FRM Exam Part I · Properties of Options

Upper and Lower Bounds on Option Prices

Updated 11 October 2026 · Fact-checked

Option bounds are the highest and lowest prices an option can have without creating a risk-free profit. For a non-dividend stock, a European call is at least max(S0 − K·e^(−rT), 0), a European put at least max(K·e^(−rT) − S0, 0). Calls cannot exceed S0. European puts cannot exceed K·e^(−rT).

Understand Upper and Lower Bounds on Option Prices

An option price is not free to be any number. If it falls outside certain limits, you can lock in a risk-free profit by trading the option against the stock and borrowing or lending. These limits are the bounds. They come from no-arbitrage logic only. You need no model of how the stock moves.

Start with the upper bounds. A call gives you the right to buy one share. It can never be worth more than the share itself, so c ≤ S0 and C ≤ S0. A put gives you the right to sell at K. The best case is that the stock falls to zero and you receive K. An American put can be exercised at once, so P ≤ K. A European put pays at most K at time T, so its value today is at most K·e^(−rT).

Now the lower bounds. Compare a call with a portfolio: buy the call, and invest K·e^(−rT) in cash. At T the portfolio is worth max(ST − K, 0) + K = max(ST, K). That is at least ST, which is what one share is worth. So call + cash ≥ share today, which gives c ≥ S0 − K·e^(−rT). A price can never be negative, so c ≥ max(S0 − K·e^(−rT), 0). The put works the same way: a put plus a share is worth max(K, ST) at T, which is at least K, so p + S0 ≥ K·e^(−rT).

Dividends change the stock side. If the stock pays dividends with a present value D before expiry, the share you hold gives up those dividends, so replace S0 with S0 − D. The lower bounds become c ≥ S0 − D − K·e^(−rT) and p ≥ K·e^(−rT) + D − S0, each floored at zero.

American options can be exercised early, so they must be worth at least their immediate exercise value. For a non-dividend stock, an American call is never exercised early, so C = c. An American put can be optimal to exercise early, so P ≥ max(K − S0, 0) and P is at least p. Put-call parity holds only for European options, so the American bracket does not come from parity directly. It comes from no-arbitrage arguments combined with parity for the European options (with C = c on a non-dividend stock): S0 − K ≤ C − P ≤ S0 − K·e^(−rT) for non-dividend stocks.

Key formulas to remember

Upper bound, calls: c ≤ S0 and C ≤ S0; Applies to European (c) and American (C) calls. Holds with or without dividends.
Upper bound, puts: p ≤ K·e^(−rT) and P ≤ K; European put is capped by the present value of the strike. American put is capped by K because it can be exercised now.
Lower bound, European call (no dividends): c ≥ max(S0 − K·e^(−rT), 0); Derived by comparing call plus cash with one share.
Lower bound, European put (no dividends): p ≥ max(K·e^(−rT) − S0, 0); Derived by comparing put plus share with cash K·e^(−rT).
Lower bounds with dividends: c ≥ max(S0 − D − K·e^(−rT), 0); p ≥ max(K·e^(−rT) + D − S0, 0); D is the present value of dividends during the option life.
American options: C ≥ max(S0 − K, 0); P ≥ max(K − S0, 0); Intrinsic value is a floor because early exercise is allowed.
American call, no dividends: C = c; Early exercise is never optimal, so the American call equals the European call.
Put-call parity bracket for American options (no dividends): S0 − K ≤ C − P ≤ S0 − K·e^(−rT); With dividends the bounds become S0 − D − K ≤ C − P ≤ S0 − K·e^(−rT).

How to solve Upper and Lower Bounds on Option Prices questions

Use this method for any bound or arbitrage question.

1Identify the option: call or put, European or American, and whether the stock pays dividends before expiry.
2Compute the present value of the strike, K·e^(−rT), using the continuously compounded rate unless the question says otherwise. If dividends exist, compute D by discounting each one.
3Write the correct lower bound from the formula sheet, using S0 − D in place of S0 when there are dividends. Floor it at zero.
4Compare the quoted price with the bound. If the price is above the bound (and below the upper bound), no arbitrage exists.
5If the price is below the lower bound, buy the underpriced option and take the offsetting position: for a call, short the share and lend K·e^(−rT); for a put, buy the share and borrow K·e^(−rT). If the stock pays dividends, the strategy must also handle D: for a call, lend K·e^(−rT) + D, because as the short seller you owe the dividends; for a put, borrow K·e^(−rT) + D, because the dividends the share pays repay the D part of the loan.
6Compute the profit as the gap between the bound and the price. This is the profit today in present-value terms.
7Check the answer: the profit should be positive and equal to bound minus price.

Quickest way: Bound check in four lines

When to use it: Use for multiple-choice questions that ask for the minimum price or whether an arbitrage exists.

Compute PV(K) = K·e^(−rT), and subtract PV of dividends from S0 if any.
For a call, take S0 − PV(K); for a put, take PV(K) − S0. If the result is negative, the bound is zero.
If the quoted price is lower, arbitrage exists and the profit today is bound minus price.
For American options, also compare with intrinsic value S0 − K or K − S0. Take the larger bound.

