A Student’s Experience

Matthew Rognlie

Econ 201FSJunior Research Seminar in Financial Econometrics

Duke University

February 3, 2010

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 1 / 24

Possible Problems with Jump Tests

Schwert (2009), in a senior thesis written through this seminar, findsthat common jump tests produce surprisingly incoherent results.

Tests devised by different authors generally identify different days as“jump days.”

Tests are not even consistent with themselves when the samplingfrequency is adjusted.

A common variant of the BNS test identifies 6.9% of days at10-minute sampling and 6.4% of days at 15-minute sampling, butonly 1.21% at both frequencies simultaneously!

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 2 / 24

Possible Problems with Jump Tests

Schwert (2009), in a senior thesis written through this seminar, findsthat common jump tests produce surprisingly incoherent results.

Tests devised by different authors generally identify different days as“jump days.”

Tests are not even consistent with themselves when the samplingfrequency is adjusted.

A common variant of the BNS test identifies 6.9% of days at10-minute sampling and 6.4% of days at 15-minute sampling, butonly 1.21% at both frequencies simultaneously!

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 2 / 24

Possible Problems with Jump Tests

Schwert (2009), in a senior thesis written through this seminar, findsthat common jump tests produce surprisingly incoherent results.

Tests devised by different authors generally identify different days as“jump days.”

Tests are not even consistent with themselves when the samplingfrequency is adjusted.

A common variant of the BNS test identifies 6.9% of days at10-minute sampling and 6.4% of days at 15-minute sampling, butonly 1.21% at both frequencies simultaneously!

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 2 / 24

Possible Problems with Jump Tests

Schwert (2009), in a senior thesis written through this seminar, findsthat common jump tests produce surprisingly incoherent results.

Tests devised by different authors generally identify different days as“jump days.”

Tests are not even consistent with themselves when the samplingfrequency is adjusted.

A common variant of the BNS test identifies 6.9% of days at10-minute sampling and 6.4% of days at 15-minute sampling, butonly 1.21% at both frequencies simultaneously!

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 2 / 24

Questions

Are these jumps real, or artifacts of the data?

If we’re detecting different jumps with different tests and at differentfrequencies, how can we know which ones to use?

How significant is the jump component of realized variation?

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 3 / 24

Questions

Are these jumps real, or artifacts of the data?

If we’re detecting different jumps with different tests and at differentfrequencies, how can we know which ones to use?

How significant is the jump component of realized variation?

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 3 / 24

Questions

Are these jumps real, or artifacts of the data?

If we’re detecting different jumps with different tests and at differentfrequencies, how can we know which ones to use?

How significant is the jump component of realized variation?

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 3 / 24

The Intraday Pattern in Volatility: An Agent of Mischief

Volatility peaks near the start of the day, reaches a midday trough,and then climbs

Pattern similar across almost all stocks

Differences in log-return distribution by a factor of nearly 2 from peakto trough

(Probably) much more significant than other changes in volatility atsuch high frequencies

This poses a problem for statistical methods that assume, at least forpractical purposes, that volatility does not change so much in a shortperiod.

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 4 / 24

The Intraday Pattern in Volatility: An Agent of Mischief

Volatility peaks near the start of the day, reaches a midday trough,and then climbs

Pattern similar across almost all stocks

Differences in log-return distribution by a factor of nearly 2 from peakto trough

(Probably) much more significant than other changes in volatility atsuch high frequencies

This poses a problem for statistical methods that assume, at least forpractical purposes, that volatility does not change so much in a shortperiod.

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 4 / 24

The Intraday Pattern in Volatility: An Agent of Mischief

Volatility peaks near the start of the day, reaches a midday trough,and then climbs

Pattern similar across almost all stocks

Differences in log-return distribution by a factor of nearly 2 from peakto trough

(Probably) much more significant than other changes in volatility atsuch high frequencies

This poses a problem for statistical methods that assume, at least forpractical purposes, that volatility does not change so much in a shortperiod.

