Stochastic Volatility Models

1
Stochastic Volatility Models

• Course Applied Econometrics
• Lecturer Zhigang Li

2
Why Study Stochastic Volatility?

• Heteroskedasticity can affect the standard error
  estimates of model parameters
• We can address this using robustness estimators.
• Risk is a key factor in many decision process
  (e.g. investment). Risk changes over time and
  this often reflect on instable variance in the
  disturbance of models with price measures as
  dependent variables.
• Capital Asset Pricing Model (CAPM) suggests a
  natural relationship between expected returns and
  volatility of returns.
• Option pricing depends critically on the risks of
  assets.
• News in the financial market may reflect on
  changes in the volatility of price measures.

3
Autoregressive Conditional Heteroskedasticity
(pp. 416 or 438)

• First-order ARCH (or ARCH(1))model
• ut2a0a1ut-12vt
• U is the error term of a typical regression model
  and v is the error term of the error term u.
• The model implies that the variance of the error
  term u is correlated over time, so called
  volatility clustering.
• ARCH is just one particular form of
  heteroskedasticity.

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Generalized ARCH Model

• GARCH(p,q)
• Ytµut
• utht1/2vt
• v is i.i.d. with standard normal distribution
• hta0S1qaiu2t-iS1pßjht-j
• GARCH(0,q) model is an ARCH (q) model.

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Why ARCH-Type Models?

• ARCH-Type models are easy to estimate and
  interpret. (There are other volatility models,
  but the ARCH-Type is generally much easier to
  implement)
• Knowledge of future risk is useful for optimal
  decision at the current stage.

6
Currency Board Reforms and Interbank Market of
Hong Kong (Tse and Yip, 2003)

• A CBS scheme is introduced in Hong Kong on 17
  October 1983, fixing the exchange rates between
  HK and US. Seven major reforms have been made
  to the CBS system since them.
• What are the effects of the reforms on the
  quality and stability of the financial sector of
  Hong Kong?
• The impact of the mean and variance of the
  interest rate differential between Hong Kong and
  the US.

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Shocks to CBS

• Oct 1983 Account Aggrangements introduced to
  limit the ability of HSBC to create money.
• Jul 1992 Liquidity Adjustment Facility
  introduced to cap the variation of interest
  rates.
• Mar 1994 Change target from interbank liquidity
  to interbank interest rate.
• Oct 1997 Financial Crisis
• Sept 1998 Reform packages

8
Empirical Strategy

• Data
• Daily observations of the HK and US interbank
  interest rates (from Datastream).
• Model
• ytd1D1td7D7tf1yt-1f1yt-pet
• st2?1D1t?7D7tae2t-1ßs2t-1

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Meteor showers or Heat Waves?(Engle et al., 1990)

• Heat waves Domestic news affects only local
  financial markets.
• Meteor showers Domestic news can affect foreign
  financial markets.
• Market failure
• Need time to absorb news into trading prices

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Test Framework

• N nonoverlapping markets within a day with market
  1 open first.
• Let ei,t be the intra-day exchange rate change
  divided by the square root of business hours in
  market i on date t.
• A modified GARCH model
• Hi,t?aißjihi,t-1S1i-1aije2j,tSinaije2j,t-1
• If aij0 for i different from j, then the heat
  waves model is supported. Otherwise, the meteor
  shower model is supported.

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Empirical Strategy

• Intra-day yen/dollar exchange rate from Oct 3,
  1985 to Sep 26, 1986.
• Four markets
• Tokyo
• Pacific
• New York
• Europe
• Three tests
• Heat waves
• Meteor showers with foreign news
• Meteor showers with country-specific news

12
Computer Exercise

• Example 12.9 (Using NYSE weekly stock returns
  data)