# Filtering Stochastic Volatility Models with Intractable Likelihoods · 2016. 5. 25. · Intro...

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Filtering Stochastic Volatility Models withIntractable Likelihoods

Katherine B. EnsorProfessor of Statistics and Director

Center for Computational Finance and Economic SystemsRice [email protected]

LEAD-Author: Emilian Vankov [email protected]: Michele Guindani

SAMSI Games and Reliability, May 20, 2016

Intro Stochastic Volatility Model with α stable noise. Filtering and Estimation Contribution - Volatility (SSM) Filtering SMC+ABC Simulation and Application In Summary

Outline

Intro

Stochastic Volatility Model with α stable noise.

Filtering and Estimation

Contribution - Volatility (SSM) Filtering

SMC+ABC

Simulation and Application

In Summary

Intro Stochastic Volatility Model with α stable noise. Filtering and Estimation Contribution - Volatility (SSM) Filtering SMC+ABC Simulation and Application In Summary

Financial Markets and Volatility

c

Source: www.vectorstock.com

Intro Stochastic Volatility Model with α stable noise. Filtering and Estimation Contribution - Volatility (SSM) Filtering SMC+ABC Simulation and Application In Summary

Typical Financial Data: Dow Jones Industrial Average

Prices

Returns

Squared Returns

What is Volatility?

Volatility is a measure of variation for price/returns of financialinstruments over time. Volatility is not directly observed. Somecommon features of volatility and prices/returns include:• Returns are serially uncorrelated but a dependent process.• Mean reversion in volatility to a long-run level.• Volatility clustering and persistence – results in high

autocorrelation.• Generally a negative correlation between prices and volatility.• Returns are asymmetric as a function of market increases or

decreases.

Since volatility is not directly observed how do we measure ormodel volatility?

VIX: Implied Volatility Index

30-day expectedvolatility, basedon the Black Sc-holes model, ofS&P 500 Index.

9000

10000

11000

12000

13000

14000

VIX [2015-06-01/2015-08-12]

Last 9500Bollinger Bands (20,2) [Upper/Lower]: 10053.639/9073.027

Volume (100,000s):814,500

5

10

15

20

25

30

Moving Average Convergence Divergence (12,26,9):MACD: -3.672Signal: -4.384

-5

0

5

Jun 012015

Jun 082015

Jun 152015

Jun 222015

Jun 292015

Jul 062015

Jul 132015

Jul 202015

Jul 272015

Aug 032015

Aug 102015

See:http://www.cboe.com/micro/vix/vixwhite.pdf

Modeling Volatility from Observed Returns

Traditional Time Series Models for Volatility:

• Conditional Heteroskedastic Volatility Models• ARCH (Engle 1982), GARCH (Bolerslev 1986), EGARCH

(Nelson 1991), etc. (in R see rugarch andrmgarch, review article Ensor and Koev(2015), realtime volatility and systemicrisk estimates NYU Stern Vlab)

• Stochastic Volatility Model (SVM) - volatility is modeledas a continuous or discrete stochastic process

• Taylor (1982) - discrete time SVM• Hull and White (1998) - continuous time SVM• In R see stochvol - relies on "anxilarity-sufficiency

interweiving strategy (ASIS)", Kastner andFrühwirth-Schnatter (2014), Yu and Meng (2011).

NYU Stern Vlab IBM | | Volatility Institute | NYU Stern

Documentation » Analysis List »

Volatility Analysis GJR-GARCH Go

INTERNATIONAL BUSINESS MACHINES CORP GJR-GARCH VOLATILITY GRAPH

Volatility Prediction for Friday, May 20, 2016: 22.08% (+0.24)

Volatility Summary Table

Closing Price: $144.93 Return: -1.65% 1 Week Pred: 22.31%

Average Week Vol: 22.33% Average Month Vol: 26.87% 1 Month Pred: 23.16%

Min Vol: 13.13% Max Vol: 77.93% 6 Months Pred: 26.71%

Average Vol: 27.86% Vol of Vol: 26.06% 1 Year Pred: 28.71%

DocumentationVolatility AnalysisGJR-GARCH

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Stochastic Volatility Models (SVM)

