Black-Scholes Model

The model's output is the theoretical option price, derived from a partial differential equation. The famous closed-form solution for a non-dividend paying stock is:

• Call Price = S * N(d1) - K * e^(-rT) * N(d2)
• Put Price = K * e^(-rT) * N(-d2) - S * N(-d1) Where S is spot price, K is strike price, T is time to expiry, r is risk-free rate, and N() is the cumulative distribution function.

The model requires five key inputs to compute a price:

• Underlying Price (S): Current spot price of the asset.
• Strike Price (K): Price at which the option can be exercised.
• Time to Expiration (T): Time until the option expires, in years.
• Risk-Free Interest Rate (r): Theoretical return of a risk-free investment (e.g., government bonds).
• Volatility (σ): The implied volatility of the underlying asset's returns, representing expected future price fluctuations.

The model enables the calculation of "Greeks", which measure the sensitivity of the option's price to various factors. Key Greeks include:

• Delta (Δ): Rate of change of option price relative to the underlying asset's price.
• Gamma (Γ): Rate of change of Delta.
• Theta (Θ): Rate of time decay of the option's price.
• Vega (ν): Sensitivity to changes in the underlying's volatility.
• Rho (ρ): Sensitivity to changes in the risk-free interest rate.

The model's precision relies on several key assumptions that define its limitations:

• Efficient Markets: The underlying asset follows a geometric Brownian motion with constant volatility and drift.
• No Dividends: The underlying pays no dividends during the option's life (later models adjusted for this).
• European Exercise: Options can only be exercised at expiration.
• No Transaction Costs: Trading the underlying incurs no fees or taxes.
• Constant Risk-Free Rate: The interest rate is known and constant.

A critical application is solving for implied volatility (IV). By inputting the market price of an option into the Black-Scholes formula and solving for volatility, traders can derive the market's expectation of future asset price fluctuation. IV is a forward-looking, market-driven metric central to options trading and the volatility smile phenomenon.

While revolutionary, the model has well-documented limitations in real markets:

• Constant Volatility Assumption: Real-world volatility is not constant; it clusters and changes (stochastic volatility).
• Log-Normal Returns: Assumes asset returns are log-normally distributed, ignoring fat tails and extreme market events (black swans).
• No Early Exercise: Does not price American-style options, which can be exercised early.
• Discontinuous Jumps: Does not account for sudden price jumps, modeled by later frameworks like Merton's jump-diffusion model.

The current market price of the underlying asset (e.g., stock, cryptocurrency). This is the most volatile input and the primary driver of an option's value. A call option increases in value as S rises, while a put option increases as S falls. For example, pricing an option on Ethereum requires the current ETH/USD spot price.

The predetermined price at which the option holder can buy (call) or sell (put) the underlying asset. The relationship between the strike price and the underlying price determines the option's moneyness:

• In-the-money (ITM): Call: S > K; Put: S < K
• At-the-money (ATM): S ≈ K
• Out-of-the-money (OTM): Call: S < K; Put: S > K OTM options are cheaper as they have only time value.

The time remaining until the option's expiry date, expressed in years. Time decay (theta) is a critical concept: an option's time value erodes as expiration approaches, all else being equal. Longer-dated options are more expensive because there is more time for the underlying price to move favorably. This input is central to the options greeks.

The theoretical return on a risk-free investment over the option's life, typically based on government bond yields (e.g., U.S. Treasury bills). This rate accounts for the time value of money. It positively affects call option prices (as buying the asset later is cheaper in present value) and negatively affects put option prices. In near-zero or crypto-native environments, this input may be set to a minimal value.

The only non-observable input, representing the expected annualized volatility of the underlying asset's returns. It measures the magnitude of price swings, not the direction. Higher volatility increases the probability of the option finishing in-the-money, thus increasing the option's premium. This is the implied volatility (IV) when the model is solved backwards from market prices, forming the core of volatility trading.

The Black-Scholes formula relies on assumptions that are often violated in real markets, especially in crypto:

• Constant volatility and interest rates.
• Lognormal distribution of asset returns (no fat tails).
• No transaction costs or arbitrage opportunities.
• European-style exercise only at expiry. These limitations led to the development of more complex models (e.g., stochastic volatility models) for pricing derivatives like Bitcoin options.

The model's outputs, known as the Greeks, are critical for risk management in DeFi options vaults and market making.

• Delta: Sensitivity of the option's price to the underlying asset's price. Used for delta-neutral hedging strategies.
• Gamma: Rate of change of Delta.
• Vega: Sensitivity to changes in implied volatility.
• Theta: Time decay of the option's value. These metrics allow automated strategies to dynamically hedge portfolios and manage exposure to different risk factors.

The Black-Scholes model makes assumptions that are often violated in crypto markets, leading to pricing inefficiencies.

• Log-Normal Distribution: Assumes asset returns are normally distributed, but crypto exhibits fat tails and extreme volatility.
• Constant Volatility: Assumes volatility is constant, whereas crypto IV is highly volatile and regime-dependent.
• Continuous Trading & No Arbitrage: Assumes perfect, continuous markets, which conflicts with blockchain finality times and liquidity fragmentation across L1s and L2s. These limitations drive innovation in alternative pricing models.

The model enables the creation of tokenized structured products that package options with other derivatives. Examples include:

• Covered Call Vaults: Users deposit an asset (e.g., ETH) and automatically sell call options against it, earning premium income. The strike price is often selected based on Delta calculations.
• Put-Selling Vaults: Users sell put options to earn premiums, often used as a yield strategy or to accumulate assets at a target price. Protocols like Ribbon Finance and Friktion (now inactive) popularized these automated vault strategies.

Reliable on-chain data is essential for the model. Oracles like Chainlink provide critical real-time inputs:

• Spot Price Feeds: For the current price of the underlying asset.
• Volatility Feeds: Some oracles calculate and publish historical or implied volatility data.
• Interest Rate Feeds: For the risk-free rate parameter, often sourced from major lending protocols like Aave or Compound. The accuracy and decentralization of these oracles directly impact the robustness of the derivative pricing.

The search for more accurate models has led to several advanced approaches in DeFi:

• Stochastic Volatility Models: Models like Heston that allow volatility to change over time.
• Local Volatility Models: Use the entire volatility surface from market prices.
• Machine Learning Models: Some protocols experiment with ML to predict option prices based on market data, moving beyond closed-form equations.
• Monte Carlo Simulations: Used for pricing exotic or path-dependent options where Black-Scholes is insufficient.