Option Pricing Model (OPM)

The Option Pricing Model (OPM) serves as the core pricing calculator integrated within the option_program. OPM validates option pricing on-chain and forms the foundation for how DeFi options are calculated on Solana.

OPM derives from the widely accepted derivatives formula used in traditional finance—the Black-Scholes Model. Both American-style and European-style options stem from this model, making it the optimal candidate for Solana's DeFi options.

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Black-Scholes Model

The Black-Scholes Model is a proven and reliable formula that traditional finance uses globally on a daily basis for derivatives pricing. However, using the Black-Scholes model within the DeFi environment presents unique challenges that require adaptation.

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Adapting Black-Scholes for DeFi

Epicentral Labs refactored the original Black-Scholes formula to conform to the Solana DeFi ecosystem. The adaptations include:

No Naked Positions

All sellers must be "cash-covered", also known as "collateralized". DeFi protocols restrict "naked" options as a preventative measure against systemic risk, ensuring all written positions are fully collateralized.

Fractionalized Options

Traders are no longer obligated to trade 100x of the underlying share per trade. The financial barrier of entry for any given contract is lowered through fractionalization.

Traders mitigate risk by encouraging capital efficiency through the use of spread strategies to create possible win-win scenarios.

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Black-Scholes Pricing Formulas

The Black-Scholes model provides mathematical formulas for calculating call and put option prices.

Call Option Price

$$C(S,t) = N(d_+)S_t - N(d_-)Ke^{-r(T-t)}$$

Put Option Price

$$P(S,t) = N(-d_+)Ke^{-r(T-t)} - N(-d_-)S_t$$

Standardized Normal Variables

The formulas use two standardized normal variables:

$$d_1 = \frac{1}{\sigma\sqrt{T-t}}\left[\ln\left(\frac{S_t}{K}\right) + \left(r + \frac{\sigma^2}{2}\right)(T-t)\right]$$ $$d_2 = d_1 - \sigma\sqrt{T-t}$$

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Notation Reference

Understanding the mathematical notation used in the Black-Scholes formulas helps traders interpret pricing calculations.

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Option Greeks

The Greeks serve as essential metrics for managing risk in options trading. They help traders understand how their portfolio value changes when specific market factors fluctuate. By analyzing each Greek independently, traders assess individual risk components and adjust their positions to maintain their target risk profile.

An option's price changes based on how traders interact with it and on market conditions. The composition or structure of an option relies heavily on Greeks. Traders use Greeks to understand how various factors affect option pricing and to make informed trading decisions.

Reference: Wikipedia - Option Greeks