time based model cfe-ercim'11_light2
Post on 05-Apr-2018
223 Views
Preview:
TRANSCRIPT
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
1/40
A new time based approach
for operational risk
Duc PHAMHI
Dept. of Financial Engineering
CFE-ERCIM 2011
CS 16: Risk management
Dec 17, 20111 Duc Pham-Hi
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
2/40
The current model in AMA Loss distribution approach
Not a market model
Not a time based model good reasons for this : rare events
Based on Monte Carlo Convolution between frequency and severity distributions
Practical unsatisfactory in agregation and scenarios
Dec 17, 20112 Duc Pham-Hi
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
3/40
Most frequent practice of operational risk For a single type of loss:
But in practice, yearly losses are given by
=
=1
)()(k
kXjL
)(p
with p ~ Poisson and Xk~ lognormal, GPD, etc.
Product of 2 distributions
Computed with FFT
Computed with Panjers formula
But in most cases with Monte Carlo
=
=1k
Dec 17, 20113 Duc Pham-Hi
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
4/40
LDA schematics
4 Dec 17, 2011Duc Pham-Hi
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
5/40
No time dimension AMA models are currently time insensitive
Switching time axis produce sameValue at risk !
They are even almost risk insensitive
Add a last loss worth a VAR
The new VAR can be moved only by a few percentdepending on the size of time series of losses
No real time risk monitoring possible !
Dec 17, 20115 Duc Pham-Hi
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
6/40
Value-at-Risk level is insensitive to risk
evolution
Illustration :
Value At Risk at 99,9 % in these 2 situations is the same !
For the bank with more risks and the bank with diminishing risks
6 Dec 17, 2011Duc Pham-Hi
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
7/40
Even worse : Regulatory time space
Regulatory capital asValue at risk computed quarterly
New losses as new data But recalculation of parameters only a yearly exercise
Qualitative data updated less frequently.
Parameters of model to evolve with enormous inertia, due tothis batch process.
Updates are slow and not welcome Many theoretical difficulties still untreated
Dec 17, 20117 Duc Pham-Hi
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
8/40
Difficulties at each step bottom-up
Local loss distribution
Total loss distribution
Risk Mitigation (1)Capital requirement
Aggregationmechanism
Duc Pham-Hi8
Data collection
Data refinement and EDA
Statistically homogeneous risk class ?
Quantification (Severity-Frequency)both LDA and SBA
Correlations /dependencies
s m ga on
Fittingmethodology
Data quality
Dec 17, 2011
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
9/40
Why Supervisors should pay attention to
modeling process
As a consequence :
Top management cares only aboutValue at Risk
Managers are not familiar with probability density
functions (Pareto, Weibull, Lognormal ) enough to careabout parameters.
Modelers are the only ones who care, but they have nomandate
Parameters do not evolve , or only as long term stairs
function.
Dec 17, 20119 Duc Pham-Hi
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
10/40
More frequent Value-at-risk watch
When in crisis, value-at-risk should be watched daily e.g. Kerviel case : top management needs ways of tracking how the
value-at-risk of not only portfolios, but also of operational risks,evolve as a time function of the unwinding of those massive illicittransactions.
, A statistical approach demand too much (scarce) data for
calibration Price of sophistication can outweight benefits (g-and-h)
Alternatives Use Lvy process Use exploratory modeling Maybe both
Dec 17, 201110 Duc Pham-Hi
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
11/40
Modelin time based o erational
risk with dynamic control
Dec 17, 2011Duc Pham-Hi11
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
12/40
Modelling Loss using Lvy process Small losses as Lognormal
Large losses as jumps
(heterogeneous Poisson) Evolution of aggregate loss is sum
If are added other business characteristics for the bank :
)exp(0, tAA BXX =
tBB LXX exp0,=
tBtAt XXZ ,, +=
Duc Pham-Hi12
a drift which is the rate of earning a subordinator representing a provisioning system that
compensates for the
cumulated loss
put together, they yield the wealth evolution equation (much like
Itos formula applied to deriveZt )
dtdR =
)(),(),(~
dzdtdzdtNdzdtN =
Dec 17, 2011
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
13/40
Incorporating time into operational risk
models Differential form for loss equation, derived from Itos formula
The banks evolution equation is ++=
< **R 1
),(~
)1()()1(R
z
z
zdzdtNedtdzzebZdZ 1
Duc Pham-Hi13
where trepresents the part of losses unabsorbed by reserves
This modelization with time flow enables risk management policy to
be formulated as an optimization problem
tt
Dec 17, 2011
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
14/40
Ex erimentation :
human fraud modeling
(not Levy based)
Dec 17, 2011Duc Pham-Hi14
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
15/40
On Fraud modeling Op risk modeling in banks needs to take new tracks
Supervisors and regulators should encourage moreexploration of human risks
Fraud
Duc Pham-Hi15
Litigations and lawsuits which are hidden and cumulative,then reach unexpected punitive levels
Time is of the essence !
But is until now absent in models
Risk policy making should be modeled too.
