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:

dS=ฮผSdt+ฯƒSdW\Large dS = \mu S dt + \sigma S dW

Where:

  • SS is the underlying asset's price.

  • ฮผฮผ is the expected return (drift) of the asset.

  • ฯƒฯƒ is the asset's volatility.

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

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

C=Sโ‹…N(d1)โˆ’Xโ‹…eโˆ’rTโ‹…N(d2)\Large C = S \cdot N(d1) - X \cdot e^{-rT} \cdot N(d2)

Where:

  • CC is the call option price.

  • SS is the current price of the underlying asset.

  • XX is the option's strike price.

  • rr is the risk-free interest rate.

  • TT is the time to expiration.

  • N(โ‹…)N(โ‹…) is the cumulative distribution function of the standard normal distribution.

  • d1=lnโก(S/X)+(r+12ฯƒ2)TฯƒT\large d1 = \frac{\ln(S/X) + (r + \frac{1}{2} \sigma^2)T}{\sigma \sqrt{T}}

  • d2=d1โˆ’ฯƒT\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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