DERIVATIVE a security whose value is driven by something elseCALL the right to BUY at a set pricePUT the right to SELL at a set pricePREMIUM the price of the right itself, paid up frontSTRIKE PRICE the price locked into the contractSESSION 18 Derivatives and Options
An option is a coupon. Not a stock, not a piece of the company, just a ticket that lets you buy or sell a stock at a price you both agreed on today, whenever you decide to use it, or let it expire. That one idea is almost the whole session.
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Background
A Security With No Value of Its Own
Every investment you've studied so far, stocks, bonds, mutual funds, has a value that comes from itself: a share of a company, a loan to a government, a basket of other investments. A derivative is different. It has no value of its own at all, its entire worth is derived from something else, an underlying stock, a commodity, an interest rate, an index.
Investors reach for derivatives for three main reasons: to hedge a position (insurance against a loss elsewhere in the portfolio), to leverage a bet (control a lot of exposure with a small amount of cash), or to speculate outright on which way a price is headed. They can trade on an exchange or directly between two parties (over the counter). And they carry real risks of their own, market risk (the underlying moves the wrong way), liquidity risk (nobody wants to buy yours when you want to sell), and leverage risk (small moves in the underlying turn into large moves in the derivative).
This session zooms in on one specific type of derivative, options, the one you're most likely to encounter as an everyday investor.
Key Terms
Vocabulary
Tap a card to flip it and reveal the definition.
Derivative
A security whose price is driven entirely by an underlying asset, rather than having value in its own right.
Underlying Security
The actual investment an option is written on. An option on Apple stock has Apple shares as its underlying.
Premium
The up-front price the option buyer pays the option seller. It's paid no matter what happens next, and it's non-refundable.
Strike Price
The fixed price written into the contract, the price at which you can buy (call) or sell (put) the underlying, no matter where the market has moved.
Expiration
The date the option stops existing. Use the right by then, or it's gone, worthless, along with whatever premium was paid for it.
Moneyness
Whether exercising the option today would turn a profit. If yes, it's "in the money." If exercising would lose money, it's "out of the money."
Call Option
The right, but not the obligation, to buy the underlying at the strike price before expiration. Gains value as the underlying rises.
Put Option
The right, but not the obligation, to sell the underlying at the strike price before expiration. Gains value as the underlying falls.
American Option
Can be exercised any time up to and including expiration. Most individual stock options trade this way.
European Option
Can only be exercised on the expiration date itself. Many stock index options trade this way.
What You'll Be Able To Do
Objectives
Define a derivative and name its main types.
Define an option, and explain the difference between a call and a put.
Identify the parts of an options contract: underlying, premium, strike price, expiration, and moneyness.
Work through the payoff of a call and a put at expiration, by hand.
Explain, in plain terms, what goes into pricing an option, without needing the formula itself.
The Derivative Family
Four Kinds You'll Hear About
Options are one member of a bigger family. Tap each card to see what makes it different.
Options
The right, but not the obligation, to buy or sell at a set price. If it goes badly, you can just walk away and lose only what you paid for it. This session's focus.
Futures & Forwards
A binding agreement to buy or sell at a set price on a set future date, no walking away. Both sides are obligated to follow through.
Swaps
An agreement between two parties to trade one stream of cash flows for another, commonly a fixed interest rate for a floating one.
CFDs
Contracts For Difference: an agreement to exchange the difference in an asset's price between when the contract opens and closes, without ever owning the asset itself.
What's an Option?
A Right, Not a Requirement
An option is the right, but not the obligation, to buy or sell an investment at a specific price. Nobody can ever force you to use it. If using it would lose you money, you simply don't, and the most you're out is what you paid to hold the right in the first place.
Think of an option like a coupon you bought at the store: "good for one share of XYZ at $50, expires in 3 months." If XYZ is selling for $80 when the coupon's about to expire, that coupon just saved you $30 a share. If XYZ is selling for $40 instead, the coupon's worthless, you just don't use it, and buy at the regular $40 price if you still want the stock.
