Putting the Power of Modern Applied

Stochastics into DFAPeter Blum 1)2) , Michel Dacorogna 2), Paul

Embrechts 1)

1)

ETH Zurich

Department of

Mathematics

CH-8092 Zurich

(Switzerland)

www.math.ethz.ch/

finance

2)

Zurich Insurance

Company

Reinsurance

CH-8022 Zurich

(Switzerland)

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.

Goal: provide some guidance on how to (re)use

stochastic models in DFA.

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 – arbitrageRisk – 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 no-arbitrage „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

to real-world data.

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 real-world probability measure: different probabilities for events.

Many models designed such that they yield explicit option prices under risk neutral measure.

Implications on models Many models in mathematical finance are designed such

thatThey 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 of real-world data.classical example: volatility smile in the Black-Scholes

model.

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, log R t T r t B T A T

T

2

2( ) / 22

( )( )( 1) 2

aba G T s

GT

GeA T

a G e G

2( 1)( )

( )( 1) 2

GT

GT

eB T

a G e G

2 22G a s

where:

CIR: Properties

One-factor model: only one source of randomness.

Nice analytical properties: explicit formulae forZero-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

True Yield Curves (CHF)

0.02

0.025

0.03

0.035

0.04

0.045

1 2 3 4 5 6 7

Time to maturity [Y]

Rat

e

Simulated yield curves using CIR (CHF)

0.0200

0.0250

0.0300

0.0350

0.0400

0.0450

1 2 3 4 5 6 7

Time to maturity [Y]

Rat

e

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!

But: What about the short-term rate?

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.

Hence: CIR can be a good model for the short-term rate!

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).

Residual uncertainty must be taken into account when evaluating final DFA results.

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.Relevance of estimates based on little data...

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)

1 21 11 12( , ) ( , ) ( , )t t t t t t t t tdr r s dt r s dW r s dW

1 22 21 22( , ) ( , ) ( , )t t t t t t t t tds r s dt r s dW r s dW

( ) ( ) ( )t t t tf r s Y X

( ) 0, ( ) 1MIN MAXY Y ( ) 0, ( ) 0t MIN t MAXX X

( ) ( )t t t tdX X dt X d B

Short-rate

Spread

Forward rate

Average yield curve shape

Stochastic deformation

Time evolution of deformation

In principle, is an infinite-dimensional process. However, it can be

boiled down to an easily tractable finite dimensional one.tX

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

financial and on the underwriting side.

Dependences: Example

02468

1012141618

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

[%]

US Interest Rate

US Inflation

02468

1012141618

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

[%]

CH Interest Rate

CH Inflation

00.10.20.30.40.50.60.70.80.9

1

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

US

D

USD per CHF

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“ outcomeseconomic 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 propertiespractical interpretation of model elements often difficult

Fundamental: by using formulae from economic theoryexplains well the „usual“ behaviour of the variablesoften suboptimal quantitative properties

Phenomenological: compromise between the twomodels designed for reflecting statistical behaviour of dataallowing nevertheless for practical interpretation

Phenomenological approach most promising for DFA.

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). Sources of inspiration: statistics (frequentist and

Bayesian), economics, information theory (Akaike...),

etc.

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 residualsMultivariate 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 tailsComputation of risk measures such as VaR or

Conditional VaR (= Expected Shortfall Expected

Policyholder Deficit)Scenario generation for simulation studiesEtc.

In general: consistent language for describing

extreme risks across various risk factors.

Multivariate extremes: setup and context As before: X1, ... , Xn ~ iid, but now: Xi n (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 foundationsFew (pre-industrial) applied studies (FX data, flood, etc.)Considerable progress expected for the next years.

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 dimensionsHigher-dimensional problems beyond the scope of

current methods

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

approaches present.)

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 (ss) distribution.ss: general class of more or less heavy-tailed

distributions; = characteristic exponent; can be estimated from data. = 2 Gaussian; = 1 Cauchy.Disadvantage: ss RV‘s in general difficult to simulate.

Take care with other heavy-tailed distributions (e.g

Student‘s t): multiperiod simulations may become

uncontrollable.

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.

Hence: Care must be taken.

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 (ss – innovations);

possibly corrected by expected forward

premiums (where available).

Regime-switching time series models, e.g.

Threshold Autoregressive (TAR or SETAR = Self-

Excited 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 and methods for parameter estimation.

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.