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 4 / 24

The Intraday Pattern in Volatility: An Agent of Mischief

Volatility peaks near the start of the day, reaches a midday trough,and then climbs

Pattern similar across almost all stocks

Differences in log-return distribution by a factor of nearly 2 from peakto trough

(Probably) much more significant than other changes in volatility atsuch high frequencies

This poses a problem for statistical methods that assume, at least forpractical purposes, that volatility does not change so much in a shortperiod.

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 4 / 24

The Intraday Pattern in Volatility: An Agent of Mischief

Volatility peaks near the start of the day, reaches a midday trough,and then climbs

Pattern similar across almost all stocks

Differences in log-return distribution by a factor of nearly 2 from peakto trough

(Probably) much more significant than other changes in volatility atsuch high frequencies

This poses a problem for statistical methods that assume, at least forpractical purposes, that volatility does not change so much in a shortperiod.

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 4 / 24

The Pattern

Intraday Volatility for JPM

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 5 / 24

Similar across many stocks...

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 6 / 24

An Established Record of Troublemaking

Van Tassel (2008) shows that the test proposed by Lee and Mykland(2008) to identify specific price changes as jumps producesinconsistent results.

The Lee-Mykland test uses a trailing window of returns to estimatelocal volatility and determine whether an individual movement is anoutlier.

The intraday volatility pattern causes problems in a predictable way.

Inconsistencies arise when we scale by returns from a different part ofthe day, where volatility is at a different level.

The trailing window does not give an unbiased measure of “localvolatility.”

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 7 / 24

An Established Record of Troublemaking

Van Tassel (2008) shows that the test proposed by Lee and Mykland(2008) to identify specific price changes as jumps producesinconsistent results.

The Lee-Mykland test uses a trailing window of returns to estimatelocal volatility and determine whether an individual movement is anoutlier.

The intraday volatility pattern causes problems in a predictable way.

Inconsistencies arise when we scale by returns from a different part ofthe day, where volatility is at a different level.

The trailing window does not give an unbiased measure of “localvolatility.”

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 7 / 24

An Established Record of Troublemaking

Van Tassel (2008) shows that the test proposed by Lee and Mykland(2008) to identify specific price changes as jumps producesinconsistent results.

The Lee-Mykland test uses a trailing window of returns to estimatelocal volatility and determine whether an individual movement is anoutlier.

The intraday volatility pattern causes problems in a predictable way.

Inconsistencies arise when we scale by returns from a different part ofthe day, where volatility is at a different level.

The trailing window does not give an unbiased measure of “localvolatility.”

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 7 / 24

An Established Record of Troublemaking

Van Tassel (2008) shows that the test proposed by Lee and Mykland(2008) to identify specific price changes as jumps producesinconsistent results.

The Lee-Mykland test uses a trailing window of returns to estimatelocal volatility and determine whether an individual movement is anoutlier.

The intraday volatility pattern causes problems in a predictable way.

Inconsistencies arise when we scale by returns from a different part ofthe day, where volatility is at a different level.

The trailing window does not give an unbiased measure of “localvolatility.”

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 7 / 24

An Established Record of Troublemaking

Van Tassel (2008) shows that the test proposed by Lee and Mykland(2008) to identify specific price changes as jumps producesinconsistent results.

The Lee-Mykland test uses a trailing window of returns to estimatelocal volatility and determine whether an individual movement is anoutlier.

The intraday volatility pattern causes problems in a predictable way.

Inconsistencies arise when we scale by returns from a different part ofthe day, where volatility is at a different level.

The trailing window does not give an unbiased measure of “localvolatility.”

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 7 / 24

Effects on Other Jump Statistics?

Since Barndorff and Nielsen-Shephard (BNS) does not examineindividual returns, the problem with Lee-Mykland is not directlyrelevant. Still...

Huang and Tauchen (2005) undertook an impressive series of MonteCarlo simulations to verify that the BNS tests actually worked, andthe results were favorable. If the BNS tests have problems, what didthe simulations miss?

They did not incorporate intraday volatility patterns, or any changesin volatility of similar magnitude at high frequencies.

Might this be our culprit? (Or at least one of them?)

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 8 / 24

Effects on Other Jump Statistics?

Since Barndorff and Nielsen-Shephard (BNS) does not examineindividual returns, the problem with Lee-Mykland is not directlyrelevant. Still...

