Bayesian modeling and analysis of stochastic volatility in finance

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

),0(~;1 ttttTttttt VNvvFYY

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?

),0(~;1 ttttt VNvvYY

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


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


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


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



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


0.1%

04-Jun-2009 09:40:00

1%


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



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


0.1%

04-Jun-2009 09:40:00

1%


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

Bayesian modeling and analysis of stochastic volatility in finance

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