a practical guide to volatility forecasting in a crisispages.stern.nyu.edu/~bkelly/volfor.pdfa...

A Practical Guide to Volatility Forecasting in aCrisis

Christian Brownlees Robert Engle Bryan Kelly

Volatility Institute @ NYU Stern

Volatilities and Correlations in Stressed MarketsApril 3, 2009

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Introduction

Setting

What is the best way to implement a recursive volatilityforecasting strategy?

Which models should we consider?

How does forecasting ability vary across different horizons?

... and ...How did these models perform in Fall ’08?Did these models predict what we have seen?

BEK (2009) 2 / 18

Introduction

Status Questionis

Relatively large literature on (volatility) forecast evaluationAndersen and Bollerslev (1998), Hansen and Lunde (2005), Hansen and Lunde

(2006), Patton (2009), Sheppard and Patton (2009)

Relatively small literature on multi–step ahead forecastingabilityChristoffersen and Diebold (2000), Ghysels et al. (2009)

In Fall ’08 big drop in forecasting performance comes fromforecasting volatility over longer horizons!

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Introduction

Approach

Detailed S&P 500 volatility forecasting exercise.We use battery of different volatility forecasting methods inorder to

1 assess which model/forecasting design option works bestand

2 analyze predictive ability across different horizons.

Summary evidence from other asset classes:Equity Sectors, International Equities, Exchange Rates

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Introduction

Findings

Identify which models and ingredients lead to successfulvolatility forecasting performance.

Best forecasting recipe persists across forecasting horizons.

The recent episodes of extreme volatilitydo not change our conclusions andmay not be as extreme as one might think.

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Forecasting Design

Forecasting Design

Methods:Model: GARCH, TGARCH, EGARCH, APARCHError Distribution: Normal or Student tEstimation Window: 2y, 4y, 8y, allEstimation Update Frequency: daily, weekly, monthly

we consider all 96 = (4× 2× 4× 3) combinations

Predictions:Horizons: 1 day, 1 week, 2 weeks, 3 weeks, 1 month

Sample Period:Forecast: January 2001 to December 2008Initial Training: January 1990 to December 2000

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Forecasting Design

Forecast Evaluation

Let σ̂2 be a variance proxy and h and the variance forecast.We evaluate forecasts using the Quasi Likelihood loss

QLike(σ̂2, h) =σ̂2

h− log

σ̂2

h− 1

We focus on predicting the cumulative τ–horizon variance

hτt =

τ∑i=1

ht+i|t

We employ both realized volatility and squared returns asproxies.

σ̂2 τrv t =

τ∑i=1

cadj

∑j

r2t+i j

σ̂2 τr2 t =

τ∑i=1

r2t+i

intra–daily return frequency: 5 minutes

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Forecasting Design

QLike Loss

QLike has several appealing properties: “robust” (Patton (2009),scale invariant, iid under correct specification (if τ = 1).

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Empirical Findings Predicting S&P500 Volatility from 2001 to 2008

Predicting S&P500 Volatility from 2001 to 2008

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Error Distribution, Estimation Window, Frequency

Error DistributionStudent t assumption doesn’t lead to better forecasts

WindowGARCH – often does well with small forecasting windows.Asymmetric Specifications – the more data the better.APARCH – poor performance with short estimationwindows.

FrequencyThe more frequent the updating, the better the predictions.

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A Closer Look at TGARCH

horizon 1 d 1 w 2 w 3 w 1 mSmall 0.2477 0.2272 0.2099 0.1954 0.1767Medium 0.2303 0.2040 0.1828 0.1636 0.1390Large 0.2338 0.2046 0.1799 0.1614 0.1386All 0.2582 0.2453 0.2374 0.2304 0.2224Monthly 0.2352 0.2133 0.1958 0.1817 0.1644Weekly 0.2459 0.2237 0.2055 0.1900 0.1706Daily 0.2464 0.2238 0.2062 0.1914 0.1726Normal 0.2440 0.2228 0.2064 0.1920 0.1737Student t 0.2410 0.2178 0.1986 0.1834 0.1647

Bigger is Better(Losses are relative to 60 days rolling variance)

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What Model?

QLike Loss – Realized VolatilityFull Sample

horizon 1 d 1 w 2 w 3 w 1 mGARCH 0.237

∗∗∗0.227∗∗∗

0.220∗∗∗

0.214∗∗∗

0.207∗∗

TGARCH 0.261 0.248∗∗

0.240∗∗∗

0.232 0.223

EGARCH 0.254∗∗

0.238∗∗

0.228∗∗

0.217∗∗

0.206

APARCH 0.277 0.259 0.250 0.240 0.229

Sept – Dec ’08horizon 1 d 1 w 2 w 3 w 1 mGARCH 2.437 2.562 2.720 2.830 2.881TGARCH 2.478 2.614 2.781 2.896 2.986EGARCH 2.500 2.585 2.618 2.591 2.551APARCH 2.485 2.598 2.739 2.831 2.873


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What Model?