Huang and Tauchen (2005) undertook an impressive series of MonteCarlo simulations to verify that the BNS tests actually worked, andthe results were favorable. If the BNS tests have problems, what didthe simulations miss?

They did not incorporate intraday volatility patterns, or any changesin volatility of similar magnitude at high frequencies.

Might this be our culprit? (Or at least one of them?)

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 8 / 24

Effects on Other Jump Statistics?

Since Barndorff and Nielsen-Shephard (BNS) does not examineindividual returns, the problem with Lee-Mykland is not directlyrelevant. Still...

Huang and Tauchen (2005) undertook an impressive series of MonteCarlo simulations to verify that the BNS tests actually worked, andthe results were favorable. If the BNS tests have problems, what didthe simulations miss?

They did not incorporate intraday volatility patterns, or any changesin volatility of similar magnitude at high frequencies.

Might this be our culprit? (Or at least one of them?)

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 8 / 24

Effects on Other Jump Statistics?

Since Barndorff and Nielsen-Shephard (BNS) does not examineindividual returns, the problem with Lee-Mykland is not directlyrelevant. Still...

Huang and Tauchen (2005) undertook an impressive series of MonteCarlo simulations to verify that the BNS tests actually worked, andthe results were favorable. If the BNS tests have problems, what didthe simulations miss?

They did not incorporate intraday volatility patterns, or any changesin volatility of similar magnitude at high frequencies.

Might this be our culprit? (Or at least one of them?)

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 8 / 24

Initial Strategy

Let’s extend basic Monte Carlo simulations to the case with intradayvolatility.

Simulate prices in a world without jumps; only movement is Brownianmotion. (Ignore drift at high frequencies.)

On data simulated with continuous prices, jump tests certainly shouldnot find many jumps.

Specifically, at the 1% significance level, a test should (ideally) detectjumps on only 1% of days.

Now, what happens when the simulated continuous pricesfollow an intraday volatility pattern?

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 9 / 24

Initial Strategy

Let’s extend basic Monte Carlo simulations to the case with intradayvolatility.

Simulate prices in a world without jumps; only movement is Brownianmotion. (Ignore drift at high frequencies.)

On data simulated with continuous prices, jump tests certainly shouldnot find many jumps.

Specifically, at the 1% significance level, a test should (ideally) detectjumps on only 1% of days.

Now, what happens when the simulated continuous pricesfollow an intraday volatility pattern?

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 9 / 24

Initial Strategy

Let’s extend basic Monte Carlo simulations to the case with intradayvolatility.

Simulate prices in a world without jumps; only movement is Brownianmotion. (Ignore drift at high frequencies.)

On data simulated with continuous prices, jump tests certainly shouldnot find many jumps.

Specifically, at the 1% significance level, a test should (ideally) detectjumps on only 1% of days.

Now, what happens when the simulated continuous pricesfollow an intraday volatility pattern?

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 9 / 24

Initial Strategy

Let’s extend basic Monte Carlo simulations to the case with intradayvolatility.

Simulate prices in a world without jumps; only movement is Brownianmotion. (Ignore drift at high frequencies.)

On data simulated with continuous prices, jump tests certainly shouldnot find many jumps.

Specifically, at the 1% significance level, a test should (ideally) detectjumps on only 1% of days.

Now, what happens when the simulated continuous pricesfollow an intraday volatility pattern?

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 9 / 24

Initial Strategy

Let’s extend basic Monte Carlo simulations to the case with intradayvolatility.

Simulate prices in a world without jumps; only movement is Brownianmotion. (Ignore drift at high frequencies.)

On data simulated with continuous prices, jump tests certainly shouldnot find many jumps.

Specifically, at the 1% significance level, a test should (ideally) detectjumps on only 1% of days.

Now, what happens when the simulated continuous pricesfollow an intraday volatility pattern?

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 9 / 24

Implementation

Our model for log-prices p(t) is just:

dp(t) = σ(t)dW (t)

We break up the day from 9:35 AM to 4:00 PM into 384 one-minuteintervals and simulate using (for i = 1, 2 . . . , 385):

p(ti+1)− p(ti ) = σiZ Z ∼ N(0, 1)

In one simulation, let σi be constant throughout the day. In the other,let it follow the average intraday volatility pattern of some stock.

