Introduction to Options

Pricing and Hedging Derivative Securities - Back, Loewenstein, and Liu

What Are Derivatives?

Derivative securities are financial instruments whose values are derived from an underlying asset, index, or rate.

  • Underlying assets can be:
    • Stocks, bonds, commodities
    • Currencies, interest rates
    • Market indices
  • Value fluctuates based on changes in the underlying
  • Used for hedging risks and speculation

Trading Venues

Exchange-Traded

  • Traded on regulated exchanges (CME, NYSE, ICE)
  • Standardized contracts
  • Centralized clearing
  • Price transparency
  • High liquidity

Over-the-Counter (OTC)

  • Private bilateral contracts
  • Customizable terms
  • Higher counterparty risk
  • Less transparency
  • Tailored solutions

Options

For the remainder of this lecture, we’ll focus on options - the most versatile derivatives.

  • Call option: Right to buy an asset at a fixed price (the strike)
  • Put option: Right to sell an asset at a fixed price (the strike)

The buyer pays the seller a premium upfront for this right.

Options are particularly important for several reasons:

  • Widely traded: Billions of contracts annually across global exchanges
  • Versatile: Can be used for speculation, hedging, or income generation
  • Building blocks: Understanding options helps with other derivatives

Option Basics: Key Terminology

Essential Terms

  • Premium: Price paid to buy the option
  • Strike Price (K): Fixed price at which option can be exercised
  • Expiration Date: When the contract ends
  • Exercise: Using the right granted by the option
  • Underlying Asset: The financial instrument the option is based on

American vs. European Options

Despite the names, both trade worldwide!

American Options

  • Can be exercised any time before expiration
  • More flexibility
  • More common on exchanges
  • Slightly more expensive

European Options

  • Can only be exercised at expiration
  • Less flexibility
  • Simpler to price
  • Common in indices and OTC contracts

Rights, Obligations, and Motivations

Option Buyers (Long)

  • Pay premium upfront
  • Have rights, no obligations
  • Choose whether to exercise
  • Max loss = premium paid
  • Motivations:
    • Speculation with leverage
    • Hedging/insurance

Option Sellers (Short)

  • Receive premium upfront
  • Have obligations
  • Must fulfill if exercised
  • Potentially unlimited losses
  • Motivation:
    • Collect premium income

Interactive Market Data

Let’s explore real option prices:

Figure 1: Real market data from CBOE via Yahoo Finance

Patterns in Option Prices: Strike Effects

Experimenting with the market data reveals consistent patterns:

Call Options:

  • Prices decrease as strike prices increase
  • Call with strike $50 > Call with strike $100
  • Why? Lower strike = more valuable right to buy

Put Options:

  • Prices increase as strike prices increase
  • Put with strike $100 > Put with strike $50
  • Why? Higher strike = more valuable right to sell

Patterns in Option Prices: Time Effects

Time to expiration matters:

  • Longer-dated options cost more than shorter-dated options
  • Why?
    • More time for favorable price movements
    • Greater flexibility (American options)
    • Higher uncertainty

This pattern holds for both calls and puts,

Intrinsic Value

Intrinsic value = What the option is worth if exercised immediately (or at expiration)

Call Option:

\[\text{Intrinsic Value} = \max(S - K, 0)\]

  • If \(S = \$50\), \(K = \$40\)
  • Intrinsic value = \(\$10\)
  • Can buy at \(\$40\), sell at \(\$50\)

Put Option:

\[\text{Intrinsic Value} = \max(K - S, 0)\]

  • If \(S = \$50\), \(K = \$60\)
  • Intrinsic value = \(\$10\)
  • Can buy at \(\$50\), sell at \(\$60\)

Moneyness: ITM, ATM, OTM

Options are classified by their relationship to the current price:

Moneyness Call Option Put Option
In the Money (ITM) \(S > K\) \(S < K\)
At the Money (ATM) \(S \approx K\) \(S \approx K\)
Out of the Money (OTM) \(S < K\) \(S > K\)
  • ITM: Positive intrinsic value
  • ATM/OTM: Zero intrinsic value

Time Value and Time Decay

Time value = Option price - Intrinsic value

  • Options are usually worth more than intrinsic value before expiration
  • Why? Potential for favorable price movements
  • Time decay: Time value decreases as expiration approaches
  • At expiration: Time value = 0, Option value = Intrinsic value

American options are always worth at least their intrinsic value (otherwise arbitrage!)

Key Insights About Options

Options differ fundamentally from other financial instruments:

  1. Non-linear payoffs: Not just multiples of the underlying
  2. Asymmetric risk/reward: Different for buyers vs. sellers
  3. Time decay: Value erodes as expiration approaches
  4. Volatility sensitivity: Option prices increase with uncertainty
  5. Multiple dimensions: Price, time, volatility all matter

These properties make options powerful but complex!

