Asymptotic Filter Behavior for High-Frequency Expert Opinions in a Market with Gaussian Drift

TL;DR

This paper analyzes how high-frequency expert opinions influence the estimation of a hidden Gaussian drift in a financial market, showing that increased opinion arrival rates lead to full information about the drift.

Contribution

It demonstrates the asymptotic behavior of the Kalman filter in a market with noisy expert opinions, proving convergence to full information as opinion frequency increases.

Findings

Conditional mean converges to the true drift

Estimation becomes consistent with high-frequency opinions

Filter asymptotics show full information in the limit

Abstract

This paper investigates a financial market where stock returns depend on a hidden Gaussian mean reverting drift process. Information on the drift is obtained from returns and expert opinions in the form of noisy signals about the current state of the drift arriving at the jump times of a homogeneous Poisson process. Drift estimates are based on Kalman filter techniques and described by the conditional mean and covariance matrix of the drift given the observations. We study the filter asymptotics for increasing arrival intensity of expert opinions and prove that the conditional mean is a consistent drift estimator, it converges in the mean-square sense to the hidden drift. Thus, in the limit as the arrival intensity goes to infinity investors have full information about the drift.

Equations

$$d{R}_t={\mu }_{t}dt+{\sigma }_{R}d{W}_t^R,$$

$$d{S}_t$$

$$\log S_t^i$$

$$=\log s_0^i+R_t^i-\frac{1}{2}\sum _{j=1}^{d_1}({\sigma }_R^{ij}{)}^{2}t,i=1,\dots ,d.$$

$$d{\mu }_t=\kappa (\overline{\mu }-{\mu }_t)dt+{\sigma }_{\mu }d{W}_t^{\mu },$$

$$\displaystyle\mu_{t}=\overline{\mu}+e^{-\kappa t}\Big{[}(\mu_{0}-\overline{\mu})+\int_{0}^{t}e^{\kappa s}\sigma_{\mu}dW^{\mu}_{s}\Big{]},\quad t\geq 0.$$

$$\mathbb{E}[{\mu }_t]$$

$$\mathrm{Cov}({\mu }_s,{\mu }_t)$$

$$Z_k={\mu }_{T_k}+{\Gamma }^{\frac{1}{2}}{\epsilon }_k,k=1,2,\dots ,$$

$$d{J}_t={\mu }_{t}dt+{\sigma }_{J}d{W}_t^J,$$

$$\@mkpream{rcll}\@arrayleft\vbox{\halign{\bgroup\@arstrut\@preamble\cr}}$$

Full text

1 Introduction

In this paper we investigate a hidden Gaussian model (HGM) for a financial market where asset prices follow a diffusion process with an unobservable Gaussian mean reverting drift modelled by an Ornstein-Uhlenbeck process. Such models are widely used in the study of portfolio optimization problems under partial information on the drift. There are two popular model classes for the drift, the above-mentioned HGM and the hidden Markov model (HMM) in which the drift process is a continuous-time Markov chain. For utility maximization problems under HGM we refer to Lakner [Lakner(1998)] and Brendle [Brendle2006] while HMMs are used in Rieder and Bäuerle [Rieder_Baeuerle2005], Sass and Haussmann [SassandHaussmann(2004)]. Both models are studied in Putschögl and Sass [PutschoeglandSass(2008)]. A generalization of these approaches and further references can be found in Björk et al. [Bjoerketal(2010)].

For solving portfolio problems under partial information the drift has to be estimated from observable quantities such as stock returns. For the above two models, HGM and HMM, the conditional distribution of the drift process given the return observations can be described completely by finite-dimensional filter processes. This allows for efficient solutions to portfolio problems including the computation of an optimal policy. For HGM and HMM finite-dimensional filters are known as the Kalman and Wonham filters, respectively, see e.g. Elliott, Aggoun and Moore [Elliottetal.(1994)], Liptser and Shiryaev [Liptser-Shiryaev].

It is well known that the drift of a diffusion process is particularly hard to estimate. Even the estimation of a constant drift would require empirical data over an extremely large time horizon, see Rogers [Rogers(2013), Chapter 4.2]. Therefore, in practice filters computed from historical price observations lead to drift estimates of quite poor precision since drifts tend to fluctuate randomly over time and drift effects are overshadowed by volatility. At the same time optimal investment strategies in dynamic portfolio optimization depend crucially on the drift of the underlying asset price process. For these reasons, practitioners also utilize external sources of information such as news, company reports, ratings or their own intuitive views on the future asset performance for the construction of optimal portfolio strategies. These outside sources of information are called expert opinions. The idea goes back to the celebrated Black-Litterman model which is an extension of the classical one-period Markowitz model, see Black and Litterman [Black_Litterman(1992)]. It uses Bayesian updating to improve drift estimates.

