Introduction to Options
A derivative security is a financial instruments whose value is derived from an underlying asset, index, or rate.
Exchange-Traded
Over-the-Counter (OTC)
For the remainder of this lecture, we’ll focus on options - the most versatile derivatives.
The buyer pays the seller a premium upfront for this right.
Options are particularly important for several reasons:
Despite the names, both trade worldwide!
American Options
European Options
Option Buyers (Long)
Option Sellers (Short)
Let’s explore real option prices:
Experimenting with the market data reveals consistent patterns:
Call Options:
Put Options:
Time to expiration matters:
This pattern holds for both calls and puts,
Intrinsic value = What the option is worth if exercised immediately (or at expiration)
Call Option:
\[\text{Intrinsic Value} = \max(S - K, 0)\]
Put Option:
\[\text{Intrinsic Value} = \max(K - S, 0)\]
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) | \(S < K\) | \(S > K\) |
Time value = Option price - Intrinsic value
American options are always worth at least their intrinsic value (otherwise arbitrage!)
Options differ fundamentally from other financial instruments:
These properties make options powerful but complex!
Most options are traded on organized exchanges:
Exchanges determine which options are available:
Unlike stocks, options have no pre-existing supply:
How open interest changes:
Example:
Understanding trading patterns:
Concentration Near Current Price:
Popularity of OTM Options:
Open interest evolves over an option’s life:
Initial Growth:
Decline Phase:
Final Settlement:
Two important ways to visualize options:
Payoff Diagram
Profit Diagram
Buying a call option (bullish strategy):
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
Buying a put option (bearish strategy):
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
Writing (selling) a call option:
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
Writing (selling) a put option:
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
| 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!
Long Positions (Buyers)
Short Positions (Sellers)
Options can be combined with each other and with the underlying asset to create sophisticated strategies:
Key technique: Portfolio payoff = sum of individual payoffs
Long asset + Long put = Portfolio insurance
Function: Portfolio insurance - downside protection while keeping upside
Minimum Value: Put strike ($95) - portfolio can never fall below this
Maximum Value: Unlimited (follows underlying asset upward)
Cost: Put premium reduces overall returns
Use case: Protecting existing stock holdings from market crashes
Long asset + Short call = Income generation with capped upside
Function: Generate premium income from existing stock holdings
Benefit: Receive call premium upfront
Cost: Cap upside potential at strike price
Trade-off: Profitable if underlying stays flat or rises moderately; regret if underlying rises substantially
Long asset + Long put + Short call = Protected range
Function: Downside protection funded by capping upside
Range: Portfolio value locked between put and call strikes
Zero-cost collar: Call premium ≈ Put premium
Use case: Risk management when you want to stay invested but limit both gains and losses
Long call (low strike) + Short call (high strike) = Bullish bet with limited risk
Function: Low-cost bullish bet with limited upside
Cost: Net premium paid (lower than buying call alone)
Profit potential: Limited to strike difference minus net premium
Use case: Moderate bullish view, want to reduce cost vs. buying call outright
Long put (high strike) + Short put (low strike) = Bearish bet with limited risk
Function: Low-cost bearish bet or downside hedge
Cost: Net premium paid (lower than buying put alone)
Profit potential: Limited to strike difference minus net premium
Use case: Moderate bearish view or portfolio insurance at reduced cost
Long call + Long put (same strike) = Bet on volatility
Function: Profit from large price movements in either direction
Market view: High volatility expected, but direction uncertain
Cost: Both call and put premiums (can be expensive)
Breakeven: Two points - one above and one below strike
Use case: Before major announcements or events with uncertain outcomes
Long 2 calls (low & high strike) + Short 2 calls (middle strike) = Bet on low volatility
Function: Profit if price stays near middle strike
Market view: Low volatility - price won’t move much
Cost: Net premium paid (relatively low)
Maximum profit: At middle strike
Use case: When you expect prices to remain stable
Experiment with your own option combinations:
For European options, a fundamental no-arbitrage relationship:
\[\text{Cash} + \text{Call} = \text{Put} + \text{Underlying}\]
More precisely:
\[\mathrm{e}^{-rT}K + C = P + S\]
where \(\mathrm{e}^{-rT}K\) = present value of strike price
Key insight: Two portfolios with identical payoffs must have identical prices
Compare payoffs at expiration:
Portfolio A: Cash + Call
If \(S_T > K\):
If \(S_T \leq K\):
Portfolio B: Put + Underlying
If \(S_T > K\):
If \(S_T \leq K\):
Identical payoffs → Identical prices!
1. Option Pricing:
2. Synthetic Instruments:
3. Arbitrage Detection:
Important restrictions:
For American options: relationship becomes an inequality
For dividend-paying assets: subtract PV of dividends from \(S\)
Should you exercise an American option early?
Option value = Intrinsic value + Time value
Early exercise destroys time value!
Only exercise early when time value = 0
For calls on non-dividend-paying assets:
Reason #1: Flexibility is valuable - “Options are better alive than dead” - Keeping your options open has value
Reason #2: Time value of money - Early exercise means paying strike now - Foregone interest on that cash
Mathematical proof: From put-call parity, \(C > S - K\) always
*Exception: May be optimal just before large dividends
For puts, time value of money works in reverse:
Why early exercise can be optimal:
Optimal when: Put is deep ITM + interest rates high + low volatility
Mathematical insight: \(P = K - S\) is possible (time value = 0)
From European put-call parity: \(C + \mathrm{e}^{-rT}K = P + S\)
For calls: Since \(P \geq 0\), \[C \geq S - \mathrm{e}^{-rT}K > S - K\] Call value strictly exceeds intrinsic value → don’t exercise early
For puts: Since \(C \geq 0\), \[P \geq \mathrm{e}^{-rT}K - S\] But intrinsic value is \(K - S\), which could be larger! \[K - S > \mathrm{e}^{-rT}K - S\] Put time value can shrink to zero → early exercise possible
Options are versatile - building blocks for complex strategies
Payoff diagrams - visualize by summing individual positions
Market patterns - OTM options popular, OI follows predictable life cycle
Put-call parity - fundamental no-arbitrage relationship
Early exercise - rarely optimal for calls, sometimes optimal for puts
Portfolio construction - combine options for specific risk/reward profiles