Repeat many, many times—get data for 100,000 simulated days or so.

Apply jump tests to the data from both simulations and see whathappens!

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 10 / 24

Implementation

Our model for log-prices p(t) is just:

dp(t) = σ(t)dW (t)

We break up the day from 9:35 AM to 4:00 PM into 384 one-minuteintervals and simulate using (for i = 1, 2 . . . , 385):

p(ti+1)− p(ti ) = σiZ Z ∼ N(0, 1)

In one simulation, let σi be constant throughout the day. In the other,let it follow the average intraday volatility pattern of some stock.

Repeat many, many times—get data for 100,000 simulated days or so.

Apply jump tests to the data from both simulations and see whathappens!

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 10 / 24

Implementation

Our model for log-prices p(t) is just:

dp(t) = σ(t)dW (t)

We break up the day from 9:35 AM to 4:00 PM into 384 one-minuteintervals and simulate using (for i = 1, 2 . . . , 385):

p(ti+1)− p(ti ) = σiZ Z ∼ N(0, 1)

In one simulation, let σi be constant throughout the day. In the other,let it follow the average intraday volatility pattern of some stock.

Repeat many, many times—get data for 100,000 simulated days or so.

Apply jump tests to the data from both simulations and see whathappens!

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 10 / 24

Implementation

Our model for log-prices p(t) is just:

dp(t) = σ(t)dW (t)

We break up the day from 9:35 AM to 4:00 PM into 384 one-minuteintervals and simulate using (for i = 1, 2 . . . , 385):

p(ti+1)− p(ti ) = σiZ Z ∼ N(0, 1)

In one simulation, let σi be constant throughout the day. In the other,let it follow the average intraday volatility pattern of some stock.

Repeat many, many times—get data for 100,000 simulated days or so.

Apply jump tests to the data from both simulations and see whathappens!

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 10 / 24

Implementation

Our model for log-prices p(t) is just:

dp(t) = σ(t)dW (t)

We break up the day from 9:35 AM to 4:00 PM into 384 one-minuteintervals and simulate using (for i = 1, 2 . . . , 385):

p(ti+1)− p(ti ) = σiZ Z ∼ N(0, 1)

In one simulation, let σi be constant throughout the day. In the other,let it follow the average intraday volatility pattern of some stock.

Repeat many, many times—get data for 100,000 simulated days or so.

Apply jump tests to the data from both simulations and see whathappens!

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 10 / 24

Basic Results

Significantly more jumps are detected when we simulate using theintraday pattern.

The extent of the effect depends on the sampling interval for thejump test.

For many stocks, the frequency of jumps detected in the simulated(jump-free) data with an intraday pattern is half or more thefrequency of jumps detected in the observed data!

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 11 / 24

Basic Results

Significantly more jumps are detected when we simulate using theintraday pattern.

The extent of the effect depends on the sampling interval for thejump test.

For many stocks, the frequency of jumps detected in the simulated(jump-free) data with an intraday pattern is half or more thefrequency of jumps detected in the observed data!

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 11 / 24

Basic Results

Significantly more jumps are detected when we simulate using theintraday pattern.

The extent of the effect depends on the sampling interval for thejump test.

For many stocks, the frequency of jumps detected in the simulated(jump-free) data with an intraday pattern is half or more thefrequency of jumps detected in the observed data!

