Fundamentals of Option Contracts and No-Arbitrage Pricing

Options are powerful because they give you a right without an obligation. In this chapter we’ll explain what options are, how their payoffs work, and why — even without a complex pricing model — we can already set firm limits on what they should be worth.

The Big Picture#

Before we build any fancy models, we need a common understanding and a few firm rules. This chapter answers two core questions: What exactly is an option? and What can we say about its price using nothing more than common sense and the idea that free money doesn’t exist? The answer is surprising — just from the contract’s structure and the rule that markets don’t allow risk‑free profits, we can find exact relationships (like put‑call parity) and strict upper and lower limits. Every pricing model you’ll meet later must follow these basics.

What Is an Option?#

An option is a contract between two people — a buyer (the holder) and a seller (the writer). The holder pays a price, called the premium, and in return gets a right, but never an obligation. The writer collects the premium and must do whatever the holder decides, if the holder uses the right.

There are two basic types:

• A call option gives the holder the right to buy an underlying asset at an agreed strike price on or before a certain expiration date.
• A put option gives the holder the right to sell the underlying asset at the strike price on or before expiration.

The underlying asset could be a stock, an index, a commodity, or a currency — the logic is the same. Because the holder can simply walk away if using the option would lose money, the option’s payoff at expiration is never negative.

Holder: The party who buys the option, pays the premium, and owns the right to exercise.
Writer: The party who sells (writes) the option, receives the premium, and must fulfil the contract if the holder exercises.

Think of a call option like a coupon that lets you buy a popular item at a fixed price, even if the market price goes up. You pay a small fee for the coupon. If the item’s price shoots up, you use the coupon and save money. If the price falls, you throw the coupon away and buy at the cheaper market price. The store that sold you the coupon is the writer — it took your fee and must sell at the agreed price if you choose to use it.

Payoff at Expiration#

Let be the price of the underlying asset at expiration. The holder will exercise only when it is profitable to do so.

For a call, the payoff is

If is above , the holder can buy at and immediately sell at , keeping the difference. If , the option is worthless.

For a put, the payoff is

The put is valuable when the asset price falls below the strike; the holder can sell the asset for even though the market price is only .

These payoff functions are the basic building blocks of every option strategy. Notice they are uneven — the holder’s loss is limited to the premium paid, while the possible gain can be large. The writer’s position is the opposite: limited profit (the premium) but possibly large losses.

📝 Section Recap: An option is a right without obligation — a call lets you buy, a put lets you sell. The payoff at expiration is for a call and for a put. The most the holder can lose is the premium.

Exercise Styles: European, American, and Bermudan#

When can the holder actually use the right? That depends on the exercise style written into the contract.

• European options can be exercised only on the expiration date itself — not a day earlier.
• American options can be exercised at any time up to and including the expiration date.
• Bermudan options are a mix: they can be exercised on a specific list of dates before expiration (for example, every quarter‑end).

Most exchange‑traded stock options are American, while many index options (like those on the S&P 500) are European. The extra freedom of American exercise means an American option is always worth at least as much as a European option that is the same in every other way, because you could simply hold it to expiration and treat it as European. In practice, exercising early is not always smart — for a call on a stock that pays no dividends, you would never exercise early because you’d lose the time value of money on the strike payment and give up the remaining protection value of the option. But the right to exercise early still has some value, especially for puts or when dividends are involved.

Think of a European option like a concert ticket that’s only valid on one specific night. An American option is like a flexible ticket that lets you attend any performance during a season — clearly the flexible ticket is worth at least as much. A Bermudan ticket lets you pick from, say, the first Saturday of each month.

📝 Section Recap: Exercise style tells you when the holder can act. European = only at expiration, American = any time until expiration, Bermudan = on specific dates. More freedom never hurts, so an American option is worth at least as much as its European twin.

