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Erschienen in:
2007 | OriginalPaper | Buchkapitel
Extreme Returns in Asset Prices
Erschienen in: Statistical Analysis of Extreme Values
Verlag: Birkhäuser Basel
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Throughout this chapter, we assume that speculative prices
s
t
like those pertaining to stocks, foreign currencies, futures etc. are evaluated at discrete times
t
= 0, 1, 2, . . ., where the periods can be days or weeks. Thus, if
s
0
is the price of an investment at time
t
= 0, then the return—the difference of prices taken relatively to the initial price—at time
T
is (
s
T
−
s
0
)/
s
0
. Our primary interest concerns daily returns under discrete compounding (arithmetic returns)
$$ \tilde r_t = \frac{{s_t - s_{t - 1} }} {{s_{t - 1} }}$$
or the daily returns under continuous compounding (log-returns)
(16.1)
$$ r_t = \log (s_t ) - \log (s_{t - 1} ).$$
These quantities are close to each other if the ratio
s
t
/
s
t
−1
is close to 1. We will focus on the latter concept. Log-returns are also called geometric returns in the financial literature.