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Putting the Power of Modern Applied Stochastics into DFA

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### Putting the Power of Modern Applied Stochastics into DFA

Peter 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 – 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 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 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 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).

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: 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...“

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.

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
- models designed for reflecting statistical behaviour of data
- allowing 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 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

- 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 foundations
- Few (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 dimensions
- Higher-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.

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