Formulating a Research Topic

Abhinay Sawant

February 18, 2009

Economics 201FS

Update from Last Time

Fixed programs: 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:

11

n

iiX

11

n

iiXn

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:

]ˆ[ pE

)1()()ˆ(

2

pVar

)ˆ()ˆ(

]ˆ[]ˆ[

21

12*

pVarpVar

pEpEt

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

]ˆ[]ˆ[ 12 pEpE

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 JumpsCompany 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 TP 0.16 1.27 1.17 Slightly More

QP 0.26 1.38 1.14

Goldman Sachs TP -0.85 -0.60 -0.50 Homogeneous:Slightly Less

QP -1.06 -0.69 -0.60

J.P. Morgan TP 2.52 0.42 1.52 Slightly More

QP 2.20 0.53 1.69

Morgan Stanley TP -0.72 0.09 0.14 Homogenous

QP -0.60 -0.07 -0.03

Homogeneity of JumpsCompany 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)

TP -1.65 -1.15 0.00 Slightly Less

QP -2.03 -1.43 -0.22

Merck (Pharmaceutical)

TP 0.66 0.59 0.59 Homogeneous: Slightly More

QP 0.37 0.31 0.80

Oracle (Software)

TP -0.94 -0.15 0.69 Homogeneous

QP -0.50 -0.06 0.48

Wal-Mart (Retail)

TP -2.28 -1.43 -0.42 Slightly Less

QP -2.62 -1.14 -0.83

Other Properties of Jumps

Other Properties of Jumps

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

Morgan Stanley data:

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?

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Alternative Research Topics

Risk Management

Incorporating time-dependent volatility for VaR model