The focus of our analysis is the Stochastic Volatility Model.

yt = exp(xt/2)vt → f (yt|xt)

xt = µ +φxt−1 +σut → q(xt|xt−1)

x0 ∼ N(µ/(1−φ), σ

2/(1−φ2))

t = 1, ...,T

• yt - returns at time t• xt - log-volatility at time t• θ = {µ,φ ,σ} are parameters• vt and ut are noise terms - generally assumed i.i.d. N(0,1)

State-Space Model (SSM) where, yt is the observation equation, andxt is the state-equation, i.e. the log volatility is an unobserved latentprocess.

SVM and the Stable DistributionAre ut and vt i.i.d. N(0,1) in practice?

• The noise of the volatility equation ut, maybe; the noise ofthe observation equation, vt, probably not.

• As early as the 1960’s the assumption of normality wasquestioned as stock returns are often skewed andheavy-tailed.(Mandelbrot (1963); Fama (1965))

• Let’s assume that errors for the return (or observation)equation vt ∼ S(α,β ,0,1), are distributed as an asymmetricα-stable distribution.

yt = exp(xt/2)vt

xt = µ +φxt−1 +σut

x0 ∼ N(µ/(1−φ), σ

2/(1−φ2))

The α - Stable Distribution: S(α,β ,δ ,γ)

• Parametric distribution that captures skewness and heavytails.

• Parameters: Stability α ∈ (0,2]; Skewness β ∈ [−1,1];Location δ ∈ (−∞,∞); Scale γ ∈ (0,∞)

• When α = 2 ⇒ N(δ ,2∗ γ)

• When α = 1 and β = 0⇒ Cauchy(δ ,γ)• Characteristic function:

φ(t)=

{exp{−γα |t|α

[1− iβ (tan πα

2 )(sign(t))]+ iδ t

}, if α 6= 1

exp{−γ|t|

[1+ iβ 2

π(sign(t))ln(t)

]+ iδ t

}, if α = 1

• The p.d.f is not available in closed form.• However, simulating random numbers is possible.

Simulation of α-stable SVM

0 100 200 300 400 500

-10

-50

510

15

Returns from Normal SVM

Time

Returns

0 100 200 300 400 500

-10

-50

510

15

Returns from Stable SVM

Time

Returns

Stochastic Volatility Models

• Filtering and Estimation for SVM

yt = exp(xt/2)vt (Observation Eq)

xt = µ +φxt−1 +σut (State Eq)

x0 ∼ N(µ/(1−φ), σ

2/(1−φ2))

t = 1, ...,T

• Quantities of interest:• Distribution of volatility from time 0 to T, given returns and

parameters:p(x0:T |y1:T ,θ)

- assumes θ is known• Joint distribution of volatility from time 0 to T and

parameters, given returns:

p(x0:T ,θ |y1:T)

- state filtering and parameter estimation

Stochastic Volatility Models - Some History onEstimation

• Normal SVM• Filtering of p(x0:T |y1:T ,θ)

• Pitt and Shephard (1999) SMC/Particle filter.• Estimation of p(x0:T ,θ |y1:T)

• Jacquier et al. (1994), Kim et al. (1998) - MCMC methods• Harvey et al. (1994) - Kalman filter + quasi-MLE

• Symmetric α-stable SVM; β ≡ 0• Filtering of p(x0:T |y1:T ,θ)

• Jasra et al. (2012) - approximate Bayesian computation(ABC) based SMC

• Estimation of p(x0:T ,θ |y1:T)• Jasra et al. (2013), Barthelme and Chopin (2014) - ABC

based MCMC

Our Stochastic Volatility Contributions

ASYMMETRIC α-stable SVM

• Filtering of p(x0:T |y1:T ,θ)• ABC based Auxiliary Particle Filter (APF-ABC)• The focus of this talk.

• Estimation of p(x0:T ,θ |y1:T)• Posterior distributions for all parameters, including the

asymmetry parameter β .• Results - submitted to Bayesian Analysis.

