Nonlinear Filtering of Asymmetric Stochastic Volatility Models and VaR Estimation

Transcription

1 Nonlinear Filtering of Asymmetric Stochastic Volatility Models and VaR Estimation Nikolay Nikolaev Goldsmiths College, University of London, UK Lilian M. de Menezes Cass Business School, City University London, UK Evgueni Smirnov Maastricht University, The Netherlands IEEE CIFEr-2014 Conference, March 27-28, London.

2 Outline Motivation: The Asymmetric Stochastic Volatility (ASV) models include 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 learning ASV models through nonlinear filtering, and shows that this makes the 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 volatility densities via recursive numerical convolution, and thus helps us to obtain efficiently the likelihood.

3 Previous Research There are several groups of approaches to ASV estimation. The first group 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 and White, 2006); Parametric Nonlinear Filtering: (Shimada and Tsukuda, 2005), (Kawakatsu, 2007), (Smith, 2009).

4 The Asymmetric ASV Model The asymmetric ASV model describes the dynamics of log-returns by: y t = ε t exp(x t /2), ε t N (0, 1) where ε t N (0, 1) is white noise, and x t denotes the log-volatility x t = log σ 2 t (σ t is the st.dev., E[y t ] = 0 and V ar[y t σ t ] = σ 2 t ) defined by the following mean-reverting process: x t = µ + ϕ(x t 1 µ) + η t, η t N (0, σ 2 η) ρ = Corr[ε t 1, η t ] where µ is the mean, ϕ is the persistence, η t is the state noise, and ρ is the correlation between the innovations ε t 1 and η t (E[ε t 1.η t ]/σ η ).

5 The Decorrelated Asymmetric ASV Model We consider the decorrelated ASV model, which using φ = ρσ η and τ 2 = (1 ρ 2 )σ 2 η, is formulated as follows (Yu, 2005): z t = φε t = φy t exp( x t /2) x t = µ + ϕ(x t 1 µ) + z t 1 + τη t, η t N (0, 1) where E[ε t 1.η t ] = 0, φ is the slope and τ 2 is the variance of the regression of x t on ε t 1.

6 Parameter Estimation Framework The maximally likely ASV parameters θ ML = { µ, ϕ, φ, τ 2} can be found with an optimizer, which involves evaluation of the likelihood. The N QF filter facilitates the calculation of the log-likelihood: T N E [log p(y 1:T θ)] = w i p(y t m i t t 1 ) t=1 where N is the number of quadrature points. The N QF filter performs time updating and measurement updating, during which the densities of interest are approximated by point masses using Gauss-Hermite quadrature weights and points.

7 Nonlinear Quadrature Filtering The NQF computes the mean of the state prior density using weights w i = W i / π and points m i t 1 = m t 1 + X i S t 1 (W i and X i determined with quadrature rules) with the following time updating equation: N m t t 1 = w i g(m i t 1) The variance of the prior state distribution is updated by: N ( ) 2 S t t 1 = w i g(m i t 1) m t t 1 + τ 2 where g(m t 1 ) = µ + ϕ(m t 1 µ) + φε t 1 is the transition function.

8 NQF Measurement Updating We derive an equation for measurement updating in one step by direct approximation of the state posterior (Kushner and Budhiraja, 200),(Zoeter et al., 2004), using again the Gaussian quadrature technique: N p(y m t = w i m i t m i t t 1 ) t t 1 N w i p(y t m i t t 1 ) The equation for updating the posterior state variance is: S t = N w i (m i t t 1 m t) 2 p(y t m i t t 1 ) N w i p(y t m i t t 1 ) where p(y t m i t t 1 ) is the sample likelihood, which in this case can be estimated using the normal probability density function.

9 Value-at-Risk Assessment The Value-at-Risk (VaR) (Jorion, 1996),(Christoffersen, 2003) for normal returns can be computed with the volatility forecasts x t obtained by NQF filtering of the ASV model as follows: V ar t (q) = q q exp(x t /2) where q q is the critical value of the Gaussian distribution. Our research performed the assessment using bootstrapped one-step ahead predictions of the volatility {x t } T +τ t=t +1 with which VaR forecasts for 95% confidence levels were calculated.

