Option Premium

The inherent, non-negative value of an option if exercised immediately. It is calculated as the difference between the underlying asset's spot price and the option's strike price.

• Call Option: Intrinsic Value = Max(0, Spot Price - Strike Price)
• Put Option: Intrinsic Value = Max(0, Strike Price - Spot Price) An option with intrinsic value is in-the-money (ITM). An option with zero intrinsic value is out-of-the-money (OTM) or at-the-money (ATM).

The portion of the premium attributable to the remaining time until the option's expiration. It represents the potential for the option to gain intrinsic value before expiry.

• Key Driver: More time until expiration equals greater time value, as there is a higher probability of a favorable price move.
• Time Decay: Time value erodes as expiration approaches, a process quantified by the Theta Greek. This decay accelerates significantly in the final weeks of an option's life.

The market's forecast of the underlying asset's future volatility, embedded in the option's price. It is the most significant component of time value.

• High IV: Indicates an expectation of large price swings, increasing the premium (and time value) for both calls and puts.
• Low IV: Suggests an expectation of price stability, decreasing premiums. The sensitivity of an option's price to changes in implied volatility is measured by Vega.

The relationship between the underlying asset's spot price and the option's strike price, which directly impacts the premium's composition.

• In-the-Money (ITM): Has intrinsic value + time value. Premium is higher.
• At-the-Money (ATM): Has the maximum time value, as the potential for moving ITM is highest. Premium consists almost entirely of time value.
• Out-of-the-Money (OTM): Has only time value. Premium is lower and more sensitive to changes in implied volatility and time decay.

The risk-free interest rate affects option premiums through the cost of carry. Higher rates increase the theoretical value of call options and decrease the value of put options.

• Call Options: A higher interest rate makes purchasing the asset today more expensive (due to financing cost), making the right to buy it in the future more valuable.
• Put Options: A higher rate makes the right to sell in the future less attractive compared to selling now and earning interest. This sensitivity is measured by Rho, though it is typically the least impactful Greek for short-dated options.

Expected dividends and the cost of holding the underlying asset (storage, funding) are factored into the option's theoretical price.

• Dividends: Expected dividends during the option's life decrease the price of call options (as the stock price drops on the ex-dividend date) and increase the price of put options.
• Cost of Carry: For commodities or assets with storage costs, these expenses are incorporated into pricing models, affecting the forward price used in valuation.

The intrinsic value is the immediate, realizable profit if the option were exercised right now. It represents the "moneyness" of the option.

• For a call option: Intrinsic Value = max(0, Underlying Price - Strike Price)
• For a put option: Intrinsic Value = max(0, Strike Price - Underlying Price) An option is in-the-money (ITM) if it has positive intrinsic value, at-the-money (ATM) if the strike equals the spot price, and out-of-the-money (OTM) if intrinsic value is zero.

Time value is the portion of the premium exceeding the intrinsic value. It represents the premium paid for the potential of future price movement before expiration. Key factors influencing time value include:

• Time to expiration: More time equals higher potential for movement (all else equal).
• Implied volatility (IV): The market's forecast of future volatility; higher IV increases time value.
• Interest rates & dividends: Affect the cost of carry for the underlying asset. Time value decays to zero as expiration approaches, a process known as theta decay.

The relationship between an option's strike price and the underlying asset's spot price determines its moneyness, which dictates the composition of its premium.

• In-the-Money (ITM): Premium = Intrinsic Value + Time Value. Has immediate exercise value.
• At-the-Money (ATM): Premium is almost entirely Time Value. Highest sensitivity to volatility (vega) and time decay (theta).
• Out-of-the-Money (OTM): Premium is 100% Time Value. No intrinsic value; pure bet on future price movement.

The Greeks are metrics that quantify how an option's premium changes in response to specific factors. They are derived from the components of the premium.

• Delta (Δ): Sensitivity to changes in the underlying price. Linked to intrinsic value.
• Gamma (Γ): Rate of change of Delta.
• Theta (Θ): Daily erosion of time value (time decay).
• Vega (ν): Sensitivity to changes in implied volatility, which directly affects time value.
• Rho (ρ): Sensitivity to interest rate changes.

Implied Volatility (IV) is the market's expectation of future volatility, derived from an option's current market price using a model like Black-Scholes. It is the primary driver of time value.

• High IV = Expensive time value (higher premium), indicating expected large price swings.
• Low IV = Cheap time value (lower premium), indicating expected calm markets. IV is often expressed as an annualized percentage and is a critical input for pricing models. Traders compare current IV to historical volatility to gauge if options are relatively cheap or expensive.

