Previous research has shown that for stock indices, the most likely time until a return of a particular size has been observed is longer for gains than for losses. We establish that this so-called gain/loss asymmetry is present also for individual stocks and show that the phenomenon is closely linked to the well-known leverage effect -- in the EGARCH model and a modified retarded volatility model, the same parameter that governs the magnitude of the leverage effect also governs the gain/loss asymmetry.
Deep Dive into Gain/loss asymmetry in time series of individual stock prices and its relationship to the leverage effect.
Previous research has shown that for stock indices, the most likely time until a return of a particular size has been observed is longer for gains than for losses. We establish that this so-called gain/loss asymmetry is present also for individual stocks and show that the phenomenon is closely linked to the well-known leverage effect – in the EGARCH model and a modified retarded volatility model, the same parameter that governs the magnitude of the leverage effect also governs the gain/loss asymmetry.
arXiv:0911.4679v2 [q-fin.ST] 25 Nov 2009
Gain/loss asymmetry in time series of individual stock prices and its relationship to
the leverage effect
Johannes Vitalis Siven∗and Jeffrey Todd Lins†
Saxo Bank A/S, Philip Heymans All´e 15, DK-2900 Hellerup, Denmark
(Dated: March 24, 2022)
Previous research has shown that for stock indices, the most likely time until a return of a
particular size has been observed is longer for gains than for losses.
We establish that this so-
called gain/loss asymmetry is present also for individual stocks and show that the phenomenon is
closely linked to the well-known leverage effect — in the EGARCH model and a modified retarded
volatility model, the same parameter that governs the magnitude of the leverage effect also governs
the gain/loss asymmetry.
PACS numbers:
Keywords: gain/loss asymmetry, leverage effect, EGARCH, retarded volatility model
Researchers have estimated empirical distributions for
first passage times of financial time series, the smallest
time interval needed for an asset to cross a fixed return
level ρ. Jensen, Johansen, and Simonsen [2] show that
for stock indices, the most likely first passage time is
shorter for ρ = −5% than for ρ = 5% — the first passage
time densities are shifted with respect to each other — a
phenomenon which they refer to as gain/loss asymmetry.
If {Xt}t≥0 denotes the logarithm of a given price pro-
cess, for instance daily closing prices of a stock or a stock
index, the first passage time τρ of the level ρ is defined
as
τρ =
min{s > 0; Xt+s −Xt ≥ρ} if ρ > 0,
min{s > 0; Xt+s −Xt ≤ρ} if ρ < 0,
and is assumed to be independent of t. The distribution
of τρ is estimated in a straightforward manner from a
time series X0, . . . , XT . Consider ρ > 0, and let t + s
be the smallest time point such that Xt+s −Xt ≥ρ, if
such a time point exists. In that case, s is viewed as an
observation of τρ. (If ρ < 0, take instead t + s such that
Xt+s −Xt ≤ρ.) Running t from 0 to T −1 gives a set
of observations from which the distribution of τρ is esti-
mated as the empirical distribution. Given the empirical
distribution, we follow Jensen et al. [2] and compute a
fit of the density function for the generalized gamma dis-
tribution. This density is plotted as a solid line together
with the empirical distribution in all figures, to guide the
eye — we do not discuss the fitted parameters, nor claim
that τρ truly follows a generalized gamma distribution.
In an unpublished working paper, Johansen, Jensen,
and Simonsen [3] demonstrate that individual stocks do
not not display a gain/loss asymmetry for ρ = ±5%, at
variance with e.g. the Dow Jones Industrial Average in-
dex. While we are able to reproduce these results, it is
not true in general that individual stocks do not display
gain/loss asymmetry.
There is an asymmetry, but for
∗Electronic address: jvs@saxobank.com
†Electronic address: jtl@saxobank.com
1
5
50
500
0.00
0.01
0.02
0.03
BA
tau
p(tau)
*
*
*
**
*
*
**
*
***
**************************************************************************
1
5
50
500
0.000
0.010
0.020
GE
tau
p(tau)
*
*
**
*
*
**
**
*
**
***********
*****
****
************************************************************
1
5
50
500
0.00
0.01
0.02
0.03
0.04
GM
tau
p(tau)
*
*
****
*
**
**
***
*******
********************************************************************
FIG. 1: Estimated distribution of the first passage time τρ
for the log price of three individual stocks: Boeing (BA),
General Electric (GE), and General Motors (GM). The graphs
correspond to ρ = +5% (stars) and ρ = −5% (rings).
1
5
50
500
0.000
0.005
0.010
0.015
BA
tau
p(tau)
*
* *
*
*
*
*
*
*
*
*
*
**
*
*****
**
*
*
*
*
**
*
********
**
*********
*
*****************************************************
1
5
50
500
0.000
0.004
0.008
0.012
GE
tau
p(tau)
*
*
*
*
*
*
*
*
*
*
****
**
**
**
**
*
**
*
****
*
*
***
*****
**
**
*****
*
***
*
**
*******************************************
1
5
50
500
0.000
0.004
0.008
GM
tau
p(tau)
*
*
**
*
*
*
*
*
*
*
*
**
*
**
*
*
**
***
**
*
*
******
********
*****
*******
********
****************************************************
FIG. 2: Estimated distribution of the first passage time τρ
for the log price of three individual stocks: Boeing (BA),
General Electric (GE), and General Motors (GM). The graphs
correspond to ρ = +5σS (stars) and ρ = −5σS (rings), where
σS is the estimated daily standard deviation of returns for
stock S.
Values: σBA = 1.9%, σGE = 1.7%, and σGM =
2.3%.
stocks one has to consider ρ of greater magnitude than
for indices. This is to be expected, since the standard
deviation of daily log returns is typically higher for indi-
vidual stocks than for indices. The choice ρ = ±5% cor-
responds to approximately ±5 daily standard deviations
for the Dow Jones index — when ρ is chosen analogously
for the individual stocks they display a clear gain/loss
asymmetry (see Figure 2).
This finding has some implications for interpreting
2
other results in the literature.
Donangelo, Jensen, Si-
monsen, and Snep
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