10 Volatility Learning from Simulated Series Returns 2 0 given NQF Volatility return mean lower upper time Figure 1. (Upper plot) Log-volatility x t = log σt 2 (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.

11 Model Parameter Estimates Parameter T = 1000 T = 2000 NQF MCMC NQF MCMC µ (0.2995) (0.3471) (0.3018) (0.3142) ϕ (0.0837) (0.1152) (0.0594) (0.1078) σ η (0.0452) (0.0725) (0.0402) (0.0715) ρ (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.

12 Returns Volatility Volatility Learning from S&P 500 Series given NQF MCMC 600 time Figure 2. (Upper plot) The considered series of returns on prices of the S&P500 stock market index recorded from January 1980 to December (Lower plot) Filtered latent volatility σ t from the series of S&P500 index returns using the NQF and MCMC algorithms.

13 Distribution of the Standardized Residuals 150 StdResid 100 Counts Bins Figure 3. Histogram of the squared standardized residuals computed with volatilities inferred by the NQF algorithm over the S&P 500 series.

14 Parameter Estimates over S&P 500 Series MCMC P F S DNF S NQF NQF St µ ϕ σ η ρ 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.

15 Diagnostics of the Standardized Residuals Skewness Kurtosis DW LB(30) l(θ) MCMC P F S DNF S NQF NQF St 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.

16 Empirical Value-at-Risk Testing Volatility returns vol CI.high CI.low time 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.

17 Results from Value-at-Risk Testing 95% daily VaR forecasts Long position FR L LR un LR ind LR cc MCMC P F S DNF S NQF NQF St Short position FR S LR un LR ind LR cc MCMC P F S DNF S NQF NQF St 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.

18 Conclusions This research found that the direct processing of nonlinear ASV models using N QF filters lead to accurate VaR forecasts, which suggests that they can be applied as practical tools for financial risk management. Current research uses the ASV volatility formulation and the N QF filter for making dynamic learning agents that trade on double auction markets using limit order books.

19 Appendix: Nonlinear Quadrature Filtering The time updating step infers the prior state density as follows: p(x t y 1:t 1 ) = p(x t x t 1 )p(x t 1 y 1:t 1 )dx t 1 N = p(x t x t 1 ) w i δ(x t 1 m i t 1)dx t 1 N = w i p(x t m i t 1) where δ is the Dirac delta function.

20 The mean of this prior density is computed by propagating the points through the transition function g( ): m t t 1 = x t p(x t y 1:t 1 )dx t = N x t w i p(x t m i t 1)dx t = N N w i x t δ(x t g(m i t 1))dx t = w i g(m i t 1) The variance of the prior state density is derived in a similar manner: S t t 1 = (x t m t t 1 ) 2 p(x t y 1:t 1 )dx t + τ 2 = = N w i (x t m t t 1 ) 2 δ(x t g(m i t 1))dx t + τ 2 N w i ( g(m i ) 2 t 1) m t t 1 + τ 2

21 The measurement updating step evaluates the posterior state density via an approximation of the prior density with deterministically chosen points, again using the Gauss-Hermite quadrature rule in the following way: m t = = = p(y t x t )p(x t y 1:t 1 ) x t p(yt x t )p(x t y 1:t 1 ) dx t p(y t x t ) N x wi δ(x t m i t t 1 ) t p(yt x t ) N wi δ(x t m i t t 1 )dx t N p(y w i m i t m i t t 1 ) t t 1 N wi p(y t m i t t 1 ) The variance of the posterior state density is obtained as follows: S t = (m i t t 1 m t) 2 p(y t x t )p(x t y 1:t 1 ) p(yt x t )p(x t y 1:t 1 ) dx t = N w i (m i t t 1 m t) 2 p(y t m i t t 1 ) N wi p(y t m i t t 1 )v

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