CFA Level I Exam · Pricing and Valuation of Options

Factors Affecting Option Value and Price Bounds

Updated 7 October 2026 · Fact-checked

Option value depends on the underlying price, exercise price, time to expiration, volatility, the risk-free rate and carry (benefits or costs of holding the underlying). Price bounds set the minimum and maximum a no-arbitrage option can trade at. To solve questions, identify call or put, European or American, then apply the direction rules and bound formulas.

Understand Factors Affecting Option Value and Bounds

An option's price has two parts: exercise value (what you would get if you exercised now) and time value (the extra amount buyers pay for the chance of a better outcome). Time value is the price minus exercise value. Exercise value is never negative, because the holder can choose not to exercise.

Six factors drive the price. A higher underlying price raises call values and lowers put values. A higher exercise price lowers call values and raises put values. Higher volatility raises both calls and puts. The holder gains from large moves in the favourable direction, while the loss is limited to the premium paid, because the payoff cannot fall below zero.

Time to expiration: for American options, more time never lowers value, for both calls and puts. For European options, more time usually raises value, but a deep in-the-money European put (or a call on an asset with large income) can be worth less with more time. Risk-free rate: a higher rate raises call values and lowers put values, because paying the exercise price later is worth more to a call buyer. Carry: income on the underlying (dividends, coupons) lowers call values and raises put values. Costs of holding the underlying (storage) do the opposite.

Bounds come from no-arbitrage. A call can never be worth more than the underlying, and a put can never be worth more than the present value of the exercise price (European) or the exercise price itself (American). For a European option, the lower bound is the greater of zero and the underlying price less the PV of the exercise price (call), or the greater of zero and the PV of the exercise price less the underlying price (put).

American options can be exercised at any time, so they are worth at least as much as otherwise identical European options. An American call on an asset with no income and no other benefits is never exercised early, so it is worth the same as the European call. An American put can be worth exercising early, especially when deep in the money.

Key formulas to remember

Exercise value of a call: max(0, S − X); S is the underlying price and X the exercise price.
Exercise value of a put: max(0, X − S); Never negative.
Time value: Option price − Exercise value; Time value is non-negative for American options. A European option can trade below its current exercise value, which gives it negative time value.
European call bounds: max(0, S₀ − X ÷ (1 + r)^T) ≤ c ≤ S₀; Lower bound for an underlying with no income. With income, subtract the present value of the benefits from S₀.
European put bounds: max(0, X ÷ (1 + r)^T − S₀) ≤ p ≤ X ÷ (1 + r)^T; With income, replace S₀ by S₀ − PV of benefits, so the lower bound is max(0, X ÷ (1 + r)^T − S₀ + PV of benefits).
American call bounds: max(0, S₀ − X) ≤ C ≤ S₀ and C ≥ c; Lower bound is at least the exercise value, and C is at least the European call value.
American put bounds: max(0, X − S₀) ≤ P ≤ X and P ≥ p; The upper bound is X because early exercise is possible.
Direction of effects (call / put): S↑: call ↑, put ↓ | X↑: call ↓, put ↑ | Volatility↑: both ↑ | Rate↑: call ↑, put ↓ | Income↑: call ↓, put ↑; Time to expiration: American both ↑ or unchanged; European usually ↑ but not always.

How to solve Factors Affecting Option Value and Bounds questions

Use this routine for any question on factors, direction of change or bounds.

1Identify the option type (call or put) and style (European or American).
2Note whether the underlying pays income or has holding costs, and whether rates or time are given.
3For a direction question, apply the factor table: start with S, X, volatility, rate, income, time.
4For a bound question, write the lower and upper bound for that option type and style.
5Compute PV of X as X ÷ (1 + r)^T. Then adjust S₀ to S₀ − PV of benefits in both lower bounds. This lowers the call bound and raises the put bound.
6Take the maximum of zero and your lower-bound expression. Do not let it go negative.
7Check whether the stated price breaks a bound. If yes, arbitrage exists; if not, the price is feasible.
8Pick the option that fits and eliminate the two others.

Quickest way: Direction table and bound check

When to use it: Use it for qualitative questions and for any question with an option price and a stated bound.

