volatility forecasting
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VOLATILITY FORECASTINGSteven Poher
Ramzi RachedRicardo Uribe
Dongting ZhengGlobal Investment Management
AGENDA
• Objective • Background Information• Forecasting Models• Data set• Methodology• Results• Conclusion
OBJECTIVE
• Objective• To establish a variance forecasting model
• Why?• Important for risk managers (VaR)
• Used to price options
• Volatility + Return = investment decision
BACKGROUND INFORMATION
• Realized / observed volatility is measured by squared returns
• Volatility displays a positive correlation with its own past
• Simple Model
• PB : Equal weights on the past m observations
m
tt Rm 1
12
12 1
FORECASTING MODELS
• More flexible model Simple GARCH or GARCH (1,1)
• Extended to Local and Global Instruments
• Models to be tested for this project• GARCH (1,1)• GARCH (1,1) + Local• GARCH (1,1) + Global• GARCH (1,1) + Local + Global
221
2ttt R
n
i
Gi
GLi
Lttt ZZR
ii1
221
2 )(
DATA SET
• Source DataStream• Period 3/27/1998 - 3/28/2008 (10 years)• Granularity 1 day
Country Index
NIKKEI 225
CAC 40
DAX 30
FTSE 100
S&P 500
DATA SET
• Local Instruments• Change in Exchange Rates
• EUR / USD / JPY / GBP
• Change in short-term interest rates• T-Bill (US) / BTAN (FR)
• Global Instruments• Change in Short-term Eurodollar rate
• Change in the Term Structure spread
METHODOLOGY
• Using EXCEL, test our 4 models for each of our 5 markets
• Use Maximum Likelihood Estimation (MLE) to estimate /
/ / EXCEL Solver
• Test the models using a regression of Squared Returns vs.
Forecasted Variance
• Discuss the statistical significance of the regression /
Select the best model for a given country
METHODOLOGY - EXAMPLEGR
Estimating GARCH(1,1) with Local and Global Instruments - German Market
DATEGERMANY
DAX 30% Change in Ū/£ Exchange Rate
Term Structure Spread Return
Squared Returns
Conditional Variance Likelihood
Without Variance TargetingMar 27, 98 5,438.86 0.084361
Mar 30, 98 5,311.90 0.44% 0.24% -2.36% 0.000558 0.000238 2.080044 0.901481
Mar 31, 98 5,396.16 0.04% 0.24% 1.57% 0.000248 0.000265 2.731626 0.000002
Apr 01, 98 5,450.07 -0.03% 0.24% 0.99% 0.000099 0.000261 3.016659 0.095203
Apr 02, 98 5,475.79 -0.37% 0.24% 0.47% 0.000022 0.000246 3.191489 0.000242
Apr 03, 98 5,530.61 -0.45% 0.24% 1.00% 0.000099 0.000226 3.058406 MLE 7,532.98 Apr 06, 98 5,595.27 -0.18% 0.25% 1.16% 0.000135 0.000216 2.988255 R2 0.1556 Apr 07, 98 5,697.96 0.14% 0.25% 1.82% 0.000331 0.000208 2.525438 F Statistic 480.62 Apr 08, 98 5,649.06 -0.56% 0.25% -0.86% 0.000074 0.000218 3.126785Apr 09, 98 5,721.73 0.02% 0.25% 1.28% 0.000163 0.000207 2.927757Apr 10, 98 5,721.73 0.00% 0.25% 0.00% 0.000000 0.000202 3.334149Apr 13, 98 5,704.71 0.00% 0.25% -0.30% 0.000009 0.000184 3.357209Apr 14, 98 5,828.16 -0.11% 0.25% 2.14% 0.000458 0.000168 2.064456Apr 15, 98 5,864.82 -0.18% 0.25% 