Contrary to the classical static one-period model, we consider a continuous-time model for asset prices where additional information in the form of expert opinions arrives repeatedly over time. Davis and Lleo [DavisandLleo(2018)] termed that approach ``Black-Litterman in Continuous-Time'' (BLCT). The first papers addressing BLCT are Frey et al. (2012) [Freyetal.(2012)] and their follow-up paper [Frey-Wunderlich-2014]. They consider an HMM for the drift and expert opinions arriving at the jump times of a Poisson process and study the maximization of expected power utility of terminal wealth. An HGM and expert opinions arriving at fixed and known times have been investigated in Gabih et al. [Gabihetal(2014)] for a market with only one risky stock, and generalized in Sass et al. [Sassetal(2017)] for markets with multiple risky stocks. Here, the authors consider maximization of logarithmic utility. Davis and Lleo [DavisandLleo(2013), DavisandLleo(2018)] consider BLCT for power utility maximization under an HGM and expert opinions arriving continuously over time. This allows for quite explicit solutions for the portfolio optimization problem. In [DavisandLleo(2018)], the authors also focus on the calibration of the model for expert opinions to real-world data.

In a recent paper Sass et al. [Sassetal(2018)] consider an HGM with expert opinions both at fixed as well as random information dates and investigate the asymptotic behavior of the filter for increasing arrival frequency of the expert opinions. They assume that a higher frequency of expert opinions is only available at the cost of accuracy. In particular, the variance of expert opinions grows linearly with the arrival frequency. This assumption reflects that it is not possible for investors to gain unlimited information in a finite time interval. Furthermore, it allows to find a certain asymptotic behavior that yields reasonable filter approximations for investors observing discrete-time expert opinions arriving with a fixed and sufficiently large intensity. The authors derive limit theorems which state that the information obtained from observing high-frequency discrete-time expert opinions is asymptotically the same as that from observing a certain diffusion process that has the same drift as the return process. The latter process can be interpreted as a continuous-time expert which permanently delivers noisy information about the drift. These so-called diffusion approximations show how the BLCT model of Davis and Lleo [DavisandLleo(2013), DavisandLleo(2018)] who work with continuous-time expert opinions can be obtained as a limit of BLCT models with discrete-time experts.

The present paper can be considered as a companion paper to the above mentioned work of Sass et al. [Sassetal(2018)]. However, contrary to [Sassetal(2018)] we assume that the expert's reliability expressed by its variance remains bounded when the arrival intensity increases. For the sake of simplicity we restrict to the case of a constant variance. This leads to a different asymptotic regime corresponding to the Law of Large Numbers while the results in [Sassetal(2018)] are in the sense of Functional Central Limit Theorems.

When the arrival intensity increases, the investor receives more and more noisy signals about the current state of the drift of the same precision. It is then expected that in the limit the drift estimate is perfectly accurate and equals the actual drift, i.e., the investor has full information about the drift. While this statistical consistency of the estimator seems to be intuitively clear a rigorous proof is an open issue and will be addressed in this paper. Gabih et al. [Gabihetal(2014)] and Sass et al. [Sassetal(2017)] provide such proof only for the case of fixed and known information dates. However, their results and methods cannot be applied to the present model with random information dates. Note that also the methods for the proof of the diffusion limits in [Sassetal(2018)] do not carry over to the present case of fixed expert's reliability. To the best of our knowledge the techniques for proving convergence constitute a new contribution to the literature. Compared to [Gabihetal(2014)] and [Sassetal(2017)] we do not only give a rigorous convergence proof but we are also able to determine the rate of convergence and give explicit bounds for the estimation error.

In this paper we concentrate on the asymptotic properties of drift estimates which are based on Kalman filter techniques and described by the conditional mean and covariance matrix of the drift given the observations. We show that for increasing arrival intensity of expert opinions, the expectation of the conditional covariance goes to zero. This implies that the conditional mean is a consistent drift estimator, it converges to the hidden drift in the mean-square sense. We expect that these convergence results carry over to the value functions of portfolio optimization problems but do not include these studies in this paper. For the maximization of expected logarithmic utility, the convergence of value functions has already been proven in Sass et al. [Sassetal(2018)]. The case of power utility will be addressed in our follow-up paper [Gabihetal(2018-2)].

