Pricing a European-Style Arithmetic Asian Option: Comparing Bootstrapping and Simulation Approaches

As of September 5, 2008, set the time to expiration and the strike price to price a European-style arithmetic Asian option on IBM stock. In general, there are no closed-form solutions for the value function of an Asian option (unlike European puts and calls under the Black–Scholes framework), so an efficient Monte Carlo pricing technique must be used instead. We compare and contrast two different simulation approaches: simulating the sample paths of geometric Brownian motion and bootstrapping sample paths directly from historical data.


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


In this Demonstration, two different approaches are taken to pricing a European-style arithmetic Asian option. Take the past 15 years of IBM stock data, assume the data are generated by a geometric Brownian motion, estimate the drift and diffusion coefficients of the geometric Brownian motion, simulate sample paths, price a European-style arithmetic Asian option along each sample path, and then generate histograms and other summary statistics.
Alternatively, take the past 15 years of IBM stock data and make no assumption about the underlying model that generates the data, other than that the log-returns of the stock data are independent and identically distributed. Simulate sample paths by directly bootstrapping from the histogram log-returns of the data, price an arithmetic Asian European option along each sample path, and then generate histograms and other summary statistics.
Consistently, the price of the option is lower under the bootstrapping approach because the bootstrapping approach effectively captures the heavy left tail and leptokurtosis found in IBM stock's historical log-returns. If you assume that IBM stock's log-returns are normally distributed, then both the left and right tails will be exponentially light, a feature that is inconsistent with historical data. Moreover, without the heavy left tail causing some of the sample paths to crash, the very positive drift term—IBM has appreciated an average of about 18% per year over the past fifteen years—will cause the geometric Brownian motion approach to overstate the value of the option.
    • 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+