Essays in Finance:
Wrong-way Risk, Jumps and Stochastic
Volatility
Dissertation zur Erlangung des akademischen Grades
Doctor rerum politicarum
der Fakultät für Wirtschaftswissenschaften
der Technischen Universität Dortmund
Janis Müller
March 11, 2019

Contents
List of Figures V
List of Tables VII
1 Introduction 3
1.1 Publication details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Wrong-way risk in tails 13
2.1 Model and data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Wrong-way risk in tails: empirical findings . . . . . . . . . . . . . . . . 19
2.3 Sensitivity analysis of the expected shortfall . . . . . . . . . . . . . . . 25
2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3 Do illiquid stocks jump more frequently? 33
3.1 Data and Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2 Liquidity and number of jumps . . . . . . . . . . . . . . . . . . . . . . 36
3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4 How does stochastic volatility influence asset prices? - A parameter-free
approach 43
4.1 Stochastic volatility under time-change . . . . . . . . . . . . . . . . . . 45
4.2 Equity premium and risk-free rate puzzle . . . . . . . . . . . . . . . . . 53
4.3 Stochastic volatility as a factor . . . . . . . . . . . . . . . . . . . . . . 55
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5 Consumption volatility ambiguity and risk premium’s time-variation 67
5.1 Measurement of ambiguity . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.2 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
III
Contents
6 Price delay and market frictions in cryptocurrency markets 75
6.1 Data and delay measures . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.3 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Appendices 81
A Supplementary material for chapter 3 83
A.1 Further jump days information . . . . . . . . . . . . . . . . . . . . . . 83
A.2 Tables for bucket analysis for split sample . . . . . . . . . . . . . . . . 85
B Appendix for chapter 4 91
B.1 Time-change of the stochastic volatility model . . . . . . . . . . . . . . 91
B.2 Time change of the risky asset . . . . . . . . . . . . . . . . . . . . . . . 92
B.3 Beta pricing model with wealth portfolio . . . . . . . . . . . . . . . . . 93
B.4 Testing anomalies and CVR . . . . . . . . . . . . . . . . . . . . . . . . 94
C Derivation of expected market return 97
Bibliography 99
IV
List of Figures
2.1 Theoretical results for the correlation when approaching the extremes . 16
2.2 Empirical correlations minus their theoretical value (European banks) . 23
2.3 Empirical correlations minus their theoretical value (American banks) . 24
2.4 Expected shortfall conditioned (European banks) . . . . . . . . . . . . 29
2.5 Expected shortfall conditioned (American banks) . . . . . . . . . . . . 30
4.1 Time-changed S&P 500 Total Return Index . . . . . . . . . . . . . . . 46
6.1 Temporal development of the three delay measures . . . . . . . . . . . 77
V

List of Tables
2.1 Descriptive statistics and correlation with the CDS indices . . . . . . . 18
2.2 Used quantile values for the thresholds in section 2.2 and section 2.3 . . 20
2.3 ESCR/ES − 1 when CDS ≥ 25 . . . . . . . . . . . . . . . . . . . . . 27
3.1 MedRV, JO and LM jump test analysis (2007-2011) . . . . . . . . . . . 38
3.2 Regression analysis (2007 - 2011) . . . . . . . . . . . . . . . . . . . . . 40
4.1 Risk aversion parameters γ and subjective discount factor β solving the
equity premium and risk-free rate puzzle . . . . . . . . . . . . . . . . . 54
4.2 Summary statistics for monthly factor percent returns . . . . . . . . . . 58
4.3 Regressions for 25 value-weight Size-Tau portfolios . . . . . . . . . . . . 62
4.4 Summary statistics for tests of five-factor and five-factor plus SVOL models 63
4.5 Regressions for 25 value-weight Size-Momentum portfolios . . . . . . . 64
5.1 Statistics for ambiguity measure . . . . . . . . . . . . . . . . . . . . . . 69
5.2 Regression with ambiguity measure . . . . . . . . . . . . . . . . . . . . 71
5.3 Regression with control variables . . . . . . . . . . . . . . . . . . . . . 73
6.1 Descriptive statistics for delay and liquidity measures . . . . . . . . . . 78
6.2 Regression results for price delay . . . . . . . . . . . . . . . . . . . . . 79
A.1 BNS tests jump days information . . . . . . . . . . . . . . . . . . . . . 84
A.2 JO and LM tests jump days information . . . . . . . . . . . . . . . . . 84
A.3 BNS jump test analysis (2007 - 2008) . . . . . . . . . . . . . . . . . . . 86
A.4 BNS jump test analysis (2009 - 2011) . . . . . . . . . . . . . . . . . . . 87
A.5 JO and LM jump test analysis (2007-2008) . . . . . . . . . . . . . . . . 88
A.6 JO and LM jump test analysis (2009 - 2011) . . . . . . . . . . . . . . . 89
VII
List of Tables
B.1 Summary statistics for tests of five-factor, five-factor plus SVOL and
five-factor plus CVR models . . . . . . . . . . . . . . . . . . . . . . . . 95
VIII
Acknowledgment
First, I thank Professor Peter N. Posch for being my supervisor and arranging a creative
environment at his chair. I also thank my research group, Johannes, Daniel, Philipp,
Vladislav, Timotheos, Gerrit and Nils for valuable discussions. Furthermore, I thank
Dirk Stemmer for giving me the opportunity to write this thesis. Finally, I thank Otilia
Pasnicu for her unconditional support.
1

1 Introduction
The main focus of this thesis is about understanding the behavior of asset prices and
asset returns regarding tail events, in the light of time-varying stochastic volatility and
with respect to market efficiency. Thus, the contribution of this thesis is twofold: The
first part deals with topics related to risk management whereas the second part deals
with topics related to asset pricing.
With new regulations like the credit valuation adjustment (CVA) the assessment
of wrong-way risk (WWR) is of utter importance. Wrong-way risk means a negative
dependence of the exposure to a counterparty on the counterparty’s credit quality.
Thus, the first paper in chapter 2 studies the co-movement of counterparty credit risk
and returns of different asset classes (equity, currency, commodity and interest rate).
Whereas there exist several articles about WWR regarding the calculation of CVA (e.g.
Glasserman and Yang, 2016; Hofer, 2016; Hull and White, 2012; Kenyon and Green, 2016;
Rosen and Saunders, 2012), there is no explicit analysis of WWR for tail events which
are of interest for calculation of, for example, margin requirements for non-centrally
cleared derivatives. This framework was released by the Basel Committee on Banking
Supervision (BCBS) and the International Organization of Securities Commissions
(IOSCO) in 2013 with revision in 2015 (Basel Committe on Banking Supervision, 2015).
The margin consists of a variation margin and an initial margin. The variation margin
is used to settle daily profits and losses of derivatives. The initial margin is an insurance
in case the counterparty defaults. It is used to cover any losses occurring from the
default event date until the close-out of the derivatives position. The focus is on the
initial margin which is exposed to WWR in extreme events, meaning major stress in the
banking sector. Using extreme value theory to model the tail of the joint distribution
of asset returns and counterparty credit risk, the impact on the risk measure expected
shortfall is analyzed when conditioning on major stress in the banking sector for a
period from December 2005 until January 2016. This is done for both the American
and the European banking sector. There are two main results: Firstly, the correlation
3
1 Introduction
between assets belonging to the four mentioned asset classes and major stress in the
European and American banking sector increases. Secondly, the expected shortfall
conditioned on stress in the banking sector is significantly higher than the unconditioned
expected shortfall. For instance, the weekly conditioned expected shortfall on a 97.5
percent level is about 2 to 440 percent higher depending on the asset.
Extreme movements in asset prices are often characterized by jumps and drying up
liquidity. The second paper aims to improve the understanding of the (unconditioned)
link between jumps and liquidity in chapter 3. Modelling price dynamics properly is of
utter importance, especially in options pricing (e.g. Merton, 1976) and risk management
(e.g. Duffie and Pan, 2001). Empirically, jump-diffusion processes lead to better results
than pure diffusion processes (Andersen et al., 2002; Bollerslev et al., 2008; Eraker
et al., 2003). Thus, this paper provides an improvement in the understanding of the
occurrence of jumps of stocks by investigating stock liquidity as a possible driver.
Liquidity is necessary to absorb temporary bulges of buy and sell orders without large
price movements (Pagano, 1989). Three different measures for liquidity, the relative
bid-ask spread, the turnover and the illiquidity measure of Amihud (2002a), and three
different kinds of jump detection tests are considered, the tests of Barndorff-Nielsen
and Shephard (2006), including modified tests by Andersen et al. (2012), as well as of
Jiang and Oomen (2008) and of Lee and Mykland (2008). Sorting US stocks in three
buckets according to their level of liquidity for the period 2007-2011, it is shown that
the number of days with jumps increases by about 10% for the relative bid-ask spread
and the turnover and about 34% for the illiquidity measure of Amihud (2002a) from
the first bucket to the third bucket considering all three jump measures for the highest
available frequency.
Apart from jumps, stochastic volatility is an important stylized fact of asset price
processes. In the third paper the dynamics of asset prices are time-changed to study
the influence of stochastic volatility on asset prices in a parameter-free approach (see
chapter 4). Stochastic volatility seems necessary to understand features of financial
markets (Bansal and Yaron, 2004; Barndorff-Nielsen and Veraart, 2013; Bollerslev
et al., 2012; Carr and Wu, 2009). However, including stochastic volatility in asset
pricing by assuming a specific process is critical. The choice of a specific process has
significant consequences for results but empirically it is doubtful if one can choose
the correct model out of all consistent alternatives (Carr and Lee, 2009). Applying
4
the time-changing technique avoids to use a specific process for volatility to study the
impact of stochastic volatility on asset prices. Firstly, formulas for the expected return
of assets and the risk-free rate are derived. It is noteworthy that the risk-free rate
becomes stochastic under the time-change. Based on the theoretical findings, stochastic
consumption volatility is explored considering prevailing puzzles. Secondly, a factor
is constructed mimicking the effect of stochastic volatility of the market portfolio on
asset prices extending the five-factor model of Fama and French (2015). Considering
anomalies targeted by existing factor models, the constructed factor especially helps to
describe cross-sectional excess returns of portfolios formed on size and momentum. This
finding indicates that the momentum effect is partly explicable by stochastic volatility.
The fourth paper deals with ambiguous volatility as an explanation for time-
variation in the market’s risk premium in chapter 5. Harvey (1989) and following
Ferson and Harvey (1991) find equity premia exhibit cyclical variation. Popular asset
pricing models allowing for variation in equity premia are habit formation (Campbell
and Cochrane, 1999) and long-run risks (Bansal and Yaron, 2004). Whereas variation
in equity premia is a consequence of variation in risk aversion in the habit model, the
long-run risk model uses stochastic consumption volatility to let equity premia vary. In
contrast to chapter 4 volatility is not only assumed to be stochastic but to be ambiguous.
Recently, Jeong et al. (2015) find ambiguity to matter empirically. Following Epstein
and Ji (2013) consumption volatility is assumed to lie in some interval and ambiguity
is defined as the width of this interval. Considering the formulas for the expected
market return and the risk-free rate, derived in a consumption-based asset pricing model
(Cochrane, 2001), the implied consumption volatility is calculated from both formulas
using a long data set for the U.S. market. Taking the difference of both resulting
implied volatilities as a proxy for ambiguity, a regression shows that variation in this
measure of ambiguity explains up to 69% of the post-war variation in the market’s risk
premium.
Finally, the fifth paper is about the currently discussed topic of market efficiency
regarding cryptocurrencies. There exist many studies about the weak-form efficiency
of Bitcoin (Urquhart, 2016; Nadarajah and Chu, 2017; Vidal-Tomás and Ibañez, 2018;
Kristoufek, 2018; Jiang et al., 2018; Bariviera, 2017; Tiwari et al., 2018; Khuntia
and Pattanayak, 2018; Alvarez-Ramirez et al., 2018) but little for the efficiency of
cryptocurrencies in general (Brauneis and Mestel, 2018; Wei, 2018). Extending the
5
1 Introduction
existing literature about the efficiency of crytptocurrencies, this stduy investigates the
average price delay of the cryptocurrency market to new information. According to
the efficient market hypothesis of Fama (1970), prices should reflect new information
instantaneously. Therefore, a delay in reflecting new information is a sign for inefficiency
of a market. Using three delay measures as given in Hou and Moskowitz (2005) it is
shown that news, affecting the cryptocurrency market, are much faster incorporated in
prices during the last three years indicating that the cryptocurrency market becomes
more efficient over time. Furthermore, the price delay is mainly driven by liquidity
which is studied in the cross-section of 75 cryptocurrencies.
6
1.1 Publication details
1.1 Publication details
Paper I (chapter 2):
Wrong-way risk in tails
Authors:
Janis Müller, Peter N. Posch
Abstract:
With new regulations like the credit valuation adjustment (CVA) the assessment of
wrong-way risk (WWR) is of utter importance. We analyze the effect of a counterparty’s
credit risk and its influence on other asset classes (equity, currency, commodity and
interest rate) in the event of extreme market movements like the counterparty’s default.
With an extreme value approach we model the tail of the joint distribution of different
asset returns belonging to the above asset classes and counterparty credit risk indicated
by changes in CDS spreads and calculate the effect on the expected shortfall when
conditioning on counterparty credit risk. We find the conditional expected shortfall
to be 2% to 440% higher than the unconditional expected shortfall depending on the
asset class. Our results give insights both for risk management as well as for setting an
initial margin for non-centrally cleared derivatives which becomes mandatory in the
European Market Infrastructure Regulation (EMIR).
Publication details:
Journal of Asset Management, 19(4) (2018): 205-215.
7
1 Introduction
Paper II (chapter 3):
Do illiquid stocks jump more frequently?
Authors:
Sebastian Kunsteller, Janis Müller, Peter N. Posch
Abstract:
We study the influence of stock liquidity on the stock price jump frequency using
intraday data of 175 US stocks during 2007-11. Grouping these stocks according to
their average liquidity we find less liquid stocks to jump more often than liquid stocks.
