Diffusion Approximations for Expert Opinions in a Financial Market with Gaussian Drift

Optimal trading strategies in dynamic portfolio optimization problems depend crucially on the drift of the underlying asset price processes. However, drift parameters are notoriously difficult to estimate from historical asset price data. Drift processes tend to fluctuate randomly over time and even if they were constant, long time series would be needed to estimate this parameter with a satisfactory degree of precision. Typically, drift effects are overshadowed by volatility. For these reasons, practitioners also incorporate external sources of information such as news, company reports, ratings or their own intuitive views when determining optimal portfolio strategies. These outside sources of information are called expert opinions. In the context of the classical one-period Markowitz model this leads to the well-known Black–Litterman approach, where return predictions are improved by means of views formulated by securities analysts, see Black and Litterman [3].

In this paper we consider a financial market where returns depend on an underlying drift process which is unobservable due to additional noise coming from a Brownian motion. The general setting has already been studied in Gabih et al. [11] for a market with only one risky asset and in Sass et al. [23] for markets with an arbitrary number of stocks. The ability to choose good trading strategies depends on how well the unobserved drift can be estimated. For estimating the hidden drift we consider the conditional distribution of the drift given the available observations, the so-called filter. The best estimate for the hidden drift process in a mean-square sense is the conditional mean of the drift given the available information. A measure for the goodness of this estimator is its conditional covariance matrix. In our setting, the filter is completely characterized by conditional mean and conditional covariance matrix since we deal with Gaussian distributions.

For investors who observe only the return process, the filter is the classical Kalman filter, see for example Liptser and Shiryaev [19]. An additional source of information is provided by expert opinions which we model as unbiased drift estimates arriving at discrete points in time. Investors who, in addition to observing the return process, have access to these expert opinions update their current drift estimates at each arrival time. These updates decrease the conditional covariance, hence they yield better estimates. This can be seen as a continuous-time version of the above mentioned static Black–Litterman approach.

We investigate in detail an investor who observes both the return process and the discrete-time expert opinions and study the asymptotic behavior of the filter when the frequency of the arrival of expert opinions tends to infinity. Sass et al. [23] and Gabih et al. [12] already addressed expert opinions which are independent of the arrival frequency and which have some minimal level of accuracy characterized by bounded covariances. In that setting, the conditional covariance of the drift estimate goes to zero as the arrival frequency goes to infinity. This implies that the conditional mean converges to the true drift process, i.e. in the limit investors have full information about the drift. Here, we study a different situation in which a higher frequency of expert opinions is only available at the cost of accuracy of the single expert opinions. In other words, as the frequency of expert opinions increases, the variance of expert opinions becomes larger. On the one hand, this assumption ensures that it is not possible for investors to gain arbitrarily much information in a fixed time interval. On the other hand, it enables us to derive a certain asymptotic behavior that yields a reasonable approximation of the filter for the investor who observes a certain, fixed number of discrete-time expert opinions. We consider two different situations, one with deterministic equidistant information dates and one with information dates that arrive randomly as the jump times of a Poisson process. For properly scaled variance of expert opinions that grows linearly with the arrival frequency we prove $\mathrm{L}\rule{0.0pt}{7.5pt}^{p}$-convergence of the conditional mean and conditional covariance matrices as the frequency of information dates goes to infinity. Our limit theorems imply that the information obtained from observing the discrete-time expert opinions is asymptotically the same as that from observing a certain diffusion process having the same drift as the return process. That process can be interpreted as a continuous-time expert who permanently delivers noisy information about the drift.

Our limit theorems allow us to derive approximations of the filter for high-frequency discrete-time expert opinions which we call diffusion approximations. These are useful since the limiting filter is easy to compute whereas the updates for the discrete-time expert opinions lead to a computationally involved filter. This is extremely helpful for deriving simplified approximate solutions of utility maximization problems. We apply our diffusion approximations to a portfolio optimization problem with logarithmic utility. Numerical simulations show that the approximation is very accurate even for a small number of expert opinions. Our rigorous $\mathrm{L}\rule{0.0pt}{7.5pt}^{p}$-convergence results of the filters however also allow to derive convergence of the value function in the more complicated problem with power utility, see Remark 5.5.

