Economics 201FS: The Basic Diffusive Model, Jumps,Variance Measures, and Noise Corrections

George Tauchen

Economics 201FS

Spring 2011

Main Points

• Basic Model without and with Jumps

• Realized Variance and Bipower Variation

• Brief Introduction to Jumps Tests

• Splitting the Quadratic Variation into Two Pieces

• Analysis of microstructure noise

• Implications for options pricing and risk management

1

1. The Continuous Time Model

Dynamics of the log price (p) process:

Continuous

dp(t) = µ(t)dt + σ(t)dw(t).

With jumps

dp(t) = µ(t)dt + σ(t)dw(t) + dJ(t),

where J(t) − J(s) =∑

s<τ≤t κ(τ).

2

Basic Diffusion

0.5 1 1.5 2 2.5

x 104

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78

80

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Simulation Generated by Euler Scheme

Generate Gaussian (normal) random variables:

Zj ∼ N(0, 1), j = 1, 2, . . . , J

M steps per day:

M = 80 ⇒ ∆ = 1/80

Standard deviation

σ = 0.011 ⇒ 1.10 percent per day

T days:

T = 1.25 × 252 ⇒ J = T × M = 315 × 80 (steps)

4

Euler Scheme (continued)

Wiener increments

dWj =√∆Zj j = 1, 2, . . . , J

Initialize p0 = log(75) ($75 per share)

Iterate:

pj = pj−1 + µ∆ + σ dWj, j = 1, 2, . . . , J

Convert to levels:

{Pj}Jj=1, Pj = e

pj j = 1, 2, , . . . , J(= M × T )

Make a nice plot.

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Euler Scheme: Time Varying VarianceWiener increments

dWj =√∆Zj j = 1, 2, . . . , J

Initialize p0 = log(75) ($75 per share)

Iterate:

pj = pj−1 + µ∆ + σj−1 dWj , j = 1, 2, . . . , J

Convert to levels:

{Pj}Jj=1, Pj = e

pj j = 1, 2, , . . . , J(= M × T )

Make a nice plot.

6

Simulation of Jump Diffusion by an Euler Scheme

Sample the number of jumps N ∼ Poisson, then draw

N jumps κk ∼ N(0, σjmp), k = 1, . . . , N , and scatter the jumps randomly.

7

Jump Diffusion

With jumps

dp(t) = µ(t)dt + σ(t)dw(t) + dJ(t),

1000 2000 3000 4000 5000 6000 7000 8000 9000 1000050

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60

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2. Observed Data

Consider, XOM September-October, 2008:

Sep 02, 2008 Sep 16, 2008 Oct 01, 2008 Oct 16, 2008 Oct 31, 200855

60

65

70

75

80

85XOM: September − October 2008

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Within-day geometric returns

rt,j = p(t − 1 + j∆) − p(t − 1 + (j − 1)∆),

for j = 1, 2, . . . ,M , integer t = 1, 2, , . . . , T .

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XOM Returns, September-October, 2008

Sep 02, 2008 Sep 16, 2008 Oct 01, 2008 Oct 16, 2008 Oct 31, 2008−10

−8

−6

−4

−2

0

2

4

6

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10XOM, Return: September − October 2008

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Default Plots from Graphics are not Acceptable

Poorly labeled axis, scales, and colors:

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0 2000 4000 6000 8000 10000 12000 14000 16000 1800055

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0 2000 4000 6000 8000 10000 12000 14000 16000 18000−10

−8

−6

−4

−2

0

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6

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Care with Data: e.g., Stock Splits

Consider Apple (APPL):

Jan 02, 2002 Apr 06, 2004 Jul 05, 2006 Oct 03, 2008 Dec 30, 2010

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100

150

200

250

300

AAPL Price

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Clearly, something is wrong, or needs attention:

Jan 02, 2002 Apr 06, 2004 Jul 05, 2006 Oct 03, 2008 Dec 30, 2010

−60

−40

−20

0

20

40

60

AAPL Returns

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Culprit

Data recording error or maybe a stock split?

