# Pricing Methodology

The Diem AMM follows the Black-Scholes pricing methodology while fetching implied volatility for strike prices through either an off-chain oracle source, or by calculating the 30-day realized volatility of the asset.

The Black-Scholes model is based on a partial differential equation (PDE) that describes the dynamics of the option price over time. By that the model assumes that the underlying asset follows the following stochastic differential equation:

$\Large dS = \mu S dt + \sigma S dW$

Where:

• $S$ is the underlying asset's price.

• $ฮผ$ is the expected return (drift) of the asset.

• $ฯ$ is the asset's volatility.

• $dW$ is a Wiener process (Brownian motion), representing random fluctuations.

The Black-Scholes formula calculates the value of a European call option as follows:

### $\Large C = S \cdot N(d1) - X \cdot e^{-rT} \cdot N(d2)$

Where:

• $C$ is the call option price.

• $S$ is the current price of the underlying asset.

• $X$ is the option's strike price.

• $r$ is the risk-free interest rate.

• $T$ is the time to expiration.

• $N(โ )$ is the cumulative distribution function of the standard normal distribution.

• $\large d1 = \frac{\ln(S/X) + (r + \frac{1}{2} \sigma^2)T}{\sigma \sqrt{T}}$

• $\large d2 = d1 - \sigma \sqrt{T}$

As the time to expiry approaches zero, the PDE used in the Black-Scholes model becomes increasingly difficult to solve due to the option price sensitivity increases in factor changes, in the underlying asset price and volatility. This results in a breakdown of the Black-Scholes modelโs assumptions, and becomes increasingly unsuitable to price very short-dated options, for that reason and to prevent buyers from purchasing premiums at zero cost, a minimum price per option is being introduced. The Diem price will be calculated based on the maximum value between the standard Black-Scholes model, and the 0.25% multiplied by the spot price of the asset, which theoretically puts a maximum of 400x per option purchased.

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