Memorise: calls like higher S, low X, high rate, low income; puts like low S, high X, low rate, high income.
Volatility helps both. Time helps American options.
For bounds, remember the upper limits: call ≤ S; put ≤ X (American) or PV(X) (European).
For lower bounds, compute max(0, ...) with PV of X.
Compare the quoted price with the bounds; eliminate options that violate them.

Common mistakes in Factors Affecting Option Value and Bounds

Saying higher volatility lowers put value because puts profit from price falls.
Students link volatility with direction instead of range.
Fix: Volatility widens the range of outcomes. The holder keeps the upside and the loss is limited to the premium, so both calls and puts gain.
Using X instead of the present value of X in European lower bounds.
The American bound uses S − X, which is easy to confuse with the European one.
Fix: For European options, discount X at the risk-free rate over the life: X ÷ (1 + r)^T.
Letting a lower bound go negative.
Students forget the max(0, ...) part.
Fix: An option cannot have a negative value, so the lower bound is zero when the expression is negative.
Saying more time always increases a European option's value.
It is true for American options, so students overgeneralise.
Fix: For European options, extra time can lower a deep in-the-money put or a call on an asset with large income. Only say 'usually higher'.
Treating a higher interest rate as bad for calls.
Students think higher rates hurt asset prices.
Fix: Holding the underlying price fixed, a higher rate cuts the present value of X. That makes calls more valuable and puts less valuable.
Ignoring income on the underlying in bound questions.
The no-income formulas are learnt first.
Fix: If the underlying pays dividends or coupons before expiry, subtract their present value from S₀ in the call lower bound and add it in the put lower bound.

Worked examples

Example 1

A European call on a non-dividend stock has an exercise price of $50 and one year to expiry. The stock trades at $54 and the risk-free rate is 4% annually. What is the minimum price the call could sell for without arbitrage? A) $4.00 B) $5.92 C) $54.00

Show the solution

Lower bound = max(0, S₀ − X ÷ (1 + r)^T).
PV of X = 50 ÷ 1.04 = 48.077.
S₀ − PV of X = 54 − 48.077 = 5.923.
The maximum of 0 and 5.923 is about 5.92.
A) $4.00 is below the bound, so it would allow arbitrage. C) $54.00 is the upper bound, not the minimum.

Answer: B

Example 2

All else equal, which change would increase the value of a European put option on a non-dividend stock? A) A decrease in volatility B) An increase in the risk-free rate C) A decrease in the underlying price

Show the solution

Volatility: lower volatility reduces the value of any option, so A is wrong.
Risk-free rate: a higher rate lowers the present value of the exercise price received, so it reduces put value. B is wrong.
Underlying price: a lower S raises the exercise value of a put, so it increases put value.

Answer: C

Exam tips

Learn the direction table cold; many questions are pure direction tests and take seconds.
For bounds, write max(0, ...) first and then fill in. It prevents negative-bound errors.
Watch for 'European' versus 'American'. The upper bound for a put changes between PV(X) and X.
If an American call has no income on the underlying, early exercise is never optimal and its value equals the European call.
On the BA II Plus (yˣ key) or HP 12C (yˣ key), compute (1 + r)^T, then divide X by it.

Practice questions from Pricing and Valuation of Options

Factors Affecting Option Value and Bounds in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Factors Affecting Option Value and Bounds: frequently asked questions

How does volatility affect call and put prices?

Higher volatility increases both call and put values. The holder benefits from large moves in the favourable direction, while the loss is limited to the premium paid.

What is the difference between American and European option value?

An American option can be exercised any time up to expiry, a European only at expiry. So an American option is worth at least as much as an otherwise identical European option. For a call on an underlying with no income, they are worth the same.

What are the minimum and maximum prices of a call option?

A European call is worth at least max(0, S₀ − PV of X) and at most S₀. For an American call, the minimum is max(0, S₀ − X) and the maximum is also S₀.

How do interest rates affect option prices?

A higher risk-free rate raises call values and lowers put values, holding other factors fixed. The reason is the lower present value of the exercise price.