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 11 / 24

Detailed Results

Table: Fraction of Days Flagged as Jumps: JPM

Test Simulated w/o P Simulated with P Observed

ZRJ,TP,0.99 .0144 .0328 .0592ZRJ,TP,0.999 .0018 .0073 .0151

ZRJ,QP,0.99 .0149 .0347 .0626ZRJ,QP,0.999 .0021 .0073 .0188

Zlog,TP,0.99 .0329 .0621 .0975Zlog,TP,0.999 .0110 .0270 .0520

Zlog,QP,0.99 .0334 .0639 .1023Zlog,QP,0.999 .0114 .0284 .0551

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 12 / 24

Detailed Results: Jiang-Oomen Test

Table: Fraction of Days Flagged as Jumps: JPM

Test Simulated w/o P Simulated with P Observed

JOdiff,0.99 .0616 .0965 .1314JOdiff,0.999 .0319 .0604 .0958

JOlog,0.99 .0457 .0761 .1102JOlog,0.999 .0181 .0400 .0626

JOratio,0.99 .0446 .0761 .1098JOratio,0.999 .0174 .0400 .0626

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 13 / 24

Results for Various Stocks

Table: Fraction of Jump Days Detected in Simulated Data withIntraday Pattern and in Observed Data, for Various Stocks

Stock ZRJ,TP,0.99 JOratio,0.99

INTC (simulated) .0302 .0669INTC (actual) .0510 .0970

JPM (simulated) .0328 .0761JPM (actual) .0592 .1098

KO (simulated) .0267 .0568KO (actual) .0578 .0971

MSFT (simulated) .0301 .0634MSFT (actual) .0572 .0979

WMT (simulated) .0262 .0600WMT (actual) .0592 .0904

XOM (simulated) .0258 .0578XOM (actual) .0477 .0888

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 14 / 24

Distribution of Test Statistic

Kernel Density Estimates for ZRJ,TP Test Statistic With and WithoutIntraday Pattern (15-minute sampling)

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 15 / 24

Why is This Happening?

Intuition: Nonparametric high-frequency jump tests generally workby comparing nearby returns to see if there are unusual differences inmagnitude.

The intraday pattern in volatility creates predictable differences inmagnitude between nearby returns.

The size of this effect depends on the sampling interval, just as wefind in the simulations. As sampling becomes fine enough thatneighboring intervals no longer have significantly different volatility,the problem goes away.

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 16 / 24

Why is This Happening?

Intuition: Nonparametric high-frequency jump tests generally workby comparing nearby returns to see if there are unusual differences inmagnitude.

The intraday pattern in volatility creates predictable differences inmagnitude between nearby returns.

The size of this effect depends on the sampling interval, just as wefind in the simulations. As sampling becomes fine enough thatneighboring intervals no longer have significantly different volatility,the problem goes away.

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 16 / 24

Why is This Happening?

Intuition: Nonparametric high-frequency jump tests generally workby comparing nearby returns to see if there are unusual differences inmagnitude.

The intraday pattern in volatility creates predictable differences inmagnitude between nearby returns.

The size of this effect depends on the sampling interval, just as wefind in the simulations. As sampling becomes fine enough thatneighboring intervals no longer have significantly different volatility,the problem goes away.

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 16 / 24

A Little Math

Consider the expression for bipower variation over finitely spaced pricedata:

BV = µ−21

(n − 1

n − 2

) n∑i=3

|ri ||ri−1| (1)

Say that the sequence of geometric returns is generated by a diffusiveprocess with deterministic volatility σ(t) and zero drift. Then we candecompose ri into the product σiZi , where Zi is a standard normal randomvariable and σi =

∫ titi−1

σ(s)ds. We now find:

BV = µ−21

(n − 1

n − 2

) n∑i=3

|Zi ||Zi−1|σiσi−1 (2)

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 17 / 24

A Little Math, Continued...

Recalling that E [|Zi |] = µ1 by definition, taking expectations we find:

E [BV ] = µ−21

(n − 1

n − 2

) n∑i=3

E [|Zi ||Zi−1|]σiσi−1 (3)

=n − 1

n − 2

n∑i=3

σiσi−1 (4)

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 18 / 24

A Little Math, Continued...

Now, noting that σiσi−1 ≤σ2

i +σ2i−1

2 , we find:

E [BV ] =n − 1

n − 2

n∑i=3

σiσi−1 (5)

≤ n − 1

n − 2

(σ2

2 + σ2n

2+

n−1∑i=3

σ2i

)(6)

=n − 1

n − 2

(E [RV ]− σ2

2 + σ2n

2

)(7)

= E [RV ] +n − 1

n − 2

(E [RV ]

n − 1− σ2

2 + σ2n

2

)(8)

Equality only holds when all σi are the same, and the inequality is morepronounced as variation in σi increases. In particular, the difference

between σiσi−1 andσ2

i +σ2i−1

2 is directly related to the magnitude of thedifference between σi and σi−1.Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 19 / 24

Conclusions from the Math

We have found:

E [BV ] ≤ E [RV ] +n − 1

n − 2

(E [RV ]

n − 1− σ2

2 + σ2n

2

)

If the σi are not all the same, the expected value of BV is less thanthe expected value of RV plus a boundary term.