Common mistakes in Upper and Lower Bounds on Option Prices

Using K instead of K·e^(−rT) in the European lower bound.
The American bound uses plain K, so the two get mixed up.
Fix: European means the strike is paid at T, so discount it. Use plain K only for American intrinsic value.
Forgetting to floor the bound at zero.
Students stop after computing S0 − K·e^(−rT), which can be negative for out-of-the-money options.
Fix: Always write max(…, 0). A negative bound means the minimum price is zero.
Ignoring dividends or subtracting them undiscounted.
Dividends look like a small detail, and the present value step is skipped.
Fix: Discount each dividend from its payment date to today at r, and subtract the total D from S0 before applying the formula.
Saying an American call on a non-dividend stock may be exercised early.
Students assume more rights always mean earlier use.
Fix: Without dividends, the call is worth more alive than exercised, because of time value and interest on K. So C = c.
Building the arbitrage with the wrong signs, or leaving out the dividends.
The call and put cases are memorised as separate recipes, and the dividend leg is forgotten when D is not zero.
Fix: Buy the cheap side, sell the expensive side. For a call below its bound: buy call, short share, lend PV(K). For a put below its bound: buy put, buy share, borrow PV(K). With dividends, add D to the loan or lending amount: lend PV(K) + D in the call case, borrow PV(K) + D in the put case. The dividends the share pays cover the D part of the put loan. The profit is then the bound minus the price.

Worked examples

Example 1

A non-dividend-paying stock trades at $40. A 6-month European call with strike $38 trades at $2.00. The continuously compounded risk-free rate is 5% per year. Find the lower bound for the call and state whether an arbitrage exists. If so, give the profit.

Show the solution

PV of strike = 38 × e^(−0.05 × 0.5) = 38 × e^(−0.025).
e^(−0.025) = 0.975310, so PV(K) = 38 × 0.975310 = 37.0618.
Lower bound = max(40 − 37.0618, 0) = 2.9382.
The quoted price $2.00 is below $2.9382, so the call is underpriced and an arbitrage exists.
Strategy: buy the call for $2.00, short the share for $40, and invest $37.0618 at the risk-free rate. Net cash today = 40 − 2.00 − 37.0618 = $0.9382.
At expiry, the $37.0618 grows to $38. If ST > 38, exercise the call, pay $38 and return the share. If ST ≤ 38, the call is not exercised. You buy the share in the market at ST and return it, and you keep 38 − ST as extra profit. Either way there is no loss.

Answer: The lower bound is $2.94 (2.9382). The call is underpriced, and the arbitrage profit today is about $0.94, with no risk.

Example 2

A stock trades at €50 and will pay a €1 dividend in 3 months. A 6-month European put has strike €52 and is priced at €2.20. The continuously compounded risk-free rate is 4% per year. Find the lower bound for the put and decide whether an arbitrage exists.

Show the solution

PV of dividend = 1 × e^(−0.04 × 0.25) = e^(−0.01) = 0.990050, so D = €0.9900.
PV of strike = 52 × e^(−0.04 × 0.5) = 52 × e^(−0.02).
e^(−0.02) = 0.980199, so PV(K) = 52 × 0.980199 = 50.9704.
Lower bound = max(PV(K) + D − S0, 0) = 50.9704 + 0.9900 − 50 = 1.9604.
The quoted price €2.20 is above €1.96, so the put does not violate its lower bound.
Upper bound: a European put cannot exceed K·e^(−rT) = €50.97. The price €2.20 is below this, so the upper bound is not violated either.

Answer: The lower bound is about €1.96. The put at €2.20 is above the lower bound and below the upper bound of €50.97, so no bound is violated and no arbitrage exists.

Exam tips

Read the first line of the question for three flags: call or put, European or American, dividends or not. These choose the formula.
Use continuous compounding for the discount factor unless the question gives a different convention. Keep four to five decimals in e^(−rT) before multiplying.
If the question asks for the minimum value of an American option, compare intrinsic value with the European bound and choose the larger.
Questions often ask for the arbitrage profit. The answer is the bound minus the quoted price, in today's money.
On a calculator, store e^(−rT) once and reuse it for both the call and put bounds.

Practice questions from Properties of Options

Upper and Lower Bounds on Option Prices: frequently asked questions

What is the lower bound for a European call option?

For a stock with no dividends, it is max(S0 − K·e^(−rT), 0). If dividends have present value D, it becomes max(S0 − D − K·e^(−rT), 0). The price cannot fall below this without allowing arbitrage.

How do I derive the lower bound of a European put?

Compare two portfolios: a put plus one share, and cash of K·e^(−rT). At expiry the first is worth max(K, ST), which is at least K, the value of the cash. So p + S0 ≥ K·e^(−rT), giving p ≥ K·e^(−rT) − S0. Floor it at zero.

Why is an American call on a non-dividend stock never exercised early?

Exercising early gives S0 − K now. Holding the call is worth at least S0 − K·e^(−rT), which is higher, and it also keeps the option's insurance value. So the American call equals the European call.

What arbitrage do I use if a call is below its lower bound?

Buy the call, short the share, and lend K·e^(−rT) at the risk-free rate. You keep the difference between the bound and the call price today. At expiry, the loan repays the strike if you need to buy back the share.