The paper is organized as follows: In Section 2 we introduce the model for our financial market including expert opinions and define information regimes for investors with different sources of information. For each of those information regimes, we state the dynamics of the corresponding conditional mean and conditional covariance process in Section LABEL:Filtering. Section LABEL:asymptotic_filter contains our main contributions and studies the asymptotic filter behavior for increasing arrival intensity of discrete-time expert opinions. First, Lemma LABEL:alpha_Properties gives an estimate for the drift term in the semimartingale representation of the conditional covariance process. Based on this estimate, Theorem LABEL:Convergence_trace shows that as the arrival intensity increases the expectation of the conditional covariance goes to zero. As a consequence, Theorem LABEL:Convergence_filter states the mean-square convergence of the conditional mean to the hidden drift. In Section LABEL:asymptotic_filter_cont, we study a related problem for continuous-time expert opinions that arises in the case of diffusion approximations of discrete-time expert opinions. Section LABEL:numerics illustrates the convergence results by some numerical experiments. In Appendix LABEL:appendix we collect some auxiliary results and technical proofs needed for our main theorems.

1.0.0.1 Notation

Throughout this paper, we use the notation $I_{d}$ for the identity matrix in $\mathbb{R}^{d\times d}$. For a symmetric and positive-semidefinite matrix $A\in\mathbb{R}^{d\times d}$ we call a symmetric and positive-semidefinite matrix $B\in\mathbb{R}^{d\times d}$ the square root of $A$ if $B^{2}=A$. The square root is unique and will be denoted by $A^{\frac{1}{2}}$.

For a vector $X$ we denote by $\left\lVert X\right\rVert$ the Euclidean norm. For a square matrix $A$ we denote by $\left\lVert A\right\rVert$ a generic matrix norm, by $\left\lVert A\right\rVert_{F}=\sqrt{\sum_{i,j}(A^{ij})^{2}}$ the Frobenius norm and by $\operatorname{tr}(A)=\sum_{i}A^{ii}$ the trace of $A$.

2 Financial Market

2.1 Price Dynamics

The setting is based on Gabih et al. [Gabihetal(2014)] and Sass et al. [Sassetal(2017), Sassetal(2018)]. For a fixed date $T>0$ representing the investment horizon, we work on a filtered probability space $(\Omega,\mathcal{G},\mathbb{G},P)$, with filtration $\mathbb{G}=(\mathcal{G}_{t})_{t\in[0,T]}$ satisfying the usual conditions. All processes are assumed to be $\mathbb{G}$-adapted.

We consider a market model for one risk-free bond with constant risk-free interest rate and $d$ risky securities whose return process $R=(R^{1},\ldots,R^{d})$ is defined by

[TABLE]

for a given $d_{1}$-dimensional $\mathbb{G}$-adapted Brownian motion $W^{R}$ with $d_{1}\geq d$. The constant volatility matrix $\sigma_{R}\in\mathbb{R}^{d\times d_{1}}$ is assumed to be such that $\Sigma_{R}:=\sigma_{R}\sigma_{R}^{\top}$ is positive definite. In this setting the price process $S=(S^{1},\ldots,S^{d})$ of the risky securities reads as

[TABLE]

Note that we can write

[TABLE]

So we have the equality $\mathbb{G}^{R}=\mathbb{G}^{\log S}=\mathbb{G}^{S}$, where for a generic process $X$ we denote by $\mathbb{G}^{X}$ the filtration generated by $X$. This is useful since it allows to work with $R$ instead of $S$ in the filtering part.

The dynamics of the drift process $\mu=(\mu_{t})_{t\in[0,T]}$ in (2.1) are given by the stochastic differential equation (SDE)

[TABLE]

where $\kappa\in\mathbb{R}^{d\times d}$, $\sigma_{\mu}\in\mathbb{R}^{d\times d_{2}}$ and $\overline{\mu}\in\mathbb{R}^{d}$ are constants such that the matrices $\kappa$ and $\Sigma_{\mu}:=\sigma_{\mu}\sigma_{\mu}^{\top}$ are positive definite, and $W^{\mu}$ is a $d_{2}$-dimensional Brownian motion independent of $W^{R}$ with $d_{2}\geq d$. Here, $\overline{\mu}$ is the mean-reversion level, $\kappa$ the mean-reversion speed and $\sigma_{\mu}$ describes the volatility of $\mu$. The initial value $\mu_{0}$ is assumed to be a normally distributed random variable independent of $W^{\mu}$ and $W^{R}$ with mean $\overline{m}_{0}\in\mathbb{R}^{d}$ and covariance matrix $\overline{q}_{0}\in\mathbb{R}^{d\times d}$ assumed to be symmetric and positive semidefinite. It is well known that SDE (2.4) has the closed-form solution

