1. Option Mechanics: Options vs. Commitments
Unlike forwards and futures, which legally bind both counterparties to execute a future transaction, an option grants its buyer a right, but not an obligation, to execute a trade. The seller (writer) of the option, however, remains legally bound to perform if the buyer chooses to exercise their right. The buyer pays an upfront, non-refundable cost called an Option Premium.
  • Call Option: The right to buy an underlying asset at a specified Strike Price (K).
  • Put Option: The right to sell an underlying asset at a specified Strike Price (K).

2. Operational Payoff Formulations
  • Long Call (Buyer of a Call Option): Profits if the market price (\(S_{t}\)) rises above the strike price (K).

    Payoff = max(0, Sₜ − K)
    Profit = max(0, Sₜ − K) − Premium

    Meaning:

    • This is a call option.
    • Sâ‚œ = asset price at expiration.
    • If Sâ‚œ > K, the option is in the money and has value; otherwise it expires worthless.
    • Profit = payoff − what you paid (premium).

  • Long Put (Buyer of a Put Option): Profits if the market price (\(S_{t}\)) falls below the strike price (K). Used by corporate treasurers to establish a price floor for asset sales.

    Payoff = max(0, K − Sₜ)
    Profit = max(0, K − Sₜ) − Premium

    Meaning:

    • This is a put option.
    • Sâ‚œ = asset price at expiration.
    • If K > Sâ‚œ, the option has value; otherwise it expires worthless.
    • Profit subtracts what you paid (the premium).

3. Option Valuation Foundations: The Black-Scholes-Merton Model

Option Premium = f(S₀, K, σ, T, r)

Where:

  • Sâ‚€ = Asset price today
  • K = Strike price
  • σ = Volatility
  • T = Time to maturity
  • r = Risk-free rate

  • The Volatility Multiplier: Volatility (σ) is the primary driver of option value. As underlying asset price volatility increases, the option premium climbs because the option buyer faces expanded upside profit potential while their downside loss remains capped at the paid premium.
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