Modelling Volatility in Financial Time Series: Daily and Intra-daily - - PowerPoint PPT Presentation

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

Siem Jan Koopman

s.j.koopman@feweb.vu.nl

Vrije Universiteit Amsterdam Tinbergen Institute

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

Data

Daily return Rn Rn = 100(ln Pn − ln Pn−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 − ln Pn,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(ln Pn,o − ln Pn−1,D).

Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 2

Data

Realised volatility is computed as ˜ σ2

n = R2 n,0 + D

• d=1

R2

n,d,

n = 1, . . . , N, but overnight return is special so it is better to take account of this: ˜ σ2

n = ˆ

σ2

• c + ˆ

σ2

co

ˆ σ2

• c

D

• d=1

R2

n,d,

where ˆ σ2

• c

=

10,000 N

N

n=1(log Pn,D − log Pn,0)2,

ˆ σ2

co

=

10,000 N

N

n=1(log Pn,0 − log Pn−1,D)2.

Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 3

Data

Implied volatility s2

n 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 a binomial method that takes into account the level and timing of dividend payments. Black-Scholes model assumption of constant volatility introduces bias into the implied volatility measure but magnitude of the bias is small for near-the-money and close-to-maturity options.

Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 4

S&P 100 Volatility

Data is based on S&P 100 stock index for the period between 6 January 1997 and 15 November 2003 (1725 observations) Summary Statistics of return and volatility time series daily return realised vol. implied vol. Rn R2

n

˜ σ2

n

log ˜ σ2

n

s2

n

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 0.004 −5.484 16.84 2.834 Maximum 5.702 80.89 15.38 2.733 50.48 3.922

Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 5

S&P 100 Volatility

Rn, R2

n, ˜

σ2

n, log ˜

σ2

n, s2 n, log s2 n (row-wise)

1997 1999 2001 2003 10 −10 −5 5 0.2 0.4 20 40 1 1997 1999 2001 2003 50 100 25 50 75 0.25 0.50 20 40 1 1997 1999 2001 2003 10 20 5 10 15 1 20 40 0.5 1.0 1997 1999 2001 2003 −5 5 −5 0.25 0.50 20 40 0.5 1.0 1997 1999 2001 2003 20 40 60 20 40 0.05 0.10 20 40 0.5 1.0 1997 1999 2001 2003 20 40 60 20 40 0.05 0.10 20 40 0.5 1.0

Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 6

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 standard Brownian 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 defined as σ2

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

t σ2(s)ds.

Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 7

OU Type Models for SV

It is established that rv ˜ σ2

n is accurate estimator of av σ2 n.

Barndorff-Nielsen and Shephard (2002) have studied the statistical properties of this estimator and its error σ2

n − ˜

σ2

• n. Also they conclude

that a model for spot volatility σ2(t) can significantly improve estimation

• f actual volatility.

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

J

• j=1

τ j(t), dτ j(t) = −λjτ j(t)dt + dzj(λjt), where zj(t) is independent Lévy process (with non-negative increments, known as a subordinator) and λj is unknown.

Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 8

OU Type Models for SV

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 j

n ≡

nh

(n−1)h τ j(t)dt, is

corr(τ j

n, 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.

Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 9

OU Type Models for SV

These convenient “BNS” results imply that τ j

n have ARMA(1,1)

representations: τ j

n+1 = wjξ + φj(τ j n − wjξ) + θjηj n−1 + ηj n,

ηj

n ∼ WN(0, σ2 ηj),

where WN(0, σ2) refers to a white noise process with zero mean and variance σ2. It follows that the autoregressive parameter φj equals e−λjh while Barndorff-Nielsen and Shephard (2003) show that θj = 1 −

• 1 − 4ϑ2

j

2ϑj , with ϑj = corr(τ j

n, τ j n+1) − φj

(1 + φ2

j) − 2φjcorr(τ j n, τ j n+1)

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

Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 10

SV Models for Daily Returns

The discrete time SV model is based on the continuous process for

• returns. By discretisation the return process at daily intervals and by

assuming an AR for log-volatility, we obtain Rn = µ + σnεn, εn ∼ NID(0, 1), σ2

n

= σ∗2 exp(hn), hn+1 = φhn + σηηn, ηn ∼ NID(0, 1), h1 ∼ NID(0, σ2

η/{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 R2

n as the

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

Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 11

Measuring Realised Volatility

High-frequency price data (tick by tick) is subject to irregularities in recording and market micro-structure. Current practice of computing realised volatility is to construct five minute returns and compute rv from these. However, such data can be messy and a regular series of daily series of 5 minute returns is not always available. Also bid-ask spreads in data can be huge (ways to capture these require a lot of extra data and modelling). Some approaches of obtaining a regular set of 5-minute return to linearly interpolate between ask-bid bounces as in Andersen, Bollerslev, Diebold and Ebens (2001). More flexible spline interpolations are used by Hansen and Lunde (2003) and Fourier methods are used by Malliavin and Mancino (2002) and Barucci and Reno (2002). First we adopt a model-based version of these interpolations.

Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 12

Spline Model in State Space Representation

Consider the smoothing problem where the log price p(t) is a continuous function of t > 0. To smooth p(t) by function µ(t), we

• bserve tick prices (bid and asks) p(ti) for i = 1, . . . , n where

0 < t1 < . . . < tn < T (ti is a tick). We can choose µ(t) to be a twice-differentiable function on (0, T) which minimises

n

• i=1

[p(ti) − µ(ti)]2 + λ T ∂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)

• =
• 1

µ(t) ν(t)

• dt + σζ
• dW(t)
• .

Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 13

Spline Model in State Space Representation

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

• µi+1

νi+1

• =
• 1

δi 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 (can be zero !). The variance of ζi, as a ratio of the variance of εi, equals q = 1/λ. This model can be used to smooth out the micro-structure in tick prices and to obtain a regular set of 5 minutes quotes from which rv can be computed, for example. In standard smoothing q (or λ) is fixed. Here, we estimate q for each day by standard maximum likelihood methods using the Kalman filter (see www.ssfpack.com). It turns out that the q estimates are very close to rv, up to a constant.

Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 14

Realised Volatility log ˜

σ2

n and estimated q’s (logged)

450 900 1350 −4 −2 2 −5.0 −2.5 0.0 2.5 0.1 0.2 0.3 0.4 20 40 0.25 0.50 0.75 1.00 450 900 1350 −10.0 −7.5 −5.0 −2.5 0.0 −10 −5 0.1 0.2 20 40 0.25 0.50 0.75 1.00

Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 15

Realised volatility: log ˜

σ2

n 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) estimated smoothing par q (in logs)

Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 16

Such results are encouraging. As we take a closer look at some daily data patterns, it becomes clear that tick prices at the opening and closure of the trading day have higher 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.

Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 17

Tick prices in one day with smoothed spline and errors

.

250 500 750 1000 1250 −1.5 −1.0 −0.5 0.0

tick price price process (spline)

250 500 750 1000 1250 −0.005 0.000 0.005 0.010

Measurement error

250 500 750 1000 1250 0.00025 0.00050 0.00075 0.00100

smoothing parameter spline

250 500 750 1000 1250 −0.10 −0.05 0.00 0.05

Price innovation

Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 18

The spline model may not be satisfactory since theoretical model would have p(t) = µ(t) + ε(t), t = t1 . . . , tn, ε(ti) ∼ N[0, σ2(ti)], with state equation dµ(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) = δiσ2

η.

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

Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 19

Tick prices in one day with theoretical price and errors

.

250 500 750 1000 1250 −1.5 −1.0 −0.5 0.0

tick price price process (spline)

250 500 750 1000 1250 −2e−11 −1e−11 1e−11 2e−11

Measurement error

250 500 750 1000 1250 0.0005 0.0010 0.0015

smoothing parameter spline

250 500 750 1000 1250 −0.10 −0.05 0.00 0.05

Price innovation

Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 20

This model produces small measurement noise but it is likeliy that there is serial correlation in the error due to market micro-structure. This can be captured by including an AR(1) component in the price equation. By the standardisation of the one-step ahead prediction errors of the decomposition model, we effectively deseasonalise the intra-day returns.

Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 21

Tick prices with predicted spline and returns

.

100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 −1

tick price predicted price

100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 5

Standardised prediction errors (RETURNS)

100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 0.0002 0.0004 0.0006

tv q

Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 22

We can now concentrate on the stochastic volatility in dµ(t)∗ = σ(t)dW(t). where µ(t)∗ refers to the process of µ(t) corrected for seasonal heteroskedasticity. Remaining dynamic volatility can be captured by the stochastic volatility model. We can model the constructed intraday returns by the discretised SV model, Ri = µ + σiεi, εi ∼ NID(0, 1), σ2

i

= σ∗2 exp(hi), hi+1 = φhi + σηηi, ηn ∼ NID(0, 1), h1 ∼ NID(0, σ2

η/{1 − φ2}),

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

Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 23

Absolute returns and estimated actual volatility

.

100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 2 4

absolute tick returns and estimated actual volatility

100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 −2 2

log volatility

Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 24

The SV model

Ri = µ + σiεi, εi ∼ NID(0, 1), σ2

i

= σ∗2 exp(hi), hi+1 = φhi + σηηi, ηn ∼ NID(0, 1), h1 ∼ NID(0, σ2

η/{1 − φ2}),

The estimates were given by ˆ φ = 0.84, ˆ σ2

η = 0.141,

ˆ σ∗2 = 0.881. Computations done in www.ssfpack.com.

Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 25

One more example: tick prices and returns

150 300 450 600 750 900 1050 1200 1350 −0.50 −0.25 0.00 0.25

tick price predicted price

150 300 450 600 750 900 1050 1200 1350 −2.5 0.0 2.5

Standardised prediction errors (RETURNS)

150 300 450 600 750 900 1050 1200 1350 0.0005 0.0010

tv q

Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 26

SV estimates

ˆ φ = 0.90, ˆ σ2

η = 0.062,

ˆ σ∗2 = 0.969

150 300 450 600 750 900 1050 1200 1350 1 2 3 150 300 450 600 750 900 1050 1200 1350 −2 −1 1

Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 27

These empirical results have motivated us to formalise this by setting up 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 that σi is standard deviation of innovations (is returns).

Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 28

Model: pi = µi + σiεi, µi+1 = µi + σiqiζi, log σi = hi = sum of ARMA components,

• σi is the (deseasonalised) standard deviation of returns as shown

in previous examples;

• qi is the seasonal volatility within the day, can be restricted to be

the same for all days;

• µi follows here a random walk, if this does not produce sufficient

smoothness to eliminate microstructure, we turn to higher-order smoothness models;

• the daily actual volatility is

i σ2 i q2 i .

Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 29