Identifying and predicting jumps in �nancial timeseries

Petros Dellaportas, UCL

with Alexopoulos (AUEB, Athens) and Papaspiliopoulos (UPF, Barcelona)

25/11/2016, Vienna

Jump identi�cation from daily close prices for the 600 stocks of theEuroStoxx 600 index (January 8 of 2007 - November 5 of 2014).

2008 2010 2012 2014

stocks

1100

200300

400500

571

Dots denote estimated probability of a jump > 0.5; Bottom graphdepicts the total number of estimated jumps

Observe only returns yt , everything else is unobserved:

yt = exp(ht/2)εt +Nt∑j=1

ξj , εt ∼ N(0, 1)

ht = µ+ φ(ht−1 − µ) + ηt , ηt ∼ N(0, σ2η)

I Nt ∼ Poisson(Λt∆t), ∆t is the distance between twosuccessive observations

I Λt ∼ gamma(1, 50) (informative) so that the probability of nojump at time t is .98

I {ξj}Ntj=1

iid∼ N(µξ, σ2ξ ), µξ ∼ N(0, 5R2), σ2ξ ∼ IG (3,R2/18)

where R is the range of the data.I h0 ∼ N(µ, σ2η/(1− φ2))I µ ∼ N(0, 10)I (φ+ 1)/2 ∼ Beta(20, 1.5) (informative)I σ2η ∼ χ21, does not bound ση away from zero a priori, see

Kastner and Frühwirth-Schnatter (2014) andFrühwirth-Schnatter and Wagner (2010).

Disentangling Volatility from Jumps

Bayesian inference with MCMC: how we sample the full conditionaldensities

I Sample simultaneously the vector of the log-volatility process

I Sample the number of jumps with rejection sampling

I Sample the parameters using interweaving; see Yu and Meng(2011) and Kastner and Frühwirth-Schnatter (2014).

I Contribution: we separate the volatility from the jump processwithout using any approximation of the model as it wasproposed by Chib et al. (2002).

Sampling the volatility process

I Denote Y = (y1, y2, . . . , yT ), θ = (µ, φ, σ2η, µξ, σ2ξ ),

H = (h1, . . . , hT ), N = (N1, . . . ,NT ) and Ξ = (ξ1, . . . , ξT ).

I Conditioning on number of jumps N and integrating out jumpsizes Ξ, sample from

p(H|θ,Y ) ∝ p(Y |H, θ)p(H|θ)

I Prior p(H) = N (H|M,Q−1)

I There are many ways to do this -here we use an idea by Titsiasthat was �rst appeared in the discussion of the RSSBdiscussion paper by Girolami and Calderhead (2011).

I The advantage is that we sample the whole vector H as ablock with one Metropolis move.

Sampling p(H |θ,Y ) ∝ p(Y |H , θ)p(H |θ)

I Current state of H is Hn. Say we wish to use slice Gibbs:

I Introduce auxiliary variables U that live in the same space asH: p(U|Hn) = N (U|Hn + δ

2∇ log p(Y |Hn), δ2 I )

I U injects Gaussian noise into Un and shifts it by(δ/2)∇ log p(Y |Hn)

I We cannot sample from p(H|U) so we use a Metropolis step:Propose H∗ from proposal q:

q(H∗|U) =1

Z(U)N (H∗|U, δ

2I )p(H∗)

= N (H∗|(I +δ

2Q)−1(U +

δ

2QM),

δ

2(I +

δ

2Q)−1).

where Z(U) =∫N (H∗|U, δ2 I )p(H∗)dH∗.

I Accept H∗ with Metropolis-Hastings probability min(1, r):

r =p(Y |H∗)p(U|H∗)p(H∗)p(Y |Hn)p(U|Hn)p(Hn)

q(Hn|U)

q(H∗|U)

=p(Y |H∗)p(U|H∗)p(H∗)p(Y |Hn)p(U|Hn)p(Hn)

1

Z(U)N (Hn|U, δ

2I )p(Hn)

1

Z(U)N (H∗|U, δ

2I )p(H∗)

=p(Y |H∗)N (U|H∗ + δ

2Gy ,

δ2I )

p(Y |Hn)N (U|Hn + δ2Gt ,

δ2I )

N (Hn|U, δ2I )

N (H∗|U, δ2I )

=p(Y |H∗)p(Y |Hn)

exp

{−(U − Hn)

TGt + (U − H∗)TGy −δ

4(||Gy ||2 − ||Gt ||2)

}where Gt = ∇ log p(Y |Hn), Gy = ∇ log p(Y |H∗) and ||Z || denotes theEuclidean norm of a vector Z .

