presentation program and dissertation topics · 2010-10-21 · program of the course in market risk...

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

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2nd Possible Topics for the Master Diploma: An Overview

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2nd Possible Topics for the Master Diploma: An Overview

3rd Some More Details about Each Topic

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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.

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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

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• 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.

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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

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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)

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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.

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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)

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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


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).

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Oil, exchange rates, inflation and economic

growth dynamics in Russia

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⇒ 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

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⇒ 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.

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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.

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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).

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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))

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Commodities Markets

Figure 2: Estimated transaction costs between the seven natural gas

spot markets pairs ($/MMBtu) (from Park et al. (2007))

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Commodities Markets


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.

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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.

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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)

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Management


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.

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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.

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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.

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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.

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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)]

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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.

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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.

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Management


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.

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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)

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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).

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Figure 4: Imitation, Herding and Rumors

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• 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?

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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)

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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.

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• 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

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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.

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presentation program and dissertation topics · 2010-10-21 · program of the course in market risk...

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