Nonparametric Bayesian volatility learning under microstructure noise

Abstract

In this work, we study the problem of learning the volatility under market microstructure noise. Specifically, we consider noisy discrete time observations from a stochastic differential equation and develop a novel computational method to learn the diffusion coefficient of the equation. We take a nonparametric Bayesian approach, where we a priori model the volatility function as piecewise constant. Its prior is specified via the inverse Gamma Markov chain. Sampling from the posterior is accomplished by incorporating the Forward Filtering Backward Simulation algorithm in the Gibbs sampler. Good performance of the method is demonstrated on two representative synthetic data examples. We also apply the method on a EUR/USD exchange rate dataset. Finally we present a limit result on the prior distribution.

Details on update steps in the Gibbs sampler

1.1 Drawing $x_{0:n}$

We first describe how to draw the state vector $x_{0:n}$ conditional on all other parameters in the model and the data $y_{1:n}$. Note that for $u_i$ in (5) we have by (4) that $u_i \sim N(0,w_i)$, where

$$\begin{aligned} \begin{aligned} w_i&= \theta _k\varDelta _i, \quad i \in [(k-1)m+1,km], \quad k=1,\ldots ,N-1,\\ w_i&= \theta _N \varDelta _i, \quad i \in [(N-1)m+1,n]. \end{aligned} \end{aligned}$$

(19)

By Eq. (4.21) in Petris et al. (2009) (we omit dependence on $\theta _{1:N}, \eta _v$ in our notation, as they stay fixed in this step),

$$\begin{aligned} p(x_{0:n} | y_{1:n})=\prod _{i=0}^n p( x_i | x_{i+1:n}, y_{1:n}), \end{aligned}$$

where the factor with $i=n$ in the product on the righthand side is the filtering density $p( x_n | y_{1:n})$. This distribution is in fact $N(\mu _n,C_n)$, with the mean $\mu _n$ and variance $C_n$ obtained from Kalman recursions

$$\begin{aligned} \mu _i=\mu _{i-1}+K_i e_i,\quad C_i=K_i \eta _v, \quad i=1,\ldots ,n. \end{aligned}$$

Here

$$\begin{aligned} K_i =\frac{C_{i-1}+w_i }{C_{i-1} + w_i + \eta _v}, \quad i=1,\ldots ,n, \end{aligned}$$

is the Kalman gain. Furthermore, $e_i=y_i-\mu _{i-1}$ is the one-step ahead prediction error, also referred to as innovation. See Petris et al. (2009), Section 2.7.2. This constitutes the forward pass of the FFBS.

Next, in the backward pass, one draws backwards in time ${\widetilde{x}}_n \sim N(\mu _n,C_n)$ and ${\widetilde{x}}_{n-1},\ldots {\widetilde{x}}_0$ from the densities $p( x_i | {\widetilde{x}}_{i+1}, y_{1:n})$ for $i=n-1,n-2,\ldots ,0$. It holds that $p( x_i | {\widetilde{x}}_{i+1:n}, y_{1:n})=p( x_i | {\widetilde{x}}_{i+1}, y_{1:n})$, and the latter distribution is $N(h_i,H_i)$, with

$$\begin{aligned} h_i=\mu _i+\frac{C_i}{C_i+w_{i+1}}({\widetilde{x}}_{i+1}-\mu _i), \quad H_i =\frac{C_i w_{i+1}}{C_i+w_{i+1}}. \end{aligned}$$

For every i, these expressions depend on a previously generated ${\widetilde{x}}_{i+1}$ and other known quantities only. The sequence ${\widetilde{x}}_0,{\widetilde{x}}_1,\ldots ,{\widetilde{x}}_n$ is a sample from $p(x_{0:n}|y_{1:n})$. See Section 4.4.1 in Petris et al. (2009) for details on FFBS.

1.2 Drawing $\eta _v$, $\theta _{1:N}$ and $\zeta _{2:N}$

Using the likelihood expression from Sect. 2.3 and the fact that $\eta _v\sim {\text {IG}}(\alpha _v,\beta _v)$, one sees that the full conditional distribution of $\eta _v$ is given by

$$\begin{aligned} \eta _v | x_{1:n}, y_{1:n} \sim {\text {IG}}\left( \alpha _v + \frac{n}{2}, \beta _v + \frac{1}{2} \sum _{i=1}^n (y_i-x_i)^2 \right) . \end{aligned}$$

Similarly, using the likelihood expression from Sect. 2.3 and the conditional distributions in (7), one sees that the full conditional distributions for $\theta _{1:N}$ are

$$\begin{aligned} \theta _1 | \zeta _2,x_{1:n}&\sim \textrm{IG}\left( \alpha _1 + \alpha +\frac{m_1}{2},\, \beta _1 + \frac{\alpha }{ \zeta _{2}} + \frac{ Z_1}{2}\right) ,\\ \theta _k | \zeta _k,\zeta _{k+1},x_{1:n}&\sim \textrm{IG}\left( 2\alpha +\frac{m_k}{2},\, \frac{\alpha }{\zeta _k}+\frac{\alpha }{\zeta _{k+1}} + \frac{ Z_k}{2}\right) , \quad k=2,\dots ,N-1, \\ \theta _N | \zeta _N,x_{1:n}&\sim \textrm{IG}\left( \alpha +\frac{m_N}{2},\, \frac{\alpha }{ \zeta _N} + \frac{ Z_N}{2}\right) . \end{aligned}$$

The full conditional distributions for $\zeta _{2:N}$ are

$$\begin{aligned} \zeta _k | \theta _k,\theta _{k-1} \sim {\text {IG}}\left( 2\alpha ,\frac{\alpha }{\theta _{k-1}}+\frac{\alpha }{ \theta _k}\right) , \quad k=2,\ldots ,N. \end{aligned}$$

1.3 Drawing $\alpha$

The unnormalised full conditional density of $\alpha$ is

$$\begin{aligned} q(\alpha ) = \pi (\alpha )\left( \frac{\alpha ^{\alpha }}{\varGamma (\alpha )}\right) ^{2(N-1)} \exp \left( -\alpha \sum _{k=2}^N \frac{1}{\zeta _k}\left( \frac{1}{\theta _{k-1}} + \frac{1}{\theta _k}\right) \right) \prod _{k=2}^N \left( \theta _{k-1} \theta _k \zeta _k^2 \right) ^{-\alpha }. \end{aligned}$$

The corresponding normalised density is nonstandard, and the Metropolis-within-Gibbs step (see, e.g., Tierney 1994) is used to update $\alpha$. The specific details are exactly the same as in Gugushvili et al. (2019b).

1.4 Gibbs sampler

Settings for the Gibbs sampler in Sect. 4 are as follows: we used a vague specification $\alpha _1,\beta _1\rightarrow 0$, and also assumed that $\log \alpha \sim N(1,0.25)$ and $\eta _v \sim {\text {IG}}(0.3,0.3)$ in Sect. 4.1. For the Heston model in Sect. 4.2 we used the specification $\eta _v \sim {\text {IG}}(0.001,0.001)$. Furthermore, we set $x_0 \sim N(0,25)$. The Metropolis-within-Gibbs step to update the hyperparameter $\alpha$ was performed via an independent Gaussian random walk proposal (with a correction as in Wilkinson (2012)) with scaling to ensure the acceptance rate of about $30-50\%$. The Gibbs sampler was run for $30 \, 000$ iterations, with the first third of the samples dropped as burn-in.