estimating models based on markov jump processes given ... · estimating models based on markov...

Estimating Models Based on Markov JumpProcesses Given Fragmented Observation Series

Markus Hahn

Johann Radon Institute for Computational and Applied Mathematics (RICAM), LinzAustrian Academy of Sciences

Joint work with S. Fruhwirth-Schnatter (JKU Linz) and J. Sass (TU Kaiserslautern)

Linz, December 2, 2008

Work funded by FWF project P17947

Introduction

Problem

I Estimation from set of observed seriesI Independent series

I Data with breaks

I Each single series is (based) Markov processI Same generator for all series

I How to estimate common generator?

I How to cope with short observation series?

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Introduction

Outline

Introduction

Markov Jump Processes

Merged Markov Jump Processes

Inference for Merged Markov Jump Processes

Generalization: Markov Switching Models

Conclusion

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Finite state Markov jump process (MJP)

I Y = (Yt)t∈[0,T ) is a continuous time Markov process

I Finite state space {1, . . . , d}

I Y is time homogeneous

I Jumps of Y are governed by rate matrix Q ∈ Rd×d

I Exponential rate of leaving state k:

λk = −Qkk =∑l 6=k

Qkl < ∞

i.e. average waiting time for leaving k is 1/λk

I Conditional transition probability:

P(Yt = l |Yt− = k,Yt 6= Yt−) = Qkl/λk

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Inference about rate matrix

I Ok occupation time of state k

I Nkl number of jumps from k to l

I Maximum likelihood estimation:

Qkl = Nkl/Ok

I Observing a path (Yt)t∈[0,T ), Ok and Nkl are sufficient forestimating Qkl

I Unbiased?

E(Qkl |Q) = ∞E(Qkl |Q,Ok > 0) = ∞

I Q is consistent, i.e.

limT→∞

P(|Qkl − Qkl | > ε) = 0

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Some Bayesian inference

I Using uninformative prior Qkl ∼ Ga(1, 0):

Qkl |Y ∼ Ga(Nkl + 1,Ok)

I Mean: (Nkl + 1)/Ok

I Variance: (Nkl + 1)/O2k , hence decreasing with 1/T

I Mode: Nkl/Ok

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Remarks on inference

I Q is well-known and looks nice

I In fact, estimation of Q is not so easy

I We need some observations for each transition k to l

I Basically, T needs to be as large as possible

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Introduction





Conclusion

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Observing a number of series of MJPs

I M series of MJPs are observed

I All Y (m) characterized by same rate matrix Q

I Series may be independent or come from data with breaks

0 0.051

2

3

0 0.051

2

3

0 0.051

2

3

0 0.051

2

3

0 0.051

2

3

Figure: Processes Y (1), . . . , Y (M)

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Observing merged MJPs

I Merged process of M single MJPs is observed

I Single MJPs characterized by same rate matrix Q

0 0.05 0.1 0.15 0.2 0.251

2

3

Figure: Merged process Y

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Merged MJPs

I Given: Observation process Y = (Yt)t∈[0,T )

I Y is concatenation of MJPs Y (1), . . . ,Y (M)

I In detail, Y (m) = (Y(m)t )t∈[0,T ), T = M T , and

Yt = Y(1)t if 0 ≤ t < T ,

Yt = Y(2)t−T if T ≤ t < 2T ,

...

Yt = Y(m)t−mT if (m − 1)T ≤ t < mT

I All Y (m) characterized by same rate matrix Q

I NB: Y itself is not Markov!

I Assumption of equal length is for notational convenience only

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Merged MJPs – Example

M = 5, d = 3, T = 0.05, T = 0.25, Q =

(−50 30 2020 −40 2030 40 −70

)

0 0.05 0.1 0.15 0.2 0.251

2

3

0 0.051

2

3

0 0.051

2

3

0 0.051

2

3

0 0.051

2

3

0 0.051

2

3

Figure: Merged process Y and single processes Y (1), . . . , Y (M)

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Introduction





Conclusion

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Splitting merged process

I Split Y into Y (1), . . . ,Y (M)

0 0.051

2

3

0 0.051

2

3

0 0.051

2

3

0 0.051

2

3

0 0.051

2

3

I Pooling:

Qkl =1

M

M∑m=1

Q(m)kl

I Problem: MLE does not exist if some O(m)k = 0;

short occupation times lead to unstable results

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Estimating directly from merged process

I Consider Y

0 0.05 0.1 0.15 0.2 0.251

2

3

I Number of jumps and occupation time:

Nkl =M∑

m=1

N(m)kl + N+

kl , where N+kl =

M−1∑m=1

I{Y

(m)T−=k, Y

(m+1)0 =l

},

Ok =M∑

m=1

O(m)k

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Estimating directly from merged process – first try

I First attempt: Qkl = Nkl/Ok

I Problem: Bias caused from “artificial” jumps N+kl /Ok

I If N+kl is observed explicitly, things are easy

I We assume N+kl cannot be observed

I Location of splitting points unknown

I Process not directly observed

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Estimating directly from merged process – bias