Trading Options on Exchanges

Most options are traded on organized exchanges:

  • Standardized contracts: Fixed strikes and expirations
  • Transparent pricing: Public order books
  • Clearinghouse guarantee: Eliminates counterparty risk
  • Most positions closed before expiration: Through offsetting trades rather than exercise

Strikes and Maturities

Exchanges determine which options are available:

  • Strike prices: Added to bracket the current market price
    • New strikes introduced as underlying price moves
    • Typically spaced at regular intervals
  • Expiration dates:
    • Weekly, monthly, and quarterly options
    • New expirations added as older ones expire
    • Availability depends on trading interest

Open Interest and Contract Creation

Unlike stocks, options have no pre-existing supply:

  • Long position: Option buyer
  • Short position: Option seller
  • Open Interest: Total number of long (= short) positions

How open interest changes:

  • New buyer + New seller → Open interest increases
  • Existing buyer sells to new buyer → Open interest unchanged
  • Existing buyer and seller both close → Open interest decreases

Volume and Open Interest Patterns

Understanding trading patterns:

Concentration Near Current Price:

  • Volume/open interest highest within 10-20% of current price
  • Far OTM options trade infrequently
  • Wider bid-ask spreads for less popular strikes

Popularity of OTM Options:

  • Lower cost → higher leverage
  • Attractive for speculation (large % returns possible)
  • Used for tail risk hedging (portfolio insurance)

The Life Cycle of Open Interest

Open interest evolves over an option’s life:

Initial Growth:

  • Starts at zero for new series
  • Grows as traders discover it
  • Peaks when several weeks/months remain

Example:

  • Day 1: Trader A buys 10 calls from Trader B → OI = 10
  • Day 2: Trader C buys 5 from Trader D → OI = 15
  • Day 3: Trader E buys 3 from Trader A → OI = 15 (unchanged)

Decline Phase:

  • Decreases as expiration approaches
  • Traders close positions
  • Risk managers avoid near-expiry options

Final Settlement:

  • ITM options: auto-exercised
  • OTM options: expire worthless

Payoff vs. Profit Diagrams

Two important ways to visualize options:

Payoff Diagram

  • Shows intrinsic value at expiration
  • Function of underlying price \(S\)
  • Ignores premium paid/received

Profit Diagram

  • Shows actual profit/loss if held to expiration
  • Payoff minus premium paid
  • Or payoff plus premium received

Long Call: Payoff and Profit

Buying a call option (bullish strategy):

Figure 2

Long Call: Key Characteristics

Market View: Bullish (expect price to rise)

Maximum Profit: Unlimited (as \(S\) increases)

Maximum Loss: Premium paid ($5 in example)

Breakeven: Strike + Premium = $105

Best for: Speculating on upside with limited downside risk

Long Put: Payoff and Profit

Buying a put option (bearish strategy):

Figure 3

Long Put: Key Characteristics

Market View: Bearish (expect price to fall)

Maximum Profit: Strike - Premium = $95 (limited by \(S \geq 0\))

Maximum Loss: Premium paid ($5)

Breakeven: Strike - Premium = $95

Best for: Portfolio insurance, speculating on downside with limited risk

Short Call: Payoff and Profit

Writing (selling) a call option:

Figure 4

Short Call: Key Characteristics

Market View: Neutral to bearish (expect price to stay flat or fall)

Maximum Profit: Premium received ($5)

Maximum Loss: Unlimited (as \(S\) increases)

Breakeven: Strike + Premium = $105

Risk: Very high! Limited upside, unlimited downside

Short Put: Payoff and Profit

Writing (selling) a put option:

Figure 5

Short Put: Key Characteristics

Market View: Neutral to bullish (expect price to stay flat or rise)

Maximum Profit: Premium received ($5)

Maximum Loss: Strike - Premium = $95 (if \(S \to 0\))

Breakeven: Strike - Premium = $95

Use case: Collect premium while willing to buy stock at strike price

Summary of Basic Option Positions

Position Market View Max Profit Max Loss Breakeven
Long Call Bullish Unlimited Premium \(K +\) Premium
Long Put Bearish \(K -\) Premium Premium \(K -\) Premium
Short Call Bearish Premium Unlimited \(K +\) Premium
Short Put Bullish Premium \(K -\) Premium \(K -\) Premium

These four positions are the building blocks for all option strategies!

Comparing Long vs. Short Positions

Long Positions (Buyers)

  • Pay premium upfront
  • Limited downside (premium)
  • Potentially large upside
  • Time decay works against you
  • Benefit from volatility

Short Positions (Sellers)

  • Receive premium upfront
  • Limited upside (premium)
  • Potentially large downside
  • Time decay works for you
  • Hurt by volatility

Next Steps in Options

Building on today’s foundation, we’ll explore:

  • Option portfolios: Protective puts, covered calls, spreads, straddles
  • Option pricing models: Black-Scholes formula, binomial trees
  • Greeks and sensitivity analysis: How option values change with inputs
  • Volatility: Historical vs. implied, volatility surfaces
  • Hedging strategies: Delta hedging, portfolio insurance