Program of the course in Market Risk Management and
possible topics for the Master Diploma
Dean Fantazzini
HSE - Moscow
Overview of the Presentation
1st Program of the course in Market Risk Management
Dean Fantazzini 2
Overview of the Presentation
1st Program of the course in Market Risk Management
2nd Possible Topics for the Master Diploma: An Overview
Dean Fantazzini 2-a
Overview of the Presentation
1st Program of the course in Market Risk Management
2nd Possible Topics for the Master Diploma: An Overview
3rd Some More Details about Each Topic
Dean Fantazzini 2-b
Program of the course in Market Risk Management
0. Brief Review of Univariate and Multivariate GARCH models
1. Market Risk Management
1.1 Risk Measures: Definitions and Properties
1.2 Standard Methods for Market Risks
1.3 Univariate Value at Risk with GARCH models
1.4 Multivariate Value at Risk with M-GARCH models
1.5 Empirical applications: Univariate T-GARCH models for Value at Risk
forecasting with European stocks.
1.6 Empirical applications: Multivariate Diagonal-VECH, Diagonal-BEKK,
CCC-GARCH and DCC models for Value at Risk forecasting.
Dean Fantazzini 3
Program of the course in Market Risk Management
2. Copula Theory
2.1 Introduction
2.2 Survey of Copula Families (Elliptical and Archimedean)
2.3 Limitations of Correlation, Tail dependence and other alternative
Dependence Measures
2.4 Estimation from market data and Simulation
2.5 Empirical applications: Bivariate copula modelling with R.
3. Advanced Market Risk Management
3.1 Multivariate Copula - GARCH models for financial returns
3.2 Market Risk Management with Multivariate Copula-GARCH models
3.3 Empirical applications with R: Multivariate Value at Risk with
Copula-GARCH models
Dean Fantazzini 4
Program of the course in Market Risk Management
• Textbooks:
– Umberto Cherubini, Elisa Luciano and Walter Vecchiato (2004), Copula
Methods in Finance, Wiley
– James D. Hamilton (1994), Time Series Analysis, Princeton University
Press
– Philippe Jorion (2007), Financial Risk Manager Handbook, Fourth
Edition, Wiley
– Ruey Tsay (2002), Analysis of Financial Time Series, Wiley
– Alexander McNeil, Rudiger Frey and Paul Embrechts (2005),
Quantitative Risk Management, Princeton University Press
• Method of Grading: Each student should take a final exam which
considers both theoretical and applied aspects.
Dean Fantazzini 5
Possible Topics for the Master Diploma:
An Overview
A general List of possible topics:
- Volatility Forecasting in Russian Markets: GARCH models vs Realized
Volatility vs Realized Range
- Optimal Capital Allocation: VaR, C-VaR, Spectral Measures and Beyond
- Oil, Exchange rates, Inflation and Economic Growth Dynamics in Russia.
- The law of One Price: Evidence from Russian Commodities Markets
- Canonical Vines Copulas for Market Risk Management
- Canonical Vines Copulas for Operational Risk Management
- Modelling of Financial bubbles and Crashes
Dean Fantazzini 6
Volatility Forecasting in Russian Markets: GARCH
models vs Realized Volatility vs Realized Range
GARCH(1,1) : σ2t = ω + α1ε2
t−1 + β1σ2t−1,
Let consider a discretely sampled ∆-period return be denoted by
yt = p(t) − p(t − ∆), and normalize the daily time interval to unity. Given a total
of nt subintervals within the day, the daily realized volatility is given by the
summation of the corresponding 1/∆ high-frequency intraday squared returns,
Realized Volatility : σRV,t =
1/∆∑j=1
y2t+j∆−1,∆ =
nt∑i=1
y2t−1,i (1)
Under this assumptions, the ex-post realized volatility is an unbiased volatility
estimator of the Integrated Volatility(IV) associated with day t dnd defined as the
integral of the instantaneous volatility over the one day integral (t − 1; t):
plim∆−→0 σRV,t =
t∫
t−1
σ2(s)ds (2)
Following Parkinson (1980), the realized range estimator for the variance is
Range : σ2RR,t =
1
4 ln 2
nt∑i=1
(ln Ht − ln Lt)2 (3)
Dean Fantazzini 7
Volatility Forecasting in Russian Markets: GARCH
models vs Realized Volatility vs Realized Range
POSSIBLE MASTER DISSERTATION OUTLINE
Step 1 : Review of the literature about Volatility Forecasting: GARCH,
Realized Volatility , Range based estimators.