The idea of a continuous-time expert is in line with Davis and Lleo [7] who study an approach called “Black–Litterman in Continuous Time” (BLCT). Our results show how the BLCT model can be obtained as a limit of BLCT models with discrete-time experts. First papers addressing BLCT are Frey et al. [9, 10] who consider an HMM for the drift and expert opinions arriving at the jump times of a Poisson process.

Convergence of the discrete-time Kalman filter to the continuous-time equivalent has been addressed in the literature, e.g. by Salgado et al. [22] or Aalto [1] for the case of deterministic information dates. Our results for that case however do not follow directly from these convergence results. The reason is that in our case a suitable continuous-time expert has to be constructed first. The discrete-time expert opinions are then not simply a discretization of the continuous-time expert. We assume that they are noisy observations of the true drift process where the noise term is correlated with the Brownian motion in the diffusion that forms the continuous-time expert. Contrary to [1, 22] we also obtain convergence results for the case where the discrete expert opinions arrive at random time points rather than on an equidistant time grid.

Coquet et al. [6] consider weak convergence of filtrations which allows to prove convergence of conditional expectations in a quite general setting. However, their results do not directly apply to our situation since the approximating sequence of filtrations in our case is not included in the limit filtration.

In the literature, diffusion approximations also appear in other contexts. They are well-known in operations research and actuarial mathematics. The basic idea is to replace a complicated stochastic process by an appropriate diffusion process which is analytically more tractable than the original process. The approach is comparable with the normal approximation of sums of random variables by the Central Limit Theorem. When looking at these sums as stochastic processes or random walks the well-known Donsker Theorem leads to an approximation by a Brownian motion.

For an introduction to diffusion approximations based on the theory of weak convergence and applications to queueing systems in heavy traffic we refer to the survey article by Glynn [13]. In risk theory the application of diffusion approximations for computing ruin probabilities goes back to Iglehart [15]. We also refer to Grandell [14, Sec. 1.2], Schmidli [24, Sec. 5.10 and 6.5] and Asmussen and Albrecher [2, Sec. V.5] as well as the references therein. Starting point is the classical Cramér–Lundberg model where the cumulated claim sizes and finally the surplus of an insurance company are modeled by a compound Poisson process. For a high intensity of the claim arrivals and small claim sizes the latter can be approximated by a Brownian motion with drift. This results from the corresponding weak convergence of the properly scaled compound Poisson processes to a Brownian motion as the intensity tends to infinity. However, these classical results for compound Poisson processes cannot be applied directly to our problem. Here, the jumps of the filter processes do not constitute a sequence of i.i.d. random variables as in the compound Poisson case. Due to the Bayesian updating of the filter at the information dates the jump size distribution depends on the value of the filter at that time. This requires special techniques for proving limit theorems from which the diffusion approximations can be derived. To the best of our knowledge these techniques constitute a new contribution to the literature.

The paper is organized as follows. In Section 2 we introduce the model for our financial market including expert opinions and define different 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 matrix. Section 3 investigates the situation where the discrete-time expert opinions arrive at deterministic equidistant time points. For an investor observing returns and discrete-time expert opinions we show convergence of the corresponding conditional mean and conditional covariance matrix to those of an investor observing the returns and the continuous-time expert. In Section 4 we prove analogous results for the situation where the time points at which expert opinions arrive are not deterministic time points but jump times of a standard Poisson process, i.e. with exponentially distributed waiting times between information dates. For the conditional mean we can then use a representation involving a Poisson random measure. When letting the intensity of the Poisson process go to infinity, we prove convergence to the same limiting filter as in the case with deterministic information dates. Section 5 provides an application of the convergence results to a utility maximization problem. For investors who maximize expected logarithmic utility of terminal wealth the optimal trading strategy depends on the conditional mean of the drift and the corresponding optimal terminal wealth is a functional of the conditional covariance matrices. That is why the convergence results from Sections 3 and 4 carry over to convergence of the corresponding value functions. Section 6 provides simulations and numerical calculations to illustrate our theoretical results. In Appendix A we collect some auxiliary results needed for the proofs of our main theorems. Appendix B gives the proofs of Theorems 3.2 and 3.3 and Appendix C those of Theorems 4.6 and 4.7.