QLike Loss – Squared ReturnsFull Sample

horizon 1 d 1 w 2 w 3 w 1 mGARCH 0.364 0.371 0.372 0.365 0.357TGARCH 0.407 0.409 0.404 0.400 0.389EGARCH 0.390 0.389 0.380 0.359 0.337APARCH 0.405 0.405 0.403 0.390 0.377

Sept – Dec ’08horizon 1 d 1 w 2 w 3 w 1 mGARCH 5.678 5.987 6.247 6.332 6.265TGARCH 5.865 6.18 6.463 6.543 6.506EGARCH 5.487 5.716 5.698 5.449 5.094APARCH 5.747 6.05 6.269 6.302 6.196


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QLike

Jan ’01 – Aug ’08 Sept – Dec ’08

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Predictive Ability Across Horizons: Patterns

Jan ’01 – Aug ’08Forecasting ability hardly deteriorates as the horizonincreases.Dispersion between different losses decreases with thehorizon.

Sept – Dec ’08Deterioration in forecasting ability is pronounced.At a 1 day horizon out–of–sample loss is not far from“normal” times.Dispersion between different losses increases with thehorizon.Even if predictive ability deteriorates, large relative gains canbe obtained by picking up the right forecasting method.

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Empirical Findings What happens to other assets?

Other Asset Classes

SPDR Equity SectorsJan ’01 – Aug ’08 Sept ’08 – Dec ’08

horizon 1d 1w 2w 3w 1m 1d 1w 2w 3w 1mGARCH 4.991 4.541 4.556 4.534 4.517 5.141 5.226 5.5 5.753 5.841TGARCH 4.972 4.526 4.542 4.520 4.503 4.995 5.027 5.272 5.484 5.548EGARCH 5.138 4.692 4.71 4.693 4.681 5.217 5.325 5.721 6.087 6.318APARCH 5.028 4.587 4.608 4.589 4.577 5.028 5.065 5.325 5.548 5.629

XLF, XLE, XLI, XLK, XLV

International Equitieshorizon 1d 1w 2w 3w 1m 1d 1w 2w 3w 1mGARCH 4.642 4.247 4.135 4.094 4.065 4.713 4.617 5.163 5.341 5.445TGARCH 4.623 4.233 4.124 4.085 4.057 4.613 4.504 5.07 5.289 5.401EGARCH 4.626 4.234 4.122 4.086 4.049 4.747 4.724 5.455 5.783 6.057APARCH 4.629 4.238 4.130 4.091 4.062 4.618 4.527 5.117 5.322 5.434

MSCIWRLD, MSCIBRIC, MSCIEM, MSCIDE, MSCIHK

FXhorizon 1d 1w 2w 3w 1m 1d 1w 2w 3w 1mGARCH 4.567 4.347 4.297 4.297 4.308 4.839 4.657 4.86 4.72 4.307TGARCH 4.566 4.346 4.296 4.294 4.305 4.827 4.649 4.857 4.719 4.31EGARCH 4.579 4.359 4.31 4.31 4.323 4.954 4.776 5.008 4.936 4.592APARCH 4.581 4.361 4.312 4.31 4.32 4.823 4.643 4.849 4.711 4.301

USD2GBP, USD2YEN, USD2EUR, USD2SFR, USD2SID

Smaller is Better

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Empirical Findings What happens to other assets?

Did we predict this?

Consider the in sample “Forward” QLike loss

QLike(σ2t+τ , ht+τ |t)

on the S&P 500 between 1927 to 2008(TGARCH / Student Innovations)

What are the means of the QLike losses accros horizons?horizon 1d 1w 2w 3w 1m1926-01 – 2008-12 2.5 2.6 2.7 2.7 2.82003-01 – 2008-08 2.4 2.5 2.5 2.5 2.52008-09 – 2008-12 2.4 2.6 3.1 3.8 5.1

How frequent are the Fall ’08 losses?horizon 1d 1w 2w 3w 1mHistorical 54.5 38.2 12.3 3.6 1.3Simulated 53.8 35.4 11.4 3.9 2.0

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Empirical Findings Conclusions

Conclusions

We’ve engaged a forecasting exercises aiming at findingsuccessful ingredients for volatility forecasting at differenthorizons with a special focus on the recent period financialdistress.

Results show thatBest forecasting recipe persists across forecasting horizons.

Recent period of financial distress has deteriorated volatilityprediction at long horizons but most ARCH specification didnot performed badly at short horizons.

BEK (2009) 18 / 18

a practical guide to volatility forecasting in a crisispages.stern.nyu.edu/~bkelly/volfor.pdfa...

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