Empirical Invariance in Stock Market and Related Problems Empirical Invariance in Stock Market and Chii-Ruey Hwang Institute of Related Problems Mathematics, Academia Sinica, Taipei TAIWAN Chii-Ruey Hwang Empirical Analysis Institute of Mathematics, Academia Sinica, Taipei Introduction and TAIWAN Discussions Mathematical Results Remarks WSAF09 References June 29 - July 3, 2009
Empirical Invariance in Stock Market and Related Problems Empirical Analysis 1 Chii-Ruey Hwang Institute of Mathematics, Introduction and Discussions Academia 2 Sinica, Taipei TAIWAN Empirical Mathematical Results 3 Analysis Introduction and Discussions Remarks 4 Mathematical Results Remarks References 5 References
Empirical Invariance in Stock Market and Related Problems Chii-Ruey This is a joint work with Lo-bin Chang, Shu-Chun Chen, Hwang Institute of Alok Goswami, Fushing Hsieh, Max Palmer. Mathematics, Academia The raw data consists of the actual trade price and volume Sinica, Taipei TAIWAN of the intraday transactions data (trades and quotes) of Empirical companies in S&P500 list from 1998 to 2007 and part of Analysis 2008. Introduction and The return process is analyzed first at five-minute, Discussions Mathematical one-minute, and 30-second intervals for a whole year. Results Remarks References
Empirical Invariance in Stock Market and Related Problems Chii-Ruey Hwang Institute of IBM is used as the base line through most of our study with Mathematics, Academia no particular reason. One may pick other base line for Sinica, Taipei TAIWAN comparison. We did try OMC (OMNICOM GP INC) which Empirical has a common stock price. The result is the same. Analysis June 26, 2009: IBM (105.68), OMC (31.78); September Introduction and 19, 2008: IBM (118.85), OMC (41.90). Discussions Mathematical Results Remarks References
Empirical Invariance in Stock Market and Related Problems Chii-Ruey Let the discrete time series of one particular stock price be Hwang Institute of denoted by { S ( t i ) , i = 0 , ..., n } with t i − t i − 1 = δ . The Mathematics, Academia return process is defined by Sinica, Taipei TAIWAN { X ( t i ) = S ( t i ) − S ( t i − 1 ) , i = 1 , ...., n } . S ( t i − 1 ) Empirical Let { V ( t i ) , i = 1 , ..., n } and { U ( t i ) , i = 1 , ..., n } be the Analysis corresponding volume process and frequency process, where Introduction and Discussions V ( t i ) and U ( t i ) denote the cumulative volume and number Mathematical of transactions (frequency) for the time period ( t i − 1 t i ]. Results Remarks References
Empirical Invariance in Stock Market and Related Problems Mark the time point t i 1 if X ( t i ) falls in a certain Chii-Ruey percentile of the returns, say the upper ten percentile, Hwang Institute of otherwise 0. The return process thus turns into a 0 − 1 Mathematics, Academia process with m = n / 10 ones. This 0 − 1 process is divided Sinica, Taipei TAIWAN into m + 1 sections consisting of runs of 0s. V ( t i ) and Empirical U ( t i ) are marked similarly. The empirical distribution of the Analysis length of runs of 0s, the waiting time of hitting a certain Introduction and percentile, plays the key role in our analysis. Discussions The empirical distributions are considered for different Mathematical Results stocks, different time units, different years from the Remarks markets. References
Empirical Invariance in Stock Market and Related Problems Chii-Ruey Hwang Institute of Mathematics, Note that for any increasing function of X ( t i ) (or V ( t i ), Academia Sinica, Taipei U ( t i )), we still have exactly the same 0 − 1 process. For TAIWAN S ( t i ) example the logarithmic return log S ( t i − 1 ) is just Empirical Analysis log( X ( t i ) + 1). Introduction and Discussions Mathematical Results Remarks References
Empirical Invariance in Stock Market and Related Problems One may use the following two criteria to measure the Chii-Ruey Hwang closeness of two distributions. Institute of Mathematics, The ROC area: Academia Sinica, Taipei TAIWAN � 1 | G ( F − 1 ( t )) − t | dt , Empirical Analysis 0 Introduction The Kolmogorov-Smirnov distance ( Sup − norm ): and Discussions Mathematical Sup x | F ( x ) − G ( x ) | . Results Remarks References
Empirical Invariance in Stock Market and Related Problems Entropy: − � p i log p i . Chii-Ruey Hwang Institute of Mathematics, Academia Volatile period is defined hierarchically: if the length of runs Sinica, Taipei TAIWAN of 0s falls in say upper ten percentile, denote that period 1 ∗ , otherwise 0 ∗ ; repeat the same procedure for the length Empirical Analysis of runs of 0 ∗ s and denote the period in the upper ten Introduction percentile 1 @ . We may regard 1 @ the volatile period. and Discussions Mathematical Results Graphs and tables. Remarks References
