1 Extreme Times in Finance J. Masoliver, M. Montero and J. Perelló Departament de Fisica Fonamental Universitat de Barcelona

2 Financial Makets: two levels of description “Microscopic” description Tick-by-tick data Continuous Time Random Walk “Mesoscopic” description Daily, weekly... data Diffusion processes Stochastic Volatility Models

3 I - CTRW formalism First developed by Montroll and Weiss (1965) Aimed to study the microstructure of random processes Applications: transport in random media, random networks, self-organized criticallity, earthquake modeling, and… now in financial markets

4 CTRW dynamics The log-return and the zero-mean return: J. Masoliver, M. Montero, G.H. Weiss Phys. Rev E 67, (2003)

5 CTRW dynamics (cont.)

6 Return distribution Renewal equation Formal solution joint distribution of increments and waiting times Objective

7 Are jumps and waiting times related to each other? a) If they are independent: b) If they are positively correlated. Some choices:

8 General Results Approach to the Gaussian density Long-tailed jump density: Lévy distribution At intermediate times: the tail behavior is given by extreme jumps Normal diffusion

9 Extreme Times At which time the return leaves a given interval [a,b] for the first time? Mean Exit Time (MET): J. Masoliver, M. Montero, J. Perelló Phys. Rev. E 71, (2005)

10 Integral Equation for the MET is the mean time between jumps. The MET does NOT depend on –the whole time distribution –the coupling between jumps and waiting times Mean First Passage Time (MFPT) to a certain critical value:

11 An exact solution Laplace (exponential) distribution: jump variance: Exact solution: Symmetrical interval For the Laplace pdf the approximate and the exact MET coincide It is also quadratic in L

12 Exponential jumps

13 Approximate solution We need to specify the jump pdf We want to get a solution as much general as possible We get an approximate solution when: –the interval L is smaller than the jump variance –jump pdf is an even function and zero-mean with scaling:

14 Models and data

15 Some Generalizations Introduction of correlations by a Markov-chain model. Assuming jumps are correlated: Integral equation for the MET: M. Montero, J. Perelló, J. Masoliver, F. Lillo, S. Miccichè, R.N. Mantegna Phys. Rev. E, 72, (2005).

16 A two-state Markov chain model r = correlation between the magnitude of two consequtive jumps Integral equationDifference equations

17 Solution mid-point: Scaling time Large values of L Stock independent

18 tick-by-tick data of 20 highly capita- lized stocks traded at the NYSE in the 4 year period 95-98; more than 12 milion transactions.

19 “Low frequency” data (daily, weekly,...) Diffusion modelsGeometric Browinian Motion (Einstein-Bachelier model) The assumption of constant volatility does not properly account for important features of the market Stochastic Volatility Models II – Stochastic Volatility models

20 Wiener processes Two-dimensional diffusions

21 1. The Ornstein-Uhlenbeck model E. Stein and J. Stein, Rev. Fin. Studies 4, 727 (1991). J. Masoliver and J. Perelló, Int. J. Theor. Appl. Finance 5, 541 (2002).

22 2. The CIR-Heston model Cox, J., Ingersoll, J., and S. Ross, (1985a), Econometrica, 53, 385 (1985). S. Heston, Rev. Fin. Studies 6, 327 (1993). A. Dragulescu and V. Yakovenko, Quant. Finance 2, 443 (2002).

23 3. The Exponential Ornstein-Uhlenbeck model J.-P. Fouque, G. Papanicolau and K. R. Sircar, Int. J. Theor. Appl. Finance 3, 101 (2002). J. Masoliver and J. Perelló, Quant. Finance (2006).

24 In SV models the volatility proces is described by a one-dimensional diffusion The OU model: The CIR-Heston model: The ExpOU model:

25 Extreme times for the volatility process The MFPT to certain level( reflecting) Averaged MFPT

26 Scaling 1 - OU model 2 - CIR-Heston model 3- ExpOU model normal level of the volatility

27 Some analytical results 1 - OU model Assymptotics

CIR-Heston model Assymptotics Kummer’s function of first kind

ExpOU model Assymptotics Kummer’s function of second kind

30 Empirical Data Financial Indices 1- DJIA: (28545 points) 2- S&P 500: (15152 points) 3- DAX: (11024 points) 4- NIKKEI: (8359 points) 5- NASDAQ: (8359 points) 6- FTSE-100: (5191 points) 7- IBEX-35: (4375 points) 8- CAC-40: (4100 points) Nomal Level (daily volatility) 1- DJIA: 0.71 % 2- S&P-500: 0.62 % 3- DAX: 0.84 % 4- NIKKEI: 0.96 % 5- NASDAQ: 0.78 % 6- FTSE-100: 0.77 % 7- IBEX-35: 0.96 % 8- CAC-40: 1.02 %

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36 Conclusions (I) The CTRW provides insight relating the market microstructure with the distributions of intraday prices and even longer-time prices. It is specially suited to treat high frequency data. It allows a thorough description of extreme times under a very general setting. MET’s do not depend on any potential coupling between waiting times and jumps. Empirical verification of the analytical estimates using a very large time series of USD/DEM transaction data. The formalism allows for generalizations to include price correlations.

37 Conclusions (II) The “macroscopic” description of the market is quite well described by SV models. Many SV models allow a analytical treatment of the MFPT. The MFPT may help to determine a suitable SV model OU and CIR-Heston models yield a quadratic behavior of the MFPT for small volatilities that is not conflicting with data. For large volatilities their exponential growth does not agree with data. In a first approximation the ExpOU model seems to agree with data for both small and large volatilities.

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39 The Laplace MET is larger than the MET when return follows a Wiener process: We conjecture that this is true in any situation: The Wiener process underestimates the MET. Practical consequences for risk control and pricing exotic derivatives. Comparison with the Wiener Process

40 The Wiener process underestimates the MET