No‑Arbitrage Bounds on Option Values#

We don’t yet have a full pricing formula, but we can already put limits on what a fair option price must be. The key is the no‑arbitrage principle: you cannot make a guaranteed profit without using your own money. If two strategies always produce the same future payoff, they must cost the same today. If an option were priced outside certain limits, a smart trader could lock in a sure profit — and in healthy markets, that opportunity would disappear quickly.

For this section we’ll assume a European option on a stock that pays no dividends, with today’s price , strike , time left until expiration , and a steady risk‑free interest rate (with continuous compounding). The present value of received at time is .

Bounds for a European Call#

1. Upper bound: A call can never be worth more than the stock itself. The right to buy a share cannot be more valuable than simply owning the share. So

2. Lower bound: The call must be worth at least its discounted “immediate exercise” value. Suppose a European call is cheaper than . Then you could do this:

   • Buy the call for .
   • Short one share (borrow and sell it, receiving ).
   • Put the leftover cash into a risk‑free bank account.

   At expiration, you’ll have in the bank. If , you exercise the call, buy the share for , and return it to close the short. Your profit is , which is positive because . If , you let the call expire, buy the share in the market for to return it, and your profit is . Since , this is at least , again positive. So you’d make a risk‑free profit. To avoid this, we must have

   Also because the payoff is never negative. So the full lower bound is

   Combining with the upper bound,

Bounds for a European Put#

1. Upper bound: A put can never be worth more than the present value of the strike, because the maximum payoff is (when the stock becomes worthless). So

2. Lower bound: Similar reasoning (or put‑call parity, which we’ll see next) gives

These limits are not just theory — they are quick checks. If you ever see a call price above the stock price or below its discounted intrinsic value, there’s a chance for risk‑free profit.

📝 Section Recap: No‑arbitrage forces option prices to stay inside definite limits. For a European call, must be between and ; for a put, between and . These limits rely solely on the idea that risk‑free profits cannot last.

Put‑Call Parity for European Options#

One of the neatest relationships in finance connects the prices of European calls and puts with the same strike and expiration. It’s called put‑call parity, and it comes directly from no‑arbitrage.

Consider two portfolios you set up today:

• Portfolio A: Buy one European call (price ) and invest in risk‑free bonds.
• Portfolio B: Buy one European put (price ) and buy one share of the underlying stock (price ).

Now look at what each portfolio is worth at expiration.

• If : Portfolio A’s call is worth , plus the bond gives . Portfolio B’s put is worthless, so you just have the share worth .
• If : Portfolio A’s call is worthless, the bond gives . Portfolio B’s put is worth , plus the share worth gives .

Both portfolios always pay exactly . Since they have the same payoff, they must cost the same today. Otherwise you could buy the cheaper one, sell the expensive one, and pocket a risk‑free profit. Therefore,

This is put‑call parity. It lets you find the fair price of a put if you know the call price (or the other way around), and any price that breaks this equation would allow a risk‑free arbitrage. For example, if , you would sell the call, borrow , buy the put, and buy the stock — the trade would give you instant cash and zero future obligation.

Parity holds exactly for European options on stocks that pay no dividends. If the stock pays dividends, we simply subtract the present value of the dividends, , from the stock price: . The idea is that owning the stock gives you the dividends, while the fake position in Portfolio A does not, so we must adjust.

Put‑call parity is a foundation. It shows that calls and puts are not independent; they are linked by the underlying asset and the risk‑free bond. Many trading plans and pricing models rely on this relationship.

📝 Section Recap: Put‑call parity states for European options. It arises because a call plus a bond perfectly copies a put plus the stock. Any price that breaks parity would allow a risk‑free arbitrage.

Summary#

We’ve built the basic terms of options and learned that even without a full pricing model, the no‑arbitrage principle gives us strong limits. An option is a right without obligation — a call to buy, a put to sell. The exercise style decides when that right can be used. Simple arbitrage arguments set upper and lower limits on option values, and put‑call parity ties the prices of calls and puts together in an exact equation. These ideas are the base on which every later valuation method, from binomial trees to the Black–Scholes formula, is built. Once you understand them deeply, the more advanced models will feel like logical next steps rather than mystery.