I The Gaussian prior terms p(Hn) and p(H∗) have been cancelled out from theacceptance probability, so their evaluation is not required: the resulting q(H∗|U)is invariant under the Gaussian prior.

I Tune δ to achieve an acceptance rate of around 50− 60%.

Rejection sampling for the number of jumpsI Target: the discrete log-concave distribution with density

p(Nt |ht ,Λt , yt , θ).I Proposal: Choose as m any point after the mode, use as a

proposal the red-dotted discrete density: it is a geometric afterm.

n

log(p(n))

mode m

Jump identi�cation from daily close prices for the 600 stocks of theEuroStoxx 600 index (January 8 of 2007 - November 5 of 2014).

2008 2010 2012 2014

stocks

1100

200300

400500

571

Dots denote estimated probability of a jump > 0.5; Bottom graphdepicts the total number of estimated jumps

Jumps prediction

I Key idea: Use a Bayesian hierarchical model for alli = 1, , . . . , p = 600 stocks to borrow strength

Nit ∼ Poisson(Λit∆it)

Λit = λ (1 + exp(−bi −WiFt))−1

λ = 0.15 is the maximum intensity of each stock, W is a p × Kmatrix of factor loadings with rows Wi and each Ft areK -dimensional time-varying independent factors

Ft = AFt−1 + et , t = 2, . . . ,T ,

et ∼ NK (0, IK ), A = diag(α1, . . . , αK ).

The full model

yit = exp(hit/2)εit +

Nit∑j=1

ξij , εit ∼ N(0, 1)

hit = µi + φi (hi ,t−1 − µi ) + ηit , ηit ∼ N(0, σ2iη)

{ξij}Nitj=1

iid∼ N(µiξ, σ2iξ), µiξ ∼ N(0, 5R2

i ), σ2iξ ∼ IG (3,R2i /18)

Nit ∼ Poisson(Λit∆it)

Λit = λ (1 + exp(−bi −WiFt))−1

Ft = AFt−1 + et

MCMC

I We integrate out the jump sizes Ξ and we target the posteriordistribution

p(θ, h,N,F |Y )

where the parameters and the data correspond to all stocks.

I p(hi |N,F , θ): Metropolis as in the 1-dim

I p(Ni |H,F , θ): rejection sampling as in the 1-dim

I p(F |H,N, θ): Label and sign switching are not taken care ofduring MCMC, see also Aÿmann et al. (2016); we chooseinformative prior distributions for the parameters of the factorprocess such that 2 jumps are expected on average every 100days. We sample all factors simultaneously based on theauxiliary Metropolis algorithm described for the 1-d case.

I p(θ|H,N,F ): we use interweaving

We choose prior distributions for the parameters b, the factorloadings in the matrix W and for the persistent parameters of thematrix A such that the induced prior (histogram) for the intensityΛit of the ith stock at time t is comparable with the Gamma(1, 50)prior (red line) used in the univariate model.

Λit

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

05

1015

2025

3035

Separating volatility and jumps: simulation resultsSimulated log returns (top) and their volatility path (bottom -blue)for 3 of the p = 300 time series of length T = 1500 with K = 2factors. Bottom, Red: posterior mean of the volatility path. Blackcircles: simulated jumps. Red and green crosses: estimatedprobability of jump greater than 50% and 70%.

Time

y 1t

0 500 1000 1500

−50

5

Time

h 1t

0 500 1000 1500

−2−1

01

Separating volatility and jumpsSimulated log returns (top) and their volatility path (bottom -blue)for 3 of the p = 300 time series of length T = 1500 with K = 2factors. Bottom, Red: posterior mean of the volatility path. Blackcircles: simulated jumps. Red and green crosses: estimatedprobability of jump greater than 50% and 70%.

Time

y 2t

0 500 1000 1500

−50

5

Time

h 2t

0 500 1000 1500

−2−1

0

Separating volatility and jumpsSimulated log returns (top) and their volatility path (bottom -blue)for 3 of the p = 300 time series of length T = 1500 with K = 2factors. Bottom, Red: posterior mean of the volatility path. Blackcircles: simulated jumps. Red and green crosses: estimatedprobability of jump greater than 50% and 70%.

Time

y 3t

0 500 1000 1500

−50

510

Time

h 3t

0 500 1000 1500

−2.5

−1.5

−0.5

0.5

Jump identi�cationBlack circles: simulated jumps for 50 of the p = 300 simulatedtimes series. Red crosses: estimated probability of at least onejump greater than 50%.

