New Development of Volatility Inference in Financial Market: Usage of High-

frequency Financial Data

Minjing Tao Department of Statistics tao@stat.fsu.edu

Ø  Volatilities, reflecting market risk, are crucial important in portfolio allocation, decision-making, and performance evaluation, etc.

Ø  High-frequency financial data provide opportunity to analyze the dynamic of financial markets, in particular, the market volatility.

Introduction

K Groups and Sub-Sampling (help to control the effect of noise)

Construct Co-Volatility and One-Scale Volatility Matrix

Define Multi-Scale Realized Volatility (MSRV) Matrix Estimator

(This MSRV estimator will minimize the effect of noise)

Impose Sparsity Condition for Large Volatility Matrix Γ Define Threshold MSRVM Estimator

(solve the high-dimensional problem)

Good for fixed (or small) number of assets

Extra work for large number of assets

Γij (τk ) = Yi τ r

k( )−Yi τ r−1k( )#$

%& Yj τ r

k( )−Yj τ r−1k( )#

$%&

r=2

τ k

∑

ΓijK =

1K

Γij (τk )

k=1

K

∑ , and ΓK = ΓijK( )

Co-volatility:

One-scale volatility: τ 1 : t1, tK+1, t2K+1,τ 2 : t2, tK+2, t2K+2,

τ k : tk, tK+k, t2K+k,

τ K : tK , t2K , t3K , Γ = am ΓKm

m=1

N

∑ +ζ ΓK1 − ΓKN( ), am = 12Km (m− N / 2−1/ 2)N(N 2 −1)

, ζ = K1KN

n(N −1)

Γijq

j=1

p

∑ ≤Ψπ n (p), for any i =1,, pΓ̂ = Γij1 Γij ≥ϖ( )( )

Comparison: performance of high-frequency estimator (MSRV) and low-frequency estimator (GARCH) Ø  GARCH model over-estimates

volatilities in most of the days. Ø  Our method, MSRV, produces

smaller estimation errors; is better than GARCH in capturing the dynamics of true volatility process.

Results

0 500 1000 1500 2000 2500−1.0

−0.5

0.0

0.5

1.0

Day

Error

GARCHMSRV

Methodology

S&P 500 Daily Volatility

Estimation Errors of MSRV and GARCH in 2500 Days