Bayesian modeling and analysis of stochastic volatility

in finance Derrick HangApril 6, 2010

Economics 201FS

Review from Last Time• Regress for Prices: Possible useful predictors of prices are lost when we take the difference between prices to obtain returns

•In general , we expect

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tttt ,0,1,0

Addressing Stationarity Concerns• Classic tests for stationarity of an AR time series (i.e. Dickey-Fuller, Phillips-Peron, etc.) test the coefficient on the lagged time-series for a unit root

• However, these tests assume a CONSTANT coefficient and have LOW POWER

• DLM allows for the possibility of “pockets of stationarity” and will reject unit root null at values close to 1

Does stationarity of a model matter if we are looking for short term forecasting?

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The Data• Jan 2, 2009 – June 31, 2009 (excluding April 10th)• Data for market hours only (no weekend) : 9:35AM – 3:55PM• 5 Minute Data (9778 total points for each dataset)

• All prices logged• 10 dependent variable (USD/variable): AUD, CHF, EUR, GBP, JPY, NZD, CAD, NOK, SGD, ZAR•12 independent variable: 10 lagged forex variables, brent oil futures, comex gold futures

• Focus on AUD, GBP, JPY, brent, gold

Back to Basics: Jump Testing• What is the relationship –if any- between jump days and periods of non-unit roots?

Back to Basics: Jump Testing

Back to Basics: Jump Testing

Back to Basics: Jump Testing

Back to Basics: Jump Testing

0.1% Significance Level = 25 / 127 (2.23%) 1% Significance Level = 6 / 127 (4.72%) 5% Significance Level = 1 / 127 (19.69%)

Back to Basics: Jump Testing

0.1% Significance Level = 5 / 127 (3.94%) 1% Significance Level = 11 / 127 (8.66%) 5% Significance Level = 24 / 127 (18.90%)

Back to Basics: Jump Testing

0.1% Significance Level = 3 / 127 (2.36%) 1% Significance Level = 15/ 127 (11.81%) 5% Significance Level = 38 / 127 (25.98%)

Time-Varying Coefficient: Lagged GBP, JPY

Time-Varying Coefficient: Brent, Gold

Time-Varying Coefficient: Lagged AUD

Significant Windows: Lagged AUD (descending order)

Start End03-Jun-2009 09:40:00 11-Jun-2009 14:55:0014-Jan-2009 09:35:00 20-Jan-2009 15:55:0018-Mar-2009 10:50:00 23-Mar-2009 15:55:0030-Mar-2009 09:35:00 02-Apr-2009 13:25:0025-Jun-2009 09:40:00 29-Jun-2009 15:55:0022-Jun-2009 09:35:00 24-Jun-2009 15:00:0008-Apr-2009 09:35:00 13-Apr-2009 12:45:0010-Mar-2009 10:25:00 12-Mar-2009 13:00:0027-Jan-2009 09:35:00 28-Jan-2009 15:55:0009-Feb-2009 09:50:00 10-Feb-2009 15:55:0019-Feb-2009 11:00:00 20-Feb-2009 14:45:0012-Jun-2009 14:25:00 16-Jun-2009 10:55:0006-Mar-2009 13:35:00 10-Mar-2009 09:50:0017-Apr-2009 14:15:00 20-Apr-2009 14:25:0030-Jan-2009 09:35:00 02-Feb-2009 09:35:00

Jump Days: AUD (5% level)

01-May-2009 09:40:0008-May-2009 09:40:0015-May-2009 09:40:0020-May-2009 09:40:0022-May-2009 09:40:0002-Jun-2009 09:40:0004-Jun-2009 09:40:0005-Jun-2009 09:40:0010-Jun-2009 09:40:0011-Jun-2009 09:40:00

08-Jan-2009 09:40:0016-Jan-2009 09:40:0028-Jan-2009 09:40:0002-Feb-2009 09:40:0003-Feb-2009 09:40:0004-Feb-2009 09:40:0023-Feb-2009 09:40:00(03-Mar-2009 09:40:00)12-Mar-2009 09:40:0023-Mar-2009 09:40:0031-Mar-2009 09:40:0001-Apr-2009 09:40:0013-Apr-2009 09:40:0015-Apr-2009 09:40:0016-Apr-2009 09:40:00