Time

stocks

125

50

1 500 1000 1500

Posterior distributions IPosterior (black lines) and prior (orange lines) distributions of thepersistent parameters of the simulated latent factors. The verticallines represent the simulated values.

0.75 0.80 0.85

05

1015

20

α1

Densi

ty

0.20 0.25 0.30 0.35 0.40 0.45

02

46

8

α2

Densi

ty

Posterior distributions for factor loadings (W )

95% credible intervals; red crosses indicate the simulated values.

−3 −2 −1 0 1 2

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Posterior distributions for factor loadings (W ) after signswitching

95% credible intervals; red crosses indicate the simulated values.

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Centered, Non Centered and Interweaving

I Simulation of p = 150 series and K = 2 factors (λ = 0.15,T = 1500). The table belows ESS per unit of time for α1,α2based on M = 3000 iterations (thinned by 10).

I ESS = M/IF where IF = γ0/s2.

I γ0 estimated spectral density of the Markov chain at zero.I s2 is the sample variance of the MCMC draws.

scheme α1 = 0.8 α2 = 0.8

A : centered 50 70B : Non centered 20 50C : Interw. (Cent.-Non Cent.) 400 200D : Interweaving (Non Cent.- Cent.) 150 60

Results on real data

I Our dataset contains daily close prices for the 600 stocks ofthe EuroStoxx 600 index from the January 8 of 2007 until theNovember 5 of 2014.

I After removing stocks with less than 1500 prices and stockswith more than 10 consecutive zero returns we applied ourMCMC algorithm on the log-returns of each one of the rest571 stocks.

I We only present here results based on subset of this dataset-running in progress! We have used K = 2 factors in theintensity process.

Identi�ed jumps

2008 2010 2012 2014

stocks

1100

200300

400500

571

Posterior (black) and prior (orange) distributions for the persistentparameters of the K = 2 factors and posterior mean of the path ofthe two latent factors (right).

0.72 0.74 0.76 0.78 0.80 0.82

05

1015

2025

α1

Density

2008 2010 2012 2014

−4−2

02

46

8

f 1

−0.60 −0.55 −0.50 −0.45

05

1015

α2

Density

2008 2010 2012 2014

−4−2

02

4

f 2

Factor 1: 95% credible intervals for each stock loading

−0.5 0.0 0.5 1.0 1.5 2.0

Factor 2: 95% credible intervals for each stock loading

−2.5 −2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0

Posterior correlationsPosterior mean correlations of jump intensities: 175 stocks of theLondon stock exchange.

0.00

0.25

0.50

0.75

1.00cor

Network based on jump intensitiesNodes represent stocks, edges are present when the posterior meancorrelation of jump intensities is > 0.9.

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southIMF

central

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central

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North

Model comparison

For two competing models M1 and M2 and out of sampleobservations yT+1, . . . , yT+n we compute the sequence of Bayesfactors

p(yT+1|y1:T ,M1)

p(yT+1|y1:T ,M2), . . . ,

p(yT+1, yT+2, . . . , yT+n|y1:T ,M1)

p(yT+1, yT+2, . . . , yT+n|y1:T ,M2),

where for every model M and j = 1, . . . , n

p(yT+1, yT+2, . . . , yT+j |y1:T ,M) =

T+j∏t=T+1

p(yt |y1:t−1,M).

Problem

Assume θ is known and equal with the posterior mean. For eacht = T + 1, . . . ,T + n, we need to compute the likelihoodincrements

p(yt |y1:t−1,M) =∫p(h0:t−1,F1:t−1|y1:t−1)p(yt |Ft , ht)p(ht |ht−1)p(Ft |Ft−1)dh0:tdF1:t

Typically the above integral is computed using SMC methods butsince in our case the likelihood function is the product of the termsp(yit |Ft , hit) it contains a lot of information for the latent state Ftresulting in poor MC estimation of the integral (Beskos et al.,2014).

Solution

By observing that for each t = T + 1, . . . ,T + n, the marginallikelihood increment p(yit |y1:t−1) of the ith stock is given by∫

p(hi(0:t−1),F1:t−1|y1:t−1)p(yit |Ft , hit)p(hit |hi(t−1))p(Ft |Ft−1)dhi(0:t)dF1:t

we use annealed importance sampling (Neal, 2001) to obtainestimates of the marginal likelihoods

p(yi(T+1)|y1:T ,M), . . . , p(yi(T+1), yi(T+2), . . . , yi(T+n)|y1:T ,M)

and we compute the predictive Bayes factors

p(yi(T+1)|y1:T ,M1)

p(yi(T+1)|y1:T ,M2), . . . ,

p(yi(T+1), yi(T+2), . . . , yi(T+n)|y1:T ,M1)

p(yi(T+1), yi(T+2), . . . , yi(T+n)|y1:T ,M2),

to compare the competing models M1 and M2.

