"
Given five inputs - spot price, strike, time to expiry, interest rate, and volatility - the formula outputs a theoretical fair value. It's the standard pricing model for European options and the basis for calculating implied volatility and the Greeks.
The Inputs
Black-Scholes and the Greeks
Play with the calculator above. Notice how the price changes when you move each slider? Those sensitivities have names - they're called the Greeks.
| Greek | What it measures |
|---|---|
| Delta | How much option price moves when spot moves $1 |
| Theta | How much option price drops each day |
| Vega | How much option price moves when IV moves 1% |
| Gamma | How much delta itself changes when spot moves |
These aren't just abstract numbers. Try it: slide Spot up slowly and watch the Call Price. That rate of change is delta.
But what is a Greek, really?
Each Greek is a slope - the steepness of a curve.
The curve shows how the option price changes as one input changes. The steeper the curve at your current position, the more sensitive the price is to that input.
- Flat curve → small Greek → price barely reacts to that input
- Steep curve → large Greek → price moves a lot when that input changes
That's all a "derivative" means in math - the slope of a curve at a point. Each Greek is just measuring slope in a different direction.
See the Greeks reference for more on each one.
Volatility (σ) is the only input that's not directly observable. You can look up S, K, T, and r - but σ must be estimated or implied from market prices. This is why implied volatility is so important.
Key Assumptions
Black-Scholes assumes:
| Assumption | Reality |
|---|---|
| European exercise only | ✓ Matches Hypercall |
| Constant volatility | ✗ Vol changes constantly |
| No dividends | ✓ Mostly true for crypto |
| Log-normal price distribution | ✗ Crypto has fat tails |
| Continuous trading | ✓ Crypto trades 24/7 |
| No transaction costs | ✗ Fees exist |
Despite these limitations, Black-Scholes remains the foundation for options pricing.
Why It Matters
- Industry standard - Everyone uses it as a baseline
- Greeks derivation - Delta, gamma, theta, vega all come from Black-Scholes
- Implied volatility - Solved by inverting Black-Scholes given market price
- Quick sanity checks - Is this option priced reasonably?
In Practice
You don't need to calculate Black-Scholes by hand. Platforms like Hypercall use it internally to:
- Display theoretical prices
- Calculate greeks
- Derive implied volatility from market prices
The model gives you a theoretical fair value. The market price may differ based on supply/demand, but Black-Scholes is the reference point.
Related:
- Option Valuation - Intrinsic vs extrinsic value
- Exercise Styles - Why European style matters for Black-Scholes