intermittent behavior of financial time series - MathFinance

Statistical Mechanics of Financial Time Series andApplications in Risk ManagementPresentation at the Math Finance WorkshopHochschule für Bankwirtschaft, Frankfurt, April 2004Dr. Peter NeuDresdner Bank AGGroup Risk ControlFrankfurtProf. Dr. Reimer KühnKing’s CollegeMathematics DepartmentLondonDisclaimer: This presentation expresses the views of the authors and does not necessarily reflect policies or risk models of Dresdner Bank AG

AGENDAGeometric Brownian motion and intermittent behavior of financial timeseries: universality, scaling, volatility correlations, non-stationarityMinimal interacting generalization of the geometric Brownian motionApplications to Risk Measurement: Market, Credit & Operational RiskPage 22 © Neu&Kühn, Math Finance Workshop · April, 1 th 2004

AGENDAGeometric Brownian motion and intermittent behavior of financial timeseries: universality, scaling, volatility correlations, non-stationarityMinimal interacting generalization of the geometric Brownian motionApplications to Risk Measurement: Market, Credit & Operational RiskPage 33 © Neu&Kühn, Math Finance Workshop · April, 1 th 2004

DO THE MODELS FOR FINANCIAL TIME SERIES WORK WELLIN PRACTICE?“Any virtue can become a vice if taken to extreme,and the models are only approximations to the complex real world.”Robert Merton, 1995“Suddenly, all the multi-factor interest rate models weremeaningless as a basis for trading decisions…Those quants hailing from physics background found the eventsreminiscent of a “phase change”, such as when liquid turns into gas.”Nicholas Dunbar, Risk Magazine 1998reporting about the Russia crisisand the LTCM bailoutPage 44 © Neu&Kühn, Math Finance Workshop · April, 1 th 2004

APPLICATION IN MARKET RISK MANAGEMENTValue-at-RiskValue-at-Risk defined w.r.t. a time horizon Dt and confidence interval q:d PV = PV(x) - PV(0)d PVValue-at-Risk defined as( δ PV ( ∆t)> VaR∆) = 1−qPrq,tVaR L0VaR xrisk factorsInterplay betweenq distribution of risk factorsq (non-linear) pricing function0xS(t + ∆t)= lnS(t)Page 66 © Neu&Kühn, Math Finance Workshop · April, 1 th 2004

IS THE GBM-MODEL CORRECT?Return Distribution Is Not GaussianCentral result of GBM-model contradicts empirical observationslogSi(t + τS ( t)i)∝⎡⎛φ ⎢⎜⎣⎝µi−σi22⎞⎟ ⋅ τ ,⎠σi⋅τ⎤⎥⎦Φ: standard normal pdf-1-2-3End-of-day DAX log-returnsbetween 14.2.1989 and13.2.2004St. NormalDAXExtreme stock price-4changes are moreln P-5frequent than-6-7predicted by the GBM-8-9-10Data Source: Dresdner Bank AG-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6Page 7Standardized log-returns7 © Neu&Kühn, Math Finance Workshop · April, 1 th 2004

HOW CAN WE DO BETTER?Physicists in academia became interested in finance because ofsimilarities to the physics of complex systems and phase transitionsQuestion raised are:q Are financial time series stationary?q Is there universality of financial time series, i.e., do prices of different assetshave the same statistical properties?q Is there a single market mechanics driving stock prices at all time scales?q Is there a stable distribution replacing the Gaussian distribution, e.g., Lévydistribution?q What is the underlying market dynamics replacing the GBM?Page 88 © Neu&Kühn, Math Finance Workshop · April, 1 th 2004

INTERMITTENT BEHAVIOR OF FINANCIAL TIME SERIESVolatility Bursts and Leverage Effect: Financial Time Series Are Not Stationary4%S&P 5002%0%-2%End-of-day log-retuns: Ln S(t+ 1 day) – lnS(t)-4%4%2%0%-2%-4%4%2%0%-2%-4%DowDAX-6%14.02.8914.08.8914.02.90Data Source: Dresdner Bank AG14.08.9014.02.9114.08.9114.02.9214.08.9214.02.9314.08.9314.02.9414.08.9414.02.9514.08.95Page 914.02.9614.08.9614.02.9714.08.9714.02.9814.08.9814.02.9914.08.9914.02.0014.08.0014.02.0114.08.0114.02.0214.08.0214.02.0314.08.039 © Neu&Kühn, Math Finance Workshop · April, 1 th 2004

