European Quant Awards 2016 - CFA Institute · 2016-11-08 · 1 European Quant Awards 2016 Detecting...
Transcript of European Quant Awards 2016 - CFA Institute · 2016-11-08 · 1 European Quant Awards 2016 Detecting...
1
European Quant Awards 2016
Detecting financial market bubbles with low-
frequency volatility models
The original research was carried out by the author during the preparation of
Master Thesis. The report is an adaptation of its main results.
2016
2
Contents
1. Introduction........................................................................... 3
2. Research and empirical evidence.......................................... 3
3. Applications and further research......................................... 7
3.1. Brief summary of main results.......................................... 7
3.2. Applications for portfolio management............................ 7
4. Appendix................................................................................ 10
4.1. Appendix 1. MIDAS regressions........................................ 10
4.2. Appendix 2. GARCH-MIDAS model.................................... 11
4.3. Appendix 3. Actual estimation results............................... 12
5. References............................................................................. 15
3
1. Introduction
Financial market bubbles are one of the main reasons behind stock market crashes
alongside with economic troubles and announcements concerning unexpected and/or adverse
political turns. Some of the most notorious and devastating market crashes were caused by
bubbles, for example the Wall Street crash of 1929, Japanese asset price bubble of 1991, the
Dot-com bubble, the housing market bubble and the corresponding crash of 2007, Chinese
stock bubble of 2007 etc. Therefore, any empirical tool that enables the stock market
participants to detect market bubbles is extremely useful and can contribute to the
methodology of market crash prediction. In this paper, we propose a new technique for market
bubble detection (and, consequently, market crash prediction) that might prove to be valuable
for stock market practitioners.
In the original research we use S&P 500 index returns from 01.01.1964 to 01.01.2014 to
detect market crashes in the aggregate US stock market, but, as it will be discussed in Section
3.2, the methodology has a high potential for extrapolation to industry stock markets and
individual stocks.
2. Research and empirical evidence
A distinctive feature of market bubbles is that they form without any solid background
other than investors’ overconfidence, or “euphoria”. Stock market volatility is generally
regarded as a measure of investors’ confidence, so it is quite natural to infer that it might
contain signs of market being overconfident prior to a bubble burst. Thus, “euphoria” can be
interpreted as a situation when stock market volatility behaves “unnaturally” – it is lower than
suggested by economic surrounding. With that in mind, we construct two volatility measures
for the same sample period of historical S&P 500 index returns:
The first measure is calculated from the GARCH-MIDAS model (see Appendices 1
and 2 for details). It has two components: the low-frequency one is modeled with
macroeconomic fundamentals and realized volatility, while the high-frequency one
follows a standard GARCH-type process. This measure represents the “true”
volatility given macroeconomic environment.
4
The second measure is taken from a simple GARCH(1,1), which is most commonly
used in financial modeling, trading, risk management etc. It represents the “actual”
volatility as it is seen by market participants.
The details of GARCH-MIDAS estimation results are shown in Table 1 and Figures 5 and 6 in
Appendix 3. Figure 1 below exhibits the estimated low-frequency volatility component for
GARCH-MIDAS (moves quarterly).
Figure 1. Low-frequency volatility estimated from GARCH-MIDAS model
In its turn, Figure 2 below plots the total volatility together with squared returns.
5
Figure 2. Squared returns and fitted volatility from the GARCH-MIDAS model
The details of GARCH(1,1) estimation results can be found in Table 2 of Appendix 3. The
series of S&P 500 logarithmic returns with superimposed standard deviations estimated from
GARCH(1,1) model is presented in Figure 3 below.
Figure 3. S&P500 log-returns with fitted GARCH(1,1)
6
Then we construct an indicator variable that we call the Volatility Ratio (VR) – the ratio of
volatility measure of GARCH(1,1) to the volatility measure of GARCH-MIDAS:
𝑉𝑅𝑡 =(𝜎𝑡
𝐺𝐴𝑅𝐶𝐻(1,1))2
(𝜎𝑡𝐺𝐴𝑅𝐶𝐻−𝑀𝐼𝐷𝐴𝑆)2
(1)
The intuition behind this action is as follows. If VR is low, then current volatility is lower
than implied by macroeconomic factors, there is growing market overconfidence. On the
contrary, if VR is high, then current volatility is higher than implied by macroeconomic factors,
the market is nervous.
