FRM Exam Part I · Options Markets

Factors Affecting Option Prices: Six Drivers and Price Bounds

Updated 11 October 2026 · Fact-checked

Six factors drive option prices: stock price, strike, time to expiry, volatility, risk-free rate and dividends. Calls gain from a higher stock price, volatility and rates, and lose from a higher strike and dividends. Puts do the opposite on price, strike, rates and dividends. Bounds such as c ≥ max(S0 − K e^(−rT), 0) stop arbitrage.

Understand Factors Affecting Option Prices

An option's price comes from its payoff at expiry and the chance of each payoff. A call pays max(S − K, 0). A put pays max(K − S, 0). Anything that raises the expected payoff, after discounting, raises the option price.

Start with the stock price and the strike. A higher stock price makes a call more valuable and a put less valuable. A higher strike does the reverse: calls fall in value and puts rise. Volatility works differently. The holder's loss on a call or put is limited to the premium, while the gain is large (unlimited for a call, up to K for a put). So more volatility raises both. Higher volatility means a wider spread of outcomes, and the holder benefits from the good side without losing more than the premium on the bad side.

Time to expiry gives more chance for the stock to move. For American options, longer maturity always adds value or leaves it unchanged, because you can exercise at any time. For European options longer maturity usually adds value, but not always. A deep in-the-money European put can be worth less with a longer life. So can a European call when a large dividend is due between the two expiry dates.

Interest rates and dividends act through the forward price of the stock. A higher risk-free rate raises the forward price and lowers the present value of the strike. This helps calls and hurts puts. Dividends lower the stock price on the ex-dividend date. This hurts calls and helps puts.

Bounds are no-arbitrage limits. If a price breaks a bound, you could lock in a risk-free profit. The bounds are tested often because they need only simple arithmetic.

Key formulas to remember

Effect summary (increase in the factor): S0 ↑: call ↑, put ↓ | K ↑: call ↓, put ↑ | σ ↑: call ↑, put ↑ | r ↑: call ↑, put ↓ | dividends ↑: call ↓, put ↑; Holds for all other factors fixed. Time to expiry is ↑ for American options; ambiguous for some European ones.
Upper bound, call: c ≤ S0 and C ≤ S0; A call cannot be worth more than the stock.
Upper bound, put: American: P ≤ K | European: p ≤ K e^(−rT); The most a put can pay is K, or K discounted for a European put.
Lower bound, European call (no dividends): c ≥ max(S0 − K e^(−rT), 0); Also the lower bound for an American call on a non-dividend stock.
Lower bound, European put (no dividends): p ≥ max(K e^(−rT) − S0, 0); American put: P ≥ max(K − S0, 0).
Lower bounds with known 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's life.
American versus European: C ≥ c and P ≥ p; The early exercise right can only add value.

How to solve Factors Affecting Option Prices questions

Use this routine for any question on factors or bounds.

1Identify whether the option is a call or a put, and whether it is European or American.
2Check for dividends. If there are any, find the present value D of the dividends during the option's life.
3For a direction question, take each factor one at a time and hold the others fixed. Use the effect table.
4For a bound question, write down the right formula: S0 − D − K e^(−rT) for a call, K e^(−rT) + D − S0 for a put.
5Compute the discount factor e^(−rT) with a calculator. Use continuous compounding unless told otherwise.
6Take the maximum with zero, since an option cannot be worth less than zero.
7Compare the quoted price with the bound. If it is below the lower bound, buy the option and short the stock (for a call) to lock in a profit.

Quickest way: Direction table and bound check

When to use it: Use when a question asks which option price rises or falls, or whether a quoted price is arbitrage-free.

Memorise the call row: S ↑, σ ↑, r ↑ help; K ↑ and dividends ↑ hurt. For time, T ↑ helps American options. For European options it is usually but not always true.
Flip the signs for S, K, r and dividends on the put row. Volatility stays positive for both.
For bounds, compute S0 − K e^(−rT) first. If it is negative, the bound is 0.
If a price is below the bound, there is arbitrage. Eliminate those answer choices.