INTERMITTENT BEHAVIOR OF FINANCIAL TIME SERIESUniversalityReturns of different stock indices have the same statistical properties420End-of-day log-returns between14.2.1989 and 13.2.2004Exponentialλ = 1.5S&PDOWDAXNIKKEIln P(Z)-2-4-6-8Data Source: Dresdner Bank AG-10-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6Normalized log-returns ZPage 1010 © Neu&Kühn, Math Finance Workshop · April, 1 th 2004

INTERMITTENT BEHAVIOR OF FINANCIAL TIME SERIESStable Lévy Distribution and Scaling at Intermediate FluctuationsMandelbrot: At intermediatefluctuations a stable (m < 2) Lévydistributions is appropriateSource: Mantegna & Stanley, Nature 376, 46 (1995)S&P 500: τ = 1 min, µ = 1.4P(Zτ)∝ Z−( 1+µ )τ,µ≈32Apparent scaling suggestsS&P 500: τ = 1 min - 16 hq same market mechanism operatingat all time scales τq single universal distribution functionPage 1111 © Neu&Kühn, Math Finance Workshop · April, 1 th 2004

INTERMITTENT BEHAVIOR OF FINANCIAL TIME SERIESSummary“Stylized” facts in financial time series:q Trading activities are very non-stationary: quiescent periods change with hectictrading activities resulting in volatility burst of financial time seriesq All empirical data have fat-tailed (leptokurtic) probability distributionq Underlying distribution of “all” stocks seems to follow an “universal” non-stablepower law in the large fluctuation regime (inverse cubic law: P(Z > x) ~ x – 3 )q Although distribution is non-stable, convergence to Gaussian is extremely slowq Amplitudes of stock returns decay algebraically (υ ~ 0.3-0.6) showing that thereis no characteristic time scale and correlations extend infinitely far in timeWhat is the underlying market dynamics replacing the GBM?The answer of this question is not truly established at presentPage 1414 © Neu&Kühn, Math Finance Workshop · April, 1 th 2004

AGENDAGeometric Brownian motion and intermittent behavior of financial timeseries: universality, scaling, volatility correlations, non-stationarityMinimal interacting generalization of the geometric Brownian motionApplications to Risk Measurement: Market, Credit & Operational RiskPage 1515 © Neu&Kühn, Math Finance Workshop · April, 1 th 2004

MINIMAL INTERACTING GENERALIZATION OF THE GBMMotivation of the ModelApparent contradiction to Central Limit Theorem: return distribution isnon-stable, nonethelessq PDF of stock indices have the same power-law behavior as have individualstocksq Power law converges very slowly (τ > 4 days) to GaussianConclusions:q Temporal and inter-asset correlation must be importantq Hypothesis that traders act as independent agent is wrongModel setting – include:q Functional dependencies between stock returnsq Incoming information, communication between traders and herdingPage 1616 © Neu&Kühn, Math Finance Workshop · April, 1 th 2004

MINIMAL INTERACTING GENERALIZATION OF THE GBMDynamicsGeometric Brownian Motiondhidt= Ii+σ ⋅η (t)iihi= logSiSi0I iσ i= µi−22Minimal interacting generalization of the Geometric Brownian Motiondhidt= Ii−κi⋅hi+ ∑Jj(j≠i)ij⋅g(hj)+σi⋅η( t)iMean revertingterm ensures longtermstability ofmarketsSource: Kühn, Kutsia & Neu, work in progresssee also: G. Iori, Int. J. Mod. Phys. C10, 1149 (1999)Functional non-linearinteraction termbetween stock pricesPage 17GBM valid ontime scalesshort comparedto t ~ 1/κ, 1/J17 © Neu&Kühn, Math Finance Workshop · April, 1 th 2004

MINIMAL INTERACTING GENERALIZATION OF THE GBMModel Settingsdhidt= Ii−κi⋅hi+ ∑Jj(j≠i)ij⋅g(hj)+σi⋅η( t)iMean reverting term: k

MINIMAL INTERACTING GENERALIZATION OF THE GBMSimulation of Incoming Negative InformationSimulated stock return distribution shows the same statisticalproperties as market stock returns and universality1,E+0101,E+00SimulationGaussExponential-2SimulationPower lawζ = 41,E-01-4GaussP(dh)1,E-02λ = 2ln P(|dh|>x)-6-81,E-03-101,E-04-121,E-05-4 -3 -2 -1 0 1 2 3 4dhSource: Kühn, Kutsia & Neu, work in progressPage 19-14-3 -2 -1 0 1 2ln (x)19 © Neu&Kühn, Math Finance Workshop · April, 1 th 2004

