1. Arthur CHARPENTIER - École d'été EURIA.
mesures de risques et dépendance
Arthur Charpentier
Université de Rennes 1 & École Polytechnique
arthur.charpentier@univ-rennes1.fr
http://blogperso.univ-rennes1.fr/arthur.charpentier/index.php/
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2. Arthur CHARPENTIER - École d'été EURIA.
Overview of the two sessions
Consider a set of risks, denoted by a random vector X = (X1 , . . . , Xd )
The interest is an agregation function of those risks g(X), where g : Rd → R, and
we wish to measure the risk of this quantity R(g(X)), for some risk measure R.
• Séance jeudi : Mesures de risques et allocation de capital
• Séance vendredi : Corrélations, copules et dépendance
how to model X?
what about diversication eects?
what is the correlation of risks in X ?
can we compare R(g(X)) and R(g(X ⊥ )) (i.e. under independence)?
what is the contribution of Xi in the overall risk?
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3. Arthur CHARPENTIER - École d'été EURIA.
Some possible motivations... in nance
Consider a set of stock prices at time T denoted X = (X1 , . . . , Xd ) and
Y = (Y1 , . . . , Yd ) the ratio of the price at time T divided by the price at time 0,
and and let g(X) denote the payo at time T of some nancial derivative,
• e.g. spread derivatives, g(x1 , x2 ) = (x1 − x2 − K)+ based on the spread
between two assets, or more generally any extreme spread options, dual
spread options, correlation options or ratio spread options,
• e.g. buttery derivatives, g(x) = (a x − K)+ , i.e. call option on a portfolio
of d assets,
• e.g. min-max derivatives or rainbow, g(x) = (min{x} − K)+ ,
g(x) = (max{x} − K)+ , i.e. call option on the minimum or maximum of d
assets,
i=i+
• e.g. Atlas derivatives, g(x) = ( i=i− Yi − K)+ , where the sum is considered
skipping the i− lowest and the d − i+ largest returns, or Himalaya
i=d
derivatives, g(x) = ( i=i+ Yi − K)+ ,
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4. Arthur CHARPENTIER - École d'été EURIA.
Some possible motivations... in environmental risks
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5. Arthur CHARPENTIER - École d'été EURIA.
Some possible motivations... in credit risk
Applications with a high number of risks can also be considered, in credit risk for
instance. Let X = (X1 , ..., Xd ) denote the vector of indicator variables,
indicating if the i-th contract defaulted during a given period of time. If a credit
derivative is based on the occurrence of k defaults among d companies, and thus,
the pricing is related to the distribution of the number of defaults, N , dened as
N = X1 + ... + Xd . Under the assumption of possible contagious risks, the
distribution of N should integrate dependencies.
CreditMetrics in 1995 suggested a Gaussian model for credit changes, based on a
∗ ∗
probit approach, Xi = 1(Xi ui ), where Xi ∼ N (0, 1).
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6. Arthur CHARPENTIER - École d'été EURIA.
0.6 Probit model in dimension 1 Probit model in dimension 2
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DEFAULT (1) DEFAULTS
0.5
2
Value of company (2)
0.4
0.3
0
0.2
!2
(2) DEFAULTS
0.1
0.0
!4
!6 !4 !2 0 2 4 6 !4 !2 0 2 4
Value of the company Value of company (1)
Figure 1: Modeling defaults based on a probit approach.
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7. Arthur CHARPENTIER - École d'été EURIA.
Some possible motivations... in risk management
Consider a set a risks X = (X1 , ..., Xd ) (returns in a portfolio, losses per line of
business, positions of nancial desks)
A classical risk measure (as in Markowitz (1959)) is the standard deviation,
2
σ(Xi ) = V ar(Xi ) = E((Xi − E(X)) ). The risk of the portfolio
S = X1 + . . . + Xd is
σ(S) = σ(X1 )2 + . . . + σ(Xd )2 + 2 r(Xi , Xj )σ(Xi )σ(Xj ).
ij
Risks are now measured using Value-at-Risk (i.e. quantiles), and there is no
relationship between q(X1 + . . . + Xd ) and q(X1 ) + . . . + q(Xd ).
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10. Arthur CHARPENTIER - École d'été EURIA.
On risk dependence in QIS's
http://www.ceiops.eu/media/files/consultations/QIS/QIS3/QIS3TechnicalSpecificationsPart1.PDF
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11. Arthur CHARPENTIER - École d'été EURIA.
On risk dependence in QIS's
http://www.ceiops.eu/media/files/consultations/QIS/QIS3/QIS3TechnicalSpecificationsPart1.PDF
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12. Arthur CHARPENTIER - École d'été EURIA.