AAPL (Yahoo Finance, Charts, click max)Splits: Jun 16, 1987 [2:1], Jun 21, 2000 [2:1], Feb 28, 2005 [2:1]

Need to adjust the data for any stock splits. The convention is to adjust backwards.

Watch it, sometimes there can be a reverse stock split such as 1:3.

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3. Variance Measures and Jump Tests

The Realized Variance

RVt =M∑j=1

r2t,j

As ∆ → 0, M → ∞, then from advanced probability theory:

RVt →∫ t

s=t−1

σ2(s) ds +

∑t−1<s≤t

κ2s

RVt converges to the integrated variance plus sum of all within-day jumps squared.

18

Bipower Variation

The Bipower variation of Barndorff-Nielsen and Shephard (2003)

BVt = µ−11

M

M − 1

M∑j=2

|rt,j||rt,j−1|

As ∆ → 0, M → ∞, and

BVt →∫ t

s=t−1

σ2(s) ds

BVt converges to the integrated variance. It is jump robust.

19

Barndorff-Nielsen and Shephard Jump Test

In the absence of jumps

RVt − BVt → 0.

The Barndorff-Nielsen and Shephard test is done by forming the z-statistic

zt =

RVt−BVtBVt

Standard Deviation

Large values of the z-statistic discredit the null hypothesis of no jumps on day t (one-sided

test).

Reject the null hypothesis of no jumps on day t if zt > z0.99, for a 1 percent level test, or

zt > z0.995 for a 0.50 percent level test. See Huang and Tauchen (2005) and Andersen,

Bollerslev, and Huang (2011) for the basic ideas elaborated.

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Continuous and Jump Variation

Model:

dp(t) = µ(t)dt + σ(t)dw(t) + dJ(t),

Split the quadratic variation into two pieces:

RVt → QVt =

∫ t

s=t−1

σ2(s) ds +

∑t−1<s≤t

κ2s

QVt = IVt + TJVt

IVt =

∫ t

s=t−1

σ2(s) ds, TJVt =

∑t−1<s≤t

κ2s

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Separate the Realized Variance

RVt = CVt + JVt

CVt = RVt I(zt ≤ c) + BVt I(zt > c)

JVt = (RVt − BVt) I(zt > c)

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Realized Variance (Annualized)

XOM 2007-2010

Jan 03, 2007 Oct 19, 2007 Aug 11, 2008 Jun 01, 2009 Mar 17, 2010 Dec 30, 20100

50

100

150

200

Annualized Realized Volatility

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Bipower Variation (Annualized)

XOM 2007-2010

Jan 03, 2007 Oct 19, 2007 Aug 11, 2008 Jun 01, 2009 Mar 17, 2010 Dec 30, 20100

50

100

150

200

Annualized Bipower Variation

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Relative Contribution of Jumps

XOM 2007-2010

A poorly scaled graphic:

Jan 03, 2007 Oct 19, 2007 Aug 11, 2008 Jun 01, 2009 Mar 17, 2010 Dec 30, 20100.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65Relative Contribution of Jumps

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Relative Contribution of Jumps

XOM 2007-2010

A reasonable well-scaled graphic:

Jan 03, 2007 Oct 19, 2007 Aug 11, 2008 Jun 01, 2009 Mar 17, 2010 Dec 30, 20100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Relative Contribution of Jumps

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Alternative Estimators of IVt

There are other possibly better ways to estimate IVt and split RVt = CVt + JVt.

Alternative jump robust measures of integrated volatility considered in (Andersen,

Dobrev, Schaumburg, 2010), among others:

Barndorff-Nielsen and Shephard Multi-power Variation:

cMP ×M

M − 2

∑j∈Dayt

|rt,j|p0|rt,j−1|p1|rt,j−2|p2

0 ≤ p0 < 1, 0 ≤ p1 < 1, 0 ≤ p2 < 1, p0 + p1 + p2 = 2.

