Recursive computation of piecewise constant volatilities

doi:10.1016/j.csda.2010.06.027

Computational Statistics & Data Analysis

Volume 56, Issue 11, November 2012, Pages 3623-3631

https://doi.org/10.1016/j.csda.2010.06.027 Get rights and content

Abstract

Returns of risky assets are often modelled as the product of a volatility function and standard Gaussian white noise. Long range data cannot be adequately approximated by simple parametric models. The choice is between retaining simple models and segmenting the data, or to use a non-parametric approach. There is not always a clear dividing line between the two approaches. In particular, modelling the volatility as a piecewise constant function can be interpreted either as segmentation based on the simple model of constant volatility, or as an approximation to the observed volatility by a simple function. A precise concept of local approximation is introduced and it is shown that the sparsity problem of minimizing the number of intervals of constancy under constraints can be solved using dynamic programming. The method is applied to the daily returns of the German DAX index. In a short simulation study it is shown that the method can accurately estimate the number of breaks for simulated data without prior knowledge of this number.

Section snippets

The problem

Let $R (t)$ be the return of some risky asset in period $t$ . For stocks with end of period price $P_{t}$ , $R (t) = ln (P_{t} / P_{t - 1})$ . In empirical finance, $R (t)$ is often decomposed as $R (t) = σ (t) \cdot Z (t), t = 1, \dots, n,$ where $Z$ is standard Gaussian white noise. The method can be adapted to other distributional assumptions such as in Curto et al. (2009). This will be briefly discussed in Section 6. The major problem is how best to model $σ (t)$ . In the enormous ARCH-class of models, for example, $σ (t)$ depends on past values of the

Minimizing the number of intervals

To define the modified problem let $I_{1}, \dots, I_{k} \subset {1, \dots, n}$ , $I_{ν} \cap I_{μ} = 0̸$ and $I_{1} \cup \dots \cup I_{k} = {1, \dots, n}$ , be the intervals where $σ (t)$ is constant, with value $σ_{I_{ν}}$ , $(σ_{I_{ν}} > 0)$ . The inequalities in Eq. (3) imply $\frac{\sum_{t \in I} R {(t)}^{2}}{χ_{| I |, \frac{1 + α_{n}}{2}}^{2}} \leq σ_{I_{ν}}^{2} \leq \frac{\sum_{t \in I} R {(t)}^{2}}{χ_{| I |, \frac{1 - α_{n}}{2}}^{2}}, \forall I \subset I_{ν}, ν = 1, \dots, k .$ A volatility function which satisfies these constraints is called locally adequate. Local adequacy is a weaker condition than (3) and it turns out that the sparsity problem can be solved for piecewise constant locally adequate volatility functions. It

Empirical volatility and dynamic programming

The use of (8) has an important drawback. As the quantiles $χ_{| I_{ν} |, 1 - \frac{α_{n}}{2}}^{2}$ are small for short intervals $I_{ν} = {s_{ν}, \dots, t_{ν},}$ the squared returns $R {(s_{ν})}^{2}, \dots, R {(t_{ν})}^{2}$ will in general be much closer to the lower bound $σ_{l}^{2} (I_{ν})$ than to the upper bound $σ_{u}^{2} (I_{ν})$ . The choice (8) will therefore in general overestimate the empirical volatilities in the interval ${s_{ν}, \dots, t_{ν}}$ . In the extreme case of $| I_{ν} | = 1$ and with $α_{n} = 0.99$ the choice (8) gives $σ_{l}^{2} (I_{ν}) \approx 0.13 R {(s_{ν})}^{2}$ and $σ_{u}^{2} (I_{ν}) \approx 25445.29 R {(s_{ν})}^{2} .$ The choice (8) is not a sensible

Minimizing the empirical quadratic deviations

In general there will not be a unique solution to the problem of minimizing the number of intervals for piecewise constant empirical volatility. A way of obtaining a unique solution is choose that minimal partition which minimizes the sum of the empirical quadratic deviations $\sum_{t = 1}^{n} {(R {(t)}^{2} - σ {(t)}^{2})}^{2} .$ Although artificial examples can be constructed where even this added restriction does not result in a unique partition this is very unlikely to happen for real data. The calculation of such a

Simulations

As mentioned in the introduction the methodology expounded above is intended to give a simple piecewise approximation to the volatility. It was not developed to detect breaks in the volatility which is a separate problem. Nevertheless it can be used to detect multiple breaks if in applications to real data sets it is kept in mind that not all breaks in the piecewise function correspond to breaks in the underlying volatility. There are many papers in the literature concerned with detecting

Extension to other distributions

So far it has been assumed that the noise $Z (t)$ in (1) is $N (0, 1)$ . This can be weakened and the $N (0, 1)$ distribution can be replaced by any other standardized (zero mean and unit variance) continuous distribution $F$ which is strictly monotone increasing. This distribution must be fully specified but this does not mean that one has to “know” the “true” distribution. An informed choice of $F$ can be based initially on the $N (0, 1)$ distribution to see how this must be altered to obtain better results. As $F$

Acknowledgements

The authors acknowledge the helpful comments and criticisms of earlier versions of the paper by two anonymous referees and the Editor. The research was supported by Deutsche Forschungsgemeinschaft (DFG). Views expressed by Christian Höhenrieder do not reflect official positions of Deutsche Bundesbank.

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^☆: The algorithms suggested here were programmed in R and are available from the authors upon request.

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Recursive computation of piecewise constant volatilities☆

Abstract

Section snippets

The problem

Minimizing the number of intervals

Empirical volatility and dynamic programming

Minimizing the empirical quadratic deviations

Simulations

Extension to other distributions

Acknowledgements

Stochastic Processes and their Applications

Computational Statistics and Data Analysis

Estimating and testing linear models with multiple structural changes

Econometrica

Consistencies and rates of convergence of jump-penalized least squares estimators

The Annals of Statistics

Modelling stock markets’ volatility using GARCH models with normal, students’ t and stable Paretian distributions

Statistical Papers

Modelling stock markets’ volatility using GARCH models with normal, students’ $t$ and stable Paretian distributions