nonlinear filtering of asymmetric stochastic volatility...

Nonlinear Filtering of Asymmetric Stochastic Volatility Models (IEEE CIFEr-2014)

Nonlinear Filtering of Asymmetric StochasticVolatility Models and VaR Estimation

Nikolay NikolaevGoldsmiths College, University of London, UK

[email protected]

Lilian M. de MenezesCass Business School, City University London, UK

[email protected]

Evgueni SmirnovMaastricht University, The Netherlands

[email protected]

IEEE CIFEr-2014 Conference, March 27-28, London.

Nikolaev, de Menezes and Smirnov


Outline

• Motivation: The Asymmetric Stochastic Volatility (ASV) modelsinclude negative correlation between the return and the volatility(leverage effect), and are suitable tools for risk management;

• Achievements: This paper develops a direct approach to learningASV models through nonlinear filtering, and shows that this makesthe ASV useful for practical Value-at-Risk (VaR) estimation;

• Contributions: We derive a Nonlinear Quadrature Filter (NQF)that evaluates the moments of the prior and posterior volatilitydensities via recursive numerical convolution, and thus helps us toobtain efficiently the likelihood.



Previous Research

There are several groups of approaches to ASV estimation. The firstgroup includes sampling approaches, like:

• Bayesian MCMC simulation: (Nakajima and Omori, 2009);

• Perticle Filtering: (Djuric, Khan and Johnston, 2012);

The second group includes deterministic approaches, like:

• QML estimation: (Harvey and Shephard, 1996).

The next group includes numerical integration approaches, like:

• Non-parametric Nonlinear Filtering (DNFS): (Clements, Hurn andWhite, 2006);

• Parametric Nonlinear Filtering: (Shimada and Tsukuda, 2005),(Kawakatsu, 2007), (Smith, 2009).



The Asymmetric ASV Model

The asymmetric ASV model describes the dynamics of log-returns by:

yt = εt exp(xt/2), εt ∼ N (0, 1)

where εt ∼ N (0, 1) is white noise, and xt denotes the log-volatilityxt = log σ2

t (σt is the st.dev., E[yt] = 0 and V ar[yt|σt] = σ2t ) defined

by the following mean-reverting process:

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

ρ = Corr[εt−1, ηt]

where µ is the mean, ϕ is the persistence, ηt is the state noise, and ρ isthe correlation between the innovations εt−1 and ηt (E[εt−1.ηt]/ση).



The Decorrelated Asymmetric ASV Model

We consider the decorrelated ASV model, which using φ = ρση andτ2 = (1 − ρ2)σ2

η, is formulated as follows (Yu, 2005):

zt = φεt = φyt exp(−xt/2)xt = µ + ϕ(xt−1 − µ) + zt−1 + τη∗

t , η∗t ∼ N (0, 1)

where E[εt−1.η∗t ] = 0, φ is the slope and τ2 is the variance of the

regression of xt on εt−1.



Parameter Estimation Framework

The maximally likely ASV parameters θML ={µ, ϕ, φ, τ2}

can be foundwith an optimizer, which involves evaluation of the likelihood. TheNQF filter facilitates the calculation of the log-likelihood:

E [log p(y1:T |θ)] =T∑

t=1

N∑i=1

wip(yt|mit|t−1)

where N is the number of quadrature points.

The NQF filter performs time updating and measurement updating,during which the densities of interest are approximated by point massesusing Gauss-Hermite quadrature weights and points.



Nonlinear Quadrature Filtering

The NQF computes the mean of the state prior density using weightswi = W i/

√π and points mi

t−1 = mt−1 + X i√

St−1 (W i and X i deter-mined with quadrature rules) with the following time updating equation:

mt|t−1 =N∑

i=1wig(mi

t−1)

The variance of the prior state distribution is updated by:

St|t−1 =N∑

i=1wi

(g(mi

t−1) − mt|t−1)2

+ τ2

where g(mt−1) = µ + ϕ(mt−1 − µ) + φεt−1 is the transition function.



NQF Measurement Updating

We derive an equation for measurement updating in one step by directapproximation of the state posterior (Kushner and Budhiraja, 200),(Zoeteret al., 2004), using again the Gaussian quadrature technique:

mt =N∑

i=1wimi

t|t−1p(yt|mi

t|t−1)∑Ni=1 wip(yt|mi

t|t−1)

The equation for updating the posterior state variance is:

St =N∑

i=1wi(mi

t|t−1 − mt)2 p(yt|mit|t−1)∑N

i=1 wip(yt|mit|t−1)

where p(yt|mit|t−1) is the sample likelihood, which in this case can be

estimated using the normal probability density function.



Value-at-Risk Assessment

The Value-at-Risk (VaR) (Jorion, 1996),(Christoffersen, 2003) for nor-mal returns can be computed with the volatility forecasts xt obtainedby NQF filtering of the ASV model as follows:

V aRt(q) = −qq exp(xt/2)

where qq is the critical value of the Gaussian distribution.

Our research performed the assessment using bootstrapped one-stepahead predictions of the volatility {x∗

t }T +τt=T +1 with which VaR forecasts

for 95% confidence levels were calculated.



Volatility Learning from Simulated Series

400 600 800-4

0

4

8

Volatility

time

return mean lower upper

400 600 800-2

0

2

Returns

given NQF

Figure 1. (Upper plot) Log-volatility xt = log σ2t (bold red curve) produced with the NQF

filter over the simulated series, and the true log-volatility curve (green curve).

(Lower plot) Confidence intervals (95%) of the predictive distribution of the returns computed

using the forecasted stochastic volatility during a forward pass with the NQF filter.



