Black-Scholes Model

Diffusal uses the Black-Scholes model for European option pricing. This page covers the mathematical formulas and Greeks calculations.

For implementation details including fixed-point arithmetic and Solidity code, see Solidity Math.

The Black-Scholes Formula

Call Option

$$C = S \cdot N(d_1) - K \cdot e^{-rT} \cdot N(d_2)$$

Put Option

$$P = K \cdot e^{-rT} \cdot N(-d_2) - S \cdot N(-d_1)$$

d1 and d2

$$d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}$$ $$d_2 = d_1 - \sigma\sqrt{T}$$

Where:

• S = Spot price (current underlying price)
• K = Strike price
• r = Risk-free interest rate
• σ (sigma) = Volatility (annualized)
• T = Time to expiry (in years)
• N(x) = Cumulative normal distribution function

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Parameters

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Greeks

The implementation calculates all primary Greeks alongside the option price.

Delta (Δ)

Rate of change of option price with respect to underlying price.

$$\Delta_{\text{call}} = N(d_1)$$ $$\Delta_{\text{put}} = N(d_1) - 1$$

Range: Calls: 0 to 1, Puts: -1 to 0

Gamma (Γ)

Rate of change of Delta with respect to underlying price.

$$\Gamma = \frac{N'(d_1)}{S \cdot \sigma \cdot \sqrt{T}}$$

Same for both calls and puts. Always positive.

Vega (ν)

Sensitivity to volatility changes.

$$\nu = S \cdot N'(d_1) \cdot \sqrt{T}$$

Same for both calls and puts. Always positive.

Theta (Θ)

Time decay - rate of change with respect to time.

$$\Theta_{\text{call}} = -\frac{S \cdot N'(d_1) \cdot \sigma}{2\sqrt{T}} - r \cdot K \cdot e^{-rT} \cdot N(d_2)$$ $$\Theta_{\text{put}} = -\frac{S \cdot N'(d_1) \cdot \sigma}{2\sqrt{T}} + r \cdot K \cdot e^{-rT} \cdot N(-d_2)$$

Usually negative (options lose value over time).

Rho (ρ)

Sensitivity to interest rate changes.

$$\rho_{\text{call}} = K \cdot T \cdot e^{-rT} \cdot N(d_2)$$ $$\rho_{\text{put}} = -K \cdot T \cdot e^{-rT} \cdot N(-d_2)$$

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Example Calculation

Inputs:

• Spot (S): $3,000
• Strike (K): $3,000
• Rate (r): 5%
• Volatility (σ): 50%
• Time (T): 30 days = 0.0822 years

Calculation:

$$d_1 = \frac{\ln(3000/3000) + (0.05 + 0.5^2/2) \times 0.0822}{0.5 \times \sqrt{0.0822}} = \frac{0 + 0.0144}{0.1434} = 0.1003$$ $$d_2 = 0.1003 - 0.1434 = -0.0431$$ $$N(d_1) = 0.5400, \quad N(d_2) = 0.4828$$ $$\text{Call} = 3000 \times 0.5400 - 3000 \times e^{-0.05 \times 0.0822} \times 0.4828 = \$176.7$$

To get this quote on-chain, call quoter.getOptionQuote() with an OptionRequest containing the pairId, strike (3000e18), expiry (30 days from now), and isCall = true. The returned quote price will be approximately 176.7e18.

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Limitations

European Options Only

The Black-Scholes model assumes:

• No early exercise (European style)
• Constant volatility
• Log-normal price distribution
• No dividends

Edge Cases

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Implementation

The Black-Scholes model is implemented in Solidity using 64.64 fixed-point arithmetic for precision:

The quoter contract provides the public API:

• DiffusalOptionsQuoter — getOptionQuote() and getOptionQuoteWithGreeks()

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Related

• Solidity Math — Fixed-point arithmetic and precision handling
• Margin System — How option prices affect margin requirements
• DiffusalOracle — Source of volatility and risk-free rate