Boundary term will be positive if the average of square volatility inthe first and last periods is less than the average of square volatilitythroughout the sample.

Since volatility tends to be higher at the beginning and end of a day,the gap between expected BV and RV over that day will be evenlarger than the first inequality suggests.

This accounts for the biased results we observed!

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 20 / 24

Conclusions from the Math

We have found:

E [BV ] ≤ E [RV ] +n − 1

n − 2

(E [RV ]

n − 1− σ2

2 + σ2n

2

)

If the σi are not all the same, the expected value of BV is less thanthe expected value of RV plus a boundary term.

Boundary term will be positive if the average of square volatility inthe first and last periods is less than the average of square volatilitythroughout the sample.

Since volatility tends to be higher at the beginning and end of a day,the gap between expected BV and RV over that day will be evenlarger than the first inequality suggests.

This accounts for the biased results we observed!

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 20 / 24

Conclusions from the Math

We have found:

E [BV ] ≤ E [RV ] +n − 1

n − 2

(E [RV ]

n − 1− σ2

2 + σ2n

2

)

If the σi are not all the same, the expected value of BV is less thanthe expected value of RV plus a boundary term.

Boundary term will be positive if the average of square volatility inthe first and last periods is less than the average of square volatilitythroughout the sample.

Since volatility tends to be higher at the beginning and end of a day,the gap between expected BV and RV over that day will be evenlarger than the first inequality suggests.

This accounts for the biased results we observed!

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 20 / 24

Conclusions from the Math

We have found:

E [BV ] ≤ E [RV ] +n − 1

n − 2

(E [RV ]

n − 1− σ2

2 + σ2n

2

)

If the σi are not all the same, the expected value of BV is less thanthe expected value of RV plus a boundary term.

Boundary term will be positive if the average of square volatility inthe first and last periods is less than the average of square volatilitythroughout the sample.

Since volatility tends to be higher at the beginning and end of a day,the gap between expected BV and RV over that day will be evenlarger than the first inequality suggests.

This accounts for the biased results we observed!

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 20 / 24

Conclusions from the Math

We have found:

E [BV ] ≤ E [RV ] +n − 1

n − 2

(E [RV ]

n − 1− σ2

2 + σ2n

2

)

If the σi are not all the same, the expected value of BV is less thanthe expected value of RV plus a boundary term.

Boundary term will be positive if the average of square volatility inthe first and last periods is less than the average of square volatilitythroughout the sample.

Since volatility tends to be higher at the beginning and end of a day,the gap between expected BV and RV over that day will be evenlarger than the first inequality suggests.

This accounts for the biased results we observed!

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 20 / 24

Biased Components of Jump Tests

Our math suggests that BV —and perhaps other similar componentsof jump tests—will be biased.

Since we simulated our data, we know what the true values of theintegrated variation and integrated quarticity are.

We can thus see the extent to which BV , TP, QP, etc. are biased inthe simulated data with and without an intraday pattern.

Table: Percentage Difference Between Average Daily Sample Statistics,from Simulated Data, and True Values of Estimated Quantities: JPM

Statistic Without Intraday Pattern With Intraday Pattern

RV 0.11% 0.09%BV 0.30% -3.77%TP 0.45% -18.82%QP 0.62% -25.51%

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 21 / 24

Biased Components of Jump Tests

Our math suggests that BV —and perhaps other similar componentsof jump tests—will be biased.

Since we simulated our data, we know what the true values of theintegrated variation and integrated quarticity are.

We can thus see the extent to which BV , TP, QP, etc. are biased inthe simulated data with and without an intraday pattern.

