Formulating a Research Topic Abhinay Sawant February 18, 2009 Economics 201FS

Update from Last Time Fixed programs: Still Coarse Sampling: Z-Scores Tri-Power Quarticity Realized Volatility Signature Plots Still Coarse Sampling: Currently, sampling frequency = 8 min Should be changed to 5 minutes in future with averaging

Homogeneity of Jumps

Homogeneity of Jumps

Homogeneity of Jumps Let X1, X2, …, Xn be Bernoulli trials with probability p of success where success is defined as a jump day Goal is to estimate p for pre-Lehman (1/1/06 – 9/12/08) and post-Lehman periods (9/15/08 – 1/7/09) Conduct a t-Test to determine if a significance difference exists in parameter p in the two periods for individual equities

Homogeneity of Jumps Assume a prior distribution ξ(θ) uniformly distributed on the interval [0, 1] Posterior distribution is a Beta distribution with the following parameters:

Homogeneity of Jumps Properties about estimated parameters can be determined as follows from Beta distribution: t-Test is used to determine if difference is significant:

Homogeneity of Jumps Morgan Stanley (Tri-Power Quarticity): Morgan Stanley (Quad-Power Quarticity): Data Set Total Days Jump Days p = 0.05 p = 0.01 p = 0.001 Entire Set 742 100 43 15 Pre-Lehman 664 92 39 14 Post-Lehman 78 8 4 1 Data Set Total Days Jump Days p = 0.05 p = 0.01 p = 0.001 Entire Set 742 107 46 17 Pre-Lehman 664 98 42 16 Post-Lehman 78 9 4 1

Homogeneity of Jumps t Statistic measures the difference Morgan Stanley t-Test: Test is inconclusive but doesn’t suggest any significant difference in proportions p = 0.05 p = 0.01 p = 0.001 TP Quarticity -0.72 0.09 0.14 QP Quarticity -0.60 -0.07 -0.03

Homogeneity of Jumps Company Name QRT p = 0.05 p = 0.01 p = 0.001 Conclusion Bank of America TP 1.78 1.67 1.39 Slightly More QP 1.35 1.55 1.22 Citigroup 0.16 1.27 1.17 0.26 1.38 1.14 Goldman Sachs -0.85 -0.60 -0.50 Homogeneous: Slightly Less -1.06 -0.69 J.P. Morgan 2.52 0.42 1.52 2.20 0.53 1.69 Morgan Stanley -0.72 0.09 0.14 Homogenous -0.07 -0.03

Homogeneity of Jumps Company Name QRT p = 0.05 p = 0.01 p = 0.001 Conclusion Caterpillar (Industrial Equipment) TP -3.84 -2.84 -1.84 Significantly Less QP -3.18 -2.91 -2.35 McDonald’s (Restaurant) -1.65 -1.15 0.00 Slightly Less -2.03 -1.43 -0.22 Merck (Pharmaceutical) 0.66 0.59 Homogeneous: Slightly More 0.37 0.31 0.80 Oracle (Software) -0.94 -0.15 0.69 Homogeneous -0.50 -0.06 0.48 Wal-Mart (Retail) -2.28 -0.42 -2.62 -1.14 -0.83

Other Properties of Jumps

Other Properties of Jumps Morgan Stanley data: Mean Daily Return (QP) p = 0.05 p = 0.01 p = 0.001 With Jumps -0.3799 -0.7061 -1.4964 Without Jumps -0.0636 -0.0698 -0.0767 Mean Realized Vol.(QP) p = 0.05 p = 0.01 p = 0.001 With Jumps 43.7430 46.2342 48.2523 Without Jumps 44.6864 44.4391 44.4636

Other Properties of Jumps

Alternative Research Topics Portfolio Risk Constructing 2-asset portfolios and how does volatility change? Realized Covariance: How does investment process change for time-dependent volatility and correlation?

Alternative Research Topics Risk Management Incorporating time-dependent volatility for VaR model