Figure 4 below shows the Volatility Ratio dynamics for the chosen sample period.
Figure 4. Volatility ratio dynamics
In these terms, we postulate the main hypothesis: the lower is VR today, the greater is the
probability of a market bubble burst tomorrow. We test this prediction using logit regression
with the lagged volatility ratio as explanatory variable. The dependent variable in the
regressions is a binary-choice variable that indicates the presence of a market crash on each
day. However, the definition of a market crash is ambiguous, so it is assumed in this paper that
the “crash” event corresponds to a daily stock index return of less than -5%. Of course, other
specifications are also possible. If our research hypothesis is correct, we expect the slope
coefficient to be significant and negative.
7
The estimation results of logit regressions can be found in Table 3 of Appendix 3. There
is significant negative relationship between lagged VR and the probability of a market crash at
5% level. This is an encouraging finding since it may be useful for market crash prediction. The
change in tomorrow’s probability of a market crash can be determined as the predicted change
in the volatility ratio (based on a 1-day ahead GARCH(1,1) and GARCH-MIDAS forecasts)
multiplied by the marginal effect of the logistic regression estimated at today’s volatility ratio:
�̂�𝑡+1 = 𝑃𝑡 +𝑒−(�̂�0+�̂�1∗𝑉𝑅𝑡) ∗ �̂�1
(1 + 𝑒−(�̂�0+�̂�1∗𝑉𝑅𝑡))2∗ (𝑉�̂�𝑡+1 − 𝑉𝑅𝑡) (2)
Here the today’s probability of crash 𝑃𝑡 is just the today’s fitted value of the logit regression.
3. Applications and further research
3.1. Brief summary of main results
The main research question was whether market crashes can be predicted using the
estimated differences in the volatility measures from different models. The answer for now is
affirmative – we have discovered a significant negative relationship between the Volatility Ratio
and the probability of a market crash. Moreover, we have found a way to quantify and measure
investors’ overconfidence and show how it leads to market crashes.
This finding speaks against the predominant theoretical concept that investors are perfectly
rational and it might provide inspiration for development of new theoretical models of financial
markets.
3.2. Applications for portfolio management
The methodology used in our research is new to academic literature. Although it was
applied to a rather specific case – the aggregate US stock market, it might be feasible to
successfully extrapolate it to other situations, for example, financial bubbles forming in a
particular industry (leading to a crash of the industry index) or bubbles concerning individual
stocks. The main issue in developing any similar model is to identify some “fundamental”, or
“true”, volatility or correlation measure to be compared to the standard measure (e.g.
8
GARCH(1,1) or DCC(1,1)). Let us consider the possible choices for individual stocks and
industrial aggregates.
In our paper we focus on S&P 500 market index, and our hypothesis is about capital market
aggregates, not individual stocks. In these terms, the diversification principle suggests that
general economic state variables provide the only source of influence on stock market return
properties. But that would not work for individual stocks, so we have to adjust our GARCH-
MIDAS model to include firm-specific fundamentals instead of macroeconomic ones. One can
extract relevant factors from the company’s financial statements. For example, financial
leverage and other ratios associated with financial risk are obvious candidates. If the
statements are published monthly, we can use the MIDAS framework (see Appendix 1) to
construct quarterly measure of low-frequency volatility including monthly lags of both RV and
the chosen ratios. In case if the statements are published quarterly, we can still use MIDAS
monthly lags for RV and leave the ratios sampled quarterly as simple regressors in the low-
frequency volatility component.
Also, it might be reasonable to augment the logit regression model with MIDAS structure.
That is, one may consider the probability of a market crash as a dependent variable sampled
weekly or monthly and the lagged Volatility Ratio as an independent variable sampled daily.
Such logit-MIDAS model will make forecasts of the probability of a market crash next
week/month given the values of Volatility Ratio on each day of the present week/month.