Table: Percentage Difference Between Average Daily Sample Statistics,from Simulated Data, and True Values of Estimated Quantities: JPM

Statistic Without Intraday Pattern With Intraday Pattern

RV 0.11% 0.09%BV 0.30% -3.77%TP 0.45% -18.82%QP 0.62% -25.51%

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 21 / 24

Biased Components of Jump Tests

Our math suggests that BV —and perhaps other similar componentsof jump tests—will be biased.

Since we simulated our data, we know what the true values of theintegrated variation and integrated quarticity are.

We can thus see the extent to which BV , TP, QP, etc. are biased inthe simulated data with and without an intraday pattern.

Table: Percentage Difference Between Average Daily Sample Statistics,from Simulated Data, and True Values of Estimated Quantities: JPM

Statistic Without Intraday Pattern With Intraday Pattern

RV 0.11% 0.09%BV 0.30% -3.77%TP 0.45% -18.82%QP 0.62% -25.51%

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 21 / 24

Biased Components of Jump Tests

Our math suggests that BV —and perhaps other similar componentsof jump tests—will be biased.

Since we simulated our data, we know what the true values of theintegrated variation and integrated quarticity are.

We can thus see the extent to which BV , TP, QP, etc. are biased inthe simulated data with and without an intraday pattern.

Table: Percentage Difference Between Average Daily Sample Statistics,from Simulated Data, and True Values of Estimated Quantities: JPM

Statistic Without Intraday Pattern With Intraday Pattern

RV 0.11% 0.09%BV 0.30% -3.77%TP 0.45% -18.82%QP 0.62% -25.51%

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 21 / 24

Bias in Bipower Variation by Sampling Interval

Negative Bias of Bipower Variation as an Estimator of IntegratedVariance: Simulated Data with Intraday Pattern

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 22 / 24

Correcting the Bias

Can we correct for the effects seen here?

Yes, somewhat.

Three basic possibilities:

(i) Scale returns to eliminate the pattern.(ii) Adjust sampling intervals so that average volatility in each interval is

roughly the same.(iii) “Transaction time” rather than clock time.

Each has strengths and weaknesses.

Lingering problem: To the extent that volatility significantly changesat high frequencies in ways that are not predictable (like the intradaypattern) and cannot be proxied, we will always have some bias.

Hard to distinguish the effects of jumps and stochastic volatility!

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 23 / 24

Conclusions

Jump tests have problems.

But if microstructure noise is limited enough that we can have a highsampling frequency (as with, say, a large composite index), they’reactually not bad!

In general, asymptotic results don’t mean that a test will worksmoothly in practice.

Always think about the effects of the diurnal volatility pattern.

If you see strange results, dig a little and see what might beresponsible.

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 24 / 24

Conclusions

Jump tests have problems.

But if microstructure noise is limited enough that we can have a highsampling frequency (as with, say, a large composite index), they’reactually not bad!

In general, asymptotic results don’t mean that a test will worksmoothly in practice.

Always think about the effects of the diurnal volatility pattern.

If you see strange results, dig a little and see what might beresponsible.

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 24 / 24

Conclusions

Jump tests have problems.

But if microstructure noise is limited enough that we can have a highsampling frequency (as with, say, a large composite index), they’reactually not bad!

In general, asymptotic results don’t mean that a test will worksmoothly in practice.

Always think about the effects of the diurnal volatility pattern.

If you see strange results, dig a little and see what might beresponsible.

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 24 / 24

Conclusions

Jump tests have problems.

But if microstructure noise is limited enough that we can have a highsampling frequency (as with, say, a large composite index), they’reactually not bad!

In general, asymptotic results don’t mean that a test will worksmoothly in practice.

Always think about the effects of the diurnal volatility pattern.

If you see strange results, dig a little and see what might beresponsible.

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 24 / 24

Conclusions

Jump tests have problems.

But if microstructure noise is limited enough that we can have a highsampling frequency (as with, say, a large composite index), they’reactually not bad!

In general, asymptotic results don’t mean that a test will worksmoothly in practice.

Always think about the effects of the diurnal volatility pattern.

If you see strange results, dig a little and see what might beresponsible.

Matthew Rognlie (Duke University) A Student’s Experience February 3, 2010 24 / 24