Step 2 : Empirical applications with Russian data
Step 3 : Perform out-of-sample backtesting analysis: Which model perform
best? Possible extensions: Combination of forecasts, Bayesian
averaging, etc.
Dean Fantazzini 8
Optimal capital allocation: VaR, C-VaR, Spectral Measures and
Beyond
Definition 1.1:
The VaR at level α is the maximus loss one could expect to lose with probability
α over a specific period of time.
Formally, given a probability level α, the VaR is the loss −γ that satisfy the
equation
F (−γ) = α ⇒ γ = −F−1(α)
that is the quantile of the loss distribution, provided that F is strictly increasing
and invertible.
Definition 1.2:
The Expected Shortfall (ESα) is the simple arithmetic mean of all the losses
that we have with probability equal or smaller than α:
ESα = −1
α
α∫
0
F−1(∆Pt)d∆Pt (4)
Dean Fantazzini 9
Optimal capital allocation: VaR, C-VaR, Spectral Measures and
Beyond
The Expected Shortfall has been defined as the simple arithmetic average of the a
worst losses. However, instead of computing the simple average, we can consider a
weighted average, thus generalizing the Expected Shortfall ⇒ Spectral Measures
POSSIBLE MASTER DISSERTATION OUTLINE
Step 1 : Review of the literature about Portfolio Management: from Markowitz till
capital allocation with spectral measures.
Step 2 : Empirical applications with Russian data
Step 3 : Verify whether an investor implementing a volatility-timing strategy with
Russian stocks would be willing to pay to switch from a daily-returns-based
estimator of the conditional variance (or a constant volatility model) to an
estimator based on intraday data, see Fleming, Kirby and Ostdiek (2001,
2003).
Dean Fantazzini 10
Oil, exchange rates, inflation and economic
growth dynamics in Russia
Dean Fantazzini 11
Oil, exchange rates, inflation and economic
growth dynamics in Russia
⇒ Threshold Cointegration! That is, we have a
• A Long-run equilibrium, but...
• ...there are transaction costs, political limits, etc, that have to be taken into
account!
Figure 3: Threshold Cointegration Model
Dean Fantazzini 12
Oil, exchange rates, inflation and economic
growth dynamics in Russia
⇒ Ito (2008) empirically investigate the effects of oil price and monetary shocks
on the Russian economy covering the period between 1997:Q1 and 2007:Q4.
His analysis leads to the finding that a 1% increase in oil prices contributes to
real GDP growth by 0.25% over the next 12 quarters, whereas that to inflation by
0.36% over the corresponding periods.
He also find that the monetary shock through interest rate channel immediately
affects real GDP and inflation as predicted by theory.
⇒ Moreover, Lescaroux and Mignon (2009) with annual data covering the period
1960-2005 found that, concerning the short term analysis, when causality exists,
it generally runs from oil prices to the other considered variables.
One of the most interesting results is that there exists a strong causality running
from oil to share prices, especially for oil-exporting countries.
As for long term analysis, the majority of long-run relationships concerns GDP,
unemployment rate and share prices.
Dean Fantazzini 13
Oil, exchange rates, inflation and economic
growth dynamics in Russia
POSSIBLE MASTER DISSERTATION OUTLINE
Step 1 : Review of the literature about Oil, inflation and economic growth
dynamics.
Step 2 : Empirical applications with Russian and World data by using
VECM, TVECM or Panel-Cointegrated models (according to the
dataset at hand)
Step 3 : Investigate the links between oil prices and a set of variables
representative of economic activity (gross domestic product, consumer
price index, household consumption expenditures, unemployment rate
and share prices) after the global financial crisis.
If Russian sectoral data are available, the analysis could be further
refined.
Dean Fantazzini 14
The law of One price: Evidence from Russian
Commodities Markets
Consider a simple two market example: the law of one price states the
commodity will have the same price at the same time in both markets when
transaction costs (all costs including trading, terminal and transportation costs)
are small enough that profitable trade is not prohibited. Otherwise, there is an
arbitrage opportunity.
Traders would be able to profit by buying the commodity in one market and
selling in the other market because of price differences in the two markets. Such
trading drives the prices in the two markets toward one price.
