Semiparametric stochastic volatility modelling using penalized splines


Stochastic volatility (SV) models mimic many of the stylized facts attributed to time series of asset returns, while maintaining conceptual simplicity. The commonly made assumption of conditionally normally distributed or Student-t-distributed returns, given the volatility, has however been questioned. In this manuscript, we introduce a novel maximum penalized likelihood approach for estimating the conditional distribution in an SV model in a nonparametric way, thus avoiding any potentially critical assumptions on the shape. The considered framework exploits the strengths both of the hidden Markov model machinery and of penalized B-splines, and constitutes a powerful alternative to recently developed Bayesian approaches to semiparametric SV modelling. We demonstrate the feasibility of the approach in a simulation study before outlining its potential in applications to three series of returns on stocks and one series of stock index returns.

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The authors would like to thank the two anonymous referees who provided useful comments on an earlier version of this manuscript.

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Correspondence to Roland Langrock.

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Appendix: HMM essentials

Appendix: HMM essentials

This appendix reviews some HMM basics. A standard \(m\)-state HMM has the same two process structure as SV models and SSMs, only that the unobserved process is a Markov chain and hence discrete-valued rather than continuous-valued. Consider an HMM with observable process \(\{ X_t \}_{t=1}^T\) and underlying Markov chain \(\{ S_t \}_{t=1}^T\). Given the current state of \(S_t\), the variable \(X_t\) is usually assumed to be conditionally independent from previous and future observations and states. The Markov chain is typically considered to be of first order, and the probabilities of transitions between the different states are summarized in the \(m \times m\) transition probability matrix \(\varvec{\Gamma }=\left( \gamma _{ij} \right) \), where \(\gamma _{ij}=\Pr \bigl (S_{t+1}=j\vert S_t=i \bigr )\), \(i,j=1,\ldots ,m\). The initial state probabilities are summarized in the vector \(\varvec{\pi }\), where \(\pi _{i} = \Pr (S_1=i)\), \(i=1,\ldots ,m\). It is usually convenient and appropriate to assume \(\varvec{\pi }\) to be the stationary distribution. For the described HMM, with observations given by \(x_1,\ldots ,x_T\) and underlying states denoted by \(s_1,\ldots ,s_T\), the likelihood is given by

$$\begin{aligned} {\mathcal {L}}^{\text {HMM}} = f(x_1, \ldots , x_T)&= \sum _{s_1=1}^m \ldots \sum _{s_T=1}^m f(x_1, \ldots , x_T | s_1, \ldots , s_T) f(s_1, \ldots , s_T) \\&= \sum _{s_1=1}^m \ldots \sum _{s_T=1}^m \pi _{s_1} \prod _{t=1}^T f (x_t | s_t) \prod _{t=2}^T \gamma _{s_{t-1},s_t}. \end{aligned}$$

In this form the likelihood involves \(m^T\) summands, which would make a numerical maximization infeasible in most cases. However, there is a much more efficient way of calculating the likelihood \({\mathcal {L}}^{\text {HMM}}\), given by a recursive scheme called the forward algorithm. To see this, we consider the vectors of forward variables, defined as \(\varvec{\alpha }_t = \bigl ( {\alpha }_t (1), \ldots , {\alpha }_t (m) \bigr )\), \(t=1,\ldots ,T\), where \({\alpha }_t (j) = f (x_1, \ldots , x_t, S_t=j)\), \(j=1,\ldots ,m\). We then have the recursion:

$$\begin{aligned} {\varvec{\alpha }}_1 = {\varvec{\pi }} {\mathbf {Q}}(x_1), \qquad {\varvec{\alpha }}_{t+1} = {\varvec{\alpha }}_{t} {\varvec{\Gamma }} {\mathbf {Q}}(x_{t+1}), \end{aligned}$$

where \({\mathbf {Q}}(x_t)= \text {diag} \bigl ( f_1 (x_{t}), \ldots , f_m (x_{t}) \big )\), with \(f_i(x_t) =f (x_{t} | S_{t}=i)\). The recursion (10) can be derived in a straightforward manner using the HMM dependence structure. The likelihood can then be written as a matrix product:

$$\begin{aligned} {\mathcal {L}}^{\text {HMM}} = \sum _{i=1}^m {\alpha }_T(i) = {\varvec{\pi }} {\mathbf {Q}}(x_1) {\varvec{\Gamma }} {\mathbf {Q}}(x_{2}) \ldots {\varvec{\Gamma }} {\mathbf {Q}}(x_{T}) {\mathbf {1}}, \end{aligned}$$

where \({\mathbf {1}}\in {\mathbb {R}}^m\) is a column vector of ones. For a missing observation \(x_t\), the associated matrix \({\mathbf {Q}}(x_{t})\) is simply replaced by the \(m\times m\) identity matrix. For more details on HMMs, see for example Zucchini and MacDonald (2009).

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Langrock, R., Michelot, T., Sohn, A. et al. Semiparametric stochastic volatility modelling using penalized splines. Comput Stat 30, 517–537 (2015).

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  • B-splines
  • Cross-validation
  • Forward algorithm
  • Hidden Markov model
  • Numerical integration
  • Penalized likelihood