When it comes to sectors, the idea has to be generalized to a multivariate case. In these
terms, one can find the DCC-MIDAS model of Colacito, Engle & Ghysels (2011) extremely useful
for this purpose. It shares a similar approach with GARCH-MIDAS, but extends its methodology
to correlation. Namely, the correlation between the two assets is multiplicatively decomposed
into high- and low-frequency components, with the former following a usual DCC structure and
the latter having a MIDAS structure. We can use monthly MIDAS lags of common industry
factors and realized correlation to construct the quarterly low-frequency correlation
component for each pair of stocks. For example, for the oil industry we can use some of the
characteristics of oil prices as factors: monthly standard deviation, monthly return etc. When a
market bubble is forming in the sector, the prices of [most of] the stocks included in this sector
are subject to substantial growth. Thus, correlation between the stocks should be abnormally
high. Analogously, we can construct a new measure of overconfidence, the Correlation Ratio,
9
for each pair of stocks. The Correlation Ratio will be defined as the ratio of correlation measure
from standard DCC(1,1) model to the correlation measure from DCC-MIDAS model. This time,
today’s increase in the correlation ratio will indicate an increase in the probability of an industry
bubble burst tomorrow, and vice versa. To test the hypothesis for the multivariate case, one
should run a logit regression for a panel of stock pairs. If the hypothesis turns out to be true,
the slope of the correlation ratio should be positive and significant.
In fact, a DCC-MIDAS model fit is a two-step procedure that includes the fit of univariate
GARCH-MIDAS models for each stock. Hence, it is capable of tracking both individual stocks and
the industry as a whole at the same time.
We believe that our results and further inferences will prove to be useful for market
practitioners in fields where market crashes are a great concern – portfolio management, risk
management and algorithmic trading. It is also worth mentioning that the suggested
methodology is rather flexible when being utilized for risk management. Every individual
investor and every investment fund has its own risk profile. By adjusting the definition of a
market crash (it was -5% or less in our research) and by varying the subjective benchmark
probability P* (if Pt+1 > P*, then short the stock/portfolio), any market practitioner can conceive
his own risk management policy.
10
4. Appendix
4.1. Appendix 1. MIDAS regressions
In this section we briefly formulate the general MIDAS regression model as it was first
published in Ghysels, Santa-Clara & Valkanov (2002). Much of the technicalities are skipped
since they can be easily found in the mentioned paper and other works of the authors, for
example in Ghysels, Santa-Clara & Valkanov (2006) and Ghysels & Valkanov (2012).
Suppose that the dependent variable 𝑌𝑡 is sampled at some fixed frequency, while 𝑋(𝑚) is
sampled m times faster. Then a simple linear MIDAS regression will have the form
𝑌𝑡 = 𝛽0 + ∑ 𝛽𝑛𝑋𝑡−𝑛/𝑚(𝑚)
𝑁
𝑛=0
+ 𝜀𝑡(𝑚)
(3)
The implementation of MIDAS models makes the researcher face a tradeoff. On the one
hand, the model does not require aggregation of higher frequency data, so it better exploits the
available information. On the other hand, this advantage comes at the cost of increasing the
number of parameters. If the number of high-frequency lags N is large, some structure has to
be imposed on the weighting polynomial βn. The authors make use of the beta weighting
scheme: 𝛽𝑛 = 𝜃 ∗ 𝜑𝑛(𝑁, 𝜔1, 𝜔2), where
𝜑𝑛(𝑁, 𝜔1, 𝜔2) =𝑓(
𝑛𝑁
, 𝜔1, 𝜔2)
∑ 𝑓(𝑗𝑁
, 𝜔1, 𝜔2)𝑁𝑗=1
, (4)
𝑓(𝑧, 𝑎, 𝑏) =𝑧𝑎−1(1 − 𝑧)𝑏−1
𝐵(𝑎, 𝑏), (5)
θ is the scaling parameter and B(a,b) is the Beta function. Such restriction imposed on
coefficients diminishes the number of parameters significantly (there are only three parameters
for each independent variable) and is capable of producing a large variety of shapes. However,
other weighting structure specifications such as exponential or Almon lag polynomial are also
possible.