On risk dependence in QIS's
http://www.ceiops.eu/media/files/consultations/QIS/QIS3/QIS3TechnicalSpecificationsPart1.PDF
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13. Arthur CHARPENTIER - École d'été EURIA.
On risk dependence in QIS's
http://www.ceiops.eu/media/files/consultations/QIS/QIS3/QIS3TechnicalSpecificationsPart1.PDF
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14. Arthur CHARPENTIER - École d'été EURIA.
On risk dependence in QIS's
http://www.ceiops.eu/media/files/consultations/QIS/QIS3/QIS3TechnicalSpecificationsPart1.PDF
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15. Arthur CHARPENTIER - École d'été EURIA.
How to capture dependence in risk models ?
Is correlation relevant to capture dependence information ?
Consider (see McNeil, Embrechts Straumann (2003)) 2 log-normal risks,
• X ∼ LN (0, 1), i.e. X = exp(X ) where X ∼ N (0, 1)
• Y ∼ LN (0, σ 2 ), i.e. Y = exp(Y ) where Y ∼ N (0, σ 2 )
Recall that corr(X , Y ) takes any value in [−1, +1].
Since corr(X, Y )=corr(X , Y ), what can be corr(X, Y ) ?
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16. Arthur CHARPENTIER - École d'été EURIA.
How to capture dependence in risk models ?
1.0
0.5
Correlation
0.0
−0.5
0 1 2 3 4 5
Standard deviation, sigma
Figure 4: Range for the correlation, cor(X, Y ), X ∼ LN (0, 1) ,Y ∼ LN (0, σ 2 ).
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17. Arthur CHARPENTIER - École d'été EURIA.
How to capture dependence in risk models ?
1.0
0.5
Correlation
0.0
−0.5
0 1 2 3 4 5
Standard deviation, sigma
Figure 5: cor(X, Y ), X ∼ LN (0, 1) ,Y ∼ LN (0, σ 2 ), Gaussian copula, r = 0.5.
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18. Arthur CHARPENTIER - École d'été EURIA.
What about regulatory technical documents ?
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19. Arthur CHARPENTIER - École d'été EURIA.
What about regulatory technical documents ?
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20. Arthur CHARPENTIER - École d'été EURIA.
What about regulatory technical documents ?
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21. Arthur CHARPENTIER - École d'été EURIA.
What about regulatory technical documents ?
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22. Arthur CHARPENTIER - École d'été EURIA.
Agenda
• General introduction
Financial risks
• Market risks
• Credit risk
• Operational risk
Risk measures and capital allocation
• Risk measures: an axiomatic introduction
• Risk measures: convexity and coherence
• Capital allocation: an axiomatic introduction
Risk measures and statistical inference
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23. Arthur CHARPENTIER - École d'été EURIA.
Agenda
• General introduction
Financial risks
• Market risks
• Credit risk
• From variance to Value-at-Risk
Risk measures and capital allocation
• Risk measures: an axiomatic introduction
• Risk measures: convexity and coherence
• Capital allocation: an axiomatic introduction
Risk measures and statistical inference
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24. Arthur CHARPENTIER - École d'été EURIA.
Some references on risk management
Föllmer , H. Schied , A. (2004). Stochastic nance. An introduction in
discrete time. Gruyter Studies in Mathematics,
Jorian , P. (2007). Value-at-Risk,
McNeil , A. Frey, R., Embrechts , P. (2005). Quantitative Risk
Management: Concepts, Techniques, and Tools. Princeton University Press,
Roncalli , T. (2004). La gestion des risques nanciers. Economica.
Basel Committee on Banking Supervision. International convergence of capital
measurement and capital standards. http://www.bis.org/publ/bcbs128.pdf
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25. Arthur CHARPENTIER - École d'été EURIA.
Introduction to risk management: crisis
a statistical model is a probability distribution constructed to enable inferences to
be drawn or decisions made from data (Jorion (2007)).
1974 Herstatt Bank, 620 million USD (⇒ Real Time Gross Settlement Systems)
1994 Metallgesellschaft, 1340 million USD (oil futures)
1994 Orange County, 1810 million USD (derivaties)
1994 Procter Gamble, 102 million USD (derivaties)
1995 Barings, 1330 million USD (fraudulent manipulations)
1997 Natwest, 127 million USD (fraudulent manipulations)
1998 LTCM ( Long Term Capital Management), 2000 million USD (liquidity crisis)
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27. Arthur CHARPENTIER - École d'été EURIA.
