Dynamic Asset Pricing Theory

By Darrell Duffie | Go to book overview
Save to active project

Numerical Methods

THIS CHAPTER REVIEWS three numerical approaches to pricing securities in a continuous-time setting: “binomial” approximation, Monte Carlo simulation, and finite-difference solution of the associated partial differential equation.

A. Central Limit Theorems

It is well known that a normal random variable can be represented as the limit of normalized sums of Bernoulli trials, that is, i.i.d. binomial random variables. This idea, a version of the Central Limit Theorem, leads to the characterization given in this section of the Black-Scholes option-pricing formula (equation [5.11]) as the limit of the binomial option-pricing formula (equation [2.16]), letting the number of trading periods per unit of time go to infinity. Aside from making an interesting connection between the discrete- and continuous-time settings, this also suggests a numerical recipe for calculating continuous-time arbitrage-free derivative security prices.

A sequence {Xn} of random variables converges in distribution to a random variable X, denoted XnX, if, for any bounded continuous function f : R R, we have E[f(Xn)] E[f(X)]. We could allow X and each of X1 ,X2,… to be defined on different probability spaces. A standard version of the Central Limit Theorem reads along the following lines. A random variable is standard normal if it has the standard normal cumulative distribution function.

Central Limit Theorem. Suppose Y1Y2, is a sequence of independent and identically distributed random variables on a probability space, each with expected


Notes for this page

Add a new note
If you are trying to select text to create highlights or citations, remember that you must now click or tap on the first word, and then click or tap on the last word.
Loading One moment ...
Project items
Cite this page

Cited page

Citations are available only to our active members.
Sign up now to cite pages or passages in MLA, APA and Chicago citation styles.

Cited page

Bookmark this page
Dynamic Asset Pricing Theory


Text size Smaller Larger
Search within

Search within this book

Look up

Look up a word

  • Dictionary
  • Thesaurus
Please submit a word or phrase above.
Print this page

Print this page

Why can't I print more than one page at a time?

While we understand printed pages are helpful to our users, this limitation is necessary to help protect our publishers' copyrighted material and prevent its unlawful distribution. We are sorry for any inconvenience.
Full screen
/ 465

matching results for page

Cited passage

Citations are available only to our active members.
Sign up now to cite pages or passages in MLA, APA and Chicago citation styles.

Cited passage

Welcome to the new Questia Reader

The Questia Reader has been updated to provide you with an even better online reading experience.  It is now 100% Responsive, which means you can read our books and articles on any sized device you wish.  All of your favorite tools like notes, highlights, and citations are still here, but the way you select text has been updated to be easier to use, especially on touchscreen devices.  Here's how:

1. Click or tap the first word you want to select.
2. Click or tap the last word you want to select.

OK, got it!

Thanks for trying Questia!

Please continue trying out our research tools, but please note, full functionality is available only to our active members.

Your work will be lost once you leave this Web page.

For full access in an ad-free environment, sign up now for a FREE, 1-day trial.

Already a member? Log in now.

Are you sure you want to delete this highlight?