I Assume Y(m)0 ∼ π, where π stationary distribution

I Extra jumps N+ do not affect distribution of occupation times andstationary distribution

I Hence, π = π and π = π(Q) is “unbiased” estimate for π

I Joining two independent stationary processes generates a jump fromk to l with probability πkπl :

P(N+

kl |π)

= Bin(M − 1, πkπl)

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Estimating directly from merged process – bias (2)

I P(N+

kl |π)

= Bin(M − 1, πkπl) is justified if

I Y (m) are independent series

I Data with breaks:Length of break τ such that ρτ (Y ) is close to zero, where

ρt(Y ) =

Pdk=1 πk(Xkk(t) − πk)Pd

k=1 πk(1 − πk)

and

Xkl(t) = P(Yt = l |Y0 = k) = exp(Q t)kl

I.e. Xkk(τ) should be close to πk

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Estimating directly from merged process – bias (3)

I Note: If π is given, N+kl and Ok are independent

I Recall: P(N+

kl |π)

= Bin(M − 1, πkπl)

I As

T

Ok

P−−−−→T→∞

1

πk,

we have

N+kl

Ok

≈ (M − 1)πkπl

πkT= πl

M − 1

T

I As π is unbiased estimate for π, these quantities can be estimatedknowing Q

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Estimating directly from merged process – correction

I 2-step construction for corrected estimate:

1) Qkl = Nkl/Ok

2) Qkl = Qkl − (M − 1) πl/T

I “Merging”

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Comparison: Splitting vs. Merging

I Variance for Splitting for Ga(1, 0) prior:

Var

(M−1

M∑m=1

Q(m)kl

∣∣∣Y (1), . . . ,Y (M)

)=

1

M2

M∑m=1

N(m)kl + 1(O

(m)k

)2I Variance for Merging for Ga(1, 0) prior:

Var(Qkl | Y

)=

∑Mm=1 N

(m)kl + 1(∑M

m=1 O(m)k

)2I For rather short single observation times T we expect Merging to

give more reliable results

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Numerical example

I M = 100, d = 3, T = 0.25, T = 25, Q =

(−100 60 4040 −70 3040 60 −100

)

I On average, about 22 jumps in each single process Y (m)

I Simulate merged data and apply Splitting and Merging

I Repeat 100 000 times and consider sampling distributions of Qkl

and Qkl

I π = (0.29 0.46 0.25) and Q = Q −

(−2.8 1.8 1.01.1 −2.1 1.01.1 1.8 −2.9

)

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Numerical example – results

−140 −120 −100 −800

0.05

0.1

50 70 900

0.05

0.1

0.15

30 40 50 600

0.1

0.2

30 40 500

0.1

0.2

−90 −80 −70 −600

0.1

0.2

20 30 400

0.1

0.2

30 40 50 600

0.1

0.2

50 70 900

0.05

0.1

0.15

−140 −120 −100 −800

0.05

0.1

Green circles: true Qkl , blue: Merging Qkl , red: Splitting Qkl

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Numerical example – remarks

I Qkl outperforms Qkl wrt. both location and dispersion

I Qkl is skewed towards over-estimation:If some O

(m)k is very small, Q

(m)kl heavily over-estimates Qkl and

also Qkl is too high

I This is the more severe, the smaller T is

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Numerical example (2)

I M = 62, d = 3, T = 1/M, T = 1, Q =

(−800 450 350100 −200 100300 400 −700

)

I On average, about 6 jumps in each single process Y (m)

I Probability that one state is visited never or only for a very shorttime is high!

I Simulate merged data and apply Splitting and Merging

I Repeat 100 000 times and consider sampling distributions of Qkl

and Qkl

I π = (0.15 0.68 0.17) and Q = Q −

(−51.8 41.4 10.49.1 −19.6 10.59.1 41.4 −50.5

)

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Numerical example (2) – results

−3,000 −2,000 −1,000 00

0.003

0.006

0 500 1000 15000

0.004

0.008

0 500 1000 15000

0.005

0.01

0 100 2000

0.02

0.04

−400 −200 00

0.01

0.02

0.03

0 100 2000

0.02

0.04

0 500 1,000 1,5000

0.005

0.01

0 500 1000 15000

0.005

0.01

−3,000 −2,000 −1,000 00

0.004

0.008

Green circles: true Qkl , blue: Merging Qkl , red: Splitting Qkl

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Introduction





Conclusion

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A continuous-time Markov switching model

I Observation process R = (Rt)t∈[0,T ] (e.g. stock returns) withdynamics

dRt = µt dt + σt dWt ,

Rt =

∫ t

0

µs ds +

∫ t

0

σs dWs

I W standard Brownian motion

I Drift and volatility jump between d levels:µt = µ(Yt), σt = σ(Yt)