Model Parameter Estimates

Parameter T = 1000 T = 2000NQF MCMC NQF MCMC

µ −9.7235 −9.2086 −9.7244 −9.2153(0.2995) (0.3471) (0.3018) (0.3142)

ϕ 0.9727 0.96537 0.9731 0.9684(0.0837) (0.1152) (0.0594) (0.1078)

ση 0.1269 0.1189 0.1227 0.1081(0.0452) (0.0725) (0.0402) (0.0715)

ρ −0.3017 −0.3546 −0.3124 0.3219(0.1243) (0.1815) (0.0954) (0.1154)

Table 1. Estimated average ASV model parameters and their RMSE errors in parentheses

obtained over 100 simulated series of different sizes, whose volatility was generated with the

particular benchmark parameters.



Volatility Learning from S&P500 Series

600 900 1200

0.5

1.0

1.5

2.0

Volatility

time

NQF MCMC

600 900 1200-4-20246

Returns

given

Figure 2. (Upper plot) The considered series of returns on prices of the S&P500 stock

market index recorded from January 1980 to December 1987.

(Lower plot) Filtered latent volatility σt from the series of S&P500 index returns using the

NQF and MCMC algorithms.



Distribution of the Standardized Residuals

-2 0 20

50

100

150

Coun

ts

Bins

StdResid

Figure 3. Histogram of the squared standardized residuals computed with volatilities inferred

by the NQF algorithm over the S&P 500 series.



Parameter Estimates over S&P500 Series

MCMC P F S DNF S NQF NQF St

µ −0.1791 −0.1344 −0.1511 −0.1453 −0.1426ϕ 0.9625 0.9692 0.9672 0.9758 0.9753ση 0.1652 0.1517 0.1554 0.1463 0.1472ρ −0.3064 −0.3051 −0.3133 −0.3224 −0.3215

Table 2. Estimated ASV model parameters using the studied algorithms over the series of

daily returns from the S&P 500 index prices recorded from 2/1/1980 to 30/12/1987.



Diagnostics of the Standardized Residuals

Skewness Kurtosis DW LB(30) l(θ)MCMC −0.0398 2.8956 1.8311 32.58 −2785P F S −0.0464 2.9423 1.8232 32.79 −2774DNF S −0.0295 2.9681 1.8266 32.58 −2759NQF −0.0287 2.9794 1.8252 32.45 −2762NQF St −0.0246 2.9831 1.8259 32.41 −2756

Table 3. Statistical diagnostics of the squared standardized residuals obtained by the studied

algorithms over the daily returns from the S&P 500 index from 2/1/1980 to 30/12/1987.



Empirical Value-at-Risk Testing

2000 2400 2800

3

6

9

12

Vola

tili

ty

time

returns vol CI.high CI.low

Figure 4. Bootsrapped 95% confidence intervals of the volatility distribution from 100 replicas

inferred by NQF St filtering using the S&P 500 series points from 1/9/2008 to 31/2/2011.

The circles show the positions of some failures to capture the returns.



Results from Value-at-Risk Testing95% daily VaR forecasts

Long positionFRL LRun LRind LRcc

MCMC 6.5425 6.2237∗ 2.8913 9.1150∗

P F S 6.7213 6.3185∗ 3.0542 9.3727∗

DNF S 5.8691 3.5216 1.8792 5.3908NQF 5.5344 3.2043 1.5471 4.7514NQF St 5.4018 3.0921 1.5125 4.6046

Short positionFRS LRun LRind LRcc

MCMC 4.6296 3.8032 2.0514 5.8546P F S 4.6517 3.9548∗ 2.1191 6.0739∗

DNF S 3.8344 2.1594 1.2647 3.4241NQF 3.9251 2.2825 1.5318 3.8143NQF St 3.9172 2.2741 1.5124 3.7875

Table 4. Results from likelihood-ratio coverage tests for adequacy of the 95% bootsrtapped

VaR estimates of the testing (out-of-sample) S&P 500 series using averaged forecasts of the

volatility from 100 replicates.Nikolaev, de Menezes and Smirnov


Conclusions

This research found that the direct processing of nonlinear ASV modelsusing NQF filters lead to accurate VaR forecasts, which suggests thatthey can be applied as practical tools for financial risk management.

Current research uses the ASV volatility formulation and the NQF fil-ter for making dynamic learning agents that trade on double auctionmarkets using limit order books.



Appendix: Nonlinear Quadrature Filtering

The time updating step infers the prior state density as follows:

p(xt|y1:t−1) =∫

p(xt|xt−1)p(xt−1|y1:t−1)dxt−1

=∫

p(xt|xt−1)N∑

i=1wiδ(xt−1 − mi

t−1)dxt−1

=N∑

i=1wip(xt|mi

t−1)

where δ is the Dirac delta function.



The mean of this prior density is computed by propagating the points throughthe transition function g(·):

mt|t−1 =∫

xtp(xt|y1:t−1)dxt

=∫

xt

N∑i=1

wip(xt|mit−1)dxt

=N∑

i=1wi

∫xtδ(xt − g(mi

t−1))dxt =N∑

i=1wig(mi

t−1)

The variance of the prior state density is derived in a similar manner:

St|t−1 =∫

(xt − mt|t−1)2p(xt|y1:t−1)dxt + τ2

=N∑

i=1wi

∫(xt − mt|t−1)2δ(xt − g(mi

t−1))dxt + τ2

=N∑

i=1wi

(g(mi

t−1) − mt|t−1)2 + τ2



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