State Filtering with Known ParametersSequential Monte Carlo

• State filtering with KNOWN parameter via SequentialMonte Carlo (SMC) (density of interest): p(x0:t|y1:t,θ)

p(x0:t|y1:t,θ) = p(x0:t−1|y1:t−1,θ)q(xt|xt−1,θ)f (yt|xt,θ)

p(yt|y1:t−1,θ)

• p(x0:t−1|y1:t−1,θ) is the distribution of interest at time t-1• Assume we have a sample of N particles that we can use

to estimate the filter distribution at time t−1; i.e.{X(i)

0:t−1, i = 1, ...,N} ∼ p(x0:t−1|y1:t−1,θ)

Sequential Monte Carlo (SMC)How do we go from time t−1 to time t or p(x0:t−1|y1:t−1,θ) to p(x0:t|y1:t,θ)?

1. Sample N particles {X(i)t , i = 1, ...,N} to get

{(X(i)0:t−1, X

(i)t ), i = 1, ...,N} ∼ g(x0:t|y1:t,θ), where g(·) is the importance

distribution2. Weight the N samples based on their importance

w(i)t ∝

p(x(i)0:t|y1:t)

g(x(i)0:t|y1:t)

∝f (yt|x(i)t )q(x(i)t |x

(i)t−1)

g(x(i)t |x(i)0:t−1,y1:t)

w(i)t−1

p(x0:t|y1:t,θ) =N

∑i=1

w(i)t δ

X(i)0:t

p(y1:T |θ) =T

∏t=1

(1N

N

∑i=1

w(i)t

)

WEIGHTS are KEY

SMC / APF - Obtaining Filtered Estimates

c

Consider allrandom variablesinvolved(X1:N

0:T ,A1:N0:T−1,θ)≡

(X0:T ,A0:T−1,θ)

Source: Modified Figure fromAndrieu et al. (2010)

Sequential Monte Carlo (SMC)

How do we choose the importance distributiong(x(i)t |x

(i)0:t−1,y1:t)?

• Typically, minimize the variance of the importance weightsby setting

g(xt|x0:t−1,y1:t) = p(xt|xt−1,yt)

• Problems:• Not available in closed form for more complicated models• In most applications g(xt|x0:t−1,y1:t) ∝ q(xt|xt−1) in which

case the ratio simplifies and thus NO DATA IS INVOLVED -i.e. proposal is chosen blindly without looking at the data;dangerous if the data is informative (e.g. outliers)

ABC based Sequential Monte Carlo (SMC+ ABC)

• Can we use the method discussed above directly with theα-stable SVM?

• Maybe, but the weights depend on the pdf of the α-stabledistribution, f (yt|xt), which is not available

• Solution - Approximate Bayesian Computation• Simulate auxiliary data, ys

t , from the model• Update the weights via a kernel density Kε(ys

t ,yt)• The SMC-ABC has the marginal target pε(x0:T |y1:T ,θ)• The bias of the SMC-ABC estimate goes to 0 as ε → 0

ABC based Sequential Monte Carlo (SMC + ABC)

• Jasra et al. (2012) advocate the use of a Uniform kerneland a proposal g(xt|x0:t−1,y1:t) = q(xt|xt−1)

• Weights are updated according to: w(i)t = IQε,yt

(yst )w

(i)t−1

• Qε,yt = {yst : ρ(ys

t ,yt)< ε} (use simulated values within acertain radius of the observed values).

• Drawbacks:• Binary weights - either 0 or 1; algorithm can collapse• Still sampling from the transition density q(xt|xt−1) ignoring

the data• Partial solution by Jasra et al. (2014)

• At each step resample the particles until N-1 of them haveNON-ZERO weight

• Solves one problem, but induces another, namely theamount of time spent at each iteration is a now a randomvariable.