11
4.2. Appendix 2. GARCH-MIDAS model
Manifold MIDAS models were applied to stock market volatility. However, we are
interested in those that focus on macroeconomic components of volatility. The GARCH-MIDAS
model was introduced by Engle, Ghysels & Sohn (2009) and represents one of the ways
decompose the stock market volatility into high- and low-frequency components, with the
former following a usual GARCH-type structure and the latter having MIDAS structure and being
driven by macroeconomic variables.
Suppose the unexpected return on day i of the quarter t is modeled as
𝑟𝑖𝑡 − 𝜇 = 𝜎𝑖𝑡 ∗ 𝜀𝑖𝑡 , (6)
where the volatility is multiplicatively decomposed into the daily component 𝑔𝑖𝑡 and the
quarterly component 𝜏𝑡, which is kept constant throughout the quarter:
𝜎𝑖𝑡 = √𝜏𝑡 ∗ 𝑔𝑖𝑡 (7)
𝑔𝑖𝑡 = (1 − 𝛼 − 𝛽) + 𝛼 ∗(𝑟𝑖−1,𝑡 − 𝜇)
2
𝜏𝑡+ 𝛽 ∗ 𝑔𝑖−1,𝑡 (8)
The constraint ω = 1 – α – β is imposed on the high-frequency component in order to
ensure that unconditional volatility is equal to its low-frequency component:
𝐸𝑡−1(𝑟𝑖𝑡 − 𝜇)2 = 𝜏𝑡𝐸𝑡−1(𝑔𝑖𝑡) = 𝜏𝑡 (9)
Therefore, the GARCH-MIDAS model relaxes the stationarity assumption of the volatility
process.
Now let us introduce macroeconomic variables into the GARCH-MIDAS model. Taking
into account the results of Engle & Rangel (2008) and Engle, Ghysels & Sohn (2009), we include
four macroeconomic variables (all sampled monthly): the inflation rate, inflation volatility,
industrial production growth rate and unemployment rate. Thus, our MIDAS structure for the
low-frequency volatility component is defined to be
ln 𝜏𝑡 = 𝑚 + 𝜑 ∑ 𝑎𝑗(3, 𝑢)𝑅𝑉𝑡−
𝑗3
3
𝑗=1
+ 𝜃1 ∑ 𝑎𝑗(3, 𝑣1)𝐼𝑃𝑡−
𝑗3
3
𝑗=1
+
+𝜃2 ∑ 𝑎𝑗(3, 𝑣2)𝜋𝑡−
𝑗3
3
𝑗=1
+ 𝜃3 ∑ 𝑎𝑗(3, 𝑣3)𝜎𝜋,𝑡−
𝑗3
3
𝑗=1
+ 𝜃4 ∑ 𝑎(3, 𝑣4)𝑢𝑡−
𝑗3
3
𝑗=1
(10)
12
We follow an exponential lag weighting scheme, so that the lag weights depend on only one
parameter:
𝑎𝑗(𝑁, 𝑢) =𝑢𝑗
∑ 𝑢𝑖𝑁𝑖=1
(11)
Like any other GARCH-type model, GARCH-MIDAS can be estimated using maximum
likelihood approach. We treat the standardized residuals as normally distributed, and the
corresponding log-likelihood function for GARCH-MIDAS is defined as
𝑙𝑟 = ln 𝐿 = −1
2∑ (ln(𝑔𝑖𝑡 ∗ 𝜏𝑡) +
(𝑟𝑖𝑡 − 𝜇)2
𝑔𝑖𝑡𝜏𝑡)
𝑇
𝑖=1
(12)
4.3. Appendix 3. Actual estimation results
Table 1. GARCH-MIDAS estimation results
Parameter Estimate Standard error 95% CI lower
bound
95% CI upper
bound
α 0.074 2.973e-05 0.074 0.074
β 0.906 4.181e-05 0.906 0.906
m -10.502 0.001 -10.504 -10.500
θ1 -26.145 0.035 -26.214 -26.076
θ2 0.439 0.010 0.419 0.459
θ3 13.204 0.027 13.151 13.257
θ4 6.039 0.013 6.014 6.064
v1 1.000 0.003 0.994 1.006
v2 1.000 0.015 0.971 1.029
v3 128996.9 3.945 128989.168 129004.632
v4 60393.83 4.805 60384.412 60403.248
ϕ 63.308 0.050 63.21 63.406
u 1.927 0.003 1.921 1.933
13
Figure 5. Plot of MIDAS lag weights for industrial production growth rate, CPI
volatility and monthly RV
Figure 6. Plot of MIDAS lag weights for unemployment rate, CPI rate and
monthly RV
14
Table 2. GARCH(1,1) estimation results
Parameter Estimated Value Standard Error
µ 4.297e-04 6.086e-05
α 0.075 0.005
β 0.922 0.005
ω 5.795e-07 2.376e-07
Table 3. Logit regression estimation results
Variable Coefficient
estimate
Standard
Error
P-value 95% CI lower
bound
95% CI upper
bound
Intercept -1.001 2.178 0.646 -5.270 3.267
Lagged VR -5.448 2.239 0.015 -9.835 -1.060
15
5. References
[1] Andersen T.G., Bollerslev T. Answering the sceptics: yes, standard volatility models do
provide accurate forecasts. International Economic Review, Vol.39 №4, 1998.
