Pricing Power Options in the Black-Scholes Model

Power options are a class of exotic options in which the payoff at expiry is related to the power of the stock price, where . For a power option on a stock with price having strike price and time to expiry , the payoff is for a call, and for a put. Within the Black–Scholes model, closed-form solutions exist for the price of power options. In this Demonstration, prices as a function of the various parameters are explored.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


Given a stock with price , fixed dividend yield , and assumed fixed interest rate , the pricing formulas for power calls and puts with strike and time to expiry are
with and . Here is the cumulative distribution function for the standard normal distribution.
Snapshot 1: in the special case , power options reduce to regular European options
Snapshot 2: for , the payoff curve for call options becomes concave, and thus the option can have negative time value (i.e., current price < payoff)
Snapshot 3: for put options, the reverse is true: the payoff curve becomes concave for
Snapshot 4: for call options, the option value becomes very large as increases
E. G. Haug, The Complete Guide to Option Pricing Formulas, New York: McGraw–Hill, 2007.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+