There are exactly two flavors:
Calls and puts form the basis of nearly every options strategy out there, whether the goal is hedging, income, or outright speculation.
One More Question First: When Can You Actually Use It?
Before going further, every option also comes stamped with a rule about when the right can be exercised. Despite the geographic-sounding names, this has nothing to do with where you live or where the option trades.
American Option
Can be exercised any time up to and including the expiration date. Most stock options traded in the U.S. are American-style. More flexible for the holder, which is part of why they tend to cost a bit more than an otherwise-identical European option.
European Option
Can only be exercised on the expiration date itself, not a day before. Many stock index options (like options on the S&P 500 index) are European-style. Less flexible, but also simpler to price, which is exactly the version the Black-Scholes formula coming up was originally built for.
Everything in the rest of this session, the worked examples, the payoff diagrams, the pricing formula, assumes the option is only being evaluated at expiration, so it applies cleanly to European-style options, and to American-style options that happen to be held until expiration anyway.
How Does a Call Option Work?
Betting the Price Goes Up
Worked Example
A stock is trading at $150. You buy a call option giving you the right to buy it for $170 a share, the strike price, sometime before the option expires.
Case 1: price rises to $200
Case 2: price stays flat at $150
Is it worth exercising?
Yes, in the money, $200 > $170
No, out of the money, $150 < $170
What you do
Exercise: pay $170, receive the share, sell it for $200
Let the option expire unused
Result
$200 − $170 = $30 profit per share
You lose the premium you paid, nothing else
That $30 is the profit on the shares themselves. The premium you paid up front for the right to do this comes out of that $30 too, so the real net profit is $30 minus whatever the premium cost.
How Does a Put Option Work?
Insurance for a Stock You Already Own
Worked Example
You already own shares trading at $150. Worried they might fall, you buy a put option giving you the right to sell them for $140 a share, the strike price, before it expires.
Case 1: price falls to $120
Case 2: price stays flat at $150
Is it worth exercising?
Yes, in the money, $140 > $120
No, out of the money, $140 < $150
What you do
Exercise: sell your shares for $140 instead of the market's $120
Let the option expire, keep holding your shares
Result
You avoided $140 − $120 = $20 of loss per share
You lose the premium you paid, nothing else
A put like this is often called a "protective put", it's less about making a profit and more about capping how much you can lose on shares you already hold, the way car insurance caps what a fender-bender costs you.
Seeing the Shape of the Bet
Payoff Diagrams: The "Hockey Stick"
Traders draw the profit or loss of an option at expiration as a simple line against the stock's price. Both shapes below use a clean example, strike price $50, premium $5, so the two charts line up and mirror each other.
Buying a Call
Buying a Put
Below the fold: max loss is always just the premium, whichever way the stock moves.
The buyer's side vs. the seller's side. Every one of these numbers describes the option buyer. The person on the other side of the trade, the option seller (or "writer"), has the mirror-image risk: capped profit (just the premium collected) and, especially for an uncovered call, exposure that can run just as unbounded as the short seller's from last session. Buying options caps your downside at the premium. Selling them uncovered does not. That distinction is worth its own session down the road.
Pricing Options
What Sets the Premium?
The premium isn't picked out of thin air. In 1973, economists Fischer Black and Myron Scholes (later joined by Robert Merton) published a formula, now called Black-Scholes, for pricing an option mathematically. It's still one of the most widely used tools in finance today, and Scholes and Merton later shared a Nobel Memorial Prize in Economic Sciences for the work.
You don't need to run the formula by hand, brokerages and options-trading platforms do that math for you automatically. What's worth knowing is which six ingredients go into it:
Current Price
Where the underlying is trading right now, relative to the strike price.
Time to Expiration
More time means more chances for the underlying to move in your favor, so more time generally means a higher premium.
Volatility
How much the underlying's price tends to swing around. Wilder swings make the option more likely to pay off big, so they raise the premium.
Strike Price
How far the strike sits from today's price. The closer an option already is to paying off, the more it costs.