Dec 17, 2011
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
16/40
Exploratory forward direct modelling
Assume an organization which produce errors, controls errors,
make mistakes or voluntary fraud, some of which accumulateuntil discovered
Experiments with computational models of 2 interacting
Duc Pham-Hi16
departments Lognormal x Poisson as usually encountered in banks model,
but add 2 parameters :
Generation of fraud rate/threshold
Detection of fraud frequency/threshold
Small, single risk models
Dec 17, 2011
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
17/40
Fat tails are
generated
naturally
200
400
600
800
1000
1200
1400
1600
140
Duc Pham-Hi17
risk prevention cancut tails even shorter
0 2 4 6 8 10 12
x 10
5
0 1 2 3 4 5 6 7
x 105
0
20
40
60
80
100
120
Dec 17, 2011
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
18/40
The rate of bursting
the bubble can becontrolled
3.5x 10
5
1
2
3
4
5
6
7x 10
5
Duc Pham-Hi181.3 1.35 1.4 1.45 1.5 1.55 1.6
x 105
0
0.5
1
1.5
2
2.5
31.3 1.35 1.4 1.45 1.5 1.55 1.6
x 105
0
Dec 17, 2011
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
19/40
Risk states: what about effects of fighting
fraud, environment of fraud?
Wealth WWealth
W+dWTransition Probability
Environt var.
)()( ttF =
Dec 17, 2011Duc Pham-Hi19
Control variables ,
=)(
0
)()(tN
jt xKLG
amountfixedHwherexHxK jj == )(),(inf)(
1)(0).()(
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
20/40
Risk Management dynamic control factors
Reducing small losses through better process management, e.g.
where t is the loss reduction factor whose cost is expense
))(()( ttF =
Duc Pham-Hi20
Reducing impact of catastrophic events
through insurance, or recovery plans, at cost where losses xj maybe capped or reduced ,
=)(
0
)()(tN
jt xKLG
1)(0).()(
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
21/40
Valuing risk management policy Wealth evolution is :
We get state of wealth at time
( ) =
+++=)(
1
0, )()()()(tN
j
jtAt xKdBXdtRdW
=
),,),(,,( ttttdWW
Duc Pham-Hi21
which leads us to formulate policy whose economic value underutility function Uis :
where t is the given of a pair ((t) , (t))
0
[ ]dttWUrtV =
0
)()exp(
Dec 17, 2011
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
22/40
Risk management
as an optimal policy search through HJB Objective is to maximize a time based value
Resolution strategies
Solve as pure Hamilton-Jacobi-Bellman through Galerkin
[ ]
= dttWUrtEV
0
)()exp(max
Duc Pham-Hi22
reminds of Basel II matrix)
Neural techniques/ supervised learning, if experience base available
and sufficient. Adaptive, Reinforcement Learning , TD learning, Q online sampling
MCMC exploratory techniques instead of backward fitting from data
= kkrJ
Dec 17, 2011
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
23/40
About solving HJB Known cases
Linear Quadratic
in Merton's problem and Black Scholes context New : with Levy processes, but only for HARA and
CARA utility function : explicit solutions
Dec 17, 2011Duc Pham-Hi23
But still, classic obstacles
Unknown P(x,y) POMDP in Markovian case
Curse of dimensionality
Too large sets { y } for eachx
Too large sets {} for eachx
Too strong nonlinearities
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
24/40
Ex lorator solution strate ies for
dynamic op risk equation
Dec 17, 2011Duc Pham-Hi24
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
25/40
Learning as quick sampling of state space
wrt. rewards To use Q-learning, (Watson) philosophy is Action-Reward :
using time based action at:
at=
(x
t)then reward the (stochastic) consequence
{ }),(),( axrEaxR =
Dec 17, 2011Duc Pham-Hi25
Classically , define value function is the total of what can be expectedin the future (here zero terminal value)
Introducing a discount rate and taking the expected value(stochastic case) :
==
0)),(,()),(,(
ttt dttxxrtxxV
==
=0
0))(,()(t
tt
txxxxrExV
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
26/40
Dual problem : solving for strategies or for
economic value of risk Rationality of the risk center is to seek maximum of value, starting from state t x0 ,
to "learn" policy maximisingV over set A of admissible actions satisfying
where V(x)is the the consequence of following policy from initial situation x[ ]
)(min)(00
*
xVxVA
=
t
Dec 17, 2011Duc Pham-Hi26
Now introduce transition mechanism between states xand y:
If we want to solve for optimal policy , it requires dealing with nonlinear equation
+=
=0
* ))(,())(,(minarg)(t
tt
txxrExxRx
=0
,,
t
tt
+= ++y
tyxtt yVPxRxV )(..),()( ,11
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
27/40
Solving for risk with Temporal Differences We turn to solving for value ( if G is terminal value)
by reasoning in terms of discrete time. Alternately, in terms of discrete states y,as possible outcomes of state x, and introducing action at :
[ ]),,()1),((min),( tuxGtxfVtxVUu
++=
+= a
yVyxPxarV )(),(),(min*
Dec 17, 2011Duc Pham-Hi27
We can iterate on V since the problem is linear.
let t be the proxy for V at time t ; we iterate thus :
As a special case, Q-learning is particularly easy to set up and is model-free
=y yxPxyxP 0),(,1),(y
+ + y tt
yyxPxrxV )(),(.),(min)(1
),().,(),(),(1 ayQyspasgasQ tt +=+
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
28/40
Forward solutions : filterin and
online learning
Dec 17, 2011Duc Pham-Hi28
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
29/40
Operational risks and expert opinions Role of scenarios
Mandatory internal surveys
Quantification ?
