realized volatility estimation (slides)

Realized Volatility Estimation

Miquel Masoliver Guillem Roig Shikhar Singla

July 1, 2014

Miquel Masoliver, Guillem Roig, Shikhar Singla Realized Volatility Estimation July 1, 2014 1 / 16

Roadmap

1 Introduction

2 Realized Volatility Estimator

3 Two-Scaled Realized Volatility Estimator

4 Realized Kernel

5 Threshold Realized Variance

6 Simulation setup

7 Data implementation

8 Conclusions


Introduction

Volatility is the main driver of portfolio construction, option pricing anddetermination of a firm’s exposure to risk

Parametric models such as the ARCH family rely upon specific distributionalassumptions, immediately calling to question its robustness

Use a non-parametric approach based on summing squares and cross-productsof intraday high-frequency returns to construct estimates of realized dailyvolatility, quadratic variation is used as an ex-post variation of asset prices

The main purpose of this study is to try to find the optimal volatilityestimator in a non-parametric framework


Realized Volatility Estimator

Estimates the quadratic variation of prices over some time interval

Discretize the time interval τ = [0, 1] into a grid of subintervalst0 = 0 ≤ · · · ≤ tn = 1 of length ti − ti−1 = ∆ = 1/n, and then set the pricesyi = yti .

RV Estimator

RV 2 =n−1∑i=1

(yi+1 − yi )2

Inconsistent and biased under market microstructure noise and price jumps


Two-Scaled Realized Volatility Estimator

Decomposes the total observed variance into two components: fundamentalprice and market microstructure noise

Sample sparsely at some lower frequency to evaluate the quadratic variationat the two frequencies

Consistent and unbiased under market microstructure noise


TSRV 2 =1

K

K∑k=1

∑ti ,ti+1∈Tk

(yti+1 − yti )2 − n

n

∑tj ,tj+1∈T

(ytj+1 − ytj )2


Sample is partitioned into K non-overlapping grids with equal number ofobservations

y1 y2 . . . y50

y51 y52 . . . y100

y101 y102 . . . y150

......

......

y951 y952 . . . y1000

Table : Sampling process


TSRV 2 =1

K

K∑k=1

∑ti ,ti+1∈Tk

(yti+1 − yti )2 − n

n

∑tj ,tj+1∈T

(ytj+1 − ytj )2


Realized Kernel

Linear function of autocovariances weighted by a specific function (Parzen)

Guarantees consistency and positive semi-definiteness

Realized Kernel Estimator

K (X ) =H∑

h=−H

k(h

H + 1)γh,

where γh is the matrix of autocovariances and and we add 2H such matrices


Threshold Realized Variance

All previous estimators are inconsistent under jumps

Detect the jumps i.e. returns above a certain threshold, and not includethose price points in volatility calculation where there is a jump

Threshold Realized Variance

TRVδ =

[T/δ]∑j=1

(yj − yj−1)21{(yj−yj−1)2≤Θ(δ)}

TRV1 =

[T/δ]∑j=1

(yj − yj−1)21{(yj−yj−1)2≤log( 1δ )√δ}

TRV2 =

[T/δ]∑j=1

(yj − yj−1)21{(yj−yj−1)2≤√δ}


Simulation setup

Prices governed by a Geometric Brownian Motion with Heston Structure forVolatility (true volatility is observable!)

Jumps generated using a compound Poisson process

Market microstructure noise produced by random draws from a normaldistribution with different variances

The number of subsamples for TSRV and number of autocovariance matricesfor RK are chosen so as to minimize the noise effect

Frobenius and Infinite norms to compare between the estimators

Frobenius and Inifinite Norm

Frobenius Norm ||A||f =

√√√√ m∑i=1

m∑j=1

|aij |2

Infinite Norm ||A||i = max|aij |.


Figure : Frobenius and Infinite norm to test the performance of the estimators on thebenchmark scenario i.e. neither jumps nor noise


Figure : Frobenius and Infinite norms to test the performance of the estimators in thepresence of jumps


Figure : Comparison of three estimators using Frobenius and Infinite norms underpresence of noise


Data implentation

Tick-by-tick prices of S&P-100 stocks executed on October 27th, 2010

Trading day of 6.5 hours is normalized into a [0,1] interval

To make the price points simultaneous for all stocks, the last recorded tradeis kept

Stocks with daily variance lower than 0.001 are excluded from the dataset

Cannot use norms to assess performance (no true volatility). Construct globalMVP portfolio and compute returns over the next month


Figure : Performance of RV, Kernel and uniformly weighted portfolio using MVP


Figure : Performance of RV and Threshold RV using MVP


Conclusions

When noise is present TSRV and Realized Kernel outperform RealizedVariance

Using the highest available frequency does not necessarily result in the bestestimates

Minimum values for the ex-post portfolios variance are found for samplingfrequencies equal to 30 seconds for RV, TRV1 and TRV2 estimators

Although TRV1 outperforms RV we cannot conclude that jumps do exist inintraday data, since both specifications behave very similarly

High-frequency data achieves better estimates than daily data


realized volatility estimation (slides)

Economy & Finance