Common mistakes in Factors Affecting Option Prices

Saying higher volatility lowers the value of a put.
Students link volatility with risk and assume risk is bad for the holder.
Fix: The holder's loss is capped at the premium while the gain is not. Higher volatility raises both calls and puts.
Using K instead of K e^(−rT) in the lower bound of a European option.
The payoff uses K, so students forget that the lower bound uses present values.
Fix: Discount the strike from expiry to today. Write S0 − K e^(−rT) every time.
Forgetting that dividends hurt calls and help puts.
Students think of dividends as good news for the stock.
Fix: The stock drops by about the dividend on the ex-date. Calls lose and puts gain. Use S0 − D in the bounds.
Saying a longer-dated European option is always worth more.
The rule is true for American options, so students generalise it.
Fix: Say it for American options only. For European options, longer maturity usually adds value, but a deep in-the-money put, or a call with a large dividend due between the two expiry dates, can be worth less.
Leaving out the zero in max(…, 0).
When the formula gives a negative number, students report it as the bound.
Fix: An option price cannot be negative. Always take the maximum with zero.

Worked examples

Example 1

A European call on a non-dividend-paying stock has S0 = $50, K = $45, T = 0.5 years and a continuously compounded risk-free rate of 4%. What is the lower bound on the call price? (A) $5.00 (B) $5.89 (C) $4.11 (D) $0.00

Show the solution

Use c ≥ max(S0 − K e^(−rT), 0).
Compute rT = 0.04 × 0.5 = 0.02.
e^(−0.02) = 0.980199.
K e^(−rT) = 45 × 0.980199 = 44.1090.
S0 − K e^(−rT) = 50 − 44.1090 = 5.8910.
This is above zero, so the bound is $5.89.

Answer: (B) $5.89 (about 5.891 before rounding).

Example 2

A European put has K = $60, T = 1 year, r = 5% continuously compounded and S0 = $55. The stock pays a $2 dividend in 6 months. Find the lower bound on the put price. Use the dividend PV at the same 5% rate.

Show the solution

Use p ≥ max(K e^(−rT) + D − S0, 0).
K e^(−rT) = 60 × e^(−0.05) = 60 × 0.951229 = 57.0737.
D = 2 × e^(−0.05 × 0.5) = 2 × e^(−0.025) = 2 × 0.975310 = 1.9506.
Sum = 57.0737 + 1.9506 = 59.0243.
Subtract S0: 59.0243 − 55 = 4.0243.
This is positive, so the bound is about $4.02.

Answer: About $4.02. Without the dividend the bound would be 57.0737 − 55 = $2.07, so the dividend raises it.

Exam tips

Learn the direction table cold. Many questions are pure recall, and the direction table settles them in seconds.
Check the option type and the style (European or American) first. Bounds and time-to-expiry rules differ.
When a question gives a dividend, use S0 − D for calls. For puts, add D to the discounted strike, giving K e^(−rT) + D − S0.
Use e^(−rT) on the calculator and keep four decimals. Answer choices are usually spaced widely enough for that.
If a quoted price is below a lower bound, name the arbitrage: buy the option and short the stock (call), or buy both put and stock (put).

Practice questions from Options Markets

Factors Affecting Option Prices 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 Prices: frequently asked questions

How does volatility affect option price?

Higher volatility raises the price of both calls and puts. The holder's downside is limited to the premium, while a wider spread of outcomes raises the chance of a large payoff.

What is the lower bound for a European call?

For a non-dividend stock, c ≥ max(S0 − K e^(−rT), 0). If dividends are paid, subtract their present value from S0 as well.

How do dividends affect option prices?

Dividends lower the stock price on the ex-dividend date. That reduces call values and increases put values, other things equal.

Does a longer time to expiry always raise option value?

For American options, yes, since the longer option holds all the rights of the shorter one. For European options it is usually true but can fail. A deep in-the-money European put can be worth less with a longer life, and so can a European call when a large dividend falls between the two expiry dates.