Small deviation in prices of the commodity, however, may exist because of
transaction costs. Transaction costs discourage traders from trading when the
possible profits are smaller than these costs.
Arbitrage opportunities occur only when the spread in prices between two
markets is larger than the transaction costs that link the markets (Goodwin and
Piggott, 2001).
Dean Fantazzini 15
The law of One price: Evidence from Russian
Commodities Markets
Figure 1: Four alternative cases of the deviations in the neutral band
associated with the law of one price (from Park et al. (2007))
Dean Fantazzini 16
The law of One price: Evidence from Russian
Commodities Markets
Figure 2: Estimated transaction costs between the seven natural gas
spot markets pairs ($/MMBtu) (from Park et al. (2007))
Dean Fantazzini 17
The law of One price: Evidence from Russian
Commodities Markets
POSSIBLE MASTER DISSERTATION OUTLINE
Step 1 : Review of the literature about the law of One price.
Step 2 : Empirical applications with Russian commodities markets by using
TVECM or time-varying TVECM models (if necessary)
Step 3 : Investigate whether there are nonlinear adjustments to the law of
one price in Russian commodities markets and whether dynamic
threshold effects relative to the base market vary depending on season,
geographical location.
Dean Fantazzini 18
Canonical Vines Copulas for Market Risk
Management
⇒ Canonical Vines provides a graphical representation of the conditional
specifications being made on a joint distribution. The multivariate
distribution is represented by the product of the marginals and edges of the
vine.
An n-dimensional vine is represented by n − 1 trees.
Tree j has n + 1 − j nodes and n − j edges.
Each edge corresponds to a pair-copula density.
Edges in tree j become nodes in tree j + 1.
Two nodes in tree j + 1 are joined by an edge if the corresponding edges in
tree j share a node.
The complete decomposition is defined by the n(n − 1)/2 edges (i.e. pair
copula densities) and the marginal densities.
Dean Fantazzini 19
Canonical Vines Copulas for Market Risk
Management
Canonical vine: A regular vine for which each tree has a unique node that
is connected to n − j edges.
D-vine: A regular vine for which no node in any tree is connected to more
than two edges.
Figure 3: Canonical vines (left) and D-vines (right)
Dean Fantazzini 20
Canonical Vines Copulas for Market Risk
Management
POSSIBLE MASTER DISSERTATION OUTLINE
Step 1 : Review of the literature about copulas in general and canonical vines
copula in particular.
Step 2 : Empirical applications with Russian stocks.
Step 3 : Investigate whether there complex copulas outperform simpler
models in market risk management by using extensive out-of-sample
backtesting with the VaR and the Expected Shortfall.
Dean Fantazzini 21
Canonical Vines Copulas for Operational Risk
Management
• What are operational risks?
The term “operational risks” is used to define all financial risks that are
not classified as market or credit risks. They may include all losses due to
human errors, technical or procedural problems etc.
→ To estimate the required capital for operational risks, the Basel
Committee on Banking supervision (1996,1998) allows for both a simple
“top-down” approach, which includes all the models which consider
operational risks at a central level, so that local Business Lines (BLs) are
not involved.
→ And a more complex “bottom- up” approach, which measures
operational risks at the BLs level, instead, and then they are aggregated,
thus allowing for a better control at the local level.
→ The methodologies we consider here belong to this second approach.
Dean Fantazzini 22
Canonical Vines Copulas for Operational Risk
Management
• The Standard Loss Distribution Approach (LDA) Approach with
Comonotonic Losses
This approach employs two types of distributions:
• The one that describes the frequency of risky events;
• The one that describes the severity of the losses
Formally, for each type of risk i = 1, . . . , R and for a given time period,
operational losses could be defined as a sum (Si) of the random number (ni) of
the losses (Xij):
Si = Xi1 + Xi2 + . . . + Xini(5)
A widespread statistical model is the actuarial model . In this model, the
probability distribution of Si is described as follows:
Fi(Si) = Fi(ni) · Fi(Xij), where
• Fi(Si) = probability distribution of the expected loss for risk i;
• Fi(ni) = probability of event (frequency) for risk i;
• Fi(Xij) = loss given event (severity) for risk i.
