OPTION

Definition

An option is a financial derivative that gives the buyer the right—but not the obligation—to buy or sell an underlying asset (like a stock, bond, commodity, or index) at a specified price (strike price) before or at a certain expiration date.

A Call Option grants the right to buy.

A Put Option grants the right to sell.

The buyer pays a premium to the seller (writer) for this right. The seller assumes the obligation if the buyer exercises the option.

Origins

Options trace back to ancient Greece, where philosopher Thales used a primitive form of options to secure olive presses. Modern options gained structure in the 17th-century Amsterdam stock exchange and evolved into regulated instruments with the establishment of the Chicago Board Options Exchange (CBOE) in 1973.

The Black-Scholes model, introduced in the same year, formalized options pricing and revolutionized derivatives trading.

Usage

Financial Industry Applications

• Speculation – Gain exposure to asset price movements with limited risk.

• Hedging – Protect portfolios from adverse price changes.

• Income Strategies – Generate income via covered calls or put writing.

• Employee Compensation – Incentivize performance with stock options.

• Real Options Analysis – Apply option theory to investment decisions (e.g., deferring or expanding projects).

How Option Works

Key Takeaway

• Options are flexible, leveraged, and risk-controlled instruments.

• They can be used for protection (hedging) or amplification (speculation).

• Offer non-linear payoffs (e.g., asymmetry in potential gain/loss).

• Require strong understanding of volatility, time decay, and greeks.

Types of Option

Context in Financial Modeling

Options are essential in:

• Valuation Models (e.g., Black-Scholes, Binomial Tree)

• Risk Management – Modeling exposure via Greeks.

• Real Options Analysis – Flexibility in capital budgeting.

• Equity Compensation Accounting – Modeled under IFRS 2 / ASC 718.

Greeks in Modeling:

• Delta: Sensitivity to price.

• Gamma: Sensitivity of delta.

• Theta: Time decay.

• Vega: Sensitivity to volatility.

• Rho: Sensitivity to interest rates.

Nuances & Complexities

• Implied Volatility (IV): Forward-looking measure from option prices—key to valuation.

• Time Decay: Option value erodes as expiration nears (Theta).

• Liquidity & Slippage: Important for execution strategies.

• Assignment Risk: Especially with American-style options.

Mathematical Formulas

1. Black-Scholes (European Call Option):

$$C = S_0 N(d_1) - Xe^{-rt} N(d_2)$$

Where:

• $S_0$: Current price

• $X$: Strike price

• $r$: Risk-free rate

• $t$: Time to expiration

• $N(d)$: Cumulative normal distribution

• $\sigma$: Volatility

2. Put-Call Parity (European Options):

$$C+X{e}^{-rt}=P+S_0$$

Related Terms

• Futures

• Warrants

• Swaps

• Volatility

• Derivatives

• Delta Hedging

• Covered Call

• Protective Put

Real-World Applications

1. Hedging Portfolio Risk

An investor holding a stock buys a put option to protect against downside risk.

2. Generating Income

A shareholder sells covered calls to earn premium income while holding the underlying stock.

3. Startup Compensation

Employees receive stock options at a fixed strike price, incentivizing long-term company value creation.

4. Real Options in Capital Budgeting

A mining company evaluates a project where it can delay development—modeled as a call option on project NPV.

References & Sources

• Black & Scholes (1973) – The Pricing of Options and Corporate Liabilities

• Hull, J. C. – Options, Futures and Other Derivatives

• CBOE (Chicago Board Options Exchange) – Market Data and Educational Guides

• IFRS 2 / ASC 718 – Equity Compensation Accounting Standards