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Putting the Power of
Modern Applied
Stochastics
into DFA
Peter
Blum
, Michel Dacorogna
, Paul
1)2)
2)
Embrechts 1)
1)
2)
ETH Zurich
Zurich Insurance Company
Department of Mathematics
Reinsurance
CH-8092 Zurich (Switzerland)
CH-8022 Zurich (Switzerland)
www.math.ethz.ch/finance
www.zurichre.com
Situation and intention

Applied stochastics provide lots of models that
lend themselves to use in DFA scenario
generation:
=> Opportunity to take profit of advanced
research.

However, DFA poses some very specific
requirements that are not necessarily met by a
given model.
=> Risk when using models uncritically.
Topics

Observations on the use of models from
mathematical finance (one discipline of applied
stochastics) in DFA

Updates on the modelling of rare and extreme
events (multivariate data and time series)

Annotated bibliography
DFA & Mathematical Finance:
Situation

DFA scenario generation requires models for
economy and assets: interest rates, stock
markets, inflation, etc.

Mathematical finance provides many such models
that can be used in DFA.

However, care must be taken because of some
particularities related to DFA.

Hereafter: some reflections...
Mathematical Finance: Background

Most models in mathematical finance were
developed for derivatives valuation. Fundamental
paradigms here:
No – arbitrage
 Risk – neutral valuation


Most models apply to one single risk factor; truly
multivariate asset models are rare.

Most models are based on Gaussian distribution
or Brownian Motion for the sake of tractability.
(However: upcoming trend towards more
advanced concepts.)
Excursion: the principle of noarbitrage

„In an efficient, liquid financial market, it is not
possible to make a profit without risk.“

No-arbitrage can be given a rigorous
mathematical formulation (assuming efficient
markets).

Asset models for derivative valuation are such
that they are formally arbitrage-free.

However, real markets have imperfections; i.e.
formally arbitrage-free models are often hard to fit
Excursion: risk-neutral valuation

In a no-arbitrage environment, the price of a
derivative security is the conditional expectation of
its terminal value under the risk-neutral probability
measure.

Risk neutral measure: probability measure under
which the asset price process is a martingale.

Risk-neutral measure is different from the realworld probability measure: different probabilities
for events.

Many models designed such that they yield
Implications on models

Many models in mathematical finance are
designed such that

They are formally aribtrage-free.

They allow for explicit solutions for option prices.

i.e. model structure often driven by mathematical
convenience.

Examples: Black-Scholes, but also Cox-Ingersoll-
Ross, HJM.

These technical restrictions can often not be
reconciled with the observed statistical properties
Consequences for DFA

Most important for DFA: Models must faithfully
reproduce the observable real-world behaviour of
the modelled assets.

Therefore: fundamental differences in paradigms
underlying the selection or construction of models.

Hence: take care when using models in DFA that
were mainly constructed for derivative pricing.

A little case study for illustration...
A little case study: CIR

Cox-Ingersoll-Ross model for short-term interest
rate r(t) and zero-coupon yields R(t,T).
dr (t )  a(b  r (t ))dt  s r (t )dZ (t )
1
R  t , T    r  t  B T   log A T  
T


2Ge( aG )T / 2
where: A(T )  

GT
(
a

G
)(
e

1)

2
G


2(eGT  1)
B(T ) 
(a  G )(eGT  1)  2G
G  a 2  2s 2
2 ab
s2
CIR: Properties

One-factor model: only one source of
randomness.

Nice analytical properties: explicit formulae for
Zero-coupon yields,
 Bond prices,
 Interest rate option prices.


(Fairly) easy to calibrate (Generalized Method of
Moments).

But: How well does CIR reproduce the behaviour
of the real-world interest rate data?
CIR: Yield Curves
Simulated yield curves using CIR (CHF)
True Yield Curves (CHF)
0.0450
0.04
0.0400
0.035
0.0350
Rate
Rate
0.045
0.03
0.0300
0.025
0.0250
0.02
0.0200
1
2
3
4
5
Time to maturity [Y]
6
7
1
2
3
4
5
Time to maturity [Y]
6
7
CIR Yield Curves: Remarks

CIR: yield curve fully determined by the short-term
rate!

Simulated curves always tend from the short-term
rate towards the long-term mean.

Hence: Insufficient reproduction of empirical
caracteristics of yield curves: e.g. humped and
inverted shapes.

From this point of view: CIR is not suitable for
DFA!
CIR: Short-term Rate (I)

Classical source: the paper by Chan, Karolyi,
Longstaff, and Sanders („CKLS“).

Evaluation based on T-Bill data from 1964 to
1989:

involving the high-rate period 1979-1982

involving possible regime switches in 1971
(Bretton-Woods) and 1979 (change of Fed policy).

Parameter estimation by classical GMM.