Auxiliary Particle Filter (APF) and ABCOur Solution

• Using ideas from Pitt and Shephard (1999) and Carpenteret al. (1999) we propose:

• Use an auxiliary density to pre-weight the samples beforeproposing based on the data

g(xt|x0:t−1,y1:t) = g(yt|ξ (xt−1))q(xt|xt−1)

• Use a Gaussian kernel to evaluate the simulation basedweights relative to the observed data (ABC); the variabilityof the kernel controls the "closeness" required

• Benefits:• Continuous weights⇒ the algorithm will not collapse• Create better proposals by considering the data• No significant increase in computational times

Auxiliary Particle Filter (APF) and ABC

• Example of choices for the auxiliary distributiong(yt|ξ (xt−1)):

• f (yt|E[xt|xt−1]) problem if there are outliers or heavy tailsand not available in our case.

• Use a t-density with heavy tails centered at E[xt|xt−1]

• In general choose densities that are more diffuse than thelikelihood and transition densities

Auxiliary Particle Filter and ABCPutting our contributions together we get the following algorithm:

APF - ABCInitialization t = 0

• Sample the initial particles (x(1)0 , ..., x(N)0 ) from the initial distribution.

For t = 1, ... ,T

1. Sample the labels (auxiliary variables) based weights that incorporateinformation about the data. (Key of APF)

2. Resample the particles using the new labels.3. Sample a new particle from the transition density of the state equation

conditioned on the resampled particles4. Update weights to account for the discrepancy between the proposal density

and the target density based on ABC (instead of likelihood)5. Normalize weights.6. Use the new weights and particles to construct empirical density of the latent

log volatility.

How well does the APF+ABC Method WorkReminder - Stochastic Volatility Models (SVM) parameterization

yt = exp(xt/2)vt

vt ∼ S(α,β ,δ ,γ)

xt = µ +φxt−1 +σut

ut ∼ N(0,1)

x0 ∼ N(

µ/(1−φ), σ2/(1−φ

2))

How well does this work?Simulation Study - Volatility Filtering

• Simulate data for t = 1, ... , 500 days from the α-stableSVM

• Parameter values: α = 1.9, β = 0.1, µ =−0.2, φ = 0.95,σ = 0.7

• Estimate the unobserved volatility with N = 5000 particles

• Auxillary particle distribution g(yt|ξ (x(i)t )) is chosen to be at-distribution with 2 degrees of freedom centered atE[xt|xt−1]

• Kε(yst ) is given by N(yt,ε = 0.25)

Simulation

−12

−8

−4

0

0 100 200 300 400 500time

log−

vola

tility Volatility

Filtered

True

APF−ABC Filtered Values

Figure: APF-ABC mean log-volatility estimate and the true statevolatility. The filtered values are averaged over 100 simulations.

Simulation - Our Improvement

ABC−APF ABC−SMC

0.9

00.9

51.0

01.0

51.1

01.1

51.2

0

Error Distributions

RM

SE

Figure: Box plots of the root-mean-square error based on 100simulations.

We can now obtain the latent volatility process with knownparameters.

How do we then obtain the posterior distribution of theparameters?• Use SMC/APF+ABC with EM algorithm to obtain MLE’s.• Use SMC/APF+ABC with MCMC to obtain posterior

distribution – Single Filter Particle Metropolis–within–Gibbs(SFPMwG).

Modeling Exchange Rate VolatilitySeries from Literature

• Exchange rate data for Brazilian Real vs United StatesDollar

• The data spans the period 1998-2004• Lombardi and Calzolari (2009) use an indirect estimation

approach• The parameters for this data are estimated to be:

α = 1.9123, µ = 0.0018, φ = 0.9985, σ = 0.1908• Assumes σy = 1, β = 1, δ = 0, γ = 1• In January 1999, the Central Bank of Brazil was forced to

abandon a managed depreciation regime by a speculativeattack triggered by a 90-days debt moratorium announcedby a provincial governor

BRL vs USD plotsSeries from Literature

0 500 1000 1500

1.5

2.0

2.5

3.0

3.5

4.0

BRLvsUSD 1998-2004

time

Exc

hang

e R

ate

0 500 1000 1500

-0.10

-0.05

0.00

0.05

0.10

BRLvsUSD 1998-2004

time

Returns

Figure: Exchange rate data for the period 1998-2004 of the BrazilianReal vs the US Dollar. Real exchange rate(left) and exchange ratereturns(right)