History of risk measures
The evolution of (analytical) Risk Management Tools (from Jorion (2007))
1938 bond duration
1952 Markowitz mean-variance framework
1963 Sharpe's single beta model
1973 Black Scholes option pricing formula
1983 RAROC, Risk Adjusted Return
1992 Stress testing
1993 Value-at-Risk (VaR)
1994 RiskMetrics
1997 CreditMetrics
1998 integration of credit and market risk
1999 coherent risk measures
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28. Arthur CHARPENTIER - École d'été EURIA.
Agenda
• General introduction
Financial risks
• Market risks
• Credit risk
• From variance to Value-at-Risk
Risk measures and capital allocation
• Risk measures: an axiomatic introduction
• Risk measures: convexity and coherence
• Capital allocation: an axiomatic introduction
Risk measures and statistical inference
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29. Arthur CHARPENTIER - École d'été EURIA.
Market risks
Classical models for stock prices,
• dynamic models ( Bachelier (1900), Black Scholes (1973)), Brownian
geometric
√
dSt = µSt dt + V St dWt ,
drift random part
where (Wt )t≥0 is a standard brownian motion,
• more advanced dynamic models ( Heston (1993)) have stochastic volatility
dS = µS dt + √V dW S
t t t t
dVt = κ(θ − Vt )dt + ξ √Vt dW V ,
t
where (WtS )t≥0 and (WtV )t≥0 are two standard brownian motions (possibly
correlated).
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30. Arthur CHARPENTIER - École d'été EURIA.
200 Stock price over 1 year, large volatility Stock price over 1 year, large volatility
200
150
150
100
100
50
50
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Time Time
Figure 6: Random generation of a stock price, dSt = µSt dt + σSt dWt .
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31. Arthur CHARPENTIER - École d'été EURIA.
Market risks
Note that continuous GARCH processes can also be considered
dS = µS dt + √V dW S
t t t t
dVt = κ(θ − Vt )dt + ξVt dW V ,
t
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32. Arthur CHARPENTIER - École d'été EURIA.
Market risks
• the capital asset pricing model ( Markowitz (1970) or the Sharpe index are
based on the mean-variance framework,
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sp/)ance
sp/)ance
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0.5
0.
0.0
0.0
!0.5
!0.
!#.0
!#.0
0 5 #0 #5 0 #0 #
ca)t!type ca)t!t+pe
Figure 7: Capital asset pricing model, the mean-variance framework.
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33. Arthur CHARPENTIER - École d'été EURIA.
How to quantify market risks : volatility
All the information about uncertainty is summarized by the volatiliy - or
variance - parameter.
Note that this is one of the drawback of the use of the Gaussian distribution.
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34. Arthur CHARPENTIER - École d'été EURIA.
A (very) short word on diversication
Naturally, in higher dimension (when dealing with multiple stocks), Gaussian
vectors are considered
2
X1 µ1 σ1 ρ1,2 σ1 σ2 ··· ρ1,d σ1 σd
2
X2 µ2 ρ2,1 σ2 σ1 σ2 ··· ρ2,d σ2 σd
X= . ∼N
. . ,
. . .
. . . .
. . . . .
2
Xd µd ρd,1 σd σ1 ρd,2 σd σ2 ··· σd
2
All the information about marginal risks is in the variances (σi ) while all the
information on the dependence is in the correlation coecients (ρi,j ).
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35. Arthur CHARPENTIER - École d'été EURIA.
On the Gaussian distribution
The Gaussian distribution is very important for many reasons,
• it is a stable distribution, i.e. it appears as a limiting distribution in the
central limit theorem: for i.i.d. Xi 's with nite variance,
√ X − E(X) L
n √ → N (0, 1).
VX
• it is an elliptic distribution, i.e. X = µ + AX 0 where A A = Σ, and where
X0 has a spheric distribution, i.e. f (x0 ) is a function of x0 x0 (spherical level
curves),
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36. Arthur CHARPENTIER - École d'été EURIA.
3 Level curves of a spherical distribution Level curves of a elliptical distribution
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2
2
1
1
0
0
−1
−1
−2
−2
−3
−3
−3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3
Figure 8: The Gaussian distribution.
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37. Arthur CHARPENTIER - École d'été EURIA.
On the Gaussian distribution
X ∼ N (µ, Σ), and if
As a consequence, if
X1 µ Σ Σ12
X= ∼ N 1 , 11
X2 µ2 Σ21 Σ22
• Xi ∼ N (µi , Σi ), for all i = 1, · · · , d,
• α X = α1 X1 + · · · + αd Xd ∼ N (α µ, α Σα),
• X 1 |X 2 = x2 ∼ N (µ1 + Σ12 Σ−1 (x2 − µ2 ), Σ1,1 − Σ12 Σ−1 Σ21 )
2,2 2,2
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