modelling volatility in financial time series: daily …modelling volatility in financial time...

Modelling Volatility in Financial Time Series:Daily and Intra-daily Data

Siem Jan Koopman

[email protected]

Vrije Universiteit Amsterdam

Tinbergen Institute

Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 1

Data

Daily return Rn

Rn = 100(lnPn − lnPn−1), n = 1, . . . , N,

where Pn is the closing asset price at trading day n.

Intraday return (5-minute) is taken between successive log prices,

Rn,d = 100(ln Pn,d − lnPn,d−1), n = 1, . . . , N, d = 1, . . . , D

where Pn,d is the asset price at trading day n and 5-minute period d.

Overnight return

Rn,o = 100(lnPn,o − lnPn−1,D).


Data

Realised volatility is computed as

σ̃2n = R2

n,0 +D∑

d=1

R2n,d, n = 1, . . . , N,

but overnight return is special so it is better to take account of this:

σ̃2n =

σ̂2oc + σ̂2

co

σ̂2oc

D∑

d=1

R2n,d,

whereσ̂2

oc = 10,000N

∑N

n=1(log Pn,D − log Pn,0)2,

σ̂2co = 10,000

N

∑N

n=1(log Pn,0 − log Pn−1,D)2.


Data

Implied volatility s2n is obtained from Chicago Board Options

Exchange Market Volatility Index (VIX), a highly liquid options market.

The VIX index is calculated from midpoint bid-ask option prices using abinomial method that takes into account the level and timing ofdividend payments.

Black-Scholes model assumption of constant volatility introduces biasinto the implied volatility measure but magnitude of the bias is small fornear-the-money and close-to-maturity options.


S&P 100 Volatility

Data is based on S&P 100 stock index for the period between 6January 1997 and 15 November 2003 (1725 observations)

Summary Statistics of return and volatility time series

daily return realised vol. implied vol.Rn R2

n σ̃2n log σ̃2

n s2n log s2

n

Mean 0.020 1.889 0.920 −0.612 26.46 3.253

Stand.Dev. 1.374 4.058 1.359 0.981 5.998 0.208

Skewness −0.122 7.918 5.109 0.245 1.266 0.744

Exc.Kurt. 5.621 110.8 39.80 0.524 1.482 0.135

Minimum −8.994 0 0.004 −5.484 16.84 2.834

Maximum 5.702 80.89 15.38 2.733 50.48 3.922


S&P 100 Volatility

Rn, R2n, σ̃2

n, log σ̃2n, s2

n, log s2n (row-wise)

1997 1999 2001 2003

0

10

−10 −5 0 5

0.2

0.4

0 20 40

0

1

1997 1999 2001 2003

50

100

0 25 50 75

0.25

0.50

0 20 40

0

1

1997 1999 2001 2003

10

20

0 5 10 150

1

0 20 40

0.5

1.0

1997 1999 2001 2003

−5

0

5

−5 0

0.25

0.50

0 20 40

0.5

1.0

1997 1999 2001 2003

20

40

60

20 40

0.05

0.10

0 20 40

0.5

1.0

1997 1999 2001 2003

20

40

60

20 40

0.05

0.10

0 20 40

0.5

1.0


Stochastic Volatility in Continuous Time

Consider spot price P (t) with return defined as

R(t) = log P (t) − log P (0), t > 0.

which follows the continuous time process

dR(t) = µ(t)dt + σ(t)dW (t), t > 0,

where µ(t) is drift process, σ(t) is spot volatility and W (t) is standardBrownian motion.Mean and variance of spot volatility are given by

E(

σ2(t))

= ξ, var(

σ2(t))

= ω2.

The actual volatility for the n-th day interval of length h is then definedas

σ2n = σ∗(hn) − σ∗ ((n − 1)h) , where σ∗(t) =

∫ t

0

σ2(s)ds.


OU Type Models for SV

It is established that rv σ̃2n is accurate estimator of av σ2

n.Barndorff-Nielsen and Shephard (2002) have studied the statisticalproperties of this estimator and its error σ2

n − σ̃2n. Also they conclude

that a model for spot volatility σ2(t) can significantly improve estimationof actual volatility.

A candidate model for σ2(t) is based on the superposition of OUprocesses τ j(t), that is

σ2(t) =

J∑

j=1

τ j(t), dτ j(t) = −λjτj(t)dt + dzj(λjt),

where zj(t) is independent Lévy process (with non-negativeincrements, known as a subordinator) and λj is unknown.



Bandorff-Nielsen and Shephard (2001, 2002):The SDE defining τ j(t) implies its acf to be

corr(

τ j(t), τ j(t + s))

= e−λj |s|.

Assume E(τ j(t)) = wjξ and var(τ j(t)) = wjω2, acf for σ2(t) is

corr(

σ2(t), σ2(t + s))

=J∑

j=1

wje−λj |s|.

