daily returns over the last 40 years Conditional variance fluctuates across time and is very persistent Impressive amount of empirical work Engle 1982 ID: 712999
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The figure displays S&P
daily
returns over the last 40 years. Conditional variance fluctuates across time and is very persistent. Impressive amount of empirical work: Engle (1982), Bollerslev (1986), Bollerslev et al. (1988), Nelson (1991), etc.
Mele
(2008): close
connection
between
aggregate
stock
market
volatility
and the business cycle.
The figure
depicts
the relation
between
volatility
and the
industrial
production
growth
rate over the last
sixty
years
.
Stock
volatility
is
largely
countercyclical
,
being
larger
in
bad
times
than
in good times.Slide5Slide6Slide7Slide8Slide9Slide10Slide11
A
B
Investor B estimates the parameters correctlyInvestor A estimates the parameters with biasSlide12
A
B
Investor B estimates the parameters correctlyInvestor A estimates the parameters with bias
Filter
(
Kalman
-
Bucy
)
=>
Difference
of
beliefs
:Slide13
Linear
Quadratic Setup
Transform into a Standard Affine SetupObtain a System of Riccati EquationsSolve by Embedding into a Matrix Riccati EquationTransform
into
a
Linear
Cauchy
Problem
Matrix
Exponential
Solution
Closed
-
Form
Solution
Cheng &
Scaillet
(MF, 2007)
Duffie (2008)
Enders &
Schmidtmann
(1992)
Polyanin
& Zaitsev (2003)
Jordan
Decomposition
Radon’s
LemmaSlide14Slide15
3
0.075
0.1
0.03
0.13
0.02Slide16Slide17Slide18Slide19Slide20Slide21Slide22