Log-Bayes factor of the univariate SV model with jumps against aunivariate SV model without jumps for 16 stocks of London. HereT = 1967 and out of sample size is n = 50. For each stock wereport estimated in-sample and out-of-sample (red circles) jumps.

8 , 0

Time

log. of

Bayes

factor

0 10 20 30 40 50

−0.5

0.00.5

1.0

5 , 1

Time

log. of

Bayes

factor

0 10 20 30 40 50

01

23

4

23 , 1

Time

log. of

Bayes

factor

0 10 20 30 40 50

−20

24

6

9 , 1

Time

log. of

Bayes

factor

0 10 20 30 40 50

050

100150

21 , 0

Time

log. of

Bayes

factor

0 10 20 30 40 50

0.51.0

1.5

9 , 0

Time

log. of

Bayes

factor

0 10 20 30 40 50

−0.6

−0.2

0.00.2

15 , 0

Time

log. of

Bayes

factor

0 10 20 30 40 50

02

46

8

4 , 0

Time

log. of

Bayes

factor

0 10 20 30 40 50

−0.1

0.00.1

0.20.3

6 , 2

Time

log. of

Bayes

factor

0 10 20 30 40 50

0.00.5

1.01.5

14 , 0

Time

log. of

Bayes

factor

0 10 20 30 40 50

−2−1

01

10 , 2

Time

log. of

Bayes

factor

0 10 20 30 40 50

01

23

45

6

11 , 0

Time

log. of

Bayes

factor

0 10 20 30 40 50

−2.5

−1.5

−0.5

13 , 0

Time

log. of

Bayes

factor

0 10 20 30 40 50

−0.6

−0.2

0.20.6

16 , 1

Time

log. of

Bayes

factor

0 10 20 30 40 50

−10

12

34

17 , 0

Time

log. of

Bayes

factor

0 10 20 30 40 50

−1.0

−0.5

0.00.5

9 , 0

Time

log. of

Bayes

factor

0 10 20 30 40 50

−2.5

−1.5

−0.5

0.5

Log-Bayes factor of the factor SV model with jumps against a SVmodel with jumps for 16 stocks of London. Here T = 1967 and outof sample size is n = 50. For each stock we report estimatedin-sample and out-of-sample jumps.

8 , 0

Time

log. of

Bayes

factor

0 10 20 30 40 50

−0.4

0.00.2

0.40.6

5 , 1

Time

log. of

Bayes

factor

0 10 20 30 40 50

0.00.2

0.40.6

0.81.0

23 , 1

Time

log. of

Bayes

factor

0 10 20 30 40 50

−0.4

0.00.2

0.4

9 , 1

Time

log. of

Bayes

factor

0 10 20 30 40 50

01

23

45

21 , 0

Time

log. of

Bayes

factor

0 10 20 30 40 50

0.00.4

0.81.2

9 , 0

Time

log. of

Bayes

factor

0 10 20 30 40 50

−0.05

0.05

0.15

0.25

15 , 0

Time

log. of

Bayes

factor

0 10 20 30 40 50

−0.2

0.00.1

0.2

4 , 0

Time

log. of

Bayes

factor

0 10 20 30 40 50

−0.35

−0.25

−0.15

−0.05

6 , 2

Time

log. of

Bayes

factor

0 10 20 30 40 50

−0.6

−0.4

−0.2

0.0

14 , 0

Time

log. of

Bayes

factor

0 10 20 30 40 50

0.00.5

1.01.5

2.0

10 , 2

Time

log. of

Bayes

factor

0 10 20 30 40 50

0.00.4

0.81.2

11 , 0

Time

log. of

Bayes

factor

0 10 20 30 40 50

0.01.0

2.03.0

13 , 0

Time

log. of

Bayes

factor

0 10 20 30 40 50

−0.2

0.00.1

0.20.3

16 , 1

Time

log. of

Bayes

factor

0 10 20 30 40 50

−0.4

−0.2

0.00.2

17 , 0

Time

log. of

Bayes

factor

0 10 20 30 40 50

0.00.5

1.01.5

2.0

9 , 0

Time

log. of

Bayes

factor

0 10 20 30 40 50

−0.2

0.00.2

0.40.6

Discussion

we have presented:

I A new algorithm for univariate SV with jumps

I A way to forecasting of jump intensities through jointmodelling of many stocks

I Evidence for better predictive performance