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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)

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putting the power of modern applied stochastics into dfa

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
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
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 & 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
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
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
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
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
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
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
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 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
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
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
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
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
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
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
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
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
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
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
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 & 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
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
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
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
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
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
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
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
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.
superposition of shocks
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
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
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).
conclusions i
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
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.