Depending on the liquidity measure the least liquid stocks exhibit on average between
10% and 34% more jumps than the most liquid stocks. Our results are robust to
different definitions of liquidity and jump measures as well prevail under different time
frequencies.
Publication details:
Applied Economics, 51(25) (2019): 2764-2769.
8
1.1 Publication details
Paper III (chapter 4):
How does stochastic volatility influence asset prices? – A parameter-
free approach
Authors:
Janis Müller, Peter N. Posch
Abstract:
We disentangle the risk of time-varying volatility and return in a consumption-based
asset pricing model by introducing stochastic volatility of consumption growth to asset
prices moving in volatility units instead of moving in time. This time-change approach
yields additional insights to risk premia’s composition. We empirically explore stochastic
volatility where it eases the risk-free rate puzzle and solves the equity premium puzzle
if people are very impatient. As a factor it significantly improves the explanation of
returns in the cross-section, solves the momentum effect among other anomalies and is
not captured by other existing factors.
Publication details:
Submitted to The Review of Financial Studies
9
1 Introduction
Paper IV (chapter 5):
Consumption volatility ambiguity and risk premium’s time-variation
Authors:
Janis Müller, Peter N. Posch
Abstract:
In a consumption based asset pricing model one can calculate the volatility of (log-
)consumption growth from the expected market return and from the risk-free rate.
We propose to use the difference between these estimates to measure ambiguity about
consumption volatility. Using a long dataset we show this measure explains up to 69%
of post-war variation in the market risk premium.
Publication details:
Finance Research Letters, Forthcoming
10
1.1 Publication details
Paper V (chapter 6):
Price delay and market frictions in cryptocurrency markets
Authors:
Gerrit Köchling, Janis Müller, Peter N. Posch
Abstract:
We study the efficiency of cryptocurrencies by measuring the price’s reaction time
to unexpected relevant information. We find the average price delay to significantly
decrease during the last three years. For the cross-section of 75 cryptocurrencies we
find delays to be highly correlated with liquidity.
Publication details:
Economics Letters, 174 (2019): 39-41
11

2 Wrong-way risk in tails
The following is based on Müller and Posch (2018b).
https://doi.org/10.1057/s41260-018-0076-9
13
2 Wrong-way risk in tails
14
15
2 Wrong-way risk in tails
16
17
2 Wrong-way risk in tails
18
19
2 Wrong-way risk in tails
20
21
2 Wrong-way risk in tails
22
23
2 Wrong-way risk in tails
24
25
2 Wrong-way risk in tails
26
27
2 Wrong-way risk in tails
28
29
2 Wrong-way risk in tails
30
31
2 Wrong-way risk in tails
32
3 Do illiquid stocks jump more frequently?
The following is based on Kunsteller et al. (2019).
https://doi.org/10.1080/00036846.2018.1558357
33
3 Do illiquid stocks jump more frequently?
34
35
3 Do illiquid stocks jump more frequently?
36
37
3 Do illiquid stocks jump more frequently?
38
39
3 Do illiquid stocks jump more frequently?
40
41
3 Do illiquid stocks jump more frequently?
42
4 How does stochastic volatility influence
asset prices? - A parameter-free approach
We explore the general influences of stochastic volatility on asset prices in a consumption-
based asset pricing framework. We do not assume a specific process for the volatility
but disentangle the risk of time-varying volatility and return risk by using time-changed
asset prices.
Commonly asset prices are viewed as evolving over time. Under the assumption of
consumption growth and asset returns to be iid. asset pricing models yield anomalies
when tested empirically (cf. Mehra and Prescott, 1985). A variety of papers target these
anomalies by extending the asset pricing model, i.e. generalizing Yt+1 in the general
stochastic discount factor presentation Mt+1 = β
(
ct+1
ct
)−γ
Yt+1 underlying most of these
models (cf. Cochrane, 2017). Bansal and Yaron (2004) introduce stochastic volatility of
the consumption growth as crucial for long-run risk models, empirically assessed by
Beeler and Campbell (2012) and Bansal et al. (2012). Many authors followed making
the case for this particular extension, eg. Carr and Wu (2009), Barndorff-Nielsen and
Veraart (2013) or Bollerslev et al. (2012). However, including stochastic volatility in
asset pricing by assuming a specific process is critical. The choice of a specific process
has significant consequences for results but empirically it is doubtful if one can choose
the correct model out of all consistent alternatives (Carr and Lee, 2009).
We propose a different approach to benefit from the extension of stochastic con-
sumption growth volatility but not having to impose a specific driving process for
consumption volatility as, for instance, in Boguth and Kuehn (2013). Instead of looking
at asset prices evolving in time, we look at their evolution in realized variance of the
(log-)consumption process. The volatility of this time-changed consumption process is
deterministic by construction but time is now stochastic. When time is stochastic, we
denote it by τ . It is a stopping time that measures how much time passes for some fixed
amount of realized variance. In the time-changed setting we derive the corresponding
43
4 How does stochastic volatility influence asset prices? - A parameter-free approach
risk-free rate and the expected return on risky assets. We find that risk premia depend
on an additional correction term, the covariance between the random time τ and an
assets return. Taking the time-changed version of the risk free-rate and the Hansen and
Jagannathan (1991) bound, we explore the influence of stochastic volatility empirically.
We find that it eases the risk-free rate puzzle and solves the equity premium puzzle if
people are impatient. Regarding factor-models we construct a factor SVOL mimicking
the effect of stochastic volatility represented by τ . It is the difference of high and low
exposure of stocks to the random time τ . SVOL delivers an expected monthly return
of 0.40% and is significant (t − statistic : 3.57) even by the standards of Harvey et al.
(2016). Sorting stocks into 25 Size-Tau portfolios, the five-factor model of Fama and
French (2015) is unable to describe the resulting monthly excess returns ranging from
0.23% to 1.22%. While the average expected return of SVOL cannot be explained
by the five-factor model, the average expected return of the factor CVR of Boguth
and Kuehn (2013) can be explained and CVR does not improve the performance of
the five-factor model considering prevailing other anomalies. In contrast, adding the
factor SVOL helps to explain cross-sectional risk premia by improving the performance
of the five-factor model. Thus, our factor contains different information than CVR
due to the differences in construction resulting from the assumed volatility process.
Considering the momentum effect of Jegadeesh and Titman (1993), the loadings on
SVOL suggest that the momentum effect is for the most part explicable by stochastic
volatility. When volatility is just a constant the relation between time and realized
variance is unique such that all formulas derived under the time-change collapse to
already known expressions.
Among the first to use a time-change to analyze financial processes, Clark (1973)
looks at future prices using a subordinated process to handle the non-normality of
observed returns. Such a time-change is also called stochastic clock, transaction clock
or business time. Time-changed Lévy processes enjoy increasing popularity in context
of option pricing (Carr and Wu, 2004), variance risk premiums (Carr et al., 2012) and
pricing non-Gaussian jump-like risks (Shaliastovich and Tauchen, 2011). Recently,
Jeong et al. (2015) use time-changed Lévy processes to address ambiguity in empirical
asset pricing. For a comprehensive overview of time-changes in asset price modeling,
we refer to Geman (2005).
This paper is organized as follows: In the following section 4.1 we give an outline of
44
4.1 Stochastic volatility under time-change
the basic idea as well as of the time-changing technique. The theoretical implications
in form of an additional risk correction for asset prices are subject of the following
section 4.1.1. A test for the usefulness of the time-changing technique in empirical
asset pricing is, firstly, given by considering the equity premium and risk-free rate
puzzle (section 4.2) and, secondly, by constructing a mimicking factor SVOL extending
existing factor-models (section 4.3). To understand the latter two sections 4.2 and 4.3
it is sufficient to skim through section 4.1.1 and skip the detailed studying of the risk
corrections for the expected risk-free rate as well as the expected return of risky assets
in section 4.1.2.
4.1 Stochastic volatility under time-change
Assuming a specific process for volatility in asset pricing models has significant conse-
quences for results but empirically it is difficult to choose the correct one. Instead, we
propose an approach without assuming a specific process for volatility. The idea is to
change perspective by looking at asset price movements with the realized variance on
the x-axis instead of time. For example, instead of observing prices daily we observe
them after some amount of variance is realized. If we assume the price process to follow
some geometric Brownian motion, the realized variance is the integrated variance of
the log-price process which is empirically given by the sum of squared log-returns.1
As an illustration we plot in figure 4.1 the S&P 500 Total Return Index from January
1998 until December 2016 in time (left) and realized variance (right).2 The time-change
stretches periods of high volatility and compresses periods of low volatility which often
coincides with downward and upward trends in stock markets.
Throughout this paper we assume asset prices and consumption to be continuous.
Volatility is assumed to be stochastic and is in general allowed to have jumps. In such
a stochastic volatility model the dynamics of consumption Ct are given by the following
1Due to micro-structure in ultra high frequency data the estimator needs to be adjusted to be robust
(c.f. At-Sahalia et al., 2011).
2We calculate the integrated variance using an EGARCH model based on daily log-returns, but one
can also calculate a model free estimator using intraday log-returns (At-Sahalia et al., 2011).
45
4 How does stochastic volatility influence asset prices? - A parameter-free approach
1000
2000
3000
4000
2000 2005 2010 2015
Time
S&
P 
50
0 
(T
R
)
1000
2000
3000
4000
0.0 0.2 0.4 0.6
Realized variance
S&
P 
50
0 
(T
R
)
Figure 4.1: S&P 500 Total Return Index (TR) from January 1998 until December
2016 plotted in time (left) and realized variance (right)
stochastic differential equation (SDE):
dCt
Ct
= μdt +
√
VtdW
c
t
dVt = αtdt + βtdW vt + ηtdJt
(4.1)
where μ is a constant and αt, βt and ηt are stochastic processes such that the SDE
is well defined. The processes W vt and W ct are standard Brownian motions with
dW vt dW
s
t = ρdt where ρ ∈ [−1, 1]. The process Jt is some pure jump process. With
ηt = 0 often used choices for αt and βt are αt = κ(θ − Vt) and βt = ξ
√
Vt as in Heston
(1993) or with βt = ξ as in Hull and White (1987). However, empirically it is difficult
to distinguish different choices of αt and βt but the selection is crucial for derivatives or
optimization (cf. Epstein and Ji, 2013). Carr and Lee (2009) criticize the selection of a
specific parametric model for the volatility process because the volatility is not directly
observable and often noisy data leads to noisy estimates.
We do not assume a specific selection for αt, βt and ηt. To state the time-changed
version of the consumption process in equation (4.1) we start by defining the following
stopping time:
τ := τv = inf{t ≥ 0;
∫ t
0
Vudu = v}. (4.2)
The idea is to observe the process Ct at the stopping time τ which is equivalent to
46
4.1 Stochastic volatility under time-change
waiting until the integrated variance exceeds some predefined level v:
Cτ = C0 exp
(
μτ − 12
∫ τ
0
Vudu +
∫ τ
0
√
VudW
c
u
)
(4.3)
Observing Cτ instead of Ct is advantageous because of the time-change for martingales
theorem according to Dambis (1965), Dubins and Schwarz (1965). The theorem as
stated and proved in Karatzas and Shreve (1991) tells us that the integral
∫ τ
0
√
VudW
c
u
is a standard Brownian motion Bcv. Using that
∫ τ
0 Vudu = v by construction of the
stopping time τ , we can restate equation (4.3) as:
Cτ = C0 exp
(
μτ − v2 + B
c
v
)
. (4.4)
A more detailed derivation can be found in the appendix B.1. From equation (4.4) we
see that a key advantage of the time-change is the separation of the risk arising from
the Brownian motion and the stochasticity of the volatility which is absorbed by the
stopping time τ . This disentanglement enables us to study the effect of return risk and
the risk of time-variation in volatility separately without assuming a specific model
for the volatility. When considering (4.1) in time, we have to deal with the integral∫ t
0
√
VudW
c
u. Separating the effect of the return risk represented by W cu and the risk
of stochastic volatility Vu is in general impractical. For instance, if we condition on
the volatility Vu to study the marginal impact of W cu, the process W cu is in general
no Brownian motion and its law is unknown if the law of Vu is unknown. Under the
time-change the separation appears as the sum of τ and Bcv, where τ captures the effects
of Vu.
4.1.1 Time-free rate and risk corrections under time-change
We derive the time-free rate as the risk-free asset under a time-change in a consumption-
based asset pricing model. It can be interpreted as the return of a zero coupon bond
which expires as soon as a predefined level of realized variance is exceeded. Thus, the
maturity of this asset is random. Given the time-free rate we derive formulas for the
risk-free rate, which is stochastic under the time-change, and for risky assets, such as
stocks.