MINIMAL INTERACTING GENERALIZATION OF THE GBMSimulation of Incoming Negative InformationSimulated stock returns show …q volatility burstsq fast decaying return correlationsq algebraically slow decaying volatilitycorrelation (υ ~ 0.6)Source: Kühn, Kutsia & Neu, work in progressPage 2020 © Neu&Kühn, Math Finance Workshop · April, 1 th 2004

AGENDAGeometric Brownian motion and intermittent behavior of financial timeseries: universality, scaling, volatility correlations, non-stationarityMinimal interacting generalization of the geometric Brownian motionApplications to Risk Measurement: Market, Credit & Operational RiskPage 2121 © Neu&Kühn, Math Finance Workshop · April, 1 th 2004

MEASUREMENT OF MARKET RISKLiquidity of Trading Position Is DecisiveStock returns change more jumpy than in a random walkq no perfect hedge possibleq impact on trading P&L hinges on liquidity of trading position− Liquid trading positions can be closed on short-term notice=> intermittent behavior has no time to impact− Illiquid trading positions or those which cannot be sold on strategic grounds arefully exposed to intermittent behaviorRegulatory capital puffer for market risks assumes market liquidationperiod of 10 days:Regulatory capital for market risk∝ average VaR 99×%,1Day10q intermittency renders the “square-root of 10” law invalid for illiquid tradingpositionsPage 2222 © Neu&Kühn, Math Finance Workshop · April, 1 th 2004

MEASUREMENT OF CREDIT RISKStructural Approach With Factor Model for Inter-asset CorrelationAsset return driven by factor noise:log A ( t ∆t)~ σ ∆t⋅η>ηi+ii= ∑ ∑2iβik⋅ Yk+ 1−βik⋅εik = 1kCommonrisk factorsfor all firmsDefault probabilities:Pr( Ai(T)< D i) PD i , TPr =( A ( T)< D Y)ii=⎛⎜ ΦΦ⎜⎜⎝Firm-specificrisk factors−1!log D( PDi,T) − ∑1−∑ βkk2ikPage 23iβikΦ is cumulative normal distributionYk⎞⎟⎟⎟⎠Monte Carlo simulationCredit-Loss distributionUnexpected LossConfidence intervalEL Value at Risk23 © Neu&Kühn, Math Finance Workshop · April, 1 th 2004

FUNCTIONAL COUPLINGS GENERATE COLLECTIVECASCADING DEFAULTSq Default indicator variableq Default dynamics driven byn ( t + ∆t)=i⎧1⎨⎩0A ( t)< DiA ( t)≥ Diii− Gaussian couplings β ik− Functional couplings w ijq Conditional default probabilitybecomes path-dependentPr( Ai(t + ∆t)< Di{ Y,n}) = ni(t + ∆ t){ Y , n}⎛⎜ Φ= Φ ⎜⎜⎝−1( PD )i,∆t+∑j1−w∑kijn ( t)−jβ2ik∑kβikYk⎞⎟⎟⎟⎠q Parameter calibration− PD i|j = transition default probabilityw ij=Φ−1−1( PD i ) − Φ ( PD )q Perform path-dependent Monte Carlosimulations: N = T/∆t time steps in riskhorizon [0,T]| jiSource: Neu & Kühn, Preprint (2004)Page 2424 © Neu&Kühn, Math Finance Workshop · April, 1 th 2004

IMPACT OF CAPITAL ALLOCATION FOR CREDIT RISKSignificant increase of economic capital due to cascading lossesLoss distribution functionEconomic Capital: EC = L 99,90%- ELPD i|j/PD i= 1.16PD i|j/PD i= 1.04100% connectivityPD i|j/PD i= 1.0020% connectivityPD i|j/PD iSource: Neu & Kühn, Preprint (2004)Page 2525 © Neu&Kühn, Math Finance Workshop · April, 1 th 2004

INTERACTING PROCESS MODEL FOR OPERATIONAL RISKLoss SeverityProcesses are designed tomutually support eachother (chain or network)low medium highP1P1P3P3P4P4P8P8P2P2P6P6Loss FrequencyP5P5P9P9P7P7low medium highSource: Kühn & Neu, Physica A 322, 650 (2003)P9P9Page 26Expert assessment or loss database:q What is the expected time period, , thatprocess i will fail for the first time?q Given that process i has failed, what is theexpected time period, , for process j tofail also?q Unconditional & transition failure probabilitywithin ∆t:PDi,∆t=∆tτq Process dynamicsiPDi |,∆ j t∆t26 © Neu&Kühn, Math Finance Workshop · April, 1 th 2004=( )+ w n ( t−1( ){ }= Φ Φ PDn(t )iτij∑n ( t + ∆t))iw ij=Φ−1−1( PD i) − Φ ( PD )| jjiijj