[2] Andersen T.G., Bollerslev T., Christoffersen P.F., Diebold F.X. Volatility and correlation
forecasting. Published in: Arrow K.J., Intriligator M.D. Handbook of economic forecasting.
Vol.1. North-Holland, 2013.
[3] Andersen T.G., Bollerslev T., Diebold F.X. Parametric and nonparametric volatility
measurement. Published in: Ait-Sahalia Y., Hansen L.P. Handbook of financial econometrics.
Vol.1. North-Holland, 2010.
[4] Andersen T.G., Davis R.A., Kreiss J.-P., Mikosch T. Handbook of financial time series.
Springer-Verlag, 2009.
[5] Baele L. Volatility spillover effects in European equity markets. The Journal of Financial and
Quantitative Analysis, Vol.40 №2, 2005.
[6] Baillie R.T., Bollerslev T., Mikkelsen H.O. Fractionally integrated generalized autoregressive
conditional heteroscedasticity. Journal of Econometrics, Vol.74, 1996.
[7] Beltratti A., Morana C. Breaks and persistency: macroeconomic causes of stock market
volatility. Journal of Econometrics, Vol.131, 2006.
[8] Bollerslev T. Generalized autoregressive conditional heteroscedasticity. Journal of
Econometrics, Vol.31, 1986.
[9] Breidt F.J., Crato N., de Lima P. The detection and estimation of long memory in stochastic
volatility. Journal of Econometrics, Vol.83, 1998.
[10] Chauvet M., Potter S. Coincident and leading indicators of the stock market. Journal of
Empirical Finance, Vol.7 №1, 2000.
[11] Chen N.-F., Roll R., Ross S.A. Economic forces and the stock market. The Journal of
Business, Vol.59 №3, 1986.
[12] Christoffersen P.F. Elements of financial risk management. Academic Press, 2nd edition,
2012.
[13] Colacito R., Engle R.F., Ghysels E. A component model for dynamic correlations. Journal of
Econometrics, Vol.164 №1, 2011.
[14] Corsi F. A simple approximate long-memory model of realized volatility. Journal of Financial
Econometrics, Vol.7, 2009.
16
[15] Davidson J. Moment and memory properties of linear conditional heteroscedasticity
models, and a new model. Journal of Business and Economic Statistics, Vol.22, 2004.
[16] Diebold F.X. Elements of forecasting. Thompson South-Western, 4th edition, 2007.
[17] Efron B., Tibshirani R. Bootstrap methods for standard errors, confidence intervals and
other measures of statistical accuracy. Statistical Science, Vol.1, №1, 1986.
[18] Efron B., Tibshirani R. An introduction to the bootstrap. Springer Science + Business Media,
1993.
[19] Engle R.F. Autoregressive conditional heteroscedasticity with estimates of the variance of
United Kingdom inflation. Econometrica, Vol.50 №4, 1982.
[20] Engle R.F. New frontiers for ARCH models. Journal of Applied Econometrics, Vol.17, 2002.