Jump Days: AUD (1% and 0.1% level)

0.1%

04-Jun-2009 09:40:00

1%

08-Jan-2009 09:40:0012-Mar-2009 09:40:0013-Apr-2009 09:40:0015-May-2009 09:40:0022-May-2009 09:40:0004-Jun-2009 09:40:00

Significant Windows: Lagged AUD, GBP, JPY (descending order)

Start End30-Mar-2009 09:35:00 01-Apr-2009 13:20:0003-Mar-2009 09:35:00 05-Mar-2009 11:05:0010-Mar-2009 12:20:00 12-Mar-2009 13:00:0026-Jun-2009 11:00:00 29-Jun-2009 15:55:0029-Jan-2009 09:45:00 30-Jan-2009 13:50:0027-Jan-2009 09:35:00 28-Jan-2009 09:55:0019-Feb-2009 14:30:00 20-Feb-2009 14:45:0017-Apr-2009 14:15:00 20-Apr-2009 14:25:0006-Mar-2009 15:50:00 09-Mar-2009 15:55:00

Jump Days: AUD (5% level)

01-May-2009 09:40:0008-May-2009 09:40:0015-May-2009 09:40:0020-May-2009 09:40:0022-May-2009 09:40:0002-Jun-2009 09:40:0004-Jun-2009 09:40:0005-Jun-2009 09:40:0010-Jun-2009 09:40:0011-Jun-2009 09:40:00

08-Jan-2009 09:40:0016-Jan-2009 09:40:0028-Jan-2009 09:40:0002-Feb-2009 09:40:0003-Feb-2009 09:40:0004-Feb-2009 09:40:0023-Feb-2009 09:40:0003-Mar-2009 09:40:0012-Mar-2009 09:40:0023-Mar-2009 09:40:0031-Mar-2009 09:40:0001-Apr-2009 09:40:0013-Apr-2009 09:40:0015-Apr-2009 09:40:0016-Apr-2009 09:40:00

Jump Days: AUD (1% and 0.1% level)

0.1%

04-Jun-2009 09:40:00

1%

08-Jan-2009 09:40:0012-Mar-2009 09:40:0013-Apr-2009 09:40:0015-May-2009 09:40:0022-May-2009 09:40:0004-Jun-2009 09:40:00

Significant Windows: Lagged AUD, Brent, Gold (descending order)

Start End30-Mar-2009 09:35:00 02-Apr-2009 09:35:0019-Mar-2009 11:10:00 23-Mar-2009 14:30:0010-Mar-2009 10:25:00 12-Mar-2009 13:00:00

Jump Days: AUD (5% level)

01-May-2009 09:40:0008-May-2009 09:40:0015-May-2009 09:40:0020-May-2009 09:40:0022-May-2009 09:40:0002-Jun-2009 09:40:0004-Jun-2009 09:40:0005-Jun-2009 09:40:0010-Jun-2009 09:40:0011-Jun-2009 09:40:00

08-Jan-2009 09:40:0016-Jan-2009 09:40:0028-Jan-2009 09:40:0002-Feb-2009 09:40:0003-Feb-2009 09:40:0004-Feb-2009 09:40:0023-Feb-2009 09:40:0003-Mar-2009 09:40:0012-Mar-2009 09:40:0023-Mar-2009 09:40:0031-Mar-2009 09:40:0001-Apr-2009 09:40:0013-Apr-2009 09:40:0015-Apr-2009 09:40:0016-Apr-2009 09:40:00

Jump Days: AUD (1% and 0.1% level)

0.1%

04-Jun-2009 09:40:00

1%

08-Jan-2009 09:40:0012-Mar-2009 09:40:0013-Apr-2009 09:40:0015-May-2009 09:40:0022-May-2009 09:40:0004-Jun-2009 09:40:00

Initial Findings• From the data so far, we have seen that the largest windows contain days declared as jump days

• Most of the time, windows that contain entire days have a “jump day” inside it

• Windows where multiple regressors are significant also contain declared jump days

• What does this all mean?

Further Research• Look to see if the relationship between large windows and jump days exist with the other dependent currency datasets

• Short term forecasts inside this windows?

• Try with different jump tests (other than Mean-adjusted TP)

• Fix bugs in code