Filtered Exchange Rate VolatilitySeries from Literature

0 500 1000 1500

-10

-8-6

-4-2

02

BRLvsUSD

time

log-vol

h.apf_abcx.abc_ukern

Figure: Filtered log-volatility for the exchange rate returns based on ABC-SMC(reddashed line) and ABC-APF(black line)

Propane Weekly Spot Prices

• Propane - clean fuel produced from natural gas or crude oil• Uses in the US - heating, agricultural and industrial sectors• Some key drivers for price changes

• Weather, inventory infrastructure and transportation

• Propane weekly spot prices for Mount Belvieu, TX for10/01/2007 - 9/28/2014

• Data Source: Energy Information Administration (EIA)• Calculate the demeaned weekly spot returns

yt = 100

(log(Pt/Pt−1)−

1T

T

∑i=1

log(Pi/Pi−1)

)

Propane Spot Returns Distribution

1

2

3

4

Moment

Mean

SD

Skewness

Kurtosis

Values

0.00

5.83

−1.44

8.90

0.000

0.025

0.050

0.075

−40 −20 0 20Returns

Den

sity

−30

−20

−10

0

10

0

Ret

urns

Figure: Histogram (left) and boxplot(right) for the demeaned weeklyspot returns for propane 10/01/2007 - 9/28/2014, Mount Belvieu TX.

• Jarque-Bera test for Gaussian returns; reject H0(p-value < 0.01)

Posterior Distribution Summary

Table: Summary of the posterior distribution for all parameters of theα-stable stochastic volatility model applied to weekly spot returns for10/01/2007 - 9/28/2014. We present the posterior mean, standarddeviation and 95% credible interval.

α β σ µ φ

Mean 1.88 −0.59 0.38 2.19 0.894SDev 0.05 0.31 0.07 0.28 0.0495% CI (1.76,1.98) (−0.98,0.22) (0.25,0.55) (1.62,2.64) (0.80,0.96)

The posterior probability that returns are left skewed is given by:

P(β < 0) = 0.948

Propane Spot Returns Distribution

0

10

20

30

2008 2010 2012 2014Year

Wee

kly

Abs

olut

e R

etur

ns

5

10

2008 2010 2012 2014Year

Wee

kly

Vol

atili

tyFigure: Absolute returns (left) and filtered volatility with 95% CI (right)from α-stable SVM for the demeaned weekly spot returns for propane10/01/2007 - 9/28/2014, Mount Belvieu TX.

Substituting the conditional mean of the posterior distributionfor the parameters, the model for Mean Adjusted PropaneReturns is given by:

yt = exp(xt/2)vt

vt ∼ S(1.88,−0.59,0,1)

xt = 2.19+0.89xt−1 +0.38ut

ut ∼ N(0,1)

x0 ∼ N(2.19/(1−0.89), 0.382/(1−0.892)

)

Forecast Distribution of Returns

The model is now fully characterized. The forecast distributionof returns conditional on parameters set to their posteriormeans is given by: VaR05=-46; CVaR05=-68

• Defined Stochastic Volatility Model which allows returns tofollow heavy-tailed and skewed distributions.

• Developed APF-ABC to obtain filtered values of the latentstate in asymmetric α-stable SVMs (and general SSM).The APF-ABC algorithm is applicable to any state-spacemodel with intractable likelihoods.

• Compared SMC-ABC and APF-ABC and appliedAPF-ABC to exchange rate series from the literature.

• Obtained the posterior distribution for the latent volatilityand parameters (methodology not discussed) forasymmetric α-stable SVMs for weekly propane spot prices.

THANK YOU FOR YOUR ATTENTION

Contact info: [email protected]; [email protected]

For detail and references see• Vankov and Ensor arXiv:1405.4323 [stat.CO]• Vankov 2015 (dissertation)• Vankov, Guindani & Ensor 2016 (under review)