European Options

European Options

European options only allow for the exercise of options at the expiry date, while American options allow for early exercise. The terms do not convey any information about where, geographically, the options are traded. The right to exercise is a complex feature of an option, and as such, European options with a fixed exercise date are dealt with first.

Standard Notation

The following notion is used:

  • cT = Price of European call at expiration
  • ST = Price of the underlying at time T
  • X = Exercise price
  • pT = Price of a European put option at expiration

Value at Expiration

European Call Options

The value of a European call option at expiration is the exercise value, which is the greater of zero or the underlying price at expiration minus the exercise price:

$$ c_T = max(0, S_T – X) $$

European Put Options

The value of a European put option at expiration is simply the opposite – the greater of zero or the exercise price minus the underlying price at expiration:

$$ p_T = max(0, X – S_T) $$


There is a European call option that gives the holder the right to buy 100 shares of company XYZ at $100 in 3 months’ time. What is the value of this call option if the spot price at expiration is $110.15?

A. $1,015

B. $0

C. -$1,015


The correct answer is A.

CT = 100 × max($0, $110.15-$100) = $100 × $10.15 = $1,015

Since we have a call option with a strike price of $100, it makes sense that if the price at expiration is higher than the strike price, our option has gained in value.

Shop CFA® Exam Prep

Offered by AnalystPrep

Featured Shop FRM® Exam Prep Learn with Us

    Subscribe to our newsletter and keep up with the latest and greatest tips for success
    Shop Actuarial Exams Prep Shop Graduate Admission Exam Prep

    Sergio Torrico
    Sergio Torrico
    Excelente para el FRM 2 Escribo esta revisión en español para los hispanohablantes, soy de Bolivia, y utilicé AnalystPrep para dudas y consultas sobre mi preparación para el FRM nivel 2 (lo tomé una sola vez y aprobé muy bien), siempre tuve un soporte claro, directo y rápido, el material sale rápido cuando hay cambios en el temario de GARP, y los ejercicios y exámenes son muy útiles para practicar.
    So helpful. I have been using the videos to prepare for the CFA Level II exam. The videos signpost the reading contents, explain the concepts and provide additional context for specific concepts. The fun light-hearted analogies are also a welcome break to some very dry content. I usually watch the videos before going into more in-depth reading and they are a good way to avoid being overwhelmed by the sheer volume of content when you look at the readings.
    Kriti Dhawan
    Kriti Dhawan
    A great curriculum provider. James sir explains the concept so well that rather than memorising it, you tend to intuitively understand and absorb them. Thank you ! Grateful I saw this at the right time for my CFA prep.
    nikhil kumar
    nikhil kumar
    Very well explained and gives a great insight about topics in a very short time. Glad to have found Professor Forjan's lectures.
    Great support throughout the course by the team, did not feel neglected
    Benjamin anonymous
    Benjamin anonymous
    I loved using AnalystPrep for FRM. QBank is huge, videos are great. Would recommend to a friend
    Daniel Glyn
    Daniel Glyn
    I have finished my FRM1 thanks to AnalystPrep. And now using AnalystPrep for my FRM2 preparation. Professor Forjan is brilliant. He gives such good explanations and analogies. And more than anything makes learning fun. A big thank you to Analystprep and Professor Forjan. 5 stars all the way!
    michael walshe
    michael walshe
    Professor James' videos are excellent for understanding the underlying theories behind financial engineering / financial analysis. The AnalystPrep videos were better than any of the others that I searched through on YouTube for providing a clear explanation of some concepts, such as Portfolio theory, CAPM, and Arbitrage Pricing theory. Watching these cleared up many of the unclarities I had in my head. Highly recommended.