In this section we investigate the asymptotic behavior of the filters for a $Z$-investor when the frequency of expert opinion arrivals goes to infinity. We consider first the case for deterministic and equidistant information dates. Therefore, let $n\in\mathbb{N}$ and $\Delta_{n}=\frac{T}{n}$. Now assume that $T_{k}=t_{k}$ for every $k=1,\dots,n$, where $(t_{k})_{k=1,\dots,n}$ is the sequence of deterministic time points $t_{k}=k\Delta_{n}$. So there are $n$ expert opinions that arrive equidistantly in the time interval $[0,T]$, the distance between two information dates being $\Delta_{n}$.

In the following, we deduce convergence results for both the conditional means and the conditional covariance matrices of the $Z$-investor when sending $n$ to infinity. Note that convergence of discrete-time filters is addressed in earlier papers, e.g. by Salgado et al. [22] or Aalto [1]. There, the authors show convergence of the discrete-time Kalman filter to the continuous-time equivalent. In Aalto [1] the discrete-time filter is based on discrete-time observations of the continuous-time observation process whereas in Salgado et al. [22] the authors approximate both the continuous-time signal and observation by discrete-time processes. Neither of these assumptions match our model for the discrete-time expert opinions which is why we need to prove convergence in the following.

We use an additional superscript $n$ to emphasize dependence on the number of expert opinions, writing for example $(Q^{Z,n}_{t})_{t\in[0,T]}$ for the conditional covariance matrix of the filter corresponding to these $n$ expert opinions. In Sass et al. [23] a convergence result is proven for the case where the expert opinions are of the form

with expert’s covariances $\Gamma_{k}^{(n)}$ that are bounded for all $n\in\mathbb{N}$ and $k=1,\dots,n$, see Theorem 3.1 in Sass et al. [23]. There it is shown that under the assumption of bounded expert’s covariances it holds

for any $t\in(0,T]$. Since $Q^{Z,n}_{t}$ is a measure for the goodness of the estimator $m^{Z,n}_{t}$, this means that the conditional mean of the $Z$-investor becomes an arbitrarily good estimator for the true state of the drift $\mu_{t}$. One can easily deduce that

for any $t\in(0,T]$. Hence, the $Z$-investor essentially approximates the fully informed $F$-investor.

This result heavily relies on the assumption that the expert covariances $\Gamma_{k}^{(n)}$ are all bounded, meaning that there is some minimal level of reliability of the experts. Here, we study a different situation where more frequent expert opinions are only available at the cost of accuracy. In other words, we assume that, as $\Delta_{n}$ goes to zero, the variance of expert opinions $Z_{k}^{(n)}$ increases. This is done for the purpose of approximating $m^{Z,n}$ and $Q^{Z,n}$ for large $n\in\mathbb{N}$ and large $\Gamma_{k}^{(n)}$. In the following we assume for the sake of simplicity that $\Gamma_{k}^{(n)}=\Gamma^{(n)}$ is not time-dependent. We then show that for properly scaled $\Gamma^{(n)}$ which grows linearly in $n$, the information obtained from observing the discrete-time expert opinions is asymptotically the same as that from observing another diffusion process. This will be the diffusion $J$ already defined in (2.1).

Recall that the matrix $\sigma_{J}\in\mathbb{R}^{d\times l}$ is exactly the volatility of the diffusion process $J$ with the dynamics

and that $\sigma_{J}$ has full rank. With $Z^{(n)}_{k}$ as defined above the discrete-time expert opinions and the continuous-time expert $J$ are obviously correlated. In fact, it holds

Further, one can easily show by using Donsker’s Theorem that the piecewise constant process $(\widetilde{J}_{t})_{t\in[0,T]}$, defined by

for all $t\in[0,T]$, converges in distribution to $J_{t}$ as $n$ goes to infinity. For our main convergence results that are given in the following, we however require stronger notions of convergence.

The following theorem now states uniform convergence of $Q^{Z,n}_{t}$ to $Q^{J}_{t}$ on $[0,T]$ for $n$ going to infinity.

The proof of Theorem 3.2 is given in Appendix B. It makes use of a discrete version of Gronwall’s Lemma for error accumulation, see Lemma A.1 in Appendix A.

Using the uniform convergence of the conditional covariance matrices $Q^{Z,n}$ to $Q^{J}$ we can also deduce convergence of the corresponding conditional mean $m^{Z,n}$ to $m^{J}$ in an $\mathrm{L}\rule{0.0pt}{7.5pt}^{p}$-sense.

