1 On the Term Structure of Model-Free Volatilities and Volatility Risk Premium Kian Guan LIM and Christopher TING Singapore Management University.

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On the Term On the Term Structure Structure

of Model-Free of Model-Free VolatilitiesVolatilities

and Volatility Risk and Volatility Risk PremiumPremiumKian Guan LIM and Christopher Kian Guan LIM and Christopher

TINGTING

Singapore Management UniversitySingapore Management University

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Motivations on VolatilityMotivations on Volatility Volatility forecast is critical to stock Volatility forecast is critical to stock

and option pricingand option pricing Historical volatilitiesHistorical volatilities Implied volatilitiesImplied volatilities Model-free volatilityModel-free volatility Term StructuresTerm Structures

Volatility RiskVolatility Risk Is it priced? Term structure?Is it priced? Term structure?

Industry DevelopmentIndustry Development CBOE VIX “Fear” gaugeCBOE VIX “Fear” gauge Variance and volatility swapsVariance and volatility swaps Options and futures on volatility indexesOptions and futures on volatility indexes

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Main ideas of paperMain ideas of paper

Present an enhanced method to Present an enhanced method to compute model-free volatility more compute model-free volatility more accuratelyaccurately

Enable the construction of the term Enable the construction of the term structure of model-free volatilities from structure of model-free volatilities from 30- to 450-day constant maturities30- to 450-day constant maturities

Explore the longer term structure of Explore the longer term structure of model-free volatilitymodel-free volatility

Gain further insight into the volatility Gain further insight into the volatility risk premiumrisk premium

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TheoryTheory

Expected variance from time 0 Expected variance from time 0 (today) to T under risk-neutral (today) to T under risk-neutral measure Qmeasure Q

The volatility The volatility is the forward-looking is the forward-looking volatility forecastvolatility forecast

Proposition 1Proposition 1

Risk-free rate r assumed constant

Put price Call price

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Their difference is second-orderTheir difference is second-order

and are not necessarily constant, but may depend on other adapted state variables

Generalized Diffusion Process

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Integrated VarianceIntegrated Variance Total variance from time 0 to Total variance from time 0 to TT

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DerivationDerivation

Log(1+Log(1+zz) < ) < zz

Necessarily positive

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A Pictorial PresentationA Pictorial Presentation Area under the two curves defined by the intersection Area under the two curves defined by the intersection

pointpoint Implied forward price Implied forward price FF00 determined by put-call parity determined by put-call parity

F0X

Option price

PutCall

D is the sum of dividends

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some technical problemssome technical problems

Discrete price gridDiscrete price grid Cannot be smaller than $0.01Cannot be smaller than $0.01

Discrete strike price intervalDiscrete strike price interval Minimum intervalMinimum interval

Limited range of strike pricesLimited range of strike prices Bid-offer spreadBid-offer spread

Use a correct price to avoid arbitrageUse a correct price to avoid arbitrage Thin volumeThin volume

Far term optionsFar term options

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Discretized ApproximationDiscretized Approximation Average of Riemann upper sum and Average of Riemann upper sum and

lower sumlower sum

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In overcoming approx due In overcoming approx due to discrete strike priceto discrete strike price

Cubic SplinesCubic Splines

No risk-free arbitrage conditions as No risk-free arbitrage conditions as constraintsconstraints

Volume as weightVolume as weight Piece-wise integration - Closed-formPiece-wise integration - Closed-form

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Range of Strike PricesRange of Strike Prices

300 400 500 600 700 800 9000

50

100

150

200

250

300

Strike Price

Opt

ion P

rice

Put Call

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Data from OptionmetricsData from Optionmetrics CBOE S&P 100 index optionCBOE S&P 100 index option

European styleEuropean style Re-introduced on July 23, 2001Re-introduced on July 23, 2001 March quarterly cycleMarch quarterly cycle Minimum strike interval 5 pointsMinimum strike interval 5 points Near term, out-of-the money, 10 or 20 points Near term, out-of-the money, 10 or 20 points

strike intervalsstrike intervals Far term, 20 points strike intervalsFar term, 20 points strike intervals 7 to 8 terms for any given trading day7 to 8 terms for any given trading day Weekly options excludedWeekly options excluded

Sample periodSample period July 23, 2001 through April 30, 2006July 23, 2001 through April 30, 2006

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Descriptive StatisticsDescriptive Statistics

Daily averageDaily average

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Interpolation and Extrapolation are Interpolation and Extrapolation are criticalcritical

The discretized approximation The discretized approximation formula leads to an upward estimate formula leads to an upward estimate of model-free volatility of model-free volatility

The market volatility indexes might The market volatility indexes might have an upward bias, especially for have an upward bias, especially for those that have a larger strike price those that have a larger strike price intervalinterval

Variance swap buyers may be Variance swap buyers may be paying for the biaspaying for the bias