Put-Call Parity is a fundamental no-arbitrage relationship that defines the price equilibrium between a European call, a European put (same strike/expiry), and the underlying asset. The formula is: Call Price + Present Value of Strike = Put Price + Underlying Price This principle ensures that the time value component is consistently priced across puts and calls with the same strike and expiration. It allows for the creation of synthetic positions (e.g., a synthetic long stock = long call + short put).

An option premium is the market price of an option contract, representing the cost for the buyer to acquire its rights and the income for the seller who assumes the obligation. It is paid upfront and is kept by the seller regardless of whether the option is exercised. The premium compensates the seller for the risk they undertake, particularly the risk of being assigned.

The premium is not arbitrary; it is calculated based on specific variables known as "The Greeks":

• Delta: Sensitivity to the underlying asset's price.
• Gamma: Rate of change of Delta.
• Vega: Sensitivity to implied volatility.
• Theta: Time decay (premium erosion as expiry approaches).
• Rho: Sensitivity to interest rates. In DeFi, protocols like Lyra and Premia use on-chain oracles and volatility surfaces to compute these values algorithmically.

In DeFi options protocols, premium settlement is trustless and immediate:

1. Buyer pays premium in the quoted asset (e.g., ETH, USDC) upon purchasing the option.
2. Premium is transferred directly to the seller's vault or smart contract.
3. Funds are locked as collateral by the seller to back the potential obligation. This automated flow eliminates counterparty risk and ensures the seller is instantly compensated for writing the option.

The option style significantly impacts premium valuation and mechanics:

• American Options: Can be exercised anytime before expiry. This additional flexibility for the buyer typically commands a higher premium.
• European Options: Can only be exercised at expiry. This restriction generally results in a lower premium. DeFi protocols implement one style or the other, affecting their risk models and capital efficiency for sellers.

The total premium is composed of two parts:

• Intrinsic Value: The immediate profit if the option were exercised now. For a call: Max(0, Spot Price - Strike Price). Only in-the-money (ITM) options have intrinsic value.
• Time Value (Extrinsic Value): The remaining premium, reflecting the probability of future price movement before expiry. It is influenced by time to expiry and implied volatility. An option's premium is always at least equal to its time value.

Implied Volatility (IV) is the market's forecast of future price swings and is the most significant factor in an option's time value. Higher expected volatility leads to higher premiums, as the probability of the option finishing in-the-money increases. DeFi protocols often use historical volatility from oracles and volatility smiles/surfaces to model IV, making premium pricing a direct function of perceived market risk.

The inherent value of an option if exercised immediately. It is calculated as the difference between the asset's spot price and the option's strike price.

• For a call option: Intrinsic Value = Max(0, Spot Price - Strike Price).
• For a put option: Intrinsic Value = Max(0, Strike Price - Spot Price). An option with intrinsic value is in-the-money (ITM). An option's premium is the sum of its intrinsic value and its time value.

The portion of an option's premium that exceeds its intrinsic value. It represents the premium paid for the probability that the option will gain intrinsic value before expiration.

• Key factors influencing time value include time to expiration, implied volatility, and interest rates.
• Time value decays as expiration approaches, a phenomenon known as theta decay. An out-of-the-money (OTM) option's premium consists entirely of time value.

A forward-looking, market-derived metric representing the expected volatility of the underlying asset over the option's life. It is a primary driver of an option's time value.

• High IV leads to higher option premiums, as the market prices in greater expected price swings.
• IV is often expressed as an annualized percentage and is a key input into option pricing models like the Black-Scholes model. Traders monitor the IV rank or IV percentile to assess if options are relatively expensive or cheap.

A set of risk measures (Delta, Gamma, Theta, Vega, Rho) that quantify how an option's premium is expected to change in response to various factors.

• Delta: Sensitivity to changes in the underlying asset's price.
• Gamma: Rate of change of Delta.
• Theta: Time decay of the option's premium.
• Vega: Sensitivity to changes in implied volatility.
• Rho: Sensitivity to changes in interest rates. These metrics are essential for managing the risk of an options portfolio.

Mathematical models used to calculate the theoretical fair value of an option premium. The most famous is the Black-Scholes-Merton model, which uses inputs like the underlying price, strike price, time to expiration, risk-free rate, and volatility.

• The Binomial options pricing model is another common approach, using a discrete-time lattice to model possible price paths.
• In DeFi, models may adapt to account for impermanent loss, liquidity pool dynamics, and the absence of a central risk-free rate.

A classification describing the relationship between an asset's spot price and an option's strike price. It directly impacts the option's intrinsic value.

• In-the-Money (ITM): Call: Spot > Strike. Put: Spot < Strike. Has intrinsic value.
• At-the-Money (ATM): Spot ≈ Strike. Premium is almost entirely time value.
• Out-of-the-Money (OTM): Call: Spot < Strike. Put: Spot > Strike. Has no intrinsic value, only time value. Moneyness is a snapshot that changes with market movements.