March 30, Preliminary Monte Carlo Investigations. Vivek Bhattacharya. Outline. Mathematical Overview. Monte Carlo. Cross Correlations

Transcription

1 March 30, 2011

2 Motivation (why spend so much time on simulations) What does corr(rj 1, RJ 2 ) really represent? Results and Graphs Future Directions

3 General Questions ( corr RJ (1), RJ (2)) = corr ( µ 2 (1) 1 r t r (1) t 1 (r (1) t ) 2, µ We have observed some qualitative facts 2 (2) 1 r t r (2) t 1 (r (2) t ) 2 Stocks in the same industry tend to have higher correlations of RJ grow stronger with time period we consider Comparing correlations from jump tests with correlations from RJ show similar clustering but no clear pattern between corr(1(jump 1 ), 1(Jump 2 )) and corr(rj (1), RJ (2) ) If BV is the jump-robust measure, Pearson correlation exceeds Kendall s τ or Spearman s ρ ).

4 General Questions How do we explain the previous observations? Are any of those observations interesting? Ways to study this: Analytic results Simulation

5 Setup We use a very simple setup to get some basic intuition ( Z (1) t Z (2) t ) N (( 0 0 ) ( 1 ρ, ρ 1 )) iid. Simulate minute-by-minute returns using the Euler scheme assuming constant volatility σ (i) Include jumps J t N(0, σjmp 2 ) as a Compound Poisson Process with parameter λ Calculate 5-minute returns and determine correlations in using RJ calculated in daily, weekly, and monthly fashion Repeat for 5000 sample paths

6 Parameters λ taken from Huang and Tauchen (2005) σ jmp chosen by eyeballing Parameter Value µ 0 σ (i) 0.01 λ 0.25 σ jmp {0, 0.001, 0.005, 0.01, 0.025, 0.05, 0.10} ρ {0, 0.15, 0.30,..., 0.75}

7 Sample Paths

8 Correlation as a Function of ρ and σ jmp As a first step, use BV and r to compute daily measures

9 Using Longer Time Periods Consider values when correlation calculated in daily vs. monthly time periods For nonzero σ jmp, correlation is actually smaller for monthly periods

10 Using Longer Time Periods (II) Result holds even if we use MinVar or MedVar MinVar is shown

11 Comparing Different Jump-Robust Measures Compare BV and MinVar MinVar correlation is larger than BV for σ jmp 0 Effect is much less pronounced for MedVar

12 Using Different Correlation Measures Noticeable difference between r and τ for BV Difference is much more slight for r and ρ (not shown)

13 Using Different Correlation Measures (II) Interestingly, Kendall s τ gives very comparable results between BV and MinVar Compare to earlier diagrams using r

14 Distributions of Sample Statistics Previous slides give some intution for how sample means behave as a function of ρ and σ jmp. Now consider distribution of the sample statistics, as measured by kernel density functions Compare the results with the studies of the mean values given in previous slides and general observations from data

15 Effect of Correlation Type on Distribution Fix ρ = Calculations from r and τ

16 Effect of Correlation Type on Distribution (II) width(τ) > width(r) width(ρ) Observation holds true across statistics (BV, MedVar, MinVar) and time periods

17 Scatterplots of Distributions Look at τ vs. r at σ jmp = 0 and σ jmp = 0.01, fixing ρ = 0.75 Graphs for MedVar and MinVar are similar, although means and variances differ slightly

18 Scatterplots of Distributions (II) Scatterplots might look like data from last time Should be careful in the comparison. Data had plots for various pairs of stocks different ρ and σ jmp, among other things

19 Distributions for Different Statistics Strengthening of observation earlier: distribution of Kendall s τ of RJ using MinVar seem similar

20 Summary for Results At least some of the observed correlation in RJ within industries can be due to correlation in underlying Brownian motion Correlation in RJ seems to decrease for longer time periods, contrary to results in data Correlation using τ has narrower distribution than that using r Distribution of τ seems independent of statistic used for RJ

21 and Autocorrelations Bollerslev, Kretschmer, Pigorsch, and Tauchen plot acf of J t log(rv t /BV t ) and find fifth order autocorrelations are significant Use S&P futures data Unsuccessfully tried to replicate result on a handful of stocks Updated some erroneous results from last time

22 Autocorrelation

23 Correlation Calculated cross correlations and 95% confidence intervals for JPM/BAC and AAPL/BAC Unsure how to intepret result

24 Stochastic Volatility Models Could generate data from dp(t) = µ dt + exp[β 0 + β 1 v(t)]dw p (t) + dl J (t) dv(t) = α v v(t) dt + dw v (t) What is the multivariate extension? Guess: dw p (t) is vector drawn from multivariate normal Perhaps something simpler (e.g., σ follows an AR(1) process) How do we generate correlated jumps?

25 Future Work Unclear how to proceed from here Trying to figure out whether corr(rj (1), RJ (2) ) returns any meaningful information Analytic results would be nice here Contemplated trying to see whether stochastic volatility models can match these moments in the data Very vague: perhaps try to use these moments as part of an estimation procedure? I would welcome comments!