CKLS‘s conclusion: CIR performs poorly for short-
rate!
CIR: Short-term rate (II)


More recent study: Dell‘Aquila, Ronchetti, and
Trojani
Evaluation on different data sets:
Same as CKLS
 Euro-mark and euro-dollar series 1975-2000


Parameter estimation by Robust GMM.

Conclusions: classical GMM leads to unreliable
estimates; CIR with parameters estimated by
robust GMM describes fairly well the data after
1982.
Methodological conclusions

Thorough statistical analysis of historical data is
crucial! Alternative estimation methods (e.g.
robust statistics) may bring better results than
classical methods.

Models may need modification to fit needs of DFA.

Careful model validation must be done in each
case.

Models that are good for other tasks are not
necessarily good for DFA (due to different
requirements).
Excursion: Robust Statistics

Methods for data analysis and inference on data
of poor quality (satisfying only weak assumptions).


Relaxed assumptions on normality.

Tolerance against outliers.
Theoretically well founded; practically well
introduced in natural and life sciences.

Not yet very popular in finance, however:
emerging use.

Especially interesting for DFA: Small Sample
Asymptotics.
An alternative model for interest rates
(I)

Due to Cont; based on a careful statistical study
of yield curves by Bouchaud et al. (nice
methodological reference)

Consequently designed for reproducing real-world
statistical behaviour of yield curves.

Can be linked to inflation and stock index models.

Theoretically not arbitrage-free. However – if well
fitted:
„as arbitrage-free as the real world...“
An alternative model for interest rates
(II)
drt  1 (rt , st )dt   11 (rt , st )dWt1   12 (rt , st )dWt 2
Short-rate
dst   2 (rt , st )dt   21 (rt , st )dWt1   22 (rt , st )dWt 2
Spread
ft ( )  rt  st Y ( )  X t ( ) 
Y ( MIN )  0, Y ( MAX )  1
X t ( MIN )  0, X t ( MAX )  0
dX t   ( X t )dt   ( X t )dB t
Forward rate
Average yield curve shape
Stochastic deformation
Time evolution of deformation
In principle, X t is an infinite-dimensional process. However, it can be
boiled down to an easily tractable finite dimensional one.
Multivariate Models: Problem
Statement

Models for single risk factors (underwriting and
financial) are available from actuarial and financial
science.

However: „The whole is more than the sum of its
parts.“ Dependences must be duly modelled.

Not modelling dependences suggests
diversification possibilities where none are
present.

Significant dependences are present on the
US Interest Rate
US Inflation
USD per CHF
19
81
19
83
19
85
19
87
19
89
19
91
19
93
19
95
19
97
19
99
USD
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
18
16
14
12
10
8
6
4
2
0
CH Interest Rate
CH Inflation
19
81
19
83
19
85
19
87
19
89
19
91
19
93
19
95
19
97
19
99
[%]
18
16
14
12
10
8
6
4
2
0
19
81
19
83
19
85
19
87
19
89
19
91
19
93
19
95
19
97
19
99
[%]
Dependences: Example
Particlular problem: integrated asset
model

An economic and investment scenario generator
for DFA (involving inflation, interest rates, stock
prices, etc.) must reflect various aspects:
marginal behaviour of the variables over time
 in particular: long-term aspects (many years ahead)
 dependences between the different variables
 „unusual“ and „extreme“ outcomes
 economic stylized facts


Hence: need for an integrated model, not just a
collection of univariate models for single risk
factors.
General modelling approaches

Statistical: by using multivariate time series
models

established standard methods, nice quantitative
properties

practical interpretation of model elements often
difficult

Fundamental: by using formulae from economic
theory


explains well the „usual“ behaviour of the variables

often suboptimal quantitative properties
Phenomenological: compromise between the two
Economic and investment models

„CIR + CAPM“ as in Dynamo

Wilkie Model in different variants (widespread in
UK)

Continuous-time models by Cairns, Chan, Smith

Random walk models with Gaussian or  - stable
innovations

Etc.: see bibliography.

None of the models outperforms the others.
Investment models: open issues

Exploration of alternative model structures

Model selection and calibration

Long-term behaviour: stability, convergence,
regime switches, drifts in parameters, etc.

Choice of initial conditions

Inclusion of rare and extremal events

Inclusion of exogeneous forecasts

Time scaling and aggregation properties

Framework for model risk management
Excursion: Model Risk Management

Qualify and (as far as possible) quantify
uncertainty as to the appropriateness of the model
in use.

Which relevant dangers are (not) reflected by the
model?

Interpretation of simulation results given model
uncertainty

Particularly important in DFA: long-term issues.

Little done on MRM in quatitative finance up to
now (exception: pure parameter risk).
Rare & extreme events: problem
statement

Rare but extreme events are one particular
danger for an insurance company.

Hence, DFA scenarios must reflect such events.

