Optimal investment with a noisy signal of future stock prices

Information is arguably a key driver in stochastic optimal control problems as policy choice strongly depends on assessments of the future evolution of a controlled system and on how these can be expected to change over time. This is particularly obvious in Finance where information drives price fluctuations and where the question how it affects investment decisions is still a topic of great interest for financial economics and mathematics alike.

In part this may be due to the many ways information can be modeled and due to the various difficulties that come with each such model choice. If, as we are setting out to do in this paper, one seeks to understand how imperfect signals on future stock price evolution can be factored into present investment decisions, one needs to specify how the signal and the stock price are connected and work out the trading opportunities afforded by the signal. For a dynamic specification of the signal and its noise such as ours, this often leads to challenging combinations of filtering and optimal control and we refer to [8, 12, 7, 20, 2] for classical and more recent work in this direction.

The present paper contributes to this literature with a simple Bachelier stock price model driven by two independent Brownian motions. The investor dynamically gets advanced knowledge about the evolution of one of these over some time period, but the noise arising from the other Brownian motion makes perfect prediction of stock price fluctuations impossible. Accounting in addition for temporary price impact from finite market depth (cf. [16]) as suggested in the Almgren-Chriss model [1], we arrive at a viable expected utility maximization problem. In fact, for exponential utility we follow an approach similar to the one of [4], where the case of perfect stock price predictions is considered, and we work out explicitly the optimal investment strategy and maximum expected utility in this case. We can even allow for dynamically varying horizons over which peeks ahead are possible, thus extending beyond the fixed horizon setting of [4] (which is still included as a special case).

Our model also complements the literature on optimal order execution where different forms of market signals have been studied. For instance, [9] add a Markovian drift to the stock price evolution to model market trends as perceived by the agent; latent factors are added in [10]. With transient price impact rather than temporary one as in our setting, [17] and [19] study such signals and [6] adds a version with non-Markovian finite variation signals. Yet another version of market signals is considered in [5] who use Meyer $\sigma$-fields to model ultra short-term signals on the jumps in the order flow.

Describing how signals are specifically processed into strategies is a challenge for each of these models. In our setting, we find that the investor should optimally weigh her noisy signals of future stock price evolution and aggregate these weights into a projection of its fluctuations. The difference of this projection with the present stock price determines the signal-based part of her investment strategy. The relevant projection weights are computed explicitly and we analyze how they depend on the signal to noise ratio. On top of that, the investor also is interested in chasing the risk premium afforded by the stock and thus aims to take her portfolio to the Merton ratio, well-known from standard expected utility maximization. In addition, we investigate the value of reductions in noise and show that it is convex in signal quality. In other words, incentives for signal improvement turn out to be skewed in favor of investors who are already good at sniffing out the market’s evolution. Finally, we also use the flexibility in the signal horizon of our model to investigate when our investor would most like to be peek ahead if she can do so only once and how much of an advantage a continually probing investor has over her more relaxed counterpart who does so only periodically.

Mathematically, our method is based on the general duality result given in [4]. Here, however, we need to understand how changes of measures can be optimized when two Brownian motions are affected. Fortunately, we again find the problem to reduce to deterministic variational problems, which, albeit more involved, remain quadratic and thus can still be solved explicitly. The time-dependent knowledge horizon adds further challenges, but also affords us the opportunity to show that the problem is continuous with respect to perturbations in signal reach.

The present paper generalizes [4] and, while similar in its line of attack, the method of solution here differs in essential ways. Indeed, both papers apply the dual approach and do not deal with the primal infinite dimensional stochastic control problem directly. But the current paper decomposes the information flow into independent parts and passes to a conditional model driven by independent Brownian motions in the ‘usual’ way. This approach does not require applying the results from the theory of Gaussian Volterra integral equations [15, 14] and readily accommodates time-dependent signal specifications and noise.

Section 2 of the paper formalizes the optimal investment problem with dynamic noisy signals mathematically. Section 3 states the main results on optimal investment strategy and utility and comments on some financial-economic implications. Section 4 is devoted to the proof of the main results.

Consider an investor who can invest over some time horizon $[0,T]$ both in the money market at zero interest (for simplicity) and in stock. The price of the stock follows Bachelier dynamics

where $S_{0}\in\mathbb{R}$ denotes the initial stock price, $\mu\in\mathbb{R}$ describes the stock’s risk premium and $\sigma\in(0,\infty)$ its volatility; $\gamma\in[-1,1]$ and $\bar{\gamma}:=\sqrt{1-\gamma^{2}}$ parameterize the correlation between the stock price process $S$ and its drivers $W$, $W^{\prime}$, two independent Brownian motions specified on a complete probability space $(\Omega,\mathcal{F},\mathbb{P})$. On top of the information flow from present and past stock prices as captured by the augmented filtration $(\mathcal{F}^{S}_{t})_{t\in[0,T]}$ generated by $S$, the investor is assumed to have access to a signal which at any time $t\in[0,T]$ allows her to deduce the future evolution of $W^{\prime}$ (but not of $W$) over a time window $[t,\tau(t)]$ where $\tau:[0,T]\to[0,T]$ is a right-continuous, nondecreasing time shift satisfying $\tau(t)\geq t$ throughout. In other words, the investor is able to partially predict the evolution of future stock prices, albeit with some uncertainty. The uncertainty is described by a noise whose variance accrues over $[t,\tau(t)]$ solely from $W$ at the rate $\sigma^{2}(1-\gamma^{2})$ and accrues from both $W$ and $W^{\prime}$ at the joint rate $\sigma^{2}$ afterwards. As a result, the investor can draw on the information flow given by the filtration

when making her investment decisions.