Interest Rates
The prevailing risk-free interest rate feeds into the formula too, a smaller effect than the other four, but a real one.
Dividend Yield
A stock paying dividends is a little less attractive to hold through a call option, since call holders don't collect those dividends, so a higher dividend yield slightly lowers a call's premium and raises a put's.
The Formula Itself
Here it is, mostly so you can see it's not magic, just algebra with a bell-curve function mixed in. This version includes dividend yield, the extension Robert Merton added shortly after the original 1973 paper.
Where C is the call price, P is the put price, S is today's stock price, X is the strike price, T is time to expiration in years, r is the risk-free interest rate, q is the dividend yield, σ (sigma) is volatility, and N() is the cumulative standard normal distribution, roughly, "the probability a randomly moving value ends up below this point." Set q to 0 and this collapses back to the original 1973 Black-Scholes formula for a non-dividend-paying stock.
Try It Yourself
The Black-Scholes Simulator
Plug in your own numbers and watch the call and put prices update. Nudge one input at a time to build intuition for which ingredients push the premium up, and which push it down.
Set the Inputs
Call Price (C)
—
Put Price (P)
—
Practice
Try It Yourself
Q1
You buy a call option, strike $75, premium $3. The stock is at $80 today. At expiration, the stock is at $95. Is the option in the money? What's the payoff per share? What's the net profit after the premium?
In the money, $95 > $75. Payoff: $95 − $75 = $20 per share. Net profit after the $3 premium: $20 − $3 = $17 per share.
Q2
Same call option, strike $75, premium $3. This time the stock ends at $70. What happens?
Out of the money, $70 < $75, exercising would mean paying more than the stock is worth. You let it expire. Total loss: the $3 premium, nothing more.
Q3
You buy a put option, strike $65, premium $2. The stock is at $60 today. At expiration, the stock is at $50. Is the option in the money? What's the payoff per share? What's the net profit after the premium?
In the money, the $65 strike is above the $50 market price. Payoff: $65 − $50 = $15 per share. Net profit after the $2 premium: $15 − $2 = $13 per share.
Q4
Same put option, strike $65, premium $2. This time the stock ends at $70. What happens?
Out of the money, the $65 strike is below the $70 market price, selling at the strike would mean giving up more than the stock is worth. You let it expire. Total loss: the $2 premium, nothing more.
Why might an investor buy a put on a stock they already own, rather than simply selling the stock outright?
DiscussionSelling locks in the outcome immediately and gives up any further upside. A protective put keeps the shares (and any dividends, voting rights, or further gains if the stock rebounds) while still putting a floor under the loss, for the cost of the premium. It's the closer-to-insurance version of the same worry.
Compare buying a call option to short selling from last session. What's different about the worst-case outcome?
DiscussionA short seller's worst case is unlimited, the stock can keep climbing forever, and so can the loss. An option buyer's worst case is always capped at the premium paid, full stop, no matter how wrong the bet turns out to be. That's a real structural advantage of buying options over short selling, though it comes at the cost of the premium itself, and of the option expiring worthless far more often than a short position gets fully wiped out.
The Bottom Line
Putting It Together
Takeaway 1
An option is a right, not an obligation. A call is the right to buy, and profits when the underlying rises. A put is the right to sell, and profits when the underlying falls.
Takeaway 2
Options are sophisticated instruments with real math behind their pricing, and they deserve careful study before real money is involved. Used well, though, they're a genuinely useful tool, for hedging existing positions, for generating income, or for speculating with a strictly limited downside.
Application: Pick a stock your team currently holds, or is watching. If you bought a call option on it today, what strike price would you choose, and why? If you bought a protective put on it instead, what would that strike price be, and why?
Enrichment: Look up the actual options chain for a real stock (most brokerage sites and Yahoo Finance publish these for free). Find a call and a put expiring in the same month. What's the premium for each? How does the premium change as you look at strikes further from today's price, and why does that match what you learned about pricing above?