Financial risk forecasting with nonlinear dynamics and support vector regression

Abstract

We propose a dynamical description of financial time series capable of making short-term prediction utilizing support vector regression on neighbourhood points. We include in our analysis estimation on the uncertainty by capturing the exogenous from historical prediction errors and adopting a probabilistic description of the prediction. Evidences from a series of backtesting using financial time series indicate that our model provides accurate description of real market data comparable with GARCH(1,1).

Keywords:

• deterministic system
• nonlinear dynamics
• support vector regression
• forecast

Notes

1 An embedding is a smooth, one-to-one transformation with a smooth inverse.

2 Supremum norm ||X_r−X_s||_sup=max(|X_r¹−X_s¹|,…,|X_r^d−X_s^d|) represents the size of a hypercube enclosing X_r and X_s.

3 According to Grassberger and Procaccia (1983), local scale ɛ can be defined through the region of small norm size where the correlation integral C_d(ɛ) satisfies a power law relation C_d(ɛ)αɛ^ν with constant ν. In the reconstructed space, correlation integral C_d(ɛ) can be calculated as the probability of finding two d-histories that are less than ɛ apart. Intuitively, local scale must also be at least an order of magnitude below the average norm size over all reconstructed vectors.

4 See Smola and Scholkopf (1998) for a complete discussion on SMO. We have developed and used here a modified version of the algorithm based on the discussion in Shevade et al (1999).

5 Another common choice of kernel function in SVR is the Gaussian function K(X_q,X_s)=exp(−||X_q−X_s||²/ω) on Euclidean norm with parameter ω. It tends to yield good performance under a general smoothness assumption on the data. The set of basis functions for Gaussian kernel is infinite and form a basis for the Hilbert space of bounded continuous functions. In our work, the performance of a Gaussian kernel is very similar to that of a polynomial kernel for financial time series.

6 In GARCH(1,1) with Gaussian innovation, we define x_t=μ+ν_tɛ_t where ɛ_t is identically and independently distributed as a standard normal distribution. The variance measures are generated through the iteration equation ν_t+1²=aν_t²+b(x_t−μ)²+c where a>0, b>0, and c>0 with a+b<1 are estimated based on maximizing the likelihood function given the sequence {x_t}_t=1ⁿ prior to the backtesting. For simplicity, the unconditional mean μ can be estimated by the historical mean as μ=E(x_t)≅(1/n)∑_t=1ⁿx_t.