Properties of Stock Options

Before you ever plug a number into Black–Scholes, arbitrage alone already pins an option's price into a narrow band. These are the model-free bounds: they hold no matter what stochastic process the stock follows, what volatility surface the desk is quoting, or whether Black–Scholes is even the right model at all. A trader who sees a quote outside this band doesn't need a pricing model to know something is wrong — they need a phone and a fast finger. Understanding these bounds, and how strike, maturity, volatility and dividends push a price around within them, is table stakes before you can sanity-check any quote on a screen — a skill this lesson builds on the market plumbing from how options actually trade.

The bounds, from pure no-arbitrage

Write c and p for the prices of European calls and puts on a non-dividend-paying stock, spot S_0, strike K, maturity T, and continuously compounded rate r.

A call can never be worth more than the stock itself: c \le S_0 — owning the right to buy is never more valuable than owning the asset outright.
A call can never be worth less than its discounted intrinsic value: c \ge \max(S_0 - Ke^{-rT},\ 0).
A put can never be worth more than the discounted strike: p \le Ke^{-rT} — the best a put can ever pay is K, received at T.
A put can never be worth less than its discounted intrinsic value: p \ge \max(Ke^{-rT} - S_0,\ 0).

Notice what's not in these formulas: no \sigma, no distributional assumption, nothing but S_0, K, T, r. They follow from the same no-arbitrage logic that proves put–call parity — build two portfolios, show one always weakly dominates the other, and any price that breaks the inequality is free money.

A non-dividend stock trades at S_0 = \$52. A six-month call struck at K = \$50 is quoted at c = \$1.50, with r = 5\%. The lower bound is

\max(S_0 - Ke^{-rT},\ 0) = \max\!\big(52 - 50e^{-0.05 \times 0.5},\ 0\big) = 52 - 48.765 = \$3.235.

The quote of \$1.50 sits well below \$3.235 — the call is mispriced too cheap. Exploit it: short the stock (receive \$52), buy the call (pay \$1.50), and invest the net \$50.50 at the risk-free rate, growing to 50.50\,e^{0.025} = \$51.775 by expiry. If the stock finishes above \$50, exercise the call to close the short for exactly \$50, banking 51.775 - 50 = \$1.775. If it finishes at or below \$50, let the call expire and buy back the stock in the market for at most \$50, banking at least the same \$1.775. Either way — a riskless profit, with zero net investment. That's what "the price must obey the bound" really means.

Watching the bounds move

The lower bound for a call, \max(S_0 - Ke^{-rT}, 0), rises toward the upper bound S_0 as the strike falls, as maturity shrinks, or as rates fall — each of those makes Ke^{-rT} smaller. Drag the sliders below to watch the actual price (whatever a model would say) get squeezed between a floor that always sits below it and a ceiling that always sits above.

What moves a price within the bounds

Four inputs, four qualitative effects — worth having cold before you ever open a pricing model:

Input rises…	Call price	Put price	Why
Strike K	falls	rises	a higher strike is a worse deal to buy at, a better one to sell at
Maturity T (American)	rises	rises	more time for the stock to move your way, and early exercise stays available
Volatility \sigma	rises	rises	both are one-sided bets — more spread in outcomes only helps the side you didn't take on
Dividends	falls	rises	a call holder misses dividends paid before exercise, so the stock (and the call's claim on it) is worth relatively less; a put holder benefits from the same effect

Volatility deserves the emphasis: a call and a put on the same strike and maturity both get more valuable when volatility rises. That surprises people who think of volatility as "risk" in the everyday sense of "bad." An option's payoff is bounded below at zero — you can never lose more than the premium — so extra spread in the outcome distribution only ever adds to the upside you're already entitled to keep, never subtracts from a floor you can't fall through.

Suppose it were — say a call traded at \$60 while the stock itself traded at \$52. Anyone in the market could simply sell the call and buy the stock instead, pocketing the \$8 difference today, while ending up holding an asset (the stock) that is worth at least as much as the liability they just wrote (the obligation to deliver a share for K if exercised) in every single future scenario. That's a riskless profit with money left over — so in a liquid, frictionless market this price simply cannot persist. The bound c \le S_0 isn't a rule someone imposed on the market; it's what "no free lunch" forces the market to obey.

These bounds are not a pricing model — they don't tell you where in the band the price sits, only that it can't leave it. A call quoted at exactly its lower bound is not "correctly priced"; it just isn't arbitrageable. Black–Scholes (or any other model) is still needed to pin down the actual number.
The lower bound uses Ke^{-rT}, the discounted strike — not K itself. Forgetting the discounting is the single most common slip when computing these by hand.

One loose end: what about early exercise?

Every bound above was proven for a European option — one exercisable only at T. An American option, exercisable at any moment up to T, must be worth at least as much as its European twin (that extra freedom can only help). But does the freedom to exercise early ever actually get used? The answer is more subtle than it looks — and turns pricing into a genuinely different kind of problem. That's the very next stop.