No-Arbitrage Bounds On Option Prices

Pricing an option properly takes a model: a tree, a formula, an assumption about how the underlying moves. But before any of that, plain logic already pins the price into a narrow band. And unlike a model, the band is not a matter of opinion. If a quote falls outside it, you do not need a view on the stock, a volatility estimate, or a single Greek. You assemble a portfolio that costs nothing today and is guaranteed to be worth something later, and you collect.

You already do this

You have never opened a car engine, and you still know two things about the price of a used car. It cannot cost more than the identical car new, because you would just buy the new one. And it cannot cost less than the scrap value of its metal, because you would buy it and weigh it in. Two bounds, no engineering.

Options work the same way, and the bounds come from the same two moves: something cannot be worth more than a thing that always pays at least as much, and something cannot be worth less than a portfolio that always pays at most as much. Everything below is those two sentences applied carefully.

The vocabulary, one symbol at a time

Let $S$ be today's price of the underlying stock, $K$ the strike of the option, $T$ the time to expiry in years, and $r$ the risk-free interest rate. Write $C$ for the price of a European call and $P$ for a European put, both struck at $K$ and expiring at $T$.

One more piece of machinery. To compare money today with money at expiry, discount it:

$$\text{PV}(K) \;=\; K\,e^{-rT}.$$

In plain English: $\text{PV}(K)$ is the amount you would need to put in the bank today so that it grows into exactly $K$ by expiry. It is the honest present-day cost of the strike, and it shows up in every bound below.

The band

$$\max\!\left(S - K e^{-rT},\; 0\right) \;\le\; C \;\le\; S.$$

Read that as two separate sentences. Upper: a call can never cost more than the stock, because the right to buy a thing cannot beat owning the thing. Lower: a call must be worth at least the stock price minus the discounted strike, because a call plus cash in the bank always finishes at least as well as owning the stock outright.

Worked example 1: collecting on a broken lower bound

The stock trades at $100, the strike is $95, the rate is 4 percent, and expiry is six months away.

1. Discount the strike: $95 \times e^{-0.04 \times 0.5} = 95 \times 0.9802 = 93.12$.
2. So the lower bound is $100 - 93.12 = 6.88$. The call must cost at least $6.88.
3. Suppose a screen shows it at $6.00. Do this: buy the call for $6, short the stock for $100. You are now holding $94 in cash, with nothing paid in.
4. Park the $94 at 4 percent for six months: $94 \times e^{0.02} = 94 \times 1.0202 = 95.90$.

Now expiry arrives, and there are only two cases.

• Stock above $95. Exercise the call, pay $95 for the share, hand it back to close the short. You keep $95.90 - 95 = \mathbf{0.90}$.
• Stock at or below $95. Let the call die, buy the share in the market for less than $95, hand it back. You keep $95.90 - S_T$, which is more than $0.90.

You put in nothing and you walk away with at least ninety cents per share, whatever the stock does. That is what "the price cannot be below the bound" means, made physical.

The put version is the mirror: $\max\!\left(Ke^{-rT} - S,\; 0\right) \le P \le K e^{-rT}$. A put cannot be worth more than the discounted strike, because the very best it can ever do is pay $K$ at expiry when the stock goes to zero.

A free result: never exercise an American call early

If the stock pays no dividend, the lower bound settles a famous question by itself. The bound says $C \ge S - Ke^{-rT}$, and whenever $r > 0$ and time remains, $S - Ke^{-rT}$ is strictly bigger than $S - K$, which is what you get by exercising. So the call is always worth more sold than exercised. In plain English: exercising early throws away the interest on the strike you have not paid yet, plus all remaining protection if the stock falls back. Dividends are the one thing that can overturn this, see American Options and Early Exercise.

Bounds across strikes

Fix the expiry and slide the strike. Three more inequalities appear, and desks check all three on live quotes.

• Monotonicity. For $K_1 < K_2$, $\;C(K_1) \ge C(K_2)$. A cheaper right to buy is worth more.
• Slope. $\;0 \le C(K_1) - C(K_2) \le (K_2 - K_1)e^{-rT}$. In plain English: raising the strike by a dollar cannot make the call cheaper by more than a discounted dollar, because the call spread's payoff never exceeds the gap between the strikes.
• Convexity. For equally spaced $K_1 < K_2 < K_3$, $\;C(K_1) - 2C(K_2) + C(K_3) \ge 0$. In plain English: the price curve must bend upward as you look across strikes. That combination of options is a butterfly, and its payoff is never negative, so its price cannot be either.

Worked example 2: the butterfly that pays you to own it

Three calls on the same expiry, strikes $95, $100 and $105, quoted at $8.00, $5.00 and $1.50.

Test the convexity bound: $8.00 - 2(5.00) + 1.50 = -0.50$. Negative, so the bound is broken. Build the butterfly anyway: buy one 95 call, sell two 100 calls, buy one 105 call. You receive $0.50 up front.

Now check what you can ever owe at expiry.

Stock at expiry	95 call	two short 100 calls	105 call	total
$90	0	0	0	0
$100	5	0	0	+5
$110	15	−20	5	0
$130	35	−60	25	0

The worst you can do is zero, and in the middle you make five dollars. You were paid fifty cents to take a position that cannot lose. Fix the middle quote to $4.00 and the arithmetic becomes $8.00 - 8.00 + 1.50 = 1.50$, a sensible cost for a structure whose best case is $5, and the arbitrage vanishes.

Drag the sliders below to see why the bound has to hold. Pick "butterfly" and move the width: the payoff line is a tent that touches zero on both sides and never dips under it. A payoff that is never negative cannot have a negative price.

What this means in practice

These inequalities are not classroom decoration. A market maker's quoting engine runs every one of them over the live book on each update, because a violated bound in your own quotes means someone is about to lift you for free. Volatility-surface fitting is subject to the same constraints: a smile that produces a negative butterfly is not a slightly wrong surface, it is an impossible one, and it will hand nonsense prices to every exotic priced off it (Convexity Of Option Prices In Strike).

The more common experience, though, is a false alarm. Nine times out of ten a screen violation is not free money, it is a stale quote on an illiquid strike, a dividend you forgot, a borrow fee that makes the short leg cost real money, or a wide market where the mid you used is not tradable. The bound is exactly right; your inputs were not.

Key terms

• Upper bound — a call cannot exceed the stock; a put cannot exceed the discounted strike.
• Lower bound — $\max(S - Ke^{-rT}, 0)$ for calls, $\max(Ke^{-rT} - S, 0)$ for puts.
• Slope bound — a call spread's price cannot exceed the discounted gap between its strikes.
• Convexity bound — the butterfly price must be non-negative; prices bend upward in strike.
• Static arbitrage — a violation you exploit once and hold to expiry, with no rehedging.