I State process Y is a MJP with state space {1, . . . , d}, Y ⊥ W

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Example

0 0.2 0.4 0.6 0.8 11

2

3State process

0 0.2 0.4 0.6 0.8 1−2

02

Drift process

0 0.2 0.4 0.6 0.8 10.1

0.15

0.2

Volatility process

0 0.2 0.4 0.6 0.8 1−0.03

0

0.03 Daily stock returns

0 0.2 0.4 0.6 0.8 10.8

1

1.2Price process

∆t = 1250 , µ = (3, 0,−2), σ = (0.20, 0.12, 0.15), Q =

(−70 40 3020 −40 2030 50 −80

)29 / 37


MSMs in Finance

I Short rate models: Elliott/Hunter/Jamieson (2001)

I Investment problems: Zhang (2001), Guo (2005)

I Risk measures for derivatives: Elliott/Siu/Chan (2008)

I Portfolio optimization: Honda (2003), Zhou/Yin (2003),Sass/Haussmann (2004), Bauerle/Rieder (2005), . . .

I Option pricing: Guo (2001), Buffington/Elliott (2002),Chan/Elliott/Siu (2005), Liu/Zhang/Yin (2006),Yao/Zhang/Zhou (2006), . . .

I . . .

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Estimation from discretely observed data

I Return process R is observable at times t = i ∆t,

Vi = ∆Ri =

∫ i ∆t

(i−1) ∆t

µs ds +

∫ i ∆t

(i−1) ∆t

σs dWs , i = 1, . . . ,H

I Vi daily stock returns

I State process Y is independent of W and not observable (hidden)

I Wanted: µ(k), σ(k), Q

I Problems:I ∆t given and fixedI Noise high compared to signalI High-frequency switching of states, i.e. λk ∆t highI Number of observations low (say, less than 5000)

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Data with breaks / merged processes

I Often, we encounter data with breaks

I Weekends for daily data, nights for intra-day data, . . .

I Observable discrete time process:V = (Vi )i=1,...,H is concatenation of V (1), . . . ,V (M)

V (m) = (V(m)i )i=1,...,H and H = M H

I Hidden continuous-time state process:Y is concatenation of Y (1), . . . ,Y (M)

Y (m) = (Y(m)t )t∈[0,T ) and T = M T

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Estimation from data with breaks

I Proceed similarly as for MJPs

I Use merged data V for estimation

I Employ arbitrary method to obtain estimates µ, σ, and Q

I µ and σ are not affected by merging, hence µ = µ, σ = σ

I Correction for Q: As described for MJPsI Point estimates: Correct Q to obtain Q

I Simulation based: Correct each sample Qj to obtain samples Qj

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Numerical example

I T = 10, H = 10 000, ∆t = 1/1000

I d = 2, Q =(−60 60

40 −40

), µ = (2 − 1), σ = (0.10 0.05),

i.e. µ∆t = (0.002 − 0.001), σ√

∆t = (0.0032 0.0016)

I M, T , H varying s.t. T = M T and H = M H

I M = 1 corresponds to one coherent seriesM = 400 corresponds to 400 series with H = 25 observations each

I Simulate merged data

I Perform method of moments-type estimation for merged data

I Repeat 1 000 times

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Numerical example – results

µ(1) µ(2) σ(1) σ(2) λ1 λ2 λ1 λ2

true 2.00 -1.00 0.100 0.050 60.0 40.0 60.0 40.0

M = 1, 2.02 -0.97 0.100 0.052 58.6 37.5 58.6 37.5τ = 10 0.11 0.05 0.002 0.002 9.8 6.5 9.8 6.5

M = 50, 2.02 -0.97 0.100 0.052 61.5 39.5 58.6 37.7τ = 0.2 0.12 0.05 0.001 0.003 10.7 6.3 10.7 6.7

M = 100, 2.03 -0.97 0.100 0.052 64.7 41.2 58.6 37.3τ = 0.1 0.11 0.05 0.001 0.003 11.6 6.4 10.5 6.8

M = 200, 2.03 -0.96 0.100 0.052 71.2 45.1 59.0 37.4τ = 0.05 0.12 0.06 0.001 0.003 15.6 8.2 10.7 6.7

M = 400, 2.06 -0.96 0.100 0.053 85.3 52.9 60.7 37.6τ = 0.025 0.13 0.06 0.002 0.003 28.4 15.1 12.6 8.0

Table: Results for MSM (T = 10, H = 10 000): mean (top), RMSE (bottom)

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Numerical example – remarks

I Estimates of µ(k), σ(k) are not affected by merging

I Quality of corrected estimates Qkl is (nearly) independent of M

I Ignoring breaks can lead to considerable bias in estimate for Q

I Method of moments-type estimation requires a lot of observations,Splitting for M > 2 not applicable;ML or Bayesian methods could be applied for M ≤ 10, but arecomputationally much more costly

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Conclusion

Conclusion

I Estimation for set of short (independent) observation series ordata containing (long) breaks

I Applicable to processes based on MJPs

I First, estimate parameters for merged seriesSecond, correct rates for bias afterwards

I Post-processing correction

I Works with arbitrary estimation approach for coherent series

I Single series need not be of same length /splitting times need not be known –only number of breaking points required

I Works similarly for discrete time processes

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