Putting the Power of Modern Applied Stochastics into DFA

1. 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

2. 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.

3. 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

4. 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...

5. 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.)

6. 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.

7. 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.

8. 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.

9. 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...

10. A little case study: CIR • Cox-Ingersoll-Ross model for short-term interest rate r(t) and zero-coupon yields R(t,T).

11. 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?

12. CIR: Yield Curves

13. 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?

14. 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!

15. 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!

16. 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.

17. 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...

18. 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...“

19. An alternative model for interest rates (II)

20. 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.

21. Dependences: Example

22. 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.

23. 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.

24. 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.

25. 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

26. 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.

27. 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

28. 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)

29. 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.

30. 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.

31. 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.

32. 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

33. 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.)

34. Heavy-tailed residuals: example • QQ normal plots of yearly inflation (Switzerland and USA) • Straight line indicates theoretical quantiles of Gaussian distribution.

35. 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. • Take care with other heavy-tailed distributions (e.g Student‘s t): multiperiod simulations may become uncontrollable.

36. 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.

37. 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.

38. 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 = Self-Excited TAR). • Non-linear time series models: ARCH or GARCH (however: more suitable for higher-frequency data).

39. 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.

40. 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.