Extreme Value Theory (EVT) is a useful tool.

C.f. Paul Embrechts‘ presentation last year.

Some complements of interest for DFA:

Time series with heavy-tailed residuals

Multivariate extensions
The classical case

X1, ... , Xn ~ iid (or stationary with additional
assumptions)

Xi  : univariate observations

Investigation of max {X1, ... , Xn}
=> Generalized Extreme Value Distribution (GEV)

Investigation of P (Xi – u  x | x > u)
(excess distribution of Xi over some threshold u)
=> Generalized Pareto Distribution (GPD)
The classical case: applications

Well established in the actuarial and financial
field:

Description of high quantiles and tails

Computation of risk measures such as VaR or
Conditional VaR (= Expected Shortfall  Expected
Policyholder Deficit)


Scenario generation for simulation studies

Etc.
In general: consistent language for describing
extreme risks across various risk factors.
Multivariate extremes: setup and
context
n

As before: X1, ... , Xn ~ iid, but now: Xi  
(multivariate)

Relevant for insurance and DFA? Yes, in some
cases, e.g.

Correlated natural perils (in the absence of suitable
CAT modelling tool coverage).


Presence of multi-trigger products in R/I
Area of active research; however, still in its
infancy:

Some publications on workable theoretical
Multivariate extremes: problems (I)

No natural order in multidimensional space:


=> no „natural“ notion of extremes
Different conceptual approaches present:

Spectral measure + tail index (think of a
transformation into polar coordinates)

Tail dependence function (= Copula transform of
joint distribution)


Both approaches are practically workable.
Generally established workable theory not yet
present.
Multivariate extremes: problems (II)

In the multivariate setup:„The Curse of
Dimensionality“

Number of data points required for obtaining „well
determined“ parameter estimates increases
dramatically with the dimension.


However, extreme events are rare by definition...
Problem perceived as tractable in „low“ dimesion
(2,3,4)

Most published studies in two dimensions

Higher-dimensional problems beyond the scope of
Time series with heay-tailed residuals

Given some time series model (e.g. AR(p)):
Xt = f (Xt-1 , Xt-2 , ... ) + t | 1 , 2 , ... ~ iid, E (i) = 0

Usually: t ~ N(0,  2) (Gaussian)

However: there are time series that cannot be
reconciled with the assumption of Gaussian
residuals (even on such high levels of time
aggregation as in DFA).

Therefore: think of heavier-tailed – also skewed –
distributions for the residuals! (Various
Heavy-tailed residuals: example

QQ normal plots of yearly inflation (Switzerland
and USA)

Straight line indicates theoretical quantiles of
Gaussian distribution.
Heavy-tailed residuals: direct
approach

Linear time series model (e.g. AR(p)), with
residuals having symmetric--stable (ss)
distribution.

ss: general class of more or less heavy-tailed
distributions;

 = characteristic exponent; can be estimated from
data.

 = 2  Gaussian;  = 1  Cauchy.

Disadvantage: ss RV‘s in general difficult to
simulate.
Superposition of shocks

Normal model with superimposed rare, but
extreme shocks:
Xt = f (Xt-1 , Xt-2 , ... ) + t + t t
 1 , 2 , ... ~ iid Bernoulli variables (occurrence of
shock)


1 , 2, ... the actual shock events
Problem: recovery of model from the shock!

Shock itself is realistic as compared to data.

But model recovers much faster/slower than actual
data.
Continuous-time approaches

„Alternatives to Brownian Motion“ (i.e. Gaussian
processes)

General Lévy processes

Continuous-time  - stable processes

Jump – diffusion processes (e.g. Brownian motion
with superimposed Poisson shock process)

Theory well understood in the univariate case.

Emerging use in finance (e.g. Morgan-Stanley)

Mutivariate case more difficult: difficulties with
correlation because second moment is infinite.
Further approaches

Heavy-tailed random walks (ss – innovations);
possibly corrected by expected forward premiums
(where available).

Regime-switching time series models, e.g.
Threshold Autoregressive (TAR or SETAR = SelfExcited TAR).

Non-linear time series models: ARCH or GARCH
(however: more suitable for higher-frequency
data).
Conclusions (I)

Applied stochastics and, in particular,
mathematical finance offer many models that are
useful for DFA.

However, before using a model, careful analysis
must be made in order to assess the
appropriateness of the model under the specific
conditions of DFA. Modifications may be
necessary.

The quality of a calibrated model crucially
depends on sensible choices of historical data
Conclusions (II)

Time dependence of and correlation between risk
factors are crucial in the multivariate and
multiperiod setup of DFA. When particularly
confronted with rare and extreme events:

Time series models with heavy tails are well
understood and lend themselves to the use in DFA.

Multivariate extreme value theory is still in its
infancy, but workable approaches can be expected
to emerge within the next few years.