In classical asset pricing the asset which does not contain any uncertainty about
47
4 How does stochastic volatility influence asset prices? - A parameter-free approach
its future return is called the risk-free asset and its return the risk-free rate. In the
time-changed setting we look at intervals of integrated variance and the risk-free rate is
not constant on such intervals. Instead, it is adjusted by an additional correction for
the stochasticity of the volatility as we show. An analogous to the risk-free asset in
time can be defined straightforwardly as the asset whose return in variance is know ex
ante, the time-free rate:
Rfvv+1 =
1
E(mv+1)
(4.5)
As in Cochrane (2001) let the stochastic discount factor be similarly given in variance
by
mv+1 = βτΔv
u′(cv+1)
u′(cv)
(4.6)
where Δv is the difference in integrated variance between v and v + 1, u is the power
utility function u(c) = c1−γ1−γ , β the subjective discount factor and τv the stopping time
as defined in (4.2). Taking the time-changed consumption process cv := Cτ as given in
equation (4.4), the stochastic discount factor can be written as
mv+1 = e−(γμ+δ)τΔv+γ
Δv
2 −γBcΔv (4.7)
where βτΔv = e−δτΔv . Starting with the expression for the time-free rate from equation
(4.5), we get
1
Rfvv+1
= E
(
e−(γμ+δ)τΔv
)
E
(
eγ
Δv
2 −γBcΔv
)
+ Cov
(
e−(γμ+δ)τΔv , eγ
Δv
2 −γBcΔv
)
= E
(
e−(γμ+δ)τΔv+γ(γ+1)
Δv
2
)
+ Cov
(
e−(γμ+δ)τΔv , eγ
Δv
2 −γBcΔv
)
(4.8)
since BcΔv is a standard Brownian motion. Looking at equation (4.8), we see some
important features of the time-free rate. If the variance is constant, i.e. there is
no time-variation in volatility, the time-free rate is equal to the risk-free rate. Let
Δv = σ2Δt, then τΔv = Δt by definition of τ and
Rfvv+1 = e(γμ+δ)Δt−γ(γ+1)
σ2
2 Δt = Rftt+1 (4.9)
48
4.1 Stochastic volatility under time-change
with Δt being the difference from time t to time t + 1. If the covariance is zero, the
time-free rate depends on the expectation over the stopping time τ only. To understand
γ and δ let us assume τΔv ∼ N (μτ , σ2τ ), which is unrealistic since τ > 0 a.s. but helps
to study the effect of the stopping time τ . The time-free rate is in this case given by
Rfvv+1 = e(γμ+δ)μτ −(γμ+δ)
2 σ
2
τ
2 −γ(γ+1)Δv2 . (4.10)
The time-free rate is lowered by precautionary savings where the last term is equivalent to
the precautionary savings term for the risk-free rate. The second term, however, captures
the volatility about the stopping time and thus the volatility of the variance process
of the stochastic volatility model in (4.1). The higher the volatility of the stopping
time, the lower the time-free rate since people try to avoid additional uncertainty.
Furthermore, the time-free rate depends on the expectation of τ which is always positive
but its influence depends on risk aversion and impatiences. When people are impatient
δ > 0 and risk averse γ > 0 then (γμ + δ) > 0 and a higher expected τ means a higher
time-free rate. In contrast, if people prefer money tomorrow to money today and are
risk loving, a higher expected τ means a lower time-free rate.
A positive covariance lowers the time-free rate and a negative covariance increases
the time-free rate. For small values we can approximate the covariance by
Cov
(
e−(γμ+δ)τΔv , eγ
Δv
2 −γBcΔv
)
≈ Cov
(
−(γμ + δ)τΔv, γΔv2 − γB
c
Δv
)
= γ(γμ + δ)Cov (τΔv, BcΔv) .
(4.11)
If people are risk averse and impatient, the sign of the covariance term depends on the
covariance of the stopping time and the Brownian motion of the consumption process.
A positive covariance means that consumption is negatively correlated with its variance
since τ is larger when the variance is low and vice versa.
4.1.2 Risk corrections
In classical asset pricing expected asset returns equal the risk-free rate plus a risk
correction which depends on the covariance of the stochastic discount factor and the
49
4 How does stochastic volatility influence asset prices? - A parameter-free approach
assets return (e.g. see Cochrane, 2001). Assuming the stochastic discount factor
and asset returns being modelled by geometric Brownian motions, the sign of this
risk correction depends on the correlation of the Brownian motions. If volatility is
stochastic, we find an additional term, the covariance between the assets return and
the stopping time τ which is linked to consumptions volatility.
Starting with 1 = E(mv+1Riv+1) and the expression for the time-free rate
Rfvv+1 = 1/E(mv+1) we get
E
(
Riv+1
)
= Rfvv+1 − Rfvv+1 Cov
(
mv+1, R
i
v+1
)
. (4.12)
With the expression for the stochastic discount factor in equation (4.7), we can approx-
imate the covariance in equation (4.12) by:
Cov
(
e−(γμ+δ)τΔv+γ
Δv
2 −γBcΔv , Riv+1
)
≈ Cov
(
−(γμ + δ)τΔv + γΔv2 − γB
c
Δv, R
i
v+1
)
= −(γμ + δ)Cov
(
τΔv, R
i
v+1
)
− γ Cov
(
BcΔv, R
i
v+1
)
. (4.13)
Thus, any assets return is equal to the time-free rate adjusted by the covariance
between the stopping time τ and the assets return as well as the driving Brownian
motion of the consumption process BcΔv. Again, if the variance is constant, i.e. τ
is constant, we arrive at the usual expression known from asset pricing in time. Let
Δv = σ2Δt, then τΔv = Δt by definition of τ and according to equation (4.9) it
holds Rfvv+1 = Rftt+1. Following the logic of equation (4.9) we have mv+1 = mt+1 and
Riv+1 = Rit+1 and we get E
(
Rit+1
)
= Rftt+1 − Rftt+1 Cov
(
mt+1, R
i
t+1
)
.
Thus equation (4.12) is a generalization of the risk correction equation known in
classical asset pricing. Further, we study the covariance given in equation (4.13) for
the expected risk-free rate as well as for the expected return of a risky asset.
Risk-free rate
First, we start with the risk-free rate. Let the risk-free rate be given by Rftt = er
ftt,
where rft is the (constant) continuous risk-free rate. In variance it simply writes as
Rftv+1 = er
ftτΔv . For small values we can do the approximation Rftv+1 ≈ rftτΔv and we
50
4.1 Stochastic volatility under time-change
get for the covariance in equation (4.13):
Cov
(
e−(γμ+δ)τΔv+γ
Δv
2 −γBcΔv , er
ftτΔv
)
≈ −(γμ + δ)rft Var (τΔv) − γrft Cov (BcΔv, τΔv) . (4.14)
Considering equation (4.12), the expected risk free-rate is equal to the time-free rate
plus some correction due to the variance of τ and the covariance of τ and BcΔv. If people
are risk averse and impatient, the expected risk-free rate is higher than the time-free
rate when the variance of the stopping time is high corresponding to a high volatility
of volatility in time. Furthermore, if people are risk averse, the expected risk-free rate
is higher than the time-free rate when the correlation of the stopping time τ and BcΔv
is positive, meaning consumption growth is high during times of low volatility and vice
versa. This is reasonable because a zero coupon bond expires at a fixed time neglecting
the level of realized variance during the lifetime of the bond. A zero coupon bond
expiring as soon as some predefined level of realized variance is exceeded hedges against
times of high volatility and low consumption growth. Thus in comparison, its price
should be higher and its return, the time-free rate, lower.
Risky assets
Denote the risky asset by St given as the solution to a stochastic volatility model such
as in equation (4.1). Since the time-change is always done according to the volatility of
the consumption process, the time-changed version of the risky asset is
Sτ = S0eμ
sτv− 12
∫ v
0 audu+
∫ v
0
√
audBsu . (4.15)
A derivation is given in the appendix B.2. The stochastic process av is the ratio of the
variance of the risky asset V st and the variance of the consumption process V ct :
av =
V sτ(v)
V cτ(v)
. (4.16)
Of course, since the risky assets variance process is different but we time change according
to the variance process of the consumption process this ratio appears naturally. Let
the stochastic discount factor again be given as in equation (4.7) and let Riv+1 = Sτ/S0.
51
4 How does stochastic volatility influence asset prices? - A parameter-free approach
Let us further assume that the ratio of the volatility of the risky asset and the volatility
of the consumption is a constant plus some independent random variable, i.e.
√
V st
V ct
= β + 
t. (4.17)
where 
t > 0 a.s. and E 
t = 0. Given the expression for the risky asset in equation
(4.15) we can approximate the covariance in equation (4.13) by
Cov
(
mv+1, R
i
v+1
)
≈ −(γμc + δ)μs Var (τΔv) − γμs Cov (BcΔv, τΔv)−
(γμc + δ)Cov
(
τΔv,
∫ Δv
0
√
audB
s
u
)
− γ Cov
(
BcΔv,
∫ Δv
0
√
audB
s
u
)
.
(4.18)
Thus, a risky asset is also influenced by the variance of τ as well as the covariance of τ
and BcΔv but additionally by two further covariances.3 Given the expression in equation
(4.17) we can simplify the additional covariances in equation 4.18. First, we get
Cov
(
τΔv,
∫ Δv
0
√
audB
s
u
)
= β Cov (τΔv, BsΔv) + Cov
(
τΔv,
∫ Δv
0

τ(u)dB
s
u
)
. (4.19)
Since 
 is independent of τ , the latter term is zero. Secondly, we get
Cov
(
BcΔv,
∫ Δv
0
√
audB
s
u
)
= E
(∫ Δv
0
β + 
τ(u)d [BcBs]u
)
= βρc,sΔv (4.20)
where we use that d [BcBs]u = ρc,sdu with correlation coefficient ρc,s and that E 
τ(u) = 0.
This is the systematic risk part in the classical asset pricing theory. For instance, if
β = σs/σc and
√
Δv = σc, i.e. volatility is constant, the covariance expression is
3Note that
Cov
(
−(γμc + δ)τΔv − γBcΔv, −
1
2
∫ v
0
audu
)
= Cov
(
−(γμc + δ)τΔv − γBcΔv, −
1
2
(
βv +
∫ v
0
τ(u)du
))
= 0
since  is independent of τ and Bc.
52
4.2 Equity premium and risk-free rate puzzle
σsσcρc,s. Summarizing equations (4.19) and (4.20), the stochastic volatility of the asset
does not influence asset prices as long as there is no persistent trend in the expectation
of the ratio in equation (4.17).
4.2 Equity premium and risk-free rate puzzle
In the previous section we show time-varying volatility in consumption-growth to
influence equity premia. Using inflation adjusted data for consumption-growth, U.S.
market returns and the risk-free rate as described in Shiller (1992) and available in
an updated version on Shiller’s homepage, we quantify the effects of the time-varying
volatility considering prevailing puzzles.
The equity premium puzzle of Mehra and Prescott (1985) and the risk-free rate
puzzle emphasized by Weil (1989) arise when consumption-growth is assumed to be
log-normally distributed. We restate the Hansen and Jagannathan (1991) bound in
the time-changed setting to determine the effect of stochastic volatility on the equity
premium puzzle. From equation (4.12) we derive
E
(
Riv+1
)
− Rfvv+1
Var (Riv+1)
= −ρmv+1,Riv+1
Var (mv+1)
E(mv+1)
, (4.21)
where ρmv+1,Riv+1 is the correlation between mv+1 and R
i
v+1. The expression R
fv
v+1 =
1/E(mv+1) is the time-free rate as shown in equation (4.5) being the return of a zero-
coupon bond expiring when the integrated variance exceeds a predefined level. With
the expression for the time-changed risk-free rate according to equation (4.12) we can
replace the time-free rate by
Rfvv+1 =
E
(
Rftv+1
)
1 − Cov
(
mv+1, R
ft
v+1
) , (4.22)
to obtain
∣∣∣∣∣∣
E
(
Riv+1 − Rftv+1
)
Var (Riv+1)
∣∣∣∣∣∣ ≤
√
Var (mv+1)
E(mv+1)
+
Cov
(
mv+1, R
ft
v+1
)
E(mv+1)
√
Var (Riv+1)
. (4.23)
53
4 How does stochastic volatility influence asset prices? - A parameter-free approach
with mv+1 = e−(γμ+δ)τΔv+γ
Δv
2 −γBcΔv . It is easy to check that the Hansen-Jagannathan
bound in general depends on the subjective discount factor β = e−δ when we consider
stochastic volatility. If volatility is constant the above expression collapses to the well
known results without β.
γ βsvrf β
ln
rf β
sv
HJ β
ln
HJ
1 1.0011 1.0001 [0, 0.3177] ∪ [1.2826, ∞) ∅
5 1.0684 1.0699 [0, 0.3625] ∪ [1.3451, ∞) ∅
10 1.1266 1.1321 [0, 0.5281] ∪ [1.3843, ∞) ∅
15 1.1486 1.1613 R+ R+
20 1.1276 1.1551 R+ R+
30 0.9490 1.0415 R+ R+
40 0.6443 0.8299 R+ R+
50 0.3352 0.5843 R+ R+
Table 4.1: Risk aversion parameters γ with corresponding intervals for the subjective
discount factor β solving the equity premium and risk-free rate puzzle; We take
consumption-growth to follow a geometric Brownian motion. The parameters βsvrf and
βlnrf are the subjective discount factors for the risk-free rate with stochastic volatility
and constant volatility, respectively. The parameters βsvHJ and βlnHJ are the subjective
discount factors for the Hansen-Jagannathan bound with stochastic volatility andd
constant volatility, respectively. We use inflation adjusted data for consumption-
growth, U.S. market returns and the risk-free rate as described in Shiller (1992) and
available in an updated version on Shiller’s homepage. The timespan is from 1889 to
2008.
Table 4.1 shows the value of the subjective discount factor β necessary to solve the
equity premium and risk-free rate puzzle, respectively. Assuming consumption-growth
follows a geometric Brownian motion with constant volatility, i.e. consumption-growth
is log-normally distributed, the subjective discount factor βlnrf is only smaller than one
for risk aversion greater than 32. With stochastic volatility the subjective discount
factor βsvrf is slightly smaller but still greater than one for risk aversion coefficients
smaller than 28. Thus, allowing for stochastic volatility for consumption-growth eases
the risk-free rate puzzle but does not solve it.
For the equity premium puzzle risk aversion must be as high as 12 in case of constant
volatility such that the Hansen-Jagannathan bound is greater than the Sharpe ratio
of the U.S. stock market which is about 0.38. In case of stochastic volatility the
Hansen-Jagannathan bound additionally depends on the subjective discount factor
54
4.3 Stochastic volatility as a factor
allowing to solve the puzzle even for low risk aversion coefficients. However, people
must be either very impatient or very patient such that the Hansen-Jagannathan bound
is sufficiently high. Indeed, the subjective discount factor must be βsvHJ ≥ 1.3451 or
βsvHJ ≤ 0.3625 having a risk aversion of five.
4.3 Stochastic volatility as a factor
Fama and French (1992) show the CAPM (Sharpe (1964), Lintner (1965) and Black
(1972)) as an one-factor model to perform poorly in describing cross-sectional average
returns. Extending the number of factors (Fama and French (1993, 2015)) improves
the performance in describing cross-sectional average returns.