[21] Engle R.F., Rangel J.G. The spline-GARCH model for low frequency volatility and its global
macroeconomic causes. Review of Financial Studies, Vol.21, 2008.
[22] Engle R.F., Lee G.G.J. A long-run and short-run component model of stock return volatility.
Published in: Engle R.F., White H. Cointegration, causality and forecasting. Oxford University
Press, 1999.
[23] Engle R.F., Ghysels E., Sohn B. Stock market volatility and macroeconomic fundamentals.
Working paper, 2009.
[24] French K.R., Schwert G.W., Stambaugh R.F. Expected stock returns and volatility. Journal
of Financial Economics, Vol.19, 1987.
[25] Ghysels E., Kvedaras V., Zemlys V. Mixed frequency data sampling regression models: the
R package midasr. Journal of Statistical Software.
[26] Ghysels E., Santa-Clara P., Valkanov R. The MIDAS touch: mixed data sampling regression
models. Working paper, UNC and UCLA, 2002.
[27] Ghysels E., Santa-Clara P., Valkanov R. Predicting volatility: getting the most out of return
data sampled at different frequencies. Journal of Econometrics, Vol.131, 2006.
[28] Ghysels E., Valkanov R. Forecasting volatility with MIDAS. Published in: Brauwens L.,
Hafner C., Laurent S. Handbook of volatility models and their applications. Wiley, 2012.
[29] Glosten L.R., Jagannathan R., Runkle D.E. On the relation between the expected value and
the volatility of the nominal excess return on stocks. The Journal of Finance, Vol.48 №5, 1993.
[30] Granger C.W.J., Engle R.F., Ding Z. A long-memory property of stock market returns and a
new model. Journal of Empirical Finance, Vol.1, 1993.
[31] Hamilton J.D. A new approach to the economic analysis of nonstationary time series and
the business cycle. Econometrica, Vol.57 №2, 1989.
17
[32] Hansen P.R., Lunde A. A forecast comparison of volatility models: does anything beat a
GARCH(1,1)? Journal of Applied Econometrics, Vol.20, 2005.
[33] Hentschel L. All in the family: nesting symmetric and asymmetric GARCH models. Journal of
Financial Economics, Vol.39, 1995.
[34] Knight J., Satchell S. Forecasting volatility in the financial markets. Butterworth-
Heinemann, 3rd edition, 2007.
[35] Kochen, T., Gilliam T. Boom, Bust, Boom. Documentary film.
[36] Minsky H.P. The Financial Instability Hypothesis. Working paper №74, 1992.
[37] Mishkin F.S., Eakins S.G. Financial Markets and Institutions. Prentice Hall, 7th edition,
2012.
[38] Morelli D. The relationship between conditional stock market volatility and conditional
macroeconomic volatility: empirical evidence based on UK data. International Review of
Financial Analysis, Vol.11, 2002.
[39] Nelson D.B. Conditional heteroscedasticity in asset returns: a new approach. Econometrica,
Vol.59 №2, 1991.
[40] Officer R.F. The variability of the market factor of the New York Stock Exchange. Journal of
Business, Vol.46, 1973.
[41] Schwert G.W. Business cycles, financial crises and stock volatility. Carnegie-Rochester
Conference Series on Public Policy №31, 1989.
[42] Schwert G.W. Why does stock market volatility change over time? The Journal of Finance,
Vol.34 №5, 1989.
[43] Schwert G.W. Stock volatility and the crash of 1987. The Review of Financial Studies, Vol.3
№1, 1990.
[44] Schwert G.W. Stock market volatility. Financial Analysts Journal, May-June 1990.
[45] Shephard N. Stochastic volatility: selected readings. Oxford University Press, 2005.
[46] Taylor S.J. Modeling stochastic volatility: a review and comparative study. Mathematical
Finance, Vol.4 №2, 1994.
[47] Tsay R.S. Analysis of financial time series. Wiley-Interscience, 2nd edition, 2005
[48] Zakharov A. Volatility and stock market crashes: implementation of low-frequency volatility
models for market crash prediction. Master Thesis under supervision of Lieven Baele, Associate
professor of finance at Tilburg University. Tilburg School of Economics and Management, 2016.