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Published in:
The Merton Problem Read chapter Read first chapter

2013 | OriginalPaper | Chapter

4. How Well Does It Work?

Author : L. C. G. Rogers

Published in: Optimal Investment

Publisher: Springer Berlin Heidelberg

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Abstract

The final chapter of the book takes a look at data, and finds virtually all of the models of the earlier part of the book to be wanting. Stylized facts of return data, well known to econometricians, are surprisingly robust across asset classes, and do not sit comfortably with the assumptions made in most of the theoretical literature.

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previous chapter Numerical Solution
Footnotes
1
... that asset dynamics are log-Brownian, with known parameters ...
 
2
.. and square-integrable. Enthusiasts for heavy-tailed distributions may look at 4.2 and declare that this shows evidence for heavy-tailed returns—as they would when looking at 4.1. But if you simulate the cumulative sum of squared heavy-tailed random variables, it looks quite unlike what we see in 4.2; the big jumps in the simulations are quite clearly visible, whereas the plots from the data do not show any noticeable discontinuities.
 
3
In the plots that follow this was calculated as \({\hat{\sigma }}_t^2 = \sum \nolimits _{j\ge 0} (1-\beta )\beta ^j \;r_{t-j}^2\) with \(\beta = 0.975\), and \(r_t \equiv \log (p_t/p_{t-1})\), where \(p_t\) is the close price on day \(t\).
 
4
Of course, we could just have a continuous time process which jumps only at integer times, but this would not be time homogeneous.
 
5
In recent years, there has been an upsurge in the study of realized variance of asset prices; an early reference is Barndorff-Nielsen and Shephard [2], a more recent survey is Shephard [37], and there have been important contributions from Aït-Sahalia, Jacod, Mykland, Zhang and many others. This literature is concerned with estimating what the quadratic variation actually was over some time period, which helps in deciding whether the asset price process has jumps, for example. However, there is no parametric model being fitted in these studies; the methodology does not claim or possess predictive power.
 
Metadata
Title
How Well Does It Work?
Author
L. C. G. Rogers
Copyright Year
2013
Publisher
Springer Berlin Heidelberg
Book
Optimal Investment

Print ISBN: 978-3-642-35201-0

Electronic ISBN: 978-3-642-35202-7

Copyright Year: 2013

DOI
https://doi.org/10.1007/978-3-642-35202-7_4