In section 4.1.2 we show stochastic volatility to have an explanatory power for
cross-sectional average returns. Now we construct a factor SVOL mimicking the effect
of stochastic volatility on asset prices following the approach of Fama and French
(1993): Sorting stocks independently into 25 Size-Tau portfolios we show the effect of
time-varying volatility on asset prices to be neither captured by the three- or five-factor
model of Fama and French (1993, 2015) nor by the momentum factor as in the factor-
model of Carhart (1997). Adding the non-redundant factor SVOL to the five-factor
model generally improves the performance in describing average returns. The resulting
six-factor model especially helps to describe the average returns on the 25 Size-Prior
2–12 portfolios indicating that a significant part of the momentum effect results from
stochastic volatility.
Following the results of section 4.1.2 we can state a beta pricing model having
one factor equal to the well-known beta factor and one factor capturing the effect of
stochastic volatility for describing an assets i excess return over the risk-free rate:
E(Riv+1 − Rftv+1) = αv+1 + βi,τ · λτ,m + βi,B · λB,m, (4.24)
where βi,τ is the covariance between the stopping time τ and the assets i return Riv+1
divided by the variance of τ and βi,B is the covariance between the driving Brownian
motion BW of the wealth portfolio and the assets i return Riv+1 divided by the variance
of BW . The variables λτ,m and λB,m as well as the intercept αv+1 are independent of
the assets return and thus equal for all assets. The stopping time τ measures the time
55
4 How does stochastic volatility influence asset prices? - A parameter-free approach
passing by until the integrated variance of the wealth portfolio passes a predefined
threshold. The stock specific parameter βi,τ =: Tau captures the effects of stochastic
volatility of the wealth portfolio on the cross-section of expected stock returns. A more
detailed derivation can be found in appendix B.3.
Data and construction of SVOL
We follow the methodology in Fama and French (1993, 2015) to construct our factor
SVOL. At the end of each June we sort stocks to two Size (Small to Big), using the
median of market capitalization, and independently to three Tau groups (Low to High),
using medians of Tau for the 30th and 70th percentiles. The intersections of the two
sorts produce 6 value-weight Size-Tau portfolios. To increase the robustness of our
factor, we exclude the stocks falling in the 85% quantile of the absolute value of Tau.4
We consider all NYSE, AMEX, and NASDAQ stocks on CRSP with share codes 10
or 11 having no missing return value for the last 12 months covering the span July
1963 – December 2016. We calculate the Tau groups based on the prior 12 months by
calculating βi,τ for each stock and sorting all stocks according to their value of βi,τ . For
the calculation of βi,τ we time-change all returns with respecto to the realized variance
of the market portfolio. We compute the daily variance of the market portfolio using
an EGARCH(1,1)5 model on daily log-returns. The integrated variance is the sum of
the daily variances. Now, we linearly approximate the return of the market and each
stock i for a fixed unit of integrated variance. There is a trade off between having
enough data points to estimate the covariance and variance for βi,τ and having at least
one observation per integrated variance unit. We find it optimal to divide the total
integrated variance by 48 resulting in 48 time-changed returns Riv+1 and realizations of
τ . Dividing by a different number does not change the results significantly. Finally, we
compute βi,τ by computing Cov(τ, Ri) and Var(τ) using the 48 time-changed returns.
For the five-factors RM − RF , SMB, HML, RMW and CMA as well as the momentum
factor MOM, we take the data available on Kenneth French’s homepage.
4The exclusion of 15% of the stocks does not change the mean of our factor (0.40 compared to 0.39
when considering all stocks) but increases its stability. We mainly exclude small and illiquid stocks
for which the time-change can be inaccurate and subsequently the calculation of βi,τ . Although we
exclude 15% of the stocks, they only represent 4.5% of the total market capitalization on average.
5We test different GARCH models and find the EGARCH model to perform best regarding AIC and
BIC. However, the choice of GARCH model has only minor effects on our results.
56
4.3 Stochastic volatility as a factor
In panel A of table 4.2 we report the average monthly return of the five-factors
RM − RF , SMB, HML, RMW and CMA, the momentum factor MOM and the factor
SVOL with standard deviation and t-statistic. The average monthly return on SVOL is
significantly different from zero (t-statistic = 3.57), even by the standards of Harvey et al.
(2016). With an average monthly return of 0.4% it is comparable to the average return
of HML with 0.37%. Panel B shows the factors correlations. In Panel C we test SVOL
for redundancy by regressing the five-factors and the five-factors and MOM on SVOL.
The intercept is in both cases highly significant and leaves a monthly return of 0.54%
and 0.39% unexplained by the five-factor model and the five-factor model plus MOM,
respectively. Therefore, SVOL helps to explain average returns when added to the
five-factor model. When regressing the five-factors and SVOL on MOM, the intercept is
significantly reduced from 0.73 (t − statistic = 4.73) to 0.40 (t − statistic = 2.48) using
the five-factors only. Considering the large loading of SVOL of 0.62 (t−statistic = 9.52)
confirms it adds significantly to the five-factor model to explain the average return of
the MOM factor. We investigate this finding further by testing the performance of
SVOL for the 25 Size-Momentum sorted portfolios which is shown in tables 4.4 and
4.5. In panel D we show the monthly average returns in excess of the one-month U.S.
Treasury bill rate for 25 value-weighted Size-Tau sorted portfolios resulting from the
intersection of sorting stocks to five Size (Small to Big) and independently to five Tau
groups (Low to High) where we exclude the stocks falling in the 85% quantile of the
absolute value of Tau. The sample is all NYSE, AMEX, and NASDAQ stocks on CRSP
with share codes 10 or 11 having no missing return value for the last 12 months. In
each column returns tend to rise from Big to Small stocks showing the well-known size
effect but the effect is weak for Low and High Tau portfolios. Average excess returns
generally increase from Low to High Tau portfolios. For Big portfolios the returns
increase monotonically from 0.23% for low Tau to 0.80% for high Tau. Considering the
portfolios for small stocks the average excess return is 0.53% for Low Tau and 0.89%
for High Tau but the highest monthly excess return is earned by the second highest
Tau (4) portfolio with 1.22%.
57
4 How does stochastic volatility influence asset prices? - A parameter-free approach
Panel A: Averages, standard deviations and t-statistics for monthly returns
RM − RF SMB HML RMW CMA MOM SV OL
Mean 0.51 0.26 0.37 0.25 0.31 0.67 0.40
Std dev. 4.42 3.04 2.82 2.23 2.01 4.22 2.81
t-Statistic 2.92 2.19 3.32 2.79 3.88 3.99 3.57
Panel B: Correlation between different factors
RM − RF 1.00 0.28 -0.26 -0.23 -0.38 -0.13 0.32
SMB 0.28 1.00 -0.08 -0.35 -0.11 -0.03 0.28
HML -0.26 -0.08 1.00 0.07 0.70 -0.18 -0.42
RMW -0.23 -0.35 0.07 1.00 -0.03 0.11 -0.37
CMA -0.38 -0.11 0.70 -0.03 1.00 -0.02 -0.30
MOM -0.13 -0.03 -0.18 0.11 -0.02 1.00 0.29
SV OL 0.32 0.28 -0.42 -0.37 -0.30 0.29 1.00
Panel C: Using other factors in regressions to explain average returns on SVOL and MOM:
Int RM − RF SMB HML RMW CMA MOM SVOL
Coef 0.54 0.08 0.11 -0.36 -0.34 0.00
t-Statistic 5.64 3.50 3.38 -7.82 -7.63 0.02
R2 0.32
Coef 0.39 0.11 0.10 -0.25 -0.39 -0.08 0.20
t-Statistic 4.31 4.89 3.18 -5.64 -9.23 -1.21 9.52
R2 0.41
Using other factors in regressions to explain average returns on MOM:
Coef 0.40 -0.18 -0.00 -0.31 0.45 0.39 0.62
t-Statistic 2.48 -4.65 -0.05 -3.98 5.86 3.52 9.52
R2 0.20
Panel D: Size-Tau sorted portfolios:
Tau → Low 2 3 4 High
Small 0.53 0.78 0.95 1.22 0.89
2 0.49 0.81 0.96 0.97 0.91
3 0.48 0.64 0.88 0.90 1.00
4 0.47 0.68 0.73 0.80 0.83
Big 0.23 0.48 0.56 0.58 0.80
Table 4.2: Summary statistics for monthly factor percent returns; July 1963–December 2016, 642 months.
RM − RF is the value-weight return on the market portfolio of all sample stocks minus the one-month
Treasury bill rate. SMB, HML, RMW, CMA and MOM are the size, book-to-market, profitability,investment
and momentum factors as in Fama and French (2015, 2016). The factor SVOL is constructed by assigning
stocks to two Size groups and also assigning stocks independently to three Tau groups, using medians of Tau
for the 30th and 70th percentiles. To increase the robustness of our factor we exclude the stocks falling in
the 85% quantile of the absolute value of Tau. In Panel A, SVOL uses the value-weighted portfolios formed
from the intersection of the Size and Tau sorts (2× 3 = 6 portfolios). Panel A shows average monthly returns
(Mean), the standard deviations of monthly returns (Std dev.) and the t-statistics for the average returns.
Panel B shows the correlations for each set of factors. Panel C shows a test for redundancy of SVOL when
using the five factors RM − RF , SMB, HML, RMW and CMA or the five factors plus MOM in regressions
to explain SVOL. Panel D shows the intersections of the two sorts Size and Tau producing 25 value-weight
Size-Tau portfolios. The table shows averages of monthly returns in excess of the one-month Treasury bill
rate.
58
4.3 Stochastic volatility as a factor
Size-Tau sorted portfolios
Although panel C of table 4.2 shows that SVOL is not a redundant factor, we test if
the five-factor model can explain returns on the 5x5 Size-Tau sorted portfolios shown in
panel D. The results are given in table 4.3 where we regress the five-factors (RM − RF ,
SMB, HML, RMW and CMA) with and without SVOL on the excess returns on the 25
Size-Tau portfolios. In panel A we report the intercepts for the five-factor model. The
intercepts are significantly negative for the low Tau portfolios and significantly positive
for the high Tau portfolios indicating the five-factor model is not able to capture the
dispersion of excess returns on the 25 Size-Tau portfolios. This holds especially for the
portfolios with the Big stocks. Adding SVOL improves the performance of the five-factor
model as given in panel B. Except for the small and low Tau portfolio all intercepts
are not significantly different from zero. The loadings of the factors HML, RMW and
CMA show they cannot explain the average returns. HML is mostly decreasing with
increasing excess returns, RMW shows no monotonic behavior and CMA is mostly
insignificant. The loading on SVOL increase with increasing excess returns and is highly
significant.
Testing other anomalies
Based on the previous results we test if SVOL can help to improve the five-factor
model to explain anomalies targeted by the five factors. Additionally, we consider
the anomalies net share issues (Ikenberry et al., 1995), accruals (Sloan, 1996) and
momentum (Jegadeesh and Titman, 1993). We follow the procedure of Fama and
French (2016) in evaluating the performance of the different factor models. The results
are shown in table 4.4. For each of the 25 portfolio sets we test the five-factor model
and the five-factor model plus SVOL. In case of the 25 Size-Tau and 25 Size-Prior 2–12
portfolios we additionally report the five-factor model plus MOM and plus MOM and
SVOL. In the first column we show the test statistic GRS of Gibbons et al. (1989)
with corresponding p-value in the second column. In the following three columns we
evaluate the capability of the models to describe the dispersion of the returns resulting
from the 25 portfolio sorts. First, we show the average absolute intercept A|ai| of the
25 portfolios. Setting the dispersion of the intercepts in relation to relative dispersion
of the portfolio excess returns r¯i is defined as the average excess return of portfolio i
59
4 How does stochastic volatility influence asset prices? - A parameter-free approach
minus the average excess return of the value-weighted market. Thus, high values of
A|ai|
A|r¯i| and
Aa2i
Ar2i
mean the dispersion of the intercepts is relatively low compared to the
portfolios average returns. The ratio As2(ai)
Aa2i
is the average squared standard error of
the intercepts divided by the average squared intercepts indicating the influence of
estimation error on the dispersion of the intercepts.
For the 25 Size-Tau sorted portfolios the five-factor model fails the GRS test (GRS =
3.08) and it is unable to explain the dispersion of average excess returns with an average
absolute intercept of 0.203, A|ai|
A|r¯i| = 0.77 and
Aa2i
Ar2i
= 0.61. Including SVOL to the
five-factor model, the resulting six-factor model passes the GRS test on a 1% level
(GRS = 1.77;p − value = 0.012). With 0.084 the average absolute intercept is less
than half the value of the five-factor model. The model describes the dispersion of
average excess well (A|ai|
A|r¯i| = 0.32;
Aa2i
Ar2i
= 0.14) and the remaining dispersion is mostly due
to estimation error (As2(ai)
Aa2i
= 0.60). Adding MOM to the five-factor model improves
the performance but is still significantly worse than the six-factor model with SVOL.
Adding both, MOM and SVOL, to the five-factor model is only a small improvement
compared to the five-factor model plus SVOL.
Although adding SVOL to the five-factor model is unable to fully describe average
excess returns on the 25 Size-Prior 2–12 portfolios it significantly improves its perfor-
mance. For instance, the average absolute intercept decreases from 0.278 to 0.168 and
the dispersion statistic Aa
2
i
Ar2i
is more than cut by half from 0.79 to 0.34. These findings
are confirmed by looking at the regression results shown in table 4.5. The loading on
SVOL is mostly significant and increases from low Prior 2–12 to high Prior 2–12. This
works best for big stocks leaving no intercept significantly different from zero but loses
traction for smaller stocks. Based on these results we conclude that the momentum
effect is for bigger stocks for the most part and for smaller stocks for some part a result
of stochastic volatility.
For the value effect (Size-B/M), operating profitability (Size-OP), investment (Size-
Inv), accruals (Size-Accruals) and net share issues (Size-Net Shares Issued) adding
SVOL to the five-factors improves the performance regarding the GRS test as well as
the dispersion measures.