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Exact versus ApproximateExact versus Approximate The difference The difference dd = = a a - - e*e*

Relative size of strike price intervalRelative size of strike price interval Relative size: 5 points/S&P 100 index levelRelative size: 5 points/S&P 100 index level dd1: sub-sample defined by relative size smaller 1: sub-sample defined by relative size smaller

than median valuethan median value dd2: sub-sample defined by relative size larger 2: sub-sample defined by relative size larger

than medianthan median

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Volatility Term StructureVolatility Term Structure Constant maturityConstant maturity Range of values over the sample period is smaller Range of values over the sample period is smaller

the longer is the constant maturitythe longer is the constant maturity

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Term Structures of Volatility Term Structures of Volatility ChangesChanges

Absolute changes of daily volatility less Absolute changes of daily volatility less drastic the longer the constant maturitydrastic the longer the constant maturity

Run Table3.m

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Term Structure of Fear Term Structure of Fear GaugesGauges

Daily change in model-free volatility Daily change in model-free volatility ttii and daily change in S&P and daily change in S&P

100 index level 100 index level LLtt are negatively correlated – volatility rises lead are negatively correlated – volatility rises lead to stock market falls. Second column is autocorrelation in to stock market falls. Second column is autocorrelation in tt

ii

Run Table6.m

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Asymmetric Correlations: Asymmetric Correlations: LLtt = c = c00+c+c11tt++

+c+c22tt--++tt

larger magnitude of equity when short-term volatility increases than when it decreases

larger magnitude of equity when long-term volatility drops than when it increases

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Change of Volatility Change of Volatility Correlations Correlations with Index with Index

ReturnsReturns Shorter constant maturityShorter constant maturity

Higher correlation with a market decline Higher correlation with a market decline Consistent with Whaley (JPM,2000) – short-Consistent with Whaley (JPM,2000) – short-

term volatility spike bad for short-term term volatility spike bad for short-term investorsinvestors

Longer constant maturityLonger constant maturity Higher correlation with a market riseHigher correlation with a market rise Quite surprisingQuite surprising

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Slope EstimateSlope Estimate

0 50 100 150 200 250 300 350 400 450 5000.03

0.031

0.032

0.033

0.034

0.035

0.036

0.037

0.038

0.039

Days to Maturity T

Model-F

ree V

ariance

2

2 = 0.032042+0.000013 x T

An example of a single day regression of (T) on T.

61.6% of all daily regressions produces positive slopes

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Term Structure of Slope Term Structure of Slope EstimatesEstimates

Upward sloping (737 estimates)Upward sloping (737 estimates)

Downward sloping (460 Downward sloping (460 estimates)estimates)

upward sloping volatility term structure corresponds with positive stock returns & lower risk

downward sloping volatility term structure corresponds with negative stock returns & higher risk

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This Figure plots the daily time series of S&P 100 index in Panel A and the slope estimates of the model-free variance term structure in Panel B. The horizon axis shows the dates in yymmdd. The sample period is from July 2001 through April 2006.

Change in slope Change in slope estimates is positively estimates is positively correlated with the correlated with the change in index level change in index level

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variance risk premiumvariance risk premium variance swap (a forward contract) buyer’svariance swap (a forward contract) buyer’s

payoff is (payoff is (RR22 - - 22) x Notional Principal) x Notional Principal

where where RR22 is realized annualized volatility is realized annualized volatility

computed over contract maturity [0,T] from computed over contract maturity [0,T] from daily sample return volatility and model-free daily sample return volatility and model-free 2 2 over [0,T] is the strike under fair valuation over [0,T] is the strike under fair valuation at start of contract t=0.at start of contract t=0.

Mean excess return (Mean excess return (RR22 - - 22)/ )/ 22 may be may be

construed as volatility risk premium that the construed as volatility risk premium that the buyer is compensated for taking the risk.buyer is compensated for taking the risk.

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Variance Swap Payoff and Return Variance Swap Payoff and Return for Buyersfor Buyers

Volatility risk premiumVolatility risk premium

increasing maturity

asymmetric larger gains at right skew and lower loss at left skew up to 180 days

$%

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ConclusionsConclusions An enhanced method to construct term structureAn enhanced method to construct term structure Hopefully a better understanding of the behavior of a long term Hopefully a better understanding of the behavior of a long term

structure of volatilitystructure of volatility Look at a number of asymmetric effectsLook at a number of asymmetric effects

Explore term structure of variance risk premiumExplore term structure of variance risk premium Validation of mean-reverting stochastic volatilityValidation of mean-reverting stochastic volatility

Uncompleted TasksUncompleted Tasks• how to check term structure using other long-life traded how to check term structure using other long-life traded

instrumentsinstruments• how to make sense of the explored term structure effects in how to make sense of the explored term structure effects in

trading and hedging strategiestrading and hedging strategies• more rigorous statistical confirmations of the term structure more rigorous statistical confirmations of the term structure

resultsresults