Dean Fantazzini 23
Canonical Vines Copulas for Operational Risk
Management
The underlying assumptions for the actuarial model are:
• the losses are random variables, independent and identically
distributed (i.i.d.);
• the distribution of ni (frequency) is independent of the distribution of
Xij (severity).
Moreover,
• The frequency can be modelled by a Poisson or a Negative Binomial
distribution.
• The severity, is modelled by a Exponential or a Pareto or a Gamma
distribution, or using the lognormal for the body of the distribution
and the EVT for the tail.
→ The distribution Fi of the losses Si for each intersection i among
business lines and event types, is then obtained by the convolution of the
frequency and severity distributions.
Dean Fantazzini 24
Canonical Vines Copulas for Operational Risk
Management
However, the analytic representation of this distribution is computationally
difficult or impossible. For this reason, this distribution is usually
approximated by Monte Carlo simulation:
→ We generate a great number of possible losses (i.e. 100.000) with
random extractions from the theoretical distributions that describe
frequency and severity. We thus obtain a loss scenario for each loss Si.
→ A risk measure like Value at Risk (VaR) or Expected Shortfall (ES) is
then estimated to evaluate the capital requirement for the loss Si.
• The VaR at the probability level α is the α-quantile of the loss
distribution for the i − th risk: V aR(Si; α) : Pr(Si ≥ V aR) ≤ α
• The Expected Shortfall at the probability level α is defined as the
expected loss for intersection i, given the loss has exceeded the VaR
with probability level α : ES(Si; α) ≡ E [Si|Si ≥ V aR(Si; α)]
Dean Fantazzini 25
Canonical Vines Copulas for Operational Risk
Management
Once the risk measures for each losses Si are estimated, the global VaR (or
ES) is usually computed as the simple sum of these individual measures:
• a perfect dependence among the different losses Si is assumed...
• ... but this is absolutely not realistic!
• If we used the Sklar’s theorem (1959) and the Frechet-Hoeffding
bounds, the multivariate distribution among the R losses would be
given by
H(S1t, . . . , SR,t) = min (F (S1,t), . . . , F (SR,t)) (6)
where H is the joint distribution of a vector of losses Sit, i = 1 . . . R,
and F (·) are the cumulative distribution functions of the losses’
marginals. Needless to say, such an assumption in quite unrealistic.
Dean Fantazzini 26
Canonical Vines Copulas for Operational Risk
Management
• Di Clemente and Romano (2004) and Fantazzini et al. (2007, 2008)
proposed to use copulas to model the dependence among
operational risk losses:
→ By using Sklar’s Theorem, the joint distribution H of a vector of losses
Si, i = 1 . . . R, is simply the copula of the cumulative distribution functions
of the losses’ marginals :
H(S1, . . . , SR) = C(F1(S1), . . . , FR(SR)) (7)
...moving to densities, we get:
h(S1, . . . , SR) = c(F1(S1), . . . , FR(SR)) · f1(S1) · . . . · fR(SR)
→ The analytic representation for the multivariate distribution of all losses
Si with copula functions is not possible, and an approximate solution with
Monte Carlo methods is necessary.
Dean Fantazzini 27
Canonical Vines Copulas for Operational Risk
Management
POSSIBLE MASTER DISSERTATION OUTLINE
Step 1 : Review of the literature about Operational Risk Management and
Copula models.
Step 2 : Empirical applications with Russian and/or other world data.
Step 3 : Investigate whether canonical vine copulas outperform simpler
models in operational risk management by using statistical tests and
(if possible) out-of-sample backtesting with the VaR and the Expected
Shortfall.
Dean Fantazzini 28
Modelling of Financial bubbles and Crashes
• The ITC “new economy” bubble (1995-2000)
• Slaving of the Fed monetary policy to the stock market descent
(2000-2003)
• Real-estate bubbles (2003-2006)
• MBS, CDOs bubble (2004-2007) and stock market bubble (2004-2007)
• Commodities and Oil bubbles (2006-2008)
• ⇒Consequences (deep loss of trust, systemic instability)
Dean Fantazzini 29
Modelling of Financial bubbles and Crashes
In summary:
• Each excess was partially “solved” by the subsequent excess... leading
to a succession of -unsustainable wealth growth -instabilities
• The present crisis+recession is the consolidation after this series of
unsustainable excesses.
• One could conclude that the extraordinary severity of this crisis is not
going to be solved by the same implicit or explicit ‘‘bubble
thinking‘‘.