The proof of Theorem 3.3 can also be found in Appendix B. Theorems 3.2 and 3.3 state that in the setting of Assumption 3.1 the filter of a $Z$-investor observing $n$ equidistant expert opinions on $[0,T]$ converges to the filter of the $J$-investor. Recalling that the $J$-investor observes the diffusion processes $R$ and $J$, this implies that the information obtained from observing the discrete-time expert opinions is for large $n$ arbitrarily close to the information that comes with observing the continuous-time diffusion-type expert $J$. This diffusion approximation of the discrete expert opinions is useful since the associated filter equations for $m^{J}$ and $Q^{J}$ are much simpler than those for $m^{Z,n}$ and $Q^{Z,n}$ which contain updates at information dates. Computing $Q^{Z,n}$ on $[0,T]$ in the multivariate case requires the numerical solution of a Riccati differential equation on each subinterval $[t_{k},t_{k+1})$. For high numbers $n$ of expert opinions this leads to very small time steps and high computing times. For computing the $J$-investor’s filter one has to find the solution to only one Riccati differential equation on $[0,T]$ for which we can use more efficient numerical solvers. We will see in Section 5 that the convergence results carry over to convergence of the value function in a portfolio optimization problem.

In this section we consider the situation where the experts’ opinions do not arrive at deterministic time points but at random information dates $T_{k}$, where the waiting times $T_{k+1}-T_{k}$ between information dates are independent and exponentially distributed with rate $\lambda>0$. Recall that we have set $T_{0}=0$ for ease of notation. The information dates can therefore be seen as the jump times of a standard Poisson process with intensity $\lambda$. In this situation, the total number of expert opinions arriving in $[0,T]$ is no longer deterministic. However, as the intensity $\lambda$ increases, expert opinions will arrive more and more frequently. So the question we address in this section is, in analogy to sending $n$ to infinity in the last section, what happens when $\lambda$ goes to infinity. We use a superscript $\lambda$ to emphasize the dependence on the intensity. The expert opinions are of the form

For constant variances $\Gamma_{k}^{(\lambda)}=\Gamma$, i.e. when there is some constant level of the expert’s reliability which does not depend on the arrival intensity $\lambda$, one can derive a similar result for the convergence to full information as in the case of deterministic information dates. This result implies that for large $\lambda$ the $Z$-investor approximates the fully informed investor. More precisely, it holds

for all $t\in(0,T]$, see Gabih et al. [12]. In contrast to the above case we now again assume that, as the frequency of expert opinions increases, the variance of the expert opinions $Z_{k}^{(\lambda)}$ also increases. As in Section 3 it will turn out that letting $\Gamma_{k}^{(\lambda)}$ grow linearly in $\lambda$ is the proper scaling for deriving diffusion limits.

Given a realization of the drift process at the random information date $T_{k}^{(\lambda)}$, the only randomness in the expert opinion comes from the Brownian motion $W^{J}$ between the deterministic times $\frac{k-1}{\lambda}$ and $\frac{k}{\lambda}$. Recall that $W^{J}$ is the Brownian motion that drives the diffusion $J$ which we interpret as our continuous expert. Hence there is a direct connection between the discrete expert opinions $Z_{k}^{(\lambda)}$ and the continuous expert.

In the following, we will omit the superscript $\lambda$ at the time points $T_{k}^{(\lambda)}$ for better readability, keeping the dependence on the intensity in mind.

The aim of this section is to determine the behavior of the conditional covariance matrix $Q^{Z,\lambda}$ and of the conditional mean $m^{Z,\lambda}$ under Assumption 4.1 when $\lambda$ goes to infinity, i.e. when expert opinions arrive more and more frequently, becoming at the same time less and less reliable. Here, it is useful to express the dynamics of $Q^{Z,\lambda}$ and $m^{Z,\lambda}$ in a way that comprises both the behavior between information dates and the jumps at times $T_{k}$. For this purpose, we work with a representation using a Poisson random measure as introduced in Cont and Tankov [5, Sec. 2.6].

The following proposition states the results we will need in the following. For a proof, see Cont and Tankov [5, Sec. 2.6.3].

Now we can use the Poisson random measure for reformulating the dynamics of $Q^{Z,\lambda}$.

The proof of Proposition 4.5 is given in Appendix A.