IMPACT OF CAPITAL ALLOCATION FOR OPERATIONAL RISKLoss dynamics characteristics:LossesTimeSource: Kühn & Neu, Physica A 322, 650 (2003)Page 2727 © Neu&Kühn, Math Finance Workshop · April, 1 th 2004

CONCLUSIONSequential “functional” correlations are a candidate to explain stylizedfacts in financial time series such asq non-stationarity: volatility burst on many scalesq universal fat (non-Gaussian) tails in probability distribution of log-returnsq fast decaying correlations of returns, but relatively slow decay of volatilitycorrelationsRelevance for Risk Management:Without sequential correlations capital allocated as risk buffer can besignificantly underestimated for illiquid positionsPage 2828 © Neu&Kühn, Math Finance Workshop · April, 1 th 2004

REFERENCESStylized facts of financial time series• J.P. Bouchaud, M. Potter, “Theory of financial risks”, Cambridge University Press (2000)• R.N. Mantegna, E.H. Stanley, “An introduction to econophysics” Cambridge University Press (2000)• J. Voit, “ The statistical mechanics of financial markets”, Springer (2003)• J.P. Bouchaud, “Elements of a theory of financial risks”, Physica A 285, 18 (2000)• E.H. Stanley, P. Gopikrishnan, V. Plerou, L.A.N. Amaral, “Quantifying fluctuations in economic systems by adapting methods fromstatistical physics”, Physica A 287, 339 (2000)• X. Xavier, P. Gopikrishnan, V. Plerou, E.H. Stanley, “A theory of power-law distributions in financial markets fluctuations”,Nature 423, 267 (2003)Credit Risk Management• R. Jarrow, F. Yu, “Counterparty risk and the pricing of defaultable securities”, Journal of Finance 56, 1765 (2001)• E. Rogge, Ph. Schönbucher, “Modelling dynamic portfolio credit risk”, Working paper (2003)• K. Giesecke, St. Weber, “Cyclical correlation, credit contagion and portfolio losses, to appear in Journal of Banking and Finance• D. Egloff, M. Leippold, P. Vanini, “A simple model for credit contagion”, Working paper (2003)• P. Neu, R. Kühn, “Credit risk enhancement in a network of interdependent firms”, Working paper (2004)Operational Risk Mangement• R. Kühn , P. Neu, “Functional correlation approach to operational risk in banking organizations”, Physica A 322, 650 (2003)• M. Leippold, P. Vanini, “The quantification of operational risk”, Working paper (2003)Page 2929 © Neu&Kühn, Math Finance Workshop · April, 1 th 2004

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Contact addresses:Dr. Peter NeuDresdner Bank AGGroup Risk Control, HH 21.OGJürgen-Ponto-Platz 160301 Frankfurt a. M.Tel: +49 (69) 263 18582Email: Peter.Neu@Dresdner-Bank.comProf. Dr. Reimer KühnKing’s CollegeMathematics DepartmentStrandLondon W2C 2LR, UKTel: +44 (20) 7848 1035Email: kuehn@mth.kcl.ac.ukPage 3131 © Neu&Kühn, Math Finance Workshop · April, 1 th 2004

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S&P 500 2 Dec 1940 to 26 Mar 20044%Daily S&P 500 log-returnfrom 2 Dec 1940 to 26 March 2004-1Daily S&P 500 log-returnsfrom 30 Dec 1927 to 26 March 20042%-20%-3-4-2%-4%-6%-8%ln P(Z)-5-6-7-8-9-10%1940194519501955196019651970197519801985199019952000-10-20 -19-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-1-012345678910111213ZPage 3333 © Neu&Kühn, Math Finance Workshop · April, 1 th 2004

MINIMAL INTERACTING GENERALIZATION OF THE GBMModel propertiesdhidt= Ii−κi⋅hi+ ∑Jj(j≠i)ij⋅g(hj)+σi⋅η( t)iDynamics is formally identical to that of a graded response neuronFor sufficient small k and J ijthe system has exponentially many metastablesolutions which produce intermittency in market dynamicsDynamics associated with fluctuations within a state is very different fromthat of transitions between different meta-stable statesDynamics between meta-stable states is associated with majorrestructurings of the whole system and is accompanied by volatility burstPage 3434 © Neu&Kühn, Math Finance Workshop · April, 1 th 2004