Median Variation:

MedianVariationt = cMedV ar ×M

M − 2

∑j∈Dayt

[median (|rt,j|, |rt,j−1|, |rt,j−2|) ]2

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More Alternative Estimators of IVt

(Andersen, Dobrev, Schaumburg, 2010):

MinRV Variation:

cMN ×M

M − 1

∑j∈Dayt

[min(|rt,j|, |rt,j−1|)]2

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Truncated Variation Estimator of IVt

Threshold Variation (Mancini, 2009):

MinRV Variation:

TVt =

M∑j=1

|rt,j|2 × I [|rt,j| ≤ cutofft ]

where I[ ] is the 0-1 indicator function (1 = true, 0 = false).

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Value of cutoff for Threshold Variation

What is the cutoff? Something like 4 (or 2, 3) standard deviations would be about right,

but what standard deviation? Papers by Ait-Sahalia, Todorov, and others provide some

guidance. A reasonable choice would be

cutofft = 4 ×√

1

MBVt−1

where BVt−1 is the bipower variation of the preceding day.

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4. Market Microstructure Noise

A better term would be trading frictions, or small deviations from the appropriate price.

From the present value model, i.e. the Gordon Growth model, the correct stock price is:

p̃t =Πe

t

ra,t − gt

New information continuously flows to the market about the three fundamentals,

Πet , ra,t, gt, and no market can keep the actual price equal to p̃t. Thus, we write

pt = p̃t + ϵt

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Implications

Note the observed geometric return over the interval t − ∆ → t is:

r∆,t = p̃t − p̃t−∆ + ϵt − ϵt−∆

r∆,t = r̃∆,t + ϵt − ϵt−∆

Var (r∆,t) = ∆σ2r̃ + 2σ

2ϵ

As ∆ → 0 the noise dominates:

lim∆↓0

Var (r∆,t) = 2σ2ϵ

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Coarse Sampling

5-Minute returns, or k-Minute returns

rt,j,k = 100 [p(t − 1 + jk/m) − p(t − 1 + (j − 1)k/m)] , j = 1, 2, . . . , end of day

Volatility signature plots are helpful.

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ALL 1997-2002

10 20 30 40 50 6025

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35

40

45

Minutes

Annualized Volatility Signature

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XOM 2007-2010:

10 20 30 40 50 6015

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25

30

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Minutes

Annualized Volatility Signature

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Sub-Sampling

RV(0)

= r29:35,9:40 + r

29:40,9:45 + · · · + r

23:54,3:59

RV(1)

= r29:36,9:41 + r

29:41,9:46 + · · · + r

23:53,3:58

...

RV(4)

= r29:39,9:44 + r

29:44,9:49 + · · · + r

23:50,3:55

RVSS =1

5×

(RV

(0)+ RV

(1)+ RV

(2)+ RV

(3)+ RV

(4))

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Pre-Averaging

See papers by Mark Podolskij and many co-authors. (Tentative) Form local averages of

the prices to wash out the noise:

r̄1 = 15 × (p9:40 + p9:41 + p9:42 + p9:43 + p9:44)−

= 15 × (p9:35 + p9:36 + p9:37 + p9:38 + p9:39)

r̄2 = 15 × (p9:45 + p9:46 + p9:47 + p9:48 + p9:49)−

= 15 × (p9:40 + p9:41 + p9:42 + p9:43 + p9:44)...

Equivalently, form local average of the returns

r̄1 = 15 × (r9:40 + r9:41 + r9:42 + r9:43 + r9:44)

r̄2 = 15 × (r9:45 + r9:46 + r9:47 + r9:48 + r9:49)...

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Bipower after Pre-Averaging

B̄V = scale factor ×M∑j=2

|r̄j||r̄j−1|

There needs to be a second adjustment to make

adj(B̄V t) →∫ t

t−1

σ2s ds

38