It follows that acf of j-th component of av, τ jn ≡

∫ nh

(n−1)hτ j(t)dt, is

corr(τ jn, τ j

n+m) =(1 − e−λjh)2

2(e−λjh − 1 + λjh)e−λjh(m−1), m = 1, 2 . . . ,

where h is the length of the day interval.



These convenient “BNS” results imply that τ jn have ARMA(1,1)

representations:

τ jn+1 = wjξ + φj(τ

jn − wjξ) + θjη

jn−1 + ηj

n, ηjn ∼ WN(0, σ2

ηj ),

where WN(0, σ2) refers to a white noise process with zero mean andvariance σ2. It follows that the autoregressive parameter φj equalse−λjh while Barndorff-Nielsen and Shephard (2003) show that

θj =1 −

√

1 − 4ϑ2j

2ϑj

, with ϑj =corr(τ j

n, τ jn+1) − φj

(1 + φ2j) − 2φjcorr(τ j

n, τ jn+1)

.

Finally, the key to modelling realised volatility in this way is set ofresults in Barndorff-Nielsen and Shephard (2001), see next slide.


SV Models for Daily Returns

The discrete time SV model is based on the continuous process forreturns. By discretisation the return process at daily intervals and byassuming an AR for log-volatility, we obtain

Rn = µ + σnεn, εn ∼ NID(0, 1),

σ2n = σ∗2 exp(hn),

hn+1 = φhn + σηηn, ηn ∼ NID(0, 1),

h1 ∼ NID(0, σ2η/{1 − φ2}),

for n = 1, . . . , N and where µ is taken to be fixed and zero.

Note that this is a non-linear state space model. Taking log R2n as the

dependent variable, the model becomes a linear non-Gaussian statespace model.Efficient estimates can be obtain by using Importance Sampling, seeSandmann and Koopman (1998).


Measuring Realised Volatility

High-frequency price data (tick by tick) is subject to irregularities inrecording and market micro-structure.

Current practice of computing realised volatility is to construct fiveminute returns and compute rv from these. However, such data can bemessy and a regular series of daily series of 5 minute returns is notalways available. Also bid-ask spreads in data can be huge (ways tocapture these require a lot of extra data and modelling).

Some approaches of obtaining a regular set of 5-minute return tolinearly interpolate between ask-bid bounces as in Andersen,Bollerslev, Diebold and Ebens (2001). More flexible splineinterpolations are used by Hansen and Lunde (2003) and Fouriermethods are used by Malliavin and Mancino (2002) and Barucci andReno (2002).

First we adopt a model-based version of these interpolations.


Spline Model in State Space Representation

Consider the smoothing problem where the log price p(t) is acontinuous function of t > 0. To smooth p(t) by function µ(t), weobserve tick prices (bid and asks) p(ti) for i = 1, . . . , n where0 < t1 < . . . < tn < T (ti is a tick). We can choose µ(t) to be atwice-differentiable function on (0, T ) which minimises

n∑

i=1

[p(ti) − µ(ti)]2 + λ

∫ T

0

[

∂2µ(t)

∂t2

]2

dt,

This problem can be represented as a state space model

p(t) = µ(t) + ε(t), t = t1 . . . , tn, ε(ti) ∼ N[0, σ2(ti)],

with state equation

d

[

µ(t)

ν(t)

]

=

[

0 1

0 0

][

µ(t)

ν(t)

]

dt + σζ

[

0

dW (t)

]

.


Spline Model in State Space Representation

In discrete time, we obtain the model pi = µi + εi with

(

µi+1

νi+1

)

=

[

1 δi

0 1

](

µi

νi

)

+

(

ξi

ζi

)

, i = 1, . . . , n,

where the disturbances are Gaussian and correlated with each other.The distance δi is for the distance in seconds between tick prices (canbe zero !). The variance of ζi, as a ratio of the variance of εi, equalsq = 1/λ.

This model can be used to smooth out the micro-structure in tick pricesand to obtain a regular set of 5 minutes quotes from which rv can becomputed, for example.

In standard smoothing q (or λ) is fixed. Here, we estimate q for eachday by standard maximum likelihood methods using the Kalman filter(see www.ssfpack.com). It turns out that the q estimates are veryclose to rv, up to a constant.


Realised Volatility log σ̃2n and estimated q’s (logged)

0 450 900 1350

−4

−2

0

2

−5.0 −2.5 0.0 2.5

0.1

0.2

0.3

0.4

0 20 40

0.25

0.50

0.75

1.00

0 450 900 1350

−10.0

−7.5

−5.0

−2.5

0.0

−10 −5 0

0.1

0.2

0 20 40

0.25

0.50

0.75

1.00


Realised volatility: log σ̃2n versus q’s (logged)

−4.0 −3.5 −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 2.5

−10

−8

−6

−4

−2

realised volatility (in logs)

estim

ated

sm

ooth

ing

par q (in

logs

)


Such results are encouraging.