The closest factor to SVOL is CVR of Boguth and Kuehn (2013). It also aims
to capture the impact of stochastic consumption volatility but its construction and
60
4.3 Stochastic volatility as a factor
underlying assumptions are different and thus its empirical performance. Boguth and
Kuehn (2013) assume a consumption based asset pricing model where the first and
second moment of consumption growth follow a Markov chain. The mean and the
volatility have two states which they switch independently and the volatility process is
additionally assumed to be independent of the Brownian motion (cf. (4.1)). In contrast,
we do not make such restrictive assumptions about the stochastic process for volatility
nor its dependence on the Brownian motion for the consumption process. Using the
time-change preserves both effects.
To test whether CVR adds additional information about expected returns, we regress
CVR on the five-factors of Fama and French (2015):
CV R = − 0.24 − 0.05(RM − RF ) − 0.27SMB − 0.11HML − 0.53RMW − 0.12CMA + et (4.25)
(−1.80) (−1.67) (−6.27) (−1.89) (−8.93) (−1.28) R2 = 0.16
The five-factor model explains about 58% of the average return of CVR. The remaining
intercept (−0.24) is insignificant (t − statistic = −1.80) such that CVR adds little
to the explanation of average returns when added to the five-factor model.6 This is
confirmed by testing the performance of CVR considering the anomalies as for SVOL
in table 4.4. The results given in the appendix B.4 show that CVR does not improve
the performance of the five-factor model regarding other anomalies. In contrast SVOL
improves the performance of the five-factor model and its average return cannot be
explained by the five-factors nor momentum. Thus, SVOL and CVR contain different
information due to the differences in the construction.
6Adding the momentum factor further decreases the intercept to −0.16 (t − statistic = −1.23).
61
4 How does stochastic volatility influence asset prices? - A parameter-free approach
R(t) − RF (t) = a + b [RM (t) − RF (t)] + sSMB(t) + hHML(t) + rRMW (t) + cCMA(t) + τSV OL(t) + e(t)
Tau → Low 2 3 4 High Low 2 3 4 High
Panel A: Five-factor intercepts: RM − RF , SMB, HML, RMW and CMA
a t(a)
Small -0.18 0.08 0.30 0.55 0.28 -1.19 0.61 2.20 3.67 1.77
2 -0.30 -0.01 0.16 0.20 0.20 -3.04 -0.18 1.82 2.11 1.87
3 -0.43 -0.18 0.05 0.09 0.27 -4.92 -2.73 0.97 1.42 3.55
4 -0.38 -0.12 -0.09 0.01 0.14 -4.20 -2.04 -1.88 0.26 2.17
Big -0.41 -0.12 0.01 0.12 0.41 -3.63 -1.89 0.15 1.91 4.73
Panel B: Six-factor coefficients: RM − RF , SMB, HML, RMW, CMA and SVOL
a t(a)
Small -0.10 0.14 0.21 0.42 0.08 -0.61 0.96 1.51 2.77 0.51
2 -0.12 0.09 0.14 0.06 -0.04 -1.23 1.07 1.51 0.63 -0.39
3 -0.13 0.00 0.08 0.00 0.05 -1.86 -0.08 1.42 0.00 0.81
4 -0.04 0.05 -0.03 -0.01 0.02 -0.53 0.95 -0.53 -0.23 0.35
Big 0.05 0.07 0.02 -0.09 0.07 0.55 1.34 0.39 -1.71 1.08
h t(h)
Small 0.09 0.28 0.19 0.15 0.15 1.23 4.04 2.77 2.01 1.93
2 0.17 0.16 0.22 0.22 0.01 3.63 3.84 5.00 4.76 0.12
3 0.11 0.13 0.18 0.15 0.08 3.10 4.51 6.98 5.16 2.57
4 0.04 0.07 0.15 0.10 0.01 1.24 2.70 6.32 3.77 0.22
Big 0.00 0.05 -0.02 0.11 0.10 0.04 1.85 -0.64 4.21 3.39
r t(r)
Small -0.27 -0.04 -0.12 -0.19 -0.32 -3.61 -0.55 -1.84 -2.65 -4.21
2 -0.27 0.05 -0.05 -0.11 -0.27 -5.83 1.18 -1.25 -2.51 -5.72
3 -0.12 -0.03 0.03 0.04 -0.18 -3.59 -1.22 1.36 1.23 -5.45
4 -0.09 0.05 0.18 0.05 -0.07 -2.69 1.91 7.53 1.87 -2.52
Big -0.13 0.11 0.21 0.12 -0.09 -3.35 4.36 8.98 4.70 -2.93
c t(c)
Small 0.05 -0.17 0.02 0.02 -0.04 0.49 -1.76 0.25 0.18 -0.41
2 -0.05 0.13 0.08 0.06 0.21 -0.72 2.20 1.37 0.96 3.11
3 0.08 0.10 0.08 0.11 0.04 1.58 2.36 2.25 2.61 0.96
4 0.12 0.19 0.14 0.14 0.00 2.53 5.50 3.98 3.54 -0.02
Big 0.07 0.14 0.14 -0.04 -0.13 1.27 3.82 4.30 -1.15 -2.94
τ t(τ)
Small -0.16 -0.10 0.17 0.24 0.37 -2.53 -1.70 2.97 3.91 5.75
2 -0.33 -0.20 0.04 0.26 0.44 -8.72 -5.71 1.22 6.89 10.89
3 -0.54 -0.32 -0.05 0.16 0.40 -19.08 -13.46 -2.16 6.60 14.64
4 -0.63 -0.31 -0.13 0.05 0.22 -23.04 -14.92 -6.21 2.21 8.67
Big -0.85 -0.34 -0.02 0.39 0.63 -25.77 -16.20 -1.13 18.42 24.64
Table 4.3: Regressions for 25 value-weight Size-Tau portfolios; July 1963 to December 2016, 642 months. At
the end of June each year, stocks are allocated to five Size groups (Small to Big) and independently to five
Tau groups (Low Tau to High Tau). The intersections of the two sorts produce 25 Size-Tau portfolios. The
LHS variables in each set of 25 regressions are the monthly excess returns on the 25 Size-Tau portfolios. The
RHS variables are the excess market return, RM − RF , the Size factor, SMB, the value factor, HML, the
profitability factor, RMW, and the investment factor, CMA, constructed using independent 2x3 sorts on Size
and each of B/M, OP, Inv. and Tau. Panel A of the table shows five-factor intercepts produced by the Mkt,
SMB, HML, RMW and CMA. Panel B shows six-factor intercepts, slopes for HML, RMW, CMA and SVOL,
and t-statistics for these coefficients.
62
4.3 Stochastic volatility as a factor
GRS p(GRS) A|ai| A|ai|A|ri|
Aa2i
Ar2
i
As2(ai)
Aa2
i
A(R2)
25 Size-Tau portfolio
3.08 0.000 0.203 0.77 0.61 0.15 0.853
SVOL 1.77 0.012 0.084 0.32 0.14 0.60 0.876
MOM 2.52 0.000 0.166 0.63 0.43 0.21 0.857
MOM SVOL 1.76 0.013 0.087 0.33 0.15 0.55 0.877
Panel A: 25 Size-Momentum portfolio
4.42 0.000 0.278 0.86 0.79 0.08 0.855
SVOL 3.59 0.000 0.168 0.52 0.34 0.18 0.865
MOM 3.63 0.000 0.116 0.36 0.14 0.23 0.921
MOM SVOL 3.37 0.000 0.118 0.37 0.14 0.24 0.922
Panel A: 25 Size-B/M portfolio
3.25 0.000 0.094 0.37 0.16 0.28 0.918
SVOL 2.88 0.000 0.084 0.33 0.13 0.35 0.920
Panel A: 25 Size-OP portfolio
2.28 0.000 0.067 0.31 0.09 0.62 0.929
SVOL 2.04 0.002 0.060 0.28 0.09 0.65 0.930
Panel A: 25 Size-Inv portfolio
3.44 0.000 0.083 0.34 0.14 0.28 0.930
SVOL 2.90 0.000 0.075 0.30 0.12 0.34 0.931
Panel A: 25 Size-Accruals portfolio
3.77 0.000 0.121 0.55 0.32 0.23 0.914
SVOL 3.11 0.000 0.111 0.50 0.27 0.28 0.916
Panel A: 25 Size-Net Shares Issued portfolio
3.52 0.000 0.100 0.32 0.15 0.36 0.893
SVOL 3.25 0.000 0.099 0.31 0.15 0.37 0.895
Table 4.4: Summary statistics for tests of five-factor and five-factor plus SVOL models; July
1963–December 2016, 642 months. The table tests the ability of five-factor and five-factor
plus SVOL models to explain monthly excess returns on 25 Size-Tau portfolios, 25 Size-
Momentum portfolios, 25 Size-B/M portfolios, 25 Size-OP portfolios, 25 Size-Inv portfolios, 25
Size-Accruals portfolios and 25 Size-Net Shares Issued portfolios. For the 25 Size-Tau and the
25 Size-Momentum portfolios, we additionally report the five-factor model plus momentum
(MOM) and SVOL and MOM. For each set of 25 regressions, the table shows the GRS statistic
testing whether the expected values of all 25 intercept estimates are zero, the average absolute
value of the intercepts, A|ai|, A|ai|A|r¯i| , the average absolute value of the intercept ai over the
average absolute value of r¯i, which is the average return on portfolio i minus the average
value-weighted market portfolio excess return, Aa
2
i
Ar¯2
i
, the average squared intercept over the
average squared value of r¯2, As
2(ai)
Aa2
i
, the average of the estimates of the variances of the
sampling errors of the estimated intercepts over Aa2i , and A(R2), the average value of the
regression R2 corrected for degrees of freedom.
63
4 How does stochastic volatility influence asset prices? - A parameter-free approach
R(t) − RF (t) = a + b [RM (t) − RF (t)] + sSMB(t) + hHML(t) + rRMW (t) + cCMA(t) + τSV OL(t) + e(t)
Panel A: Five-factor intercepts: RM − RF , SMB, HML, RMW and CMA
a t(a)
Small -0.73 -0.21 0.06 0.25 0.59 -4.94 -2.7 1.02 3.90 6.30
2 -0.63 -0.20 -0.02 0.17 0.49 -4.56 -2.49 -0.43 3.19 5.64
3 -0.43 -0.19 -0.11 -0.06 0.51 -2.82 -2.35 -1.78 -1.01 5.38
4 -0.46 -0.23 -0.13 0.03 0.42 -2.78 -2.43 -1.95 0.51 4.05
Big -0.40 -0.12 -0.17 -0.04 0.32 -2.48 -1.23 -2.78 -0.60 3.06
Panel B: Six-factor coefficients: RM − RF , SMB, HML, RMW, CMA and SVOL
a t(a)
Small -0.62 -0.10 0.10 0.20 0.40 -4.09 -1.33 1.59 3.01 4.48
2 -0.42 -0.05 0.04 0.18 0.34 -3.05 -0.70 0.80 3.20 4.04
3 -0.20 -0.07 0.01 -0.04 0.36 -1.32 -0.82 0.16 -0.69 3.88
4 -0.21 -0.05 -0.02 0.05 0.25 -1.29 -0.57 -0.34 0.81 2.46
Big -0.17 0.04 -0.09 -0.07 0.13 -1.05 0.44 -1.46 -1.10 1.33
h t(h)
Small 0.40 0.35 0.32 0.22 0.03 5.42 9.46 10.5 6.89 0.61
2 0.28 0.27 0.23 0.18 -0.09 4.27 7.09 8.52 6.70 -2.10
3 0.27 0.27 0.24 0.21 -0.13 3.69 6.98 8.19 6.98 -2.98
4 0.32 0.18 0.21 0.09 -0.13 4.05 4.10 6.84 3.08 -2.57
Big 0.27 0.17 0.11 0.00 -0.10 3.42 3.72 3.74 -0.10 -1.95
r t(r)
Small -0.56 -0.05 0.12 0.13 0.01 -7.75 -1.31 4.15 4.12 0.21
2 -0.48 0.05 0.17 0.17 0.03 -7.32 1.30 6.26 6.53 0.63
3 -0.50 0.06 0.17 0.30 0.11 -6.92 1.44 5.85 10.05 2.58
4 -0.53 0.06 0.21 0.32 0.14 -6.83 1.48 6.66 10.50 2.90
Big -0.30 0.02 0.17 0.29 0.23 -3.97 0.33 5.62 9.42 4.70
c t(c)
Small -0.47 -0.07 0.01 0.11 0.08 -4.57 -1.35 0.28 2.44 1.31
2 -0.49 -0.10 0.02 0.08 0.01 -5.23 -1.95 0.61 2.24 0.21
3 -0.47 -0.10 0.01 0.09 0.04 -4.52 -1.85 0.32 2.12 0.59
4 -0.43 0.05 0.05 0.15 0.10 -3.86 0.77 1.13 3.46 1.46
Big -0.43 -0.04 0.00 0.21 -0.01 -3.87 -0.61 0.05 4.74 -0.13
τ t(τ)
Small -0.21 -0.19 -0.07 0.10 0.35 -3.52 -6.35 -2.68 3.86 9.66
2 -0.40 -0.27 -0.13 -0.01 0.27 -7.20 -8.55 -5.64 -0.37 7.93
3 -0.43 -0.23 -0.22 -0.03 0.27 -7.04 -7.13 -9.20 -1.36 7.27
4 -0.46 -0.33 -0.20 -0.04 0.31 -6.95 -8.95 -7.50 -1.44 7.67
Big -0.43 -0.30 -0.15 0.06 0.34 -6.65 -7.76 -6.08 2.34 8.32
Table 4.5: Regressions for 25 value-weight Size-Momentum portfolios; July 1963 to December 2016, 642
months. At the end of June each year, stocks are allocated to five Size groups (Small to Big) using NYSE
market cap breakpoints. Stocks are allocated independently to five Momentum groups (Low Tau to High Tau),
again using NYSE breakpoints. The intersections of the two sorts produce 25 Size-Momentum portfolios. The
LHS variables in each set of 25 regressions are the monthly excess returns on the 25 Size-Momentum portfolios.