In the following, we give convergence results in analogy to those in Theorems 3.2 and 3.3 stating that the conditional covariance matrix and the conditional mean of the $Z$-investor converge to the conditional covariance matrix and conditional mean of the $J$-investor as $\lambda$ goes to infinity. In the setting with deterministic equidistant information dates in Section 3 the conditional covariance matrices were deterministic. For the conditional means we proved $\mathrm{L}\rule{0.0pt}{7.5pt}^{p}$-convergence. Due to the joint Gaussian distribution of the conditional means it was enough to prove $\mathrm{L}^{2}$-convergence and use a result from Rosiński and Suchanecki [21] to generalize to $\mathrm{L}\rule{0.0pt}{7.5pt}^{p}$-convergence. In the setting of this section with random information dates, the conditional covariance matrices of the $Z$-investor are random and the joint distribution of the conditional means is no longer Gaussian. Therefore, the generalization mentioned above does not apply here. Hence, we directly prove $\mathrm{L}\rule{0.0pt}{7.5pt}^{p}$-convergence in the following. The next theorem states $\mathrm{L}\rule{0.0pt}{7.5pt}^{p}$-convergence of $Q^{Z,\lambda}$ to $Q^{J}$ on $[0,T]$ as $\lambda$ goes to infinity.

The proof of Theorem 4.6 is given in Appendix C. It is based on applying Gronwall’s Lemma in integral form which we recall in Lemma A.5. We also prove $\mathrm{L}\rule{0.0pt}{7.5pt}^{p}$-convergence of the conditional means.

The proof of Theorem 4.7 can be found in Appendix C.

Theorems 4.6 and 4.7 show that under Assumption 4.1, the filter of the $Z$-investor converges to the filter of the $J$-investor. These are the analogous results to those in Section 3 where we have assumed deterministic and equidistant information dates. Here, we see that the convergence result also holds for non-deterministic information dates $T_{k}$ being defined as the jump times of a standard Poisson process, i.e. where the time between information dates is exponentially distributed with a parameter $\lambda>0$. When sending $\lambda$ to infinity, the frequency of expert opinions goes to infinity.

Again, as for the case with deterministic information dates, the assumption that $Z_{k}^{(\lambda)}$ is given as in (4.2) is only needed for the proof of Theorem 4.7. For the proof of Theorem 4.6 it is sufficient to assume that the experts’ covariance matrices are of the form $\Gamma_{k}^{(\lambda)}=\Gamma^{(\lambda)}=\lambda\sigma_{J}\sigma_{J}^{\top}$.

The above theorems provide a useful diffusion approximation since the filter of the $J$-investor is easier to compute than the filter of the $Z$-investor for which there are updates at each information date. Further, the conditional covariance $Q^{J}$ is deterministic and can be computed offline in advance while $Q^{Z,\lambda}$ is a stochastic process that has to be updated when a new expert opinion arrives. For high-frequency expert opinions one may simplify the computation of $m^{Z,\lambda}$ by replacing the exact conditional covariance $Q^{Z,\lambda}$ by its diffusion approximation $Q^{J}$. Given the discrete-time expert’s covariance matrix $\Gamma$ and the arrival intensity $\lambda$ the volatility $\sigma_{J}$ is chosen such that $\sigma_{J}\sigma_{J}^{\top}=\lambda^{-1}\Gamma$.

Even more important are the benefits from the simpler filter equations if we consider utility maximization problems for financial markets with partial information and discrete-time expert opinions. See the next section for an application to logarithmic utility and Remark 5.5 as well as Kondakji [17, Ch. 7,8] for the more involved power utility case where closed-form expressions for the optimal strategies are available for the $J$-investor but not for the $Z$-investor.

As an application of the convergence results from the last two sections we now consider a portfolio optimization problem in our financial market. For the sake of convenience, we assume here that the interest rate $r$ of the risk-free asset is equal to zero. However, the results below can easily be extended to a market model with $r\neq 0$.

An investor’s trading in the market can be described by a self-financing trading strategy $(\pi_{t})_{t\in[0,T]}$ with values in $\mathbb{R}^{d}$. Here, $\pi_{t}^{i}$, $i=1,\dots,d$, is the proportion of wealth that is invested in asset $i$ at time $t$. The corresponding wealth process $(X^{\pi}_{t})_{t\in[0,T]}$ is then governed by the stochastic differential equation

with initial capital $X^{\pi}_{0}=x_{0}>0$. An investor’s trading strategy has to be adapted to her investor filtration. To ensure strictly positive wealth, we also impose some integrability constraint on the trading strategies. Then we denote by