What Are Derivatives? Options, Puts, Calls, and Why They Matter

Chapter 2 is where Wilmott introduces the main character of the book: options. Also known as derivatives or contingent claims. No heavy math here yet, just definitions, jargon, and some clever strategies people use to make (or lose) money.

Options were first traded on an exchange on April 26, 1973, at the Chicago Board Options Exchange (CBOE). Initially there were just calls on 16 stocks. Puts did not even show up until 1977. Today there are over 50 exchanges worldwide trading these things.

The Basic Idea: Rights Without Obligations

Remember from Chapter 1 how forwards and futures obligate you to buy or sell? Options are different. An option gives you the right but not the obligation to trade at a pre-agreed price.

Call Options

A call option gives you the right to buy an asset at a specific price (the strike price) on a specific date (the expiry date).

Say Microsoft stock is at $24.50 and you buy a call with strike $25, expiring in one month. If Microsoft stays at $24.50, you let the option expire. Why would you pay $25 for something worth $24.50? But if Microsoft shoots up to $29, you exercise your option, pay $25, get a stock worth $29. That is $4 profit.

The payoff at expiry is:

$\max(S - E, 0)$

Where S is the stock price and E is the strike. You only exercise when the stock is above the strike.

Put Options

A put option is the opposite. It gives you the right to sell at the strike price. You buy a put when you think the stock will fall.

The payoff:

$\max(E - S, 0)$

You exercise when the stock is below the strike.

The Jargon You Need to Know

Options come with a lot of terminology. Here is the essential vocabulary:

• Premium: what you pay to buy the option
• Underlying: the asset the option is based on
• Strike/Exercise price: the price at which you can buy (call) or sell (put)
• Expiry: when the option dies
• Intrinsic value: what the option would be worth if exercised right now
• Time value: any extra value above intrinsic value (because there is still time for things to change)
• In the money: option has positive intrinsic value
• Out of the money: option has zero intrinsic value
• At the money: strike is close to current asset price
• Long: you own it
• Short: you sold it (and owe obligations)

Payoff Diagrams: Seeing Your Bets

One of the most useful tools in options is the payoff diagram. You plot the value of your option at expiry against the underlying price. Call payoff looks like a hockey stick going up to the right. Put payoff looks like a hockey stick going up to the left.

You can also draw profit diagrams that subtract the premium you paid. This shows you the break-even point, the asset price where you start making money.

Writing Options: The Other Side

For every option bought, someone has to write (sell) it. The writer receives the premium upfront but takes on the obligation. The writer of a call must deliver the stock if the buyer exercises. The writer of a put must buy the stock.

The risk is asymmetric. The buyer can only lose the premium. The writer can lose much, much more. That is why exchanges require margin deposits from option writers, both an initial margin and a maintenance margin that must be kept topped up.

Poor understanding of margin requirements has led to some spectacular financial disasters. Wilmott mentions Metallgesellschaft and Long Term Capital Management (LTCM) as examples.

What Determines an Option’s Price?

Before expiry, the option has a value that depends on several factors:

• Asset price: higher asset means higher call value, lower put value
• Time to expiry: generally more time means more value (more time for things to happen)
• Strike price: higher strike means lower call value, higher put value
• Interest rates: affects the time value of money in the payoff
• Dividends: affect the asset behavior during the option’s life
• Volatility: the big one. More randomness means more option value

Volatility is the most important and hardest to estimate. It measures how much the asset price bounces around. The more volatile the asset, the more valuable the option, because there is a bigger chance of ending up deep in the money.

European vs. American Options

European options can only be exercised at expiry. American options can be exercised any time before expiry. American options give you more rights, so they are worth at least as much as European ones.

Bermudan options are a hybrid, allowing exercise on specific dates. (Bermuda is between America and Europe. Get it?)

Put-Call Parity: A Fundamental Relationship

This is one of the most important results in Chapter 2. If you buy a call and sell a put with the same strike and expiry, your payoff at expiry is exactly:

$S(T) - E$

That is the same as owning the stock minus a fixed amount of cash. So the prices must satisfy:

$C - P = S(t) - E \cdot e^{-r(T-t)}$

This is put-call parity. It holds regardless of what model you use. It is model-independent. If it does not hold, there is a riskless arbitrage opportunity. Someone will exploit it, and prices will adjust.

Binary Options: All or Nothing

Binary (or digital) options have a discontinuous payoff. A binary call pays $1 if the asset is above the strike at expiry, and zero otherwise. No gradual increase, just on or off.

Binary call plus binary put with same strike always equals $1, since one of them always pays off. Simple and clean.

Option Strategies: Getting Creative

Here is where things get interesting. You can combine options to create custom payoff profiles:

Bull Spread

Buy a call at strike 100, sell a call at strike 120. You profit from a moderate rise but cap your upside at 20. Cheaper than buying a single call outright.

Bear Spread

Buy a put at strike 120, sell a put at strike 100. You profit from a moderate decline.

Straddle

Buy a call AND a put at the same strike. The payoff is V-shaped. You win if the stock moves a lot in either direction, and lose if it stays flat. Great for when you expect big news but do not know which way it will push the price.

Strangle

Like a straddle but with different strikes. Even cheaper, but the stock needs to move more before you start making money.

Butterfly Spread

Three different strikes, buying the outer two and selling two at the middle. Small payoff if the asset stays near the middle strike. You bet on the stock not moving much.

Risk Reversal

Long a call above the spot, short a put below the spot. The value is usually small and is linked to market expectations about volatility behavior.

Calendar Spread

Options with different expiry dates. Useful when you have a view on both the timing and direction of a move.

Leverage: The Double-Edged Sword

One of the main reasons people trade options is gearing (or leverage). Wilmott gives a nice example: stock at $666, call option at $39 with strike $680. If the stock goes to $730, buying the stock gives you a 9.6% return. Buying the call option gives you 28.2% return. Much more bang for your buck.

But the downside is just as amplified. If the stock stays at $666, you keep your stock but your option is a total loss. 100% gone.

The Takeaway

Chapter 2 is really a vocabulary chapter. You learn what calls and puts are, how payoff diagrams work, what the key terms mean, and how traders combine simple options into complex strategies. No pricing formulas yet, just the building blocks.

The two most important things to take away: first, put-call parity gives us a model-independent relationship between calls and puts. Second, volatility is the most important and hardest-to-pin-down factor in option pricing. Both ideas will be central to everything that follows.

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