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Possible Research InterestsKyu Won Choi

Econ 201FSFebruary 16, 2011

+GARCH model + Realized Variation Measures

Combining realized variation measures based on high-frequency data with more traditional GARCH type models

Some Examples Realized GARCH Models HEAVY models Multiplicative Error Model HYBRID GARCH Models Generalized Expected RV (GERV) models HARG-RV models Other multi-period forecasts joint models

+ High Frequency Data

Realized Measures based on high frequency data Valuable predictors of future volatility

Realized Variance (most commonly used) Bi-power Variance Realized Kernel

High frequency data is crucial Volatility is highly persistent The more accurate measure of a current volatility, the better

able to forecast volatility Evaluation of volatility forecast models

accurate proxy when comparing volatility models Close analysis of announcements and the effects

+ Standard GARCH Model

Yt+1 = t+1+ t+1 where t+1 ~ WN (0, 2t+1)

ARMA(1,1) t+1 = 0 + 1 Yt + t

GARCH (1,1) 2t+1 = + 2

t + 2

t

Conditional mean t+1 = E [Yt+1 Ft]

Conditional variance 2t+1 = Var [Yt+1 Ft]

Ft as filtration Represents all information available at time t Generally exclusively by past returns consisted of sparse daily

data i.e. opening and closing only Ft = (yt, yt-1, y1)

Consisting of high frequency information is useful such as 30-min intraday transaction prices, bid/ask quotes,

etc

+ Adding Realized Measures of Volatility: GARCH-X Model

Since Ft RM = (RM t, yt, RM t-1, yt-1, y1) ≠ Ft ,

2t+1 = Var [Yt+1 Ft] ≠ Var [Yt+! Ft

RM] = 2t+1

RM

When Realized Measures (such as RV and BV are included), becomes insignificant ( 0, > 0) 2

t+1RM = + 2

t + 2

t + RVt

Estimating a GARCH model with additional realized measures of volatility based on high-frequency data Now the Ft

RM includes greater set of data Including variable that adds predictive power Realized measures can improve the empirical fit

+ GRAPH illustrated in the class GARCH model is sensitive to rapid volatility change (jump)

Slow at “catching up”: longer time periods (around 3 months)

to reach the new volatility GARCH-X model within a few days

+ GARCH-X Model

Two different methods depending on the number of latent volatility variables

Parallel GARCH structure For each realized measure, additional GARCH-type model

(latent volatility process) is introduced Multiplicative Error Model (MEM) High-frequency based Volatility Model (HEAVY)

Realized Measures Similar to the traditional GARCH Realized GARCH model with a single latent volatility factor

Connected to conditional variance of returns

+ Parallel GARCH Structure

MEM and HEAVY models digress from the traditional GARCH Which uses only a single latent volatility factor

HEAVY model by Shephard and Sheppard (2010) Realized kernel (RK)

Multiplicative Error Model (MEM) by Engel (2002) In addition to squared returns, Two realized measures

Intraday range (high minus low) Realized variance

+Realized GARCH Measurement equation that ties the realized measure to the

conditional variance of returns where ut ~ iid (0, 2

u) and zt ~ iid(0,1)

RMt = + ht + (zt) + ut

Second volatility factor ht = var (yt Ft-1)

Ft-1 = (yt-1,RMt-1,yt-2,RMt-2.....)

(zt): leverage condition Dependence between returns and future volatility Phenomenon is referred as leverage effect expected leverage is zero whenever zt has mean zero and

unit variance (zt) = 1a1(zt) + + kak(zt) where Eak(zt) = 0 for k

News impact curve: how positive and negative shocks to the price affect future volatility

+Linear Realized GARCH (1,1) model Simplest GARCH (1,1) equation

rt : return

xt : realized measure of volatility

zt ~ iid(0,1) ut ~ iid (0, 2u)

ht = var (rt Ft-1)

Where Ft-1 = (rt-1,xt-1,rt-2,xt-2.....) Last equation relates observed realized measure to

the latent volatility: measurement equation Leverage function

+Log-Linear Realized GARCH

Key variable of interest: conditional variance ht

Log-Linear GARCH (p, q)

Automatically ensures positive variance

Preserves the ARMA structure that characterizes some of the standard GARCH models Conditions zt = rt/ht

1/2 ~ iid(0,1) and ut ~ iid(0, 2u)

Example: GARCH (1,1) ht-1 and r2t-1

Then log ht ~ AR(1) and log xt ~ ARMA(1,1)

+Key Models (Hansen, Huang, Shek, 2011)

+HYBRID GARCH

High Frequency Data-Based Projection-Driven GARCH Volatility driven by HYBRID processes

Vt+1t = + tt-1 + Ht where Ht is HYBRID process

Volatility process need not be defined to be conditional variance of returns Tomorrow’s expected volatility using intra-daily returns Next three days volatility forecasting with past daily data

Three broad classes of HYBRID processes Parameter-free process purely data driven

Structural HYBRIDS assuming an underlying high frequency data structure

HYBRID filter processes

+ The Practical Application

Out-of-sample forecasting Risk Measurement & Management Asset Pricing Portfolio Allocation Option Pricing

+ Work To Do & Further Interests

Use the data and compare various GARCH +RM Observe the positive and negative sides of each

Multivariate GARCH models & Realized GARCH framework: multi-factor structure (multi-period forecasting) m realized measures and k latent volatility variables

Presence of jumps in the price process Information about forecasting volatility Inclusion of a jump robust realized measure

Extent to which microstructure effects are relevant for the forecasting problem using realized measures that are robust to microstructure effects

+References

Realized GARCH: A Joint Model for Returns and Realized Measures of Volatility (Hansen, Huang, Shek, 2010)

Forecasting Volatility using High Frequency Data (Hansen, Lunde, 2011)

The Class of HYBRID-GARCH Models (Chen, Ghysels,Wang, 2011)

Exchange Rate Returns Standardized by Realized Volatility are (Nearly) Gaussian (Torben G. Andersen, Bollerslev, Diebold, Labys, 2000)