• ⇒ “The problems that we have created cannot be solved at the level of
thinking that created them” (Albert Einstein).
Dean Fantazzini 30
Modelling of Financial bubbles and Crashes
Figure 4: Imitation, Herding and Rumors
Dean Fantazzini 31
Modelling of Financial bubbles and Crashes
Dean Fantazzini 32
Modelling of Financial bubbles and Crashes
• A financial collapse has never happened when things look bad.
• Macroeconomic flows look good before crashes.
• Before every collapse, the majority of analysts say the economy is in
the best of all worlds.
• Everything looks rosy, stock markets go up...
• Macroeconomic flows (output, employment, etc.) appear to be
improving further and further.
• A crash catches most people by surprise.
• The good times are extrapolated linearly into the future.
• Is it not perceived as senseless by most people in a time of general
euphoria to talk about crash and depression?
Dean Fantazzini 33
Modelling of Financial bubbles and Crashes
The upswing usually starts with an opportunity - new markets, new technologies
or some dramatic political change - and investors looking for good returns.
• It proceeds through the euphoria of rising prices, particularly of assets, while
an expansion of credit inflates the bubble.
• In the manic phase, investors scramble to get out of money and into illiquid
things such as stocks, commodities, real estate or tulip bulbs
• Ultimately, the markets stop rising and people who have borrowed heavily
find themselves overstretched. This is ’distress’, which generates unexpected
failures, followed by ’revulsion’ or ’discredit’.
• The final phase is a self-feeding panic, where the bubble bursts:...
• ...people of wealth and credit scramble to unload whatever they have bought
at greater and greater losses, and cash becomes king.
Charles Kindleberger, Manias, Panics and Crashes (1978)
Dean Fantazzini 34
Modelling of Financial bubbles and Crashes
Is it possible to model bubbles and financial crashes?
• Economic Literature: Abreu and Brunnermeier (2003), Gurkaynak
(2008), and references therein.
Pro: A lot of theory with a high level of sophistication, and many ex-post
empirical analysis.
Cons: NO ex-ante forecasts. Moreover, Gurkaynak (2008) found that for
each paper that finds evidence of bubbles, there is another one that fits the
data equally well without allowing for a bubble, so that it is not possible to
distinguish bubbles from time-varying fundamentals!
• Physics Literature: Johansen et al. (2000), Sornette (2003a,b), Zhou and
Sornette (2003, 2006, 2007, 2008). Lin et al.(2009), Jiang et al. (2010).
Pro: Good level of sophistication, and several ex-ANTE successful forecasts
(around 60-65 % success rate). Recently, a scientific platform has been set
up at the ETH - Zurich, called Financial Crisis Observatory and aimed at
‘‘testing and quantifying rigorously the hypothesis that financial markets
exhibit a degree of inefficiency and a potential for predictability, especially
during regimes when bubbles develop‘‘,
Cons: Some ex-ante wrong forecasts, too few economic theory.
Dean Fantazzini 35
Modelling of Financial bubbles and Crashes
• Professional Technical Analysis for Crashes: Hindenburg Omen
The Hindenburg Omen is a combination of technical factors that attempt to
measure the health of the NYSE, and to signal increased probability of a
stock market crash.
Pro: the probability of a market move greater than 5% to the downside after
a confirmed Hindenburg Omen was 77%, and usually takes place within the
next forty days. The probability of a panic sellout was 41% and the
probability of a major stock market crash was 24%.
Every NYSE crash since 1985 has been preceded by a Hindenburg Omen.
Of the previous 25 confirmed signals only two (8%) have failed to predict at
least mild (2.0% to 4.9%) declines.
Cons: NO economic theory. Because of the specific and seemingly random
nature of the Hindenburg Omen criteria, the phenomenon may be simply a
case of overfitting! The Omen is at best an imperfect technical indicator that
is a work in progress
http://en.wikipedia.org/wiki/Hindenburg Omen
Dean Fantazzini 36
Modelling of Financial bubbles and Crashes
POSSIBLE MASTER DISSERTATION OUTLINE
Step 1 : Review of the literature about Financial Bubbles and Crashes.
Step 2 : Empirical applications with Russian and other Emerging Markets stock
indexes.
Step 3 : Investigate which approach perform best in out-of-sample backtesting, and
verify whether the different approaches can be combined.
Dean Fantazzini 37