The RHS variables are the excess market return, RM − RF , the Size factor, SMB, the value factor, HML, the
profitability factor, RMW, and the investment factor, CMA, constructed using independent 2x3 sorts on Size
and each of B/M, OP, Inv. and Tau. Panel A of the table shows five-factor intercepts produced by the Mkt,
SMB, HML, RMW and CMA. Panel B shows six-factor intercepts, slopes for HML, RMW, CMA and SVOL,
and t-statistics for these coefficients.
64
4.4 Conclusion
4.4 Conclusion
We use a time-changing technique to show the fundamental influence of stochastic
volatility for a consumption-based asset pricing model. Letting the stochastic discount
factor being the subjectively discounted marginal isoelastic utility, we show that if the
consumption process has a time-varying volatility, there is an additional risk correction
not only for risky assets but also for the risk-free rate. Indeed, an asset that performs
relatively well in times of highly volatile consumption growth has lower expected returns
than an asset that performs relatively poor in times of high consumption growth
volatility. Additionally, investors demand a higher expected return if consumption
growth is low in times of high consumption growth volatility. These results hold for
both, risky assets and the risk-free rate.
Furthermore, we test the influence of stochastic volatility on the risk-free rate and
the equity premium puzzle. The risk-free rate is lower than compared to assuming
consumption growth being log-normally distributed. However, we still need a high risk
aversion to get a subjective discount factor below one. Therefore, stochastic volatility
for consumption growth eases the risk-free rate puzzle but does not solve it. The results
for the equity premium puzzle, however, are promising: If volatility is assumed to be
time-varying the Hansen-Jagannathan bound also depends on the subjective discount
factor. Thus, the equity premium puzzle can be solved even for low one digit risk
aversions if investors are just impatient enough.
Finally, we test if the effects of stochastic volatility are included in the Fama and
French (2015) five-factor model or in the four-factor model of Carhart (1997). Construct-
ing a mimicking factor for stochastic volatility SVOL, we find that this effect is neither
included in the five-factor model nor in the five factor-model plus the momentum factor.
A sorting of 25 portfolios according to size and the the exposure to τ shows a wide range
of monthly excess returns of 0.23% up to 1.22%. The five-factor model with and without
the momentum factor can neither jointly explain the excess returns according to the
GRS test nor can explain the dispersion of returns. Adding the mimicking factor SVOL
to the five-factor model, the resulting six-factor model describes the excess returns well.
To show the usefulness of the additional factor SVOL we also check other anomalies. For
the 25 Size-Prior 2-12 portfolios (momentum) the six-factor model performs significantly
better than the five-factor model according to the GRS-test as well as to the measures
65
4 How does stochastic volatility influence asset prices? - A parameter-free approach
for dispersion. Although the intercepts are insignificant for bigger stocks the six-factor
model has still difficulties to explain the excess returns on the small portfolios. However,
based on the loadings of SVOL we conclude that the momentum effect is mainly a
result of the effects of stochastic volatility. For the value effect (Size-B/M), operating
profitability (Size-OP), investment (Size-Inv) and accruals (Size-Accruals) adding SVOL
to the five factors slightly improves the performance regarding the GRS test as well as
the measures for dispersion. Summarizing, our time-changing technique is a helpful
tool for theoretical as well as empirical asset pricing allowing to disentangle the risk of
time-varying volatility and return risk without assuming difficult and complex models
being intractable for empirical research and too restrictive for theoretical work.
66
5 Consumption volatility ambiguity and risk
premium’s time-variation
The following is based on Müller and Posch (2018a).
https://doi.org/10.1016/j.frl.2018.08.016 https://doi.org/10.1016/j.frl.2018.08.016
67
5 Consumption volatility ambiguity and risk premium’s time-variation
68
69
5 Consumption volatility ambiguity and risk premium’s time-variation
70
71
5 Consumption volatility ambiguity and risk premium’s time-variation
72
73
5 Consumption volatility ambiguity and risk premium’s time-variation
74
6 Price delay and market frictions in
cryptocurrency markets
The following is based on Köchling et al. (2019).
https://doi.org/10.1016/j.econlet.2018.10.025
75
6 Price delay and market frictions in cryptocurrency markets
76
77
6 Price delay and market frictions in cryptocurrency markets
78
79
6 Price delay and market frictions in cryptocurrency markets
80
Appendices
81

A Supplementary material for chapter 3
Additionally to the analysis of the whole sample, we also do robustness checks to our
results when using periods of high and low volatility since the jump detection tests
are sensitive to volatility. We split the sample from 2007-2011 in the period 2007-2008
and 2009-2011, where the first period is a period with high volatility and 2009-2011
a period of low volatility. Furthermore, we use additional jump detection tests: The
quadpower (QPV) and tripower (TPV) variation of Barndorff-Nielsen and Shephard
(2006), the MinRV of Andersen et al. (2012) as well as another configuration of the
JO test with MPVi(4, 6). As shown in tables A.3, A.4, A.5 and A.6 these robustness
checks lead to the same results as the analysis provided in section 3.2.1.
A.1 Further jump days information
In section 3.2.2 we argue that a higher turnover ratio is associated with larger jumps
for all jump tests. Indeed, comparing the average jump size for the different liquidity
buckets reveals that the sorting regarding the turnover leads to a higher average jump
size for liquid stocks than for illiquid stocks shown in tables A.1 and A.2. When sorting
regarding the bid-ask spread and ILLIQ yields exactly the opposite. The more liquid
a stock, the lower its average jump size. This behavior of the turnover ratio could give
further insights in the discussion whether it is a measure of liquidity or uncertainty as
studied in Barinov (2014).
83
A Supplementary material for chapter 3
QPV TPV medRV minRV
rel. Bid-Ask 1min 5 min 15 min 1 min 5 min 15 min 1 min 5 min 15 min 1 min 5 min 15 min
ΔJumps in %
Bucket (1) 0.51 0.90 1.13 0.52 0.90 1.14 0.66 1.02 1.23 0.67 1.03 1.31
Bucket (2) 0.63 1.06 1.34 0.65 1.07 1.33 0.84 1.21 1.52 0.84 1.18 1.58
Bucket (3) 0.80 1.32 1.67 0.81 1.34 1.67 1.06 1.58 1.92 1.08 1.64 2.25
Turnover
ΔJumps in %
Bucket (1) 0.78 1.34 1.63 0.80 1.36 1.62 1.01 1.54 1.87 1.02 1.55 2.02
Bucket (2) 0.61 1.04 1.35 0.62 1.05 1.34 0.81 1.22 1.51 0.82 1.20 1.60
Bucket (3) 0.54 0.89 1.16 0.55 0.91 1.18 0.71 1.03 1.29 0.74 1.09 1.45
ILLIQ
ΔJumps in %
Bucket (1) 0.52 0.93 1.19 0.54 0.93 1.19 0.66 1.03 1.28 0.67 1.02 1.32
Bucket (2) 0.61 1.05 1.32 0.62 1.06 1.30 0.80 1.20 1.48 0.81 1.20 1.59
Bucket (3) 0.80 1.29 1.62 0.81 1.31 1.64 1.10 1.58 1.93 1.13 1.64 2.23
Table A.1: BNS tests jump days information; This table displays information about jumps
detected by the four BNS tests from 2007 until 2011 at the one-, five- and fifteen-minute
frequency. The jump information is partitioned into three buckets based on their liquidity
level during the sample period. Bucket (1) contains the most liquid and bucket (3) the most
illiquid stocks. Jump sizes are given by the mean of all logarithmic returns in % within a
bucket and are denoted by "ΔJumps".
JO test LM test
p6 p4
rel. Bid-Ask 1min 5 min 15 min 1 min 5 min 15 min 1 min 5 min 15 min
ΔJumps in %
Bucket (1) 0.68 0.87 0.97 0.74 0.95 1.04 0.67 1.27 2.17
Bucket (2) 0.83 1.05 1.17 0.91 1.14 1.25 0.83 1.51 2.58
Bucket (3) 1.10 1.39 1.46 1.19 1.52 1.59 1.19 2.11 3.40
Turnover
ΔJumps in %
Bucket (1) 1.02 1.33 1.42 1.11 1.46 1.52 1.04 1.90 3.15
Bucket (2) 0.83 1.06 1.16 0.92 1.14 1.25 0.85 1.52 2.56
Bucket (3) 0.75 0.92 1.02 0.78 1.00 1.10 0.79 1.44 2.47
ILLIQ
ΔJumps in %
Bucket (1) 0.68 0.91 1.01 0.73 0.98 1.07 0.68 1.31 2.34
Bucket (2) 0.81 1.02 1.12 0.88 1.11 1.20 0.81 1.46 2.43
Bucket (3) 1.13 1.39 1.50 1.25 1.54 1.63 1.21 2.14 3.41
Table A.2: JO and LM tests jump days information; This table displays information about
jumps detected by the two JO tests and the LM test from 2007 until 2011 at the one-, five-
and fifteen-minute frequency. The jump information is partitioned into three buckets based on
their liquidity level during the sample period. Bucket (1) contains the most liquid and bucket
(3) the most illiquid stocks. Jump sizes are given by the mean of all logarithmic returns in %
within a bucket and are denoted by "ΔJumps".
84
A.2 Tables for bucket analysis for split sample
A.2 Tables for bucket analysis for split sample
85
A Supplementary material for chapter 3
Table
A
.3:BN
S
jum
p
test
analysis
(2007
-2008)
Thistable
sum
m
arizesthe
m
ean
num
berofa
stock’sjum
p
daysduring
the
tim
e
period
from
2007
until2008
(504
trading
days)according
to
theirliquidity
bucketforthe
differentBN
S
tests.The
jum
p
daysare
calculated
on
observationsbased
on
a
one-,five-and
fifteen-m
inute
tim
e
fram
e,respectively.
Each
stock
is
categorized
by
its
liquidity
levelw
hich
is
given
by
the
m
ean
ofrelative
bid-ask
spread,turnover
and
I
L
L
I
Q
during
this
period.
Bucket
(1)
offers
the
highest
degree
ofliquidity
and
bucket
(3)
the
lowest.
A
dditionally,the
p-values
of
an
A
N
O
VA
analysis
are
reported.
T
he
p-values
ofTukey’s
H
SD
test
to
highlight
the
differences
between
the
buckets
are
only
reported
if
there
is
a
significant
p-value
ofthe
A
N
O
VA
test.
N
is
the
num
ber
ofstocks
w
ith
an
available
liquidity
m
easure
and
jum
p
days
for
each
year
in
the
period.
***,**,*
denote
statisticalsignificance
at
the
1%
,5%
and
10%
level,respectively.
Q
PV
T
PV
m
edRV
m
inRV
rel.
B
id-A
sk
1
m
in
5
m
in
15
m
in
1
m
in
5
m
in
15
m
in
1
m
in
5
m
in
15
m
in
1
m
in
5
m
in
15
m
in
Bucket
(1)
142.2
38.0
19.7
123.0
34.1
17.9
69.9
27.8
15.4
38.8
14.9
4.3
Bucket
(2)
150.2
38.0
19.0
130.8
33.9
17.7
69.6
27.6
15.2
41.7
14.6
5.2
Bucket
(3)
172.5
43.5
22.0
152.9
37.5
19.9
82.7
28.4
16.7
50.4
15.2
6.1
A
N
O
VA
0.1803
0.1738
0.007 ∗∗∗
0.142
0.288
0.014 ∗∗
0.1513
0.615
0.0025 ∗∗∗
0.0563 ∗
0.6384
0.0001 ∗∗∗
(1)-(2)
0.9803
0.9951
0.9706
0.8255
0.1336
(1)-(3)
0.02
0.0242
0.0047
0.0543
0.00004
(2)-(3)
0.0124
0.0319
0.0098
0.1894
0.0248
N
130
130
130
130
130
130
130
130
130
130
130
130
Turnover
Bucket
(1)
147.5
40.8
21.4
130.5
35.6
19.4
71.7
27.4
16.1
43.4
14.7
5.7
Bucket
(2)
150.3
37.5
18.6
130.4
33.6
16.9
71.4
27.6
15.2
41.2
14.9
4.9
Bucket
(3)
186.4
44.5
22.3
164.0
38.8
20.8
83.9
29.9
16.5
48.6
15.8
5.2
A
N
O
VA
0.0369 ∗∗
0.2108
0.0954 ∗
0.0487 ∗∗
0.267
0.0521
∗
0.1473
0.3063
0.3947
0.2856
0.5967
0.1775
(1)-(2)
0.985
0.1091
0.9999
0.0944
(1)-(3)
0.0541
0.9441
0.0828
0.9987
(2)-(3)
0.0794
0.203
0.0809
0.0828
N
142
141
137
142
141
137
142
141
137
142
141
137
I
L
L
I
Q
Bucket
(1)
116.0
32.9
18.1
101.0
29.8
16.7
60.2
25.7
14.6
33.9
13.4
4.5
Bucket
(2)
149.3
38.3
20.0
128.7
33.9
18.6
72.7
27.5
16.7
42.5
15.2
4.8
Bucket
(3)
220.4
51.9
24.1
196.6
44.6
21.8
94.5
31.7
16.6
57.1
16.8
6.5
A
N
O
VA
0.0000 ∗∗∗
0.0000 ∗∗∗
0.0000 ∗∗∗
0.0000 ∗∗∗
0.0000 ∗∗∗
0.0000 ∗∗∗
0.0000 ∗∗∗
0.0000 ∗∗∗
0.0001 ∗∗∗
0.0000 ∗∗∗
0.0005 ∗∗∗
0.0000 ∗∗∗
(1)-(2)
0.0556
0.2931
0.3996
0.0989
0.3183
0.3257
0.1456
0.4393
0.0544
0.1154
0.1322
0.8086
(1)-(3)
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0001
0.0000
0.0003
0.0000
(2)-(3)
0.0000
0.0002
0.0000
0.0000
0.0003
0.0001
0.0043
0.0038
0.0894
0.0031
0.0903
0.0000
N
142
141
137
142
141
137
142
141
137
142
141
137
86
A.2 Tables for bucket analysis for split sample
Ta
bl
e
A
.4
:B
N
S
ju
m
p
te
st
an
al
ys
is
(2
00
9
-2
01
1)
Th
is
ta
bl
e
su
m
m
ar
ize
st
he
m
ea
n
nu
m
be
ro
fa
st
oc
k’
sj
um
p
da
ys
du
rin
g
th
e
tim
e
pe
rio
d
fro
m
20
09
un
til
20
11
(7
56
tr
ad
in
g
da
ys
)a
cc
or
di
ng
to
th
eir
liq
ui
di
ty
bu
ck
et
fo
rt
he
di
ffe
re
nt
BN
S
te
st
s.