As we take a closer look at some daily data patterns, it becomes clearthat tick prices at the opening and closure of the trading day havehigher variation than during the main trading hours.

Therefore we extend the spline model with different q’s within the day:we allow smoothing parameter to be a spline itself !

Let’s look at an example of one day of tick prices.


Tick prices in one day with smoothed spline and errors

.

0 250 500 750 1000 1250

−1.5

−1.0

−0.5

0.0

tick price price process (spline)

0 250 500 750 1000 1250

−0.005

0.000

0.005

0.010 Measurement error

0 250 500 750 1000 1250

0.00025

0.00050

0.00075

0.00100 smoothing parameter spline

0 250 500 750 1000 1250

−0.10

−0.05

0.00

0.05Price innovation


The spline model may not be satisfactory since theoretical modelwould have

p(t) = µ(t) + ε(t), t = t1 . . . , tn, ε(ti) ∼ N [0, σ2(ti)],

with state equationdµ(t) = σ(t)dW (t).

In discrete time, we obtain the model

pi = µi + εi, µi+1 = µi + qiσηηi,

where ηi is WN with var(ηi) = δiσ2η.

At the opening and closure of the trading day, qi is higher: volatilityseasonality.


Tick prices in one day with theoretical price and errors

.

0 250 500 750 1000 1250

−1.5

−1.0

−0.5

0.0tick price price process (spline)

0 250 500 750 1000 1250

−2e−11

−1e−11

0

1e−11

2e−11Measurement error

0 250 500 750 1000 1250

0.0005

0.0010

0.0015

smoothing parameter spline

0 250 500 750 1000 1250

−0.10

−0.05

0.00

0.05

Price innovation


This model produces small measurement noise but it is likeliy thatthere is serial correlation in the error due to market micro-structure.This can be captured by including an AR(1) component in the priceequation.

By the standardisation of the one-step ahead prediction errors of thedecomposition model, we effectively deseasonalise the intra-dayreturns.


Tick prices with predicted spline and returns

.

0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300

−1

0tick price predicted price

0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300

0

5 Standardised prediction errors (RETURNS)

0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300

0.0002

0.0004

0.0006tv q


We can now concentrate on the stochastic volatility indµ(t)∗ = σ(t)dW (t). where µ(t)∗ refers to the process of µ(t) correctedfor seasonal heteroskedasticity. Remaining dynamic volatility can becaptured by the stochastic volatility model.

We can model the constructed intraday returns by the discretised SVmodel,

Ri = µ + σiεi, εi ∼ NID(0, 1),

σ2i = σ∗2 exp(hi),

hi+1 = φhi + σηηi, ηn ∼ NID(0, 1),

h1 ∼ NID(0, σ2η/{1 − φ2}),

However, we deal with tick returns rather than day returns.The SV model can be generalised by formulating hi as a sum ofstationary ARMA components.


Absolute returns and estimated actual volatility

.

0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300

2

4

absolute tick returns and estimated actual volatility

0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300

−2

0

2log volatility


The SV model

Ri = µ + σiεi, εi ∼ NID(0, 1),

σ2i = σ∗2 exp(hi),

hi+1 = φhi + σηηi, ηn ∼ NID(0, 1),

h1 ∼ NID(0, σ2η/{1 − φ2}),

The estimates were given by

φ̂ = 0.84, σ̂2η = 0.141, σ̂∗2 = 0.881.

Computations done in www.ssfpack.com.


One more example: tick prices and returns

0 150 300 450 600 750 900 1050 1200 1350

−0.50

−0.25

0.00

0.25tick price predicted price

0 150 300 450 600 750 900 1050 1200 1350

−2.5

0.0

2.5Standardised prediction errors (RETURNS)

0 150 300 450 600 750 900 1050 1200 1350

0.0005

0.0010 tv q


SV estimates

φ̂ = 0.90, σ̂2η = 0.062, σ̂∗2 = 0.969

0 150 300 450 600 750 900 1050 1200 1350

1

2

3

0 150 300 450 600 750 900 1050 1200 1350

−2

−1

0

1


These empirical results have motivated us to formalise this by settingup a model for tick prices that requires no pre-filtering of the tick data.In discrete form, the model will possibly be

pi = µi + σiεi,

µi+1 = µi + σiqiζi,

log σi = hi = sum of ARMA components,

Note that σi is common to both εi and ζi which effectively means thatσi is standard deviation of innovations (is returns).


Model:pi = µi + σiεi,

µi+1 = µi + σiqiζi,

log σi = hi = sum of ARMA components,

• σi is the (deseasonalised) standard deviation of returns as shownin previous examples;

• qi is the seasonal volatility within the day, can be restricted to bethe same for all days;

• µi follows here a random walk, if this does not produce sufficientsmoothness to eliminate microstructure, we turn to higher-ordersmoothness models;

• the daily actual volatility is∑

i σ2i q2

i .


modelling volatility in financial time series: daily …modelling volatility in financial time...

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