0.63% 0.000039 0.000192 3.257141Apr 16, 98 5,760.89 0.75% 0.25% -1.79% 0.000320 0.000179 2.501362Apr 17, 98 5,701.19 -0.47% 0.25% -1.04% 0.000109 0.000195 3.074001Apr 20, 98 5,905.46 -0.55% 0.22% 3.52% 0.001239 0.000189 0.087533Apr 21, 98 5,886.94 -0.72% 0.22% -0.31% 0.000010 0.000279 3.154881Apr 22, 98 5,851.94 0.07% 0.22% -0.60% 0.000036 0.000259 3.141278Apr 23, 98 5,710.36 -0.17% 0.22% -2.45% 0.000600 0.000238 1.994152Apr 24, 98 5,604.24 -0.05% 0.22% -1.88% 0.000352 0.000268 2.536585Apr 27, 98 5,560.93 -0.08% 0.27% -0.78% 0.000060 0.000273 3.074337Apr 28, 98 5,467.96 0.12% 0.27% -1.69% 0.000284 0.000253 2.660274Apr 29, 98 5,570.09 0.16% 0.27% 1.85% 0.000342 0.000254 2.545737Apr 30, 98 5,566.62 0.08% 0.27% -0.06% 0.000000 0.000259 3.208788May 01, 98 5,614.38 -1.09% 0.27% 0.85% 0.000073 0.000236 3.102717May 04, 98 5,834.77 0.00% 0.30% 3.85% 0.001483 0.000232 0.067145May 05, 98 5,772.30 -0.64% 0.30% -1.08% 0.000116 0.000336 2.908149May 06, 98 5,796.04 -0.23% 0.30% 0.41% 0.000017 0.000318 3.081275May 07, 98 5,751.14 -0.93% 0.30% -0.78% 0.000060 0.000290 3.049123May 08, 98 5,813.58 -0.19% 0.30% 1.08% 0.000117 0.000277 2.966504May 11, 98 5,880.87 0.00% 0.25% 1.15% 0.000132 0.000261 2.952416May 12, 98 5,838.03 0.12% 0.25% -0.73% 0.000053 0.000249 3.123407May 13, 98 5,915.37 0.17% 0.25% 1.32% 0.000173 0.000230 2.893011May 14, 98 5,897.80 -0.22% 0.25% -0.30% 0.000009 0.000224 3.262508May 15, 98 5,921.45 0.10% 0.25% 0.40% 0.000016 0.000205 3.287909May 18, 98 5,840.49 -0.10% 0.24% -1.38% 0.000190 0.000188 2.866553May 19, 98 5,906.52 -0.10% 0.24% 1.12% 0.000126 0.000187 3.035028May 20, 98 6,089.75 -0.30% 0.24% 3.06% 0.000933 0.000181 0.815444May 21, 98 6,132.91 -0.53% 0.24% 0.71% 0.000050 0.000245 3.136732May 22, 98 6,192.65 0.05% 0.24% 0.97% 0.000094 0.000229 3.066475
RESULTS
Best models for each country
Country Model R2
GARCH + % Change in Term Structure Spread (G)
1.21 %
GARCH + % Change in €/£ (L) 14.12 %
GARCH + % Change in €/£ (L) + % Change in Term Structure Spread (G)
15.56 %
GARCH (1,1) + % Change in $/£ (L) + % Change in ST Eurodollar (G)
14.23 %
GARCH 10.58 %
RESULTS
• Best model for German Market
• R2 of 15.56%
• Final equation• Simple GARCH +
• % Change in €/£ Exchange (L) +
• % Change in Term Structure Spread (G)
1322
12 000242.0095203.0901481.0084361.0000002.0 GL
ttt ZR
RESULTS
DE MARKET - SIMPLE GARCH + Ū/£ + Term Structure Spread
0.0000
0.0004
0.0008
0.0012
0.0016
0.0020
0.0024
Mar98
Sep98
Mar99
Sep99
Mar00
Sep00
Mar01
Sep01
Mar02
Sep02
Mar03
Sep03
Mar04
Sep04
Mar05
Sep05
Mar06
Sep06
Mar07
Sep07
Realized Forecasts
CONCLUSION
• No universal model• Different countries = different models• Good proxy for DE / Bad for JP
• GARCH could also be extended• Leverage effects• Day-of-week effects• Jumps
• Economic intuition & reality check
QUESTIONS?
THANK YOU
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