Empirical Invariance in Stock Market and Related Problems We study data from an empirical point of view without Chii-Ruey assuming any model by looking at simple attributes. Our Hwang Institute of approach is to describe these attributes using as little Mathematics, Academia information as possible. Sinica, Taipei TAIWAN Empirical We do find an empirical invariance for the real stock prices. Analysis What are the mathematics and financial dynamics driving Introduction and this invariance are still not clear. And when the returns Discussions follow a L´ evy process, we prove the invariance distribution Mathematical Results being geometric. The invariance property for the fractional Remarks Brownian motion is yet to be proved. References
Empirical Invariance in Stock Market and Related Problems More precisely, the stock price S ( t ) follows Chii-Ruey Hwang Institute of Mathematics, S ( t ) = S (0) exp Z ( t ) , Academia Sinica, Taipei TAIWAN where Z ( t ) is a L´ evy process or Empirical Analysis S ( t ) = S (0) exp( µ t − σ 2 2 t 2 H + σ B H ( t )) , Introduction and Discussions where B H is a fractional Brownian motion with parameter Mathematical Results H . Remarks References
Empirical Invariance in Stock Market and Related Problems Chii-Ruey Hwang However both invariances are different to each other and Institute of Mathematics, are different from the one from the real data empirically. Academia Sinica, Taipei TAIWAN Empirical invariance is also observed for the volume process Empirical Analysis and the frequency process. The theoretical counterpart is Introduction yet to be proposed. The volatile periods of the return, the and Discussions volume and the frequency are highly correlated. Mathematical Results Remarks References
Empirical Invariance in Stock Market and Related Problems A L´ evy process is a continuous-time stochastic process Z ( t ) Chii-Ruey with stationary independent increments. Hwang Institute of Mathematics, Academia Sinica, Taipei A fractional Brownian motion with parameter H in (0 , 1) TAIWAN is a continuous-time Gaussian process B H starting at zero Empirical with mean zero and covariance function Analysis Introduction E ( B H ( s ) B H ( t )) = 1 and 2( | s | 2 H + | t | 2 H − | s − t | 2 H ) . Discussions Mathematical Results H = 1 2 is the Brownian motion. Remarks References
Empirical Invariance in Stock Market and Related Problems Chii-Ruey Hwang For any non-overlapping intervals ( t 0 , t 1 ) · · · ( t n − 1 , t n ), Institute of Mathematics, Z ( t 1 ) − Z ( t 0 ) , · · · , Z ( t n ) − Z ( t n − 1 ) are independent. And Academia Sinica, Taipei the distribution of Z ( t ) − Z ( s ) depends only on t − s . TAIWAN Empirical Analysis These two processes are generalizations of the Brownian Introduction motion, one keeps the stationary independent increments and Discussions and the other one the stationary Gaussian increments. Mathematical Results Remarks References
Empirical Invariance in Stock Market and Related Problems Chii-Ruey Hwang For each stock the empirical distribution of the waiting time Institute of Mathematics, to hit the upper (lower) ten percentile of the returns is Academia Sinica, Taipei considered. Most of the empirical distributions are close to TAIWAN each other under two different comparison criteria, ROC Empirical Analysis area and Kolmogorov-Smirnov distance. Comparisons are Introduction done across stocks, years, different time units. This may be and Discussions regarded as an empirical invariance. Mathematical Results Remarks References
Empirical Invariance in We carry out a similar analysis when the returns are finite Stock Market and Related sequence of i.i.d. random variables, e.g. from L´ evy process. Problems The corresponding empirical distributions which are the Chii-Ruey Hwang Institute of same as those from finite sequence of exchangeable random Mathematics, Academia variables converge completely to a geometric distribution Sinica, Taipei TAIWAN G ( x ). This is a law of large numbers, but the limit is already an invariance. How about the corresponding Empirical Analysis Kolmogorov theorem and Donsker’s theorem? Introduction and √ nSup x | P n ( x ) − G ( x ) | converges weakly to Discussions Mathematical Results Sup x | B ( G ( x )) | , Remarks √ n ( P n ( x ) − G ( x )) converges weakly to B ( G ( x )) , References where B ( t ) is a Brownian bridge ( W ( t ) − tW (1)) in [0 , 1]?
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