Th
e
ju
m
p
da
ys
ar
e
ca
lcu
la
te
d
on
ob
se
rv
at
io
ns
ba
se
d
on
a
on
e-
,fi
ve
-a
nd
fif
te
en
-m
in
ut
e
tim
e
fra
m
e,
re
sp
ec
tiv
el
y.
Ea
ch
st
oc
k
is
ca
te
go
riz
ed
by
its
liq
ui
di
ty
le
ve
lw
hi
ch
is
gi
ve
n
by
th
e
m
ea
n
of
re
la
tiv
e
bi
d-
as
k
sp
re
ad
,t
ur
no
ve
r
an
d
I
L
L
I
Q
du
rin
g
th
is
pe
rio
d.
Bu
ck
et
(1
)
off
er
s
th
e
hi
gh
es
t
de
gr
ee
of
liq
ui
di
ty
an
d
bu
ck
et
(3
)
th
e
lo
we
st
.
A
dd
iti
on
al
ly
,t
he
p-
va
lu
es
of
an
A
N
O
VA
an
al
ys
is
ar
e
re
po
rt
ed
.
T
he
p-
va
lu
es
of
Tu
ke
y’
s
H
SD
te
st
to
hi
gh
lig
ht
th
e
di
ffe
re
nc
es
be
tw
ee
n
th
e
bu
ck
et
s
ar
e
on
ly
re
po
rt
ed
if
th
er
e
is
a
sig
ni
fic
an
t
p-
va
lu
e
of
th
e
A
N
O
VA
te
st
.
N
is
th
e
nu
m
be
r
of
st
oc
ks
w
ith
an
av
ai
la
bl
e
liq
ui
di
ty
m
ea
su
re
an
d
ju
m
p
da
ys
fo
r
ea
ch
ye
ar
in
th
e
pe
rio
d.
**
*,
**
,*
de
no
te
st
at
ist
ic
al
sig
ni
fic
an
ce
at
th
e
1%
,5
%
an
d
10
%
le
ve
l,
re
sp
ec
tiv
el
y.
Q
PV
T
PV
m
ed
RV
m
in
RV
re
l.
B
id
-A
sk
1
m
in
5
m
in
15
m
in
1
m
in
5
m
in
15
m
in
1
m
in
5
m
in
15
m
in
1
m
in
5
m
in
15
m
in
Bu
ck
et
(1
)
14
1.
9
48
.5
27
.7
12
3.
4
43
.5
25
.0
78
.3
39
.5
23
.1
41
.9
20
.7
8.
0
Bu
ck
et
(2
)
18
6.
7
52
.2
30
.4
16
3.
8
45
.9
28
.4
82
.9
40
.8
25
.2
46
.4
22
.3
7.
6
Bu
ck
et
(3
)
34
2.
6
79
.0
37
.4
31
2.
1
68
.3
33
.3
10
2.
0
42
.3
26
.9
59
.0
21
.3
8.
5
A
N
O
VA
0.
00
00
∗∗
∗
0.
00
00
∗∗
∗
0.
00
09
∗∗
∗
0.
00
00
∗∗
∗
0.
00
00
∗∗
∗
0.
00
18
∗∗
∗
0.
03
56
∗∗
0.
5
0.
04
63
∗∗
0.
02
22
∗∗
0.
50
56
0.
38
35
(1
)-
(2
)
0.
19
27
0.
77
58
0.
72
8
0.
21
72
0.
85
76
0.
45
26
0.
87
67
0.
55
54
0.
75
13
(1
)-
(3
)
0.
00
00
0.
00
00
0.
00
11
0.
00
00
0.
00
00
0.
00
14
0.
03
84
0.
03
64
0.
02
08
(2
)-
(3
)
0.
00
00
0.
00
00
0.
01
19
0.
00
00
0.
00
00
0.
04
52
0.
11
79
0.
30
01
0.
11
73
N
13
3
13
3
13
3
13
3
13
3
13
3
13
3
13
3
13
3
13
3
13
3
13
3
Tu
rn
ov
er
Bu
ck
et
(1
)
21
0.
4
60
.0
31
.0
18
8.
8
53
.5
28
.1
80
.6
40
.5
25
.1
46
.8
21
.5
7.
1
Bu
ck
et
(2
)
21
4.
4
58
.7
32
.0
19
1.
4
51
.3
29
.1
86
.9
40
.9
24
.3
49
.0
21
.6
8.
2
Bu
ck
et
(3
)
28
7.
6
70
.3
36
.7
25
8.
3
59
.9
32
.9
10
8.
5
41
.3
27
.6
59
.8
22
.0
9.
3
A
N
O
VA
0.
02
57
∗∗
0.
20
12
0.
24
95
0.
03
44
∗∗
0.
31
18
0.
30
78
0.
03
3∗
∗
0.
93
9
0.
11
1
0.
18
98
0.
94
59
0.
06
69
∗
(1
)-
(2
)
0.
99
12
0.
99
55
0.
83
13
0.
63
14
(1
)-
(3
)
0.
04
16
0.
05
54
0.
03
34
0.
05
51
(2
)-
(3
)
0.
05
66
0.
06
84
0.
12
74
0.
32
47
N
14
5
14
5
14
3
14
5
14
5
14
3
14
5
14
5
14
3
14
5
14
5
14
3
I
L
L
I
Q
Bu
ck
et
(1
)
15
1.
9
49
.1
29
.2
13
3.
0
43
.7
26
.5
75
.2
38
.1
23
.6
41
.8
20
.8
7.
6
Bu
ck
et
(2
)
20
1.
5
53
.3
30
.3
17
7.
5
47
.2
28
.2
80
.1
40
.6
25
.0
45
.1
21
.1
8.
1
Bu
ck
et
(3
)
36
2.
1
87
.3
40
.2
33
1.
0
74
.4
35
.6
12
1.
2
44
.2
28
.4
69
.1
23
.3
8.
9
A
N
O
VA
0.
00
00
∗∗
∗
0.
00
00
∗∗
∗
0.
00
02
∗∗
∗
0.
00
00
∗∗
∗
0.
00
00
∗∗
∗
0.
00
07
∗∗
∗
0.
00
00
∗∗
∗
0.
02
63
∗∗
0.
00
16
∗∗
0.
00
03
∗∗
∗
0.
13
8
0.
20
74
(1
)-
(2
)
0.
15
01
0.
77
05
0.
81
52
0.
17
9
0.
76
24
0.
65
0.
88
54
0.
50
45
0.
38
0.
88
96
(1
)-
(3
)
0.
00
00
0,
00
00
0.
00
03
0.
00
00
0.
00
00
0.
00
07
0.
00
01
0.
02
0.
00
11
0.
00
07
(2
)-
(3
)
0.
00
00
0.
00
00
0.
00
28
0.
00
00
0.
00
00
0.
01
34
0.
00
04
0.
24
35
0.
05
82
0.
00
32
N
14
5
14
5
14
3
14
5
14
5
14
3
14
5
14
5
14
3
14
5
14
5
14
3
87
A Supplementary material for chapter 3
Table
A
.5:JO
and
LM
jum
p
test
analysis
(2007-2008)
Thistable
sum
m
arizesthe
m
ean
num
berofa
stock’sjum
p
daysduring
the
tim
e
period
from
2007
until2008
(504
trading
days)according
to
their
liquidity
bucket
for
the
two
JO
tests
and
the
LM
test.
T
he
jum
p
days
are
calculated
on
observations
based
on
a
one-,five-and
fifteen-m
inute
tim
e
fram
e,respectively.
Each
stock
is
categorized
by
its
liquidity
levelw
hich
is
given
by
the
m
ean
ofrelative
bid-ask
spread,turnover
and
I
L
L
I
Q
during
this
period.
Bucket
(1)
offers
the
highest
degree
ofliquidity
and
bucket
(3)
the
lowest.
A
dditionally,
the
p-values
ofan
A
N
O
VA
analysis
are
reported.
T
he
p-values
ofTukey’s
H
SD
test
to
highlight
the
differences
between
the
buckets
are
only
reported
ifthere
is
a
significantp-value
ofthe
A
N
O
VA
test.
N
is
the
num
ber
ofstocks
with
an
available
liquidity
m
easure
and
jum
p
days
for
each
year
in
the
period.
***,**,*
denote
statisticalsignificance
at
the
1%
,5%
and
10%
level,respectively.
JO
test
LM
test
p6
p4
rel.
B
id-A
sk
1m
in
5
m
in
15
m
in
1
m
in
5
m
in
15
m
in
1
m
in
5
m
in
15
m
in
Bucket
(1)
95.4
85.8
110.0
67.5
60.0
75.5
216.1
50.5
11.9
Bucket
(2)
98.8
86.6
103.8
69.0
61.6
71.8
214.9
45.3
9.8
Bucket
(3)
102.2
89.4
102.2
69.5
61.4
70.3
221.9
49.0
11.4
A
N
O
VA
0.461
0.2513
0.1716
0.8982
0.6003
0.286
0.7693
0.0361 ∗∗
0.0267 ∗∗
(1)-(2)
0.9788
0.9748
(1)-(3)
0.072
0.0523
(2)-(3)
0.0519
0.0398
N
130
130
130
130
130
130
130
130
125
Turnover
Bucket
(1)
89.0
84.9
102.7
59.9
57.8
72.0
204.0
48.5
12.0
Bucket
(2)
101.1
87.7
103.8
71.3
62.4
70.1
220.4
47.9
10.3
Bucket
(3)
108.8
93.0
114.7
74.7
63.8
78.2
230.6
51.2
11.3
A
N
O
VA
0.0004 ∗∗∗
0.1846
0.139
0.0008 ∗∗∗
0.166
0.1266
0.00205 ∗∗
0.7745
0.1108
(1)-(2)
0.0379
0.0125
0.1919
(1)-(3)
0.0003
0.0009
0.0161
(2)-(3)
0.2704
0.6805
0.5347
N
142
141
137
142
141
137
142
132
125
I
L
L
I
Q
Bucket
(1)
87.4
78.5
105.3
62.1
56.2
72.9
195.0
45.3
11.5
Bucket
(2)
101.3
89.4
110.8
72.4
63.8
75.3
225.2
51.1
10.3
Bucket
(3)
110.4
97.9
104.7
71.2
64.2
71.7
235.1
51.1
11.9
A
N
O
VA
0.0000 ∗∗∗
0.0000 ∗∗∗
0.0023 ∗∗∗
0.0248 ∗∗
0.0004 ∗∗∗
0.0092 ∗∗∗
0.0001 ∗∗∗
0.0000 ∗∗∗
0.0001 ∗∗∗
(1)-(2)
0.0121
0.0021
0.2153
0.033
0.0063
0.5762
0.0029
0.0053
0.9142
(1)-(3)
0.0000
0.0000
0.0014
0.0752
0.0007
0.0073
0.0001
0.0000
0.0026
(2)-(3)
0.1481
0.0032
0.1281
0.9567
0.7367
0.0879
0.5239
0.0057
0.0082
N
142
141
137
142
141
137
142
132
125
88
A.2 Tables for bucket analysis for split sample
Ta
bl
e
A
.6
:J
O
an
d
LM
ju
m
p
te
st
an
al
ys
is
(2
00
9
-2
01
1)
Th
is
ta
bl
e
su
m
m
ar
ize
st
he
m
ea
n
nu
m
be
ro
fa
st
oc
k’
sj
um
p
da
ys
du
rin
g
th
e
tim
e
pe
rio
d
fro
m
20
09
un
til
20
11
(7
56
tr
ad
in
g
da
ys
)a
cc
or
di
ng
to
th
ei
r
liq
ui
di
ty
bu
ck
et
fo
r
th
e
tw
o
JO
te
st
s
an
d
th
e
LM
te
st
.
T
he
ju
m
p
da
ys
ar
e
ca
lc
ul
at
ed
on
ob
se
rv
at
io
ns
ba
se
d
on
a
on
e-
,fi
ve
-a
nd
fif
te
en
-m
in
ut
e
tim
e
fr
am
e,
re
sp
ec
tiv
el
y.
Ea
ch
st
oc
k
is
ca
te
go
riz
ed
by
its
liq
ui
di
ty
le
ve
lw
hi
ch
is
gi
ve
n
by
th
e
m
ea
n
of
re
la
tiv
e
bi
d-
as
k
sp
re
ad
,t
ur
no
ve
r
an
d
I
L
L
I
Q
du
rin
g
th
is
pe
rio
d.
Bu
ck
et
(1
)
off
er
s
th
e
hi
gh
es
t
de
gr
ee
of
liq
ui
di
ty
an
d
bu
ck
et
(3
)
th
e
lo
we
st
.
A
dd
iti
on
al
ly
,
th
e
p-
va
lu
es
of
an
A
N
O
VA
an
al
ys
is
ar
e
re
po
rt
ed
.
T
he
p-
va
lu
es
of
Tu
ke
y’
s
H
SD
te
st
to
hi
gh
lig
ht
th
e
di
ffe
re
nc
es
be
tw
ee
n
th
e
bu
ck
et
s
ar
e
on
ly
re
po
rt
ed
if
th
er
e
is
a
sig
ni
fic
an
tp
-v
al
ue
of
th
e
A
N
O
VA
te
st
.