[TABLE]

This is a Gaussian process and known as Ornstein-Uhlenbeck process. It has mean value and covariance function

[TABLE]

2.2 Expert Opinions

We assume that investors observe the return process $R$ but they neither observe the factor process $\mu$ nor the Brownian motion $W^{R}$. They do however know the model parameters such as $\sigma_{R},\kappa,\overline{\mu},\sigma_{\mu}$ and the distribution $\mathcal{N}(\overline{m}_{0},\overline{q}_{0})$ of the initial value $\mu_{0}$. Information about the drift $\mu$ can be drawn from observing the returns $R$. A special feature of our model is that investors may also have access to additional information about the drift in form of expert opinions such as news, company reports, ratings or their own intuitive views on the future asset performance. The expert opinions provide noisy signals about the current state of the drift arriving at discrete points in time $T_{k}$. We model these expert opinions by a marked point process $(T_{k},Z_{k})_{k}$, so that at $T_{k}$ the investor observes the realization of a random vector $Z_{k}$ whose distribution depends on the current state $\mu_{T_{k}}$ of the drift process. The arrival dates $T_{k}$ are modelled as jump times of a standard Poisson process with intensity $\lambda>0$, independent of both the Brownian motions $W^{R},W^{J}$ and the initial value of the drift $\mu_{0}$, so that the timing of the information arrival does not carry any useful information about the drift. For the sake of convenience we also write $T_{0}:=0$ although no expert opinion arrives at time $t=0$.

The signals or ``the expert views'' at time $T_{k}$ are modelled by $\mathbb{R}^{d}$-valued Gaussian random vectors $Z_{k}=(Z_{k}^{1},\cdots,Z_{k}^{d})^{\top}$ with

[TABLE]

where the matrix $\Gamma\in\mathbb{R}^{d\times d}$ is symmetric and positive definite. Further, $(\varepsilon_{k})_{k\geq 1}$ is a sequence of independent standard normally distributed random vectors, i.e., $\varepsilon_{k}\sim\mathcal{N}(0,I_{d})$. It is also independent of the Brownian motions $W^{R},W^{\mu}$, the initial value $\mu_{0}$ and the arrival dates $(T_{k})_{k\geq 1}$. That means that, given $\mu_{T_{k}}$, the expert opinion $Z_{k}$ is $\mathcal{N}(\mu_{T_{k}},\Gamma)$-distributed. So, $Z_{k}$ can be considered as an unbiased estimate of the unknown state of the drift at time $T_{k}$. The matrix $\Gamma$ is a measure of the expert's reliability. In a model with $d=1$ risky asset $\Gamma$ is just the variance of the expert's estimate of the drift at time $T_{k}$: the larger $\Gamma$ the less reliable is the expert.

Note that one may also allow for relative expert views where experts give an estimate for the difference in the drift of two stocks instead of absolute views. This extension is studied in Schöttle et al. [Schoettleetal.(2010)] where the authors show how to switch between these two models for expert opinions by means of a pick matrix.

Finally, we introduce expert opinions arriving continuously over time. This is motivated by the results of Sass et al. [Sassetal(2018)]. There the authors consider the information drawn from observing certain sequences of expert opinions and show that for a large number of expert opinions it is essentially the same as the information resulting from observing another diffusion process. The interpretation of that diffusion process is an expert providing continuous-time estimates about the state of the drift. Let this estimate be given by the diffusion process

[TABLE]

where $W^{J}$ is an $d_{3}$-dimensional Brownian motion with $d_{3}\geq d$ that is independent of all other Brownian motions in the model and of the information dates $T_{k}$. The constant matrix $\sigma_{J}\in\mathbb{R}^{d\times d_{3}}$ is assumed to be such that $\Sigma_{J}:=\sigma_{J}\sigma_{J}^{\top}$ is positive definite.

2.3 Investor Filtration

We consider various types of investors with different levels of information. The information available to an investor is described by the investor filtration $\mathbb{F}^{H}=(\mathcal{F}^{H}_{t})_{t\in[0,T]}$. Here, $H$ denotes the information regime for which we consider the cases $H=R,Z,J,F$, where

[TABLE]