Nonparametric Bayesian Volatility Estimation

Abstract

Given discrete time observations over a fixed time interval, we study a nonparametric Bayesian approach to estimation of the volatility coefficient of a stochastic differential equation. We postulate a histogram-type prior on the volatility with piecewise constant realisations on bins forming a partition of the time interval. The values on the bins are assigned an inverse Gamma Markov chain (IGMC) prior. Posterior inference is straightforward to implement via Gibbs sampling, as the full conditional distributions are available explicitly and turn out to be inverse Gamma. We also discuss in detail the hyperparameter selection for our method. Our nonparametric Bayesian approach leads to good practical results in representative simulation examples. Finally, we apply it on a classical data set in change-point analysis: weekly closings of the Dow-Jones industrial averages.

Bayesian wavelet de-noising with the caravan prior

“…The hyperparameter a controls the amount of smoothing in the gamma chain, with small values corresponding to less smoothing; we assume a ∼ Gamma(a a , b a ). For a statistical use of inverse gamma chains outside the sparsity context see, e.g., Gugushvili et al (2018a), Gugushvili et al (2019) and Gugushvili et al (2018b).…”

Nonparametric Bayesian Volatility Learning Under Microstructure Noise

“…Over short time intervals [t i−1 , t i ], the term ti ti−1 s(t)dW t in (3), roughly speaking, will dominate the term ti ti−1 b(t, X t )dt, as the former scales as ∆t i , whereas the latter as √ ∆t i (due to the properties of the Wiener process paths). As our emphasis is on learning s rather than b, following Gugushvili et al [2017], Gugushvili et al [2018a] we act as if the process X had a zero drift, b ≡ 0. A similar idea is often used in frequentist volatility estimation procedures in the high frequency financial data setting; see Mykland and Zhang [2012], Section 2.1.5 for an intuitive exposition.…”

“…For the measurement error variance η v , we assume a priori η v ∼ IG(α v , β v ). The construction of the prior for s is more complex and follows Gugushvili et al [2018a], that in turn relies on Cemgil and Dikmen [2007]. Fix an integer m < n. Then we have a unique decomposition…”

“…We also endow α with a prior distribution and assume log α ∼ N (a, b), with hyperparameters a ∈ R, b > 0 chosen so as to render the hyperprior on α diffuse. As explained in Gugushvili et al [2017], Gugushvili et al [2018a], the hyperparameter N (or equivalently m) can be considered both as a smoothing parameter and the resolution at which one wants to learn the volatility function. Obviously, given the limited amount of data, this resolution cannot be made arbitrarily fine.…”

“…Obviously, given the limited amount of data, this resolution cannot be made arbitrarily fine. On the other hand, as shown in Gugushvili et al [2018a] (see also Gugushvili et al [2018b]), inference with the IGMC prior is quite robust with respect to a wide range of values of N , as the corresponding Bayesian procedure has an additional regularisation parameter α that is learned from the data. Statistical optimality results in Munk and Schmidt-Hieber [2010] suggest that in our setting N should be chosen considerably smaller than in the case of an SDE observed without noise (that was studied via the IGMC prior in Gugushvili et al [2018a]).…”