How LDA and scenarios mix
Duc Pham-Hi29
Often labeled bayesian approach Its rather belief networks
Another variant, Bayesian networks Essentially a factor network
Quantified through exposure, occurrences, severity parameters
Dec 17, 2011
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
30/40
Positioning bayesian and non-bayesian
inferences
Exploration (previously seen) requires online states sampling
Q learning (Watson)
Reinforced learning (Barto & Sutton)
Temporal Differences (Tsitsiklis & Van Roy)
Duc Pham-Hi30
Too little data ? Learn from simulation
Interacting Particle Systems (N. Shephard & Flury)
but if and only if model has no bias
Dec 17, 2011
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
31/40
Transpose real time filtering techniques. One of the easier ways in filtering is to try some moving average
with or without weighting systems that let one attribute a decaying role to
older data as one goes back in time.
One of the more sophisticated filters is the Kalman filter.
considers the whole distribution of probability,
if it can be assumed that it is Gaussian, and that the distortion is linear.
But in risk situations that are extreme, neither of these assumptions holds.
Sequential Monte Carlo (SMC), or Interacting Particle Filters (IPS).
has been applied to Probabilistic Robotics [Thrun][De Freitas] or signal
processing [Doucet]
Dec 17, 201131 Duc Pham-Hi
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
32/40
An example of IPS tracking a noisy evolution
Dec 17, 2011Duc Pham-Hi32
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
33/40
Getting proactivity into equations of
operational Losses
just feed the projection of new losses into the usual batch process for theValue-at-risk
However, as already discussed, this process suffers from too muchinertia.
A more proactive way to take into account new data : Bayesian blending of
expert based opinions and hard, collected loss data. This method has long been discussed, as various implementations of
credibility theory were proposed. One of the more rigoroustreatments of the topic is by [Lambrigger ShevchenkoWuthrich].
Even if expert opinions are off target, they bring about a quickerconvergence of the inferred parameters to the real hidden values.[Peters]
Dec 17, 201133 Duc Pham-Hi
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
34/40
Lambrigger-Shevchenko-Wuthrich approach Give weights to sources of information
)()1()()()(exp212int1
xFwwxFwxFwxFertext
++=
)()( =PoissonNd
Given a set of observations at timeJ,N = (N1 ,, NJ )
If furthermore we have a set of expert opinions = ( 1 ,, )
)(~
=
J
j
jNPN1
)()()(
Dec 17, 201134 Duc Pham-Hi
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
35/40
Theorem For a Poisson-Gamma-Gamma model,
( ))exp(
)2(2
)/(,
1
2/)1(
=
+
+
KN
/1 +=dd
Nj
0
1
+= J
=m
m
+
+ =0
2/)/1(
12
1)( dueuzK uuz
Dec 17, 201135 Duc Pham-Hi
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
36/40
Computing loss by L-S-W
Posterior of , given N and , has a Generalized Inverse
Gaussian (GIG) distribution Algorithm
1. Simulate from GIG accordin to revious
2. Simulate N from Poisson ()
And obtain the empirical distribution for the number of losses
Or use MCMC method
Dec 17, 201136 Duc Pham-Hi
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
37/40
e.g. Poisson-Gamma
Poisson ( = 0.6) ( 0 = 3.4 ; 0 = 0.15) = 4Vs. Maximum likelihood
Dec 17, 2011Duc Pham-Hi37
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
38/40
To conclude
AMA op risk models as of today are not
Forward looking Time Risk sensitive
Dec 17, 2011Duc Pham-Hi38
There are ways to model time into them And there are ways to solve these models
The Regulator should encourage these efforts
T h t t t ti li k b t IPS d
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
39/40
Tech note: tentative links between IPS and
expert opinion
Linking Resampling IPS, classic Importance Sampling (IS), Sequential MCafter a spectrum of variants has been introduced to minimize the effects ofdegeneration in particles.
They can both be viewed as changes of measure in Markov chains but, inthe case of IPS, implicit Feynman-Kac change of measure is lessdemandin on data than the im licit Girsanov chan e of measure in IS.
In the field of Finance, there has been a first application into Credit Risk[Carmona]. [Crepey] investigated, as a consequence, the advantages of IPSover IS in Contagion cases in Credit portfolio risks.
[Shephard] showed that IPS can be very useful for learning out of simulated
samples, thus reducing the need for real data. This is a great remedy tothe scarcity of data in extreme losses in operational risks.
Dec 17, 201139 Duc Pham-Hi
-
8/2/2019 Time Based Model CFE-ERCIM'11_light2
40/40
Thank ou !
phamhi@ece.fr
Dec 17, 2011Duc Pham-Hi40
top related