N
is
th
e
nu
m
be
r
of
st
oc
ks
wi
th
an
av
ai
la
bl
e
liq
ui
di
ty
m
ea
su
re
an
d
ju
m
p
da
ys
fo
r
ea
ch
ye
ar
in
th
e
pe
rio
d.
**
*,
**
,*
de
no
te
st
at
ist
ic
al
sig
ni
fic
an
ce
at
th
e
1%
,5
%
an
d
10
%
le
ve
l,
re
sp
ec
tiv
el
y.
JO
te
st
LM
te
st
p6
p4
re
l.
B
id
-A
sk
1
m
in
5
m
in
15
m
in
1
m
in
5
m
in
15
m
in
1
m
in
5
m
in
15
m
in
Bu
ck
et
(1
)
12
0.
4
12
5.
7
15
7.
5
88
.0
92
.0
11
0.
1
30
7.
3
70
.1
12
.5
Bu
ck
et
(2
)
12
9.
4
13
1.
6
16
8.
3
90
.1
92
.8
11
6.
3
32
3.
7
76
.7
13
.5
Bu
ck
et
(3
)
12
5.
9
13
5.
7
16
5.
1
75
.7
87
.1
11
4.
9
29
7.
2
75
.2
15
.0
A
N
O
VA
0.
30
95
0.
02
9∗
∗
0.
15
42
0.
00
35
∗∗
∗
0.
12
97
0.
35
62
0.
08
92
∗
0.
04
98
∗∗
0.
00
17
∗∗
∗
(1
)-
(2
)
0.
25
2
0.
87
75
0.
36
31
0.
49
06
0.
91
42
(1
)-
(3
)
0.
02
25
0.
02
04
0.
68
22
0.
03
85
0.
00
26
(2
)-
(3
)
0.
51
3
0.
00
49
0.
07
53
0.
35
95
0.
00
82
N
13
3
13
3
13
3
13
3
13
3
13
3
13
3
13
3
13
3
Tu
rn
ov
er
Bu
ck
et
(1
)
11
9.
3
13
5.
8
16
3.
6
79
.7
93
.1
11
4.
0
30
5.
5
78
.6
14
.2
Bu
ck
et
(2
)
12
4.
8
12
6.
3
16
1.
5
84
.6
89
.5
11
3.
3
30
9.
7
71
.1
12
.9
Bu
ck
et
(3
)
13
4.
3
13
4.
1
16
6.
1
88
.0
88
.6
11
5.
8
30
6.
5
71
.7
14
.4
A
N
O
VA
0.
02
46
∗∗
0.
02
96
∗∗
0.
10
11
0.
17
85
0.
26
4
0.
26
04
0.
93
23
0.
02
5∗
∗
0.
39
49
(1
)-
(2
)
0.
56
83
0.
03
34
0.
06
44
(1
)-
(3
)
0.
01
93
0.
88
95
0.
03
61
(2
)-
(3
)
0.
19
88
0.
10
62
0.
97
27
N
14
5
14
5
14
3
14
5
14
5
14
3
14
4
13
8
13
4
I
L
L
I
Q
Bu
ck
et
(1
)
11
3.
1
12
3.
7
16
2.
9
80
.3
87
.9
11
2.
6
28
6.
6
67
.1
12
.6
Bu
ck
et
(2
)
12
8.
0
13
3.
0
16
3.
6
87
.8
92
.9
11
4.
7
32
0.
9
77
.6
13
.3
Bu
ck
et
(3
)
13
7.
4
13
9.
7
16
4.
7
84
.0
90
.4
11
5.
9
31
4.
2
76
.8
15
.6
A
N
O
VA
0.
00
00
∗∗
∗
0.
00
01
∗∗
∗
0.
42
02
0.
24
1
0.
22
73
0.
18
7
0.
00
73
∗∗
∗
0.
00
11
∗∗
∗
0.
00
19
∗∗
∗
(1
)-
(2
)
0.
01
37
0.
02
82
0.
00
86
0.
02
6
0.
83
26
(1
)-
(3
)
0.
00
00
0.
00
00
0.
04
37
0.
00
1
0.
00
23
(2
)-
(3
)
0.
17
75
0.
16
0.
82
68
0.
52
26
0.
01
37
N
14
5
14
5
14
3
14
5
14
5
14
3
14
4
13
8
13
4
89

B Appendix for chapter 4
B.1 Time-change of the stochastic volatility model
Here we derive the time-changed version of Ct to show the principle mechanism. We
start with
d ln(Ct) = μdt +
√
VtdW
c
t −
1
2Vtdt (B.1)
which is equivalent to writing
ln(Ct) = ln(C0) + μt +
∫ t
0
√
VudW
c
u −
1
2
∫ t
0
Vudu. (B.2)
Now we replace t by the stopping time τv = inf{t ≥ 0; [M ]t ≥ v} with Mt =
∫ t
0
√
VudW
c
u.
Thus the stopping time is defined as τv = inf{t ≥ 0; ∫ t0 Vudu ≥ v} and the stochastic
process Cτ(v) writes as
ln(Cτ(v)) = ln(C0) + μτ(v) +
∫ τ(v)
0
√
VudW
c
u −
1
2
∫ τ(v)
0
Vudu. (B.3)
According to the Dambis, Dubins & Schwarz Theorem as stated in Karatzas and Shreve
(1991) we have
Mτ(v) =
∫ τ(v)
0
√
VudW
c
u =: Bcv (B.4)
where Bcv is a standard Brownian motion. With
∫ τ(v)
0 Vudu = v by construction of the
stopping time τ(v), the differentiate expression of (B.3) writes
ln(Cτ(v)) = μτv + Bcv −
v
2 . (B.5)
91
B Appendix for chapter 4
This is the time-changed version Cτ stated in (4.4).
B.2 Time change of the risky asset
To derive the time-changed version of the risky asset, we start with
d ln(Sst ) = μsdt +
√
V st dW
s
t −
1
2V
s
t dt
⇔ ln(Sst ) = ln(Ss0) + μst +
∫ t
0
√
V su dW
s
u −
1
2
∫ t
0
V su du. (B.6)
We define M st =
∫ t
0
√
V cu dW
s
u where V c is the variance process of the consumption process.
Then the stopping time for the time-change is given by τv = inf{t ≥ 0; ∫ t0 V cu du ≥ v}
and it holds
∫ τ(v)
0
√
V su dW
s
u =
∫ τ(v)
0
√
V su
V cu
d
∫ u
0
√
V cy dW
s
y
=
∫ τ(v)
0
√
V su
V cu
dM su. (B.7)
Taking the differentiate representation we get
√√√√V sτ(v)
V cτ(v)
dM sτ(v) =
√√√√V sτ(v)
V cτ(v)
dBsv (B.8)
where Bsv is a standard Brownian motion according to the Dambis, Dubins & Schwarz
Theorem. With equation (B.3) and av =
V s
τ(v)
V c
τ(v)
it holds
ln(Ssτ(v)) = μsτv −
1
2
∫ v
0
audu +
∫ v
0
√
audB
s
u. (B.9)
This is the stated time-changed version of the risky asset.
92
B.3 Beta pricing model with wealth portfolio
B.3 Beta pricing model with wealth portfolio
Recall, that any assets expected return can be written as
E(Riv+1) = R
fv
v+1 − Rfvv+1 Cov(mv+1, Riv+1). (B.10)
We linearize the stochastic discount factor doing a Taylor approximation mv+1 =
av+1 + bv+1RWv+1 where RWv+1 is the return on the wealth portfolio. Using the above
equation to the expected return on the wealth portfolio gives
E(RWv+1) = R
fv
v+1 − Rfvv+1 Cov(av+1 + bv+1RWv+1, RWv+1)
= Rfvv+1 − Rfvv+1bv+1 Var(RWv+1) (B.11)
and solving for the constant bv+1 we get
bv+1 = −E(R
W
v+1 − Rfvv+1)
Rfvv+1 Var(RWv+1)
. (B.12)
With a positive expected excess return on the wealth portfolio and a positive time-free
rate the constant bv+1 is always negative. Assuming that the wealth portfolio follows a
geometric Brownian motion with stochastic volatility, i.e.
RWt = RW0 eμ
W t− 12
∫ t
0 V
W
u du+
∫ t
0
√
V Wu dW
W
u , the time-changed process is given by
RWv = RW0 eμ
W τv− v2+BWv with τv = inf{t ≥ 0; ∫ t0 V Wu du ≥ v}. If the exponent is suffi-
ciently small we can approximate the covariance by
Cov(RWv+1, Riv+1) ≈
Cov(τv+1, Riv+1)
Var(τv+1)
· μ
W Var(τv+1)
E(mv+1)
+ Cov(B
W
v+1, R
i
v+1)
Var(BWv+1)
· Var(B
W
v+1)
E(mv+1)
.
(B.13)
With the expression for the time-free rate
Rfvv+1 =
E
(
Rftv+1
)
1 − Cov
(
mv+1, R
ft
v+1
) , (B.14)
93
B Appendix for chapter 4
we get the beta pricing model
E(Riv+1 − Rftv+1) = αv+1 + βi,τ · λτ,m + βi,BW · λBW ,m, (B.15)
where αv+1 = Rftv+1
Cov(mv+1,Rftv+1)
1−Cov(mv+1,Rftv+1)
, λτ,m and λBW ,m are equal for all assets. The factor
βi,τ is the covariance between the stopping time τ and the assets i return Riv+1 divided
by the variance of τ and βi,BW is the covariance between the driving Brownian motion
BWv of the wealth portfolio and the assets i return Riv+1 divided by the variance of BWv
which is just v since BWv is a standard Brownian motion.
B.4 Testing anomalies and CVR
94
B.4 Testing anomalies and CVR
GRS p(GRS) A|ai| A|ai|A|ri|
Aa2i
Ar2
i
As2(ai)
Aa2
i
A(R2)
Panel A: 25 Size-Tau portfolio
2.69 0.000 0.213 0.71 0.50 0.16 0.857
SVOL 1.44 0.079 0.075 0.25 0.09 0.86 0.881
CVR 2.73 0.000 0.218 0.73 0.51 0.16 0.859
Panel A: 25 Size-Momentum portfolio
4.22 0.000 0.284 0.82 0.70 0.09 0.854
SVOL 3.43 0.000 0.161 0.46 0.29 0.23 0.865
CVR 4.12 0.000 0.273 0.78 0.63 0.10 0.857
Panel A: 25 Size-B/M portfolio
2.97 0.000 0.099 0.35 0.14 0.30 0.919
SVOL 2.54 0.000 0.089 0.32 0.11 0.39 0.921
CVR 2.92 0.000 0.100 0.36 0.14 0.29 0.919
Panel A: 25 Size-OP portfolio
1.92 0.005 0.075 0.30 0.08 0.60 0.930
SVOL 1.67 0.023 0.064 0.26 0.08 0.66 0.931
CVR 1.94 0.004 0.075 0.30 0.09 0.58 0.931
Panel A: 25 Size-Inv portfolio
3.32 0.000 0.087 0.31 0.12 0.30 0.931
SVOL 2.77 0.000 0.077 0.28 0.10 0.38 0.932
CVR 3.27 0.000 0.088 0.32 0.12 0.30 0.932
Panel A: 25 Size-Accruals portfolio
4.01 0.000 0.135 0.51 0.26 0.23 0.916
SVOL 3.32 0.000 0.122 0.46 0.22 0.30 0.917
CVR 4.06 0.000 0.133 0.51 0.26 0.23 0.916
Panel A: 25 Size-Net Shares Issued portfolio
3.27 0.000 0.103 0.30 0.13 0.37 0.900
Tau 3.11 0.000 0.106 0.30 0.14 0.37 0.902
CVR 3.21 0.000 0.105 0.30 0.13 0.37 0.901
Table B.1: Summary statistics for tests of five-factor, five-factor plus SVOL and five-factor
plus CVR models; July 1964–January 2010-December, 564 months. The table tests the ability
of five-factor, five-factor plus SVOL and five-factor plus CVR models to explain monthly
excess returns on 25 Size-Tau portfolios, 25 Size-Momentum portfolios, 25 Size-B/M portfolios,
25 Size-OP portfolios, 25 Size-Inv portfolios, 25 Size-Accruals portfolios and 25 Size-Net
Shares Issued portfolios. The factor CVR is taken from Oliver Boguth’s homepage as used in
Boguth and Kuehn (2013). For each set of 25 regressions, the table shows the GRS statistic
testing whether the expected values of all 25 intercept estimates are zero, the average absolute
value of the intercepts, A|ai|, A|ai|A|r¯i| , the average absolute value of the intercept ai over the
average absolute value of r¯i, which is the average return on portfolio i minus the average
value-weighted market portfolio excess return, Aa
2
i
Ar¯2
i
, the average squared intercept over the
average squared value of r¯2, As
2(ai)
Aa2
i
, the average of the estimates of the variances of the
sampling errors of the estimated intercepts over Aa2i , and A(R2), the average value of the
regression R2 corrected for degrees of freedom.
95

C Derivation of expected market return
As given in (Cochrane, 2001) the expected market return can be written as
ERmt =
1
Emt
− 1
Emt
Cov(mt, Rmt )
= 1
Emt
− 1
Emt
√
Var(mt)Var(Rmt )ρm,Rm (C.1)
where ρm,Rm is the correlation between the market return and the stochastic discount
factor mt = β
(
ct+1
ct
)−γ
with subjective discount factor β = e−δ, risk aversion γ and
consumption ct. Assuming consumption growth to be lognormal we get for the mean
and the variance of mt:
Emt = e−δ−γμ+
γ2
2 σ
2
t (C.2)
Var(mt) = e−2δ−2γμ+γ
2σ2t
(
eγ
2σ2t − 1
)
. (C.3)
Inserting equations (C.2) and (C.3) in (C.1) leads to the formula given in equation
(5.2).
97

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