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