In case $\gamma=\pm 1$ all noise is wiped out from the stock price signal and so the investor gets perfect knowledge of some future prices, affording her obvious arbitrage opportunities. But even for the complementary case $\gamma\in(-1,1)$ it is easy to check that $S$ is not a semimartingale and, by the Fundamental Theorem of Asset Pricing of [11], there is a free lunch with vanishing risk (and in fact even a strong arbitrage as can be shown by a Borel-Cantelli argument similar to the one in [18]). As a consequence, we need to curb the investor’s trading capabilities in order to maintain a viable financial model. In line with the economic view of the role and effect of arbitrageurs, we choose to accomplish this by taking into account the market impact from the investor’s trades. These cause execution prices for absolutely continuous changes $d\Phi_{t}=\phi_{t}dt$ in the investor’s position to be given by

So, when marking to market her position $\Phi_{t}=\Phi_{0}+\int_{0}^{t}\phi_{s}ds$ in the stock accrued by time $t\in[0,T]$, the investor will consider her net profit to be

As a consequence, the investor will have to choose her turnover rates from the class of admissible strategies

Assuming for convenience constant absolute risk aversion $\alpha\in(0,\infty)$, the investor would then seek to

The paper’s main results are collected in the following theorem.

Let us briefly collect some financial-economic observations from this result. First, similar to the case without signal noise discussed in [4], an average of future price proxies $\widehat{S}_{t,h}$ is formed, but here weights are time-inhomogenous. In fact, the weights used in this averaging are determined in terms of the function $\Upsilon$ and interestingly depend on the peek-ahead horizon $\tau$ only through its present value $\tau(t)$. As a consequence, when deciding about the time $t$ turnover rate $\widehat{\phi}_{t}$, the investor does not care if her information horizon $\tau$ is going to stall or to jump all the way to $T$ soon. The reason for this is the investor’s exponential utility which makes investment decisions insensitive to present wealth and thus does not require planning ahead. It would be interesting to see how this changes with different utility functions. Unfortunately, our method to obtain an explicit solution to the optimal investment problem strongly depends on our choice of exponential utility, leaving this a challenge for future research.

Second, it is interesting to assess the impact of signal noise on investment decisions. Here, we observe that the emphasis $\Upsilon_{t}(0)/\Upsilon_{t}(\tau(t)-t)$ that the projection $\bar{S}_{t}$ puts on the last learned signal $\widehat{S}_{t,\tau(t)-t}$ decreases when noise gets stronger due to an increase in $\overline{\gamma}$. This reflects the investor’s reduced trust in this most noisy of her signals when noise becomes more prominent.

Third, we can assess how the presence of noise affects the value of the signal $W^{\prime}$. To this end, let us consider the certainty equivalent

that an agent, who can eliminate a fraction $\gamma^{2}$ from the variance in stock price noise $\gamma W^{\prime}+\sqrt{1-\gamma^{2}}W$ by observing part of $W^{\prime}$, gains compared to her peer without that privilege (whence her terminal wealth $\tilde{V}_{T}$ is determined by $\tilde{S}_{t}=S_{0}+\mu t+\sigma W_{t}$, $t\in[0,T]$, instead of $S$). Contributions to this quantity from times long before the investment horizon ($T-t$ large) when the signal $W^{\prime}_{t}$ has been received for a long time ($t-\tau^{-1}(t))$ large), depend on $\gamma$ by a factor of about $\gamma^{2}/(\sqrt{1-\gamma^{2}}+1)$. This factor increases from 0 to 1 as $\gamma$ increases from 0 (no signal) to 1 (noiseless signal), with the steepest increase at $\gamma=1$, indicating limited, but ever higher returns from noise reductions. Conversely, an increase in noise is making itself felt the most when the signal is already quite reliable.

Finally, the flexibility concerning the form of the peek-ahead length given by $\tau:[0,T]\to[0,T]$ affords one to account for periods when predictions are harder to make over periods where they are easier. Moreover, this flexibility also allows us to shed light on some natural questions such as the following ones:

1. (i)

   When best to peek ahead? Suppose that the investor has an opportunity to choose a (deterministic) moment of time $\mathbb{T}\in[0,T]$ when, for one time only, she can peek ahead over some period $\Delta>0$ into the future. What time $\mathbb{T}$ should she choose? We can formalize this as an optimization problem over the family of time changes

   $$\tau_{\mathbb{T}}(t)=\left.\begin{cases}t,&\text{for }t\leq\mathbb{T}\\ (t\vee(\mathbb{T}+\Delta))\wedge T,&\text{otherwise}\\ \end{cases}\right\},\ \ \ \mathbb{T}\in[0,T].$$

   Clearly, the corresponding inverse function is given by

   $$\tau^{-1}_{\mathbb{T}}(t)=\left.\begin{cases}\mathbb{T},&\text{for }\mathbb{T}\leq t\leq\mathbb{T}+\Delta\\ t,&\text{otherwise}\\ \end{cases}\right\}.$$

   Hence, in view of (3.8) we need to maximize over $\mathbb{T}\in[0,T]$ the value of

   $$\int_{\mathbb{T}}^{(\mathbb{T}+\Delta)\wedge T}\frac{dt}{\overline{\gamma}\coth\left(\overline{\gamma}\sqrt{\rho}\left(t-\mathbb{T}\right)\right)+\tanh\left(\sqrt{\rho}\left(T-t\right)\right)}\rightarrow\max.$$

   Observe that

   	$\displaystyle\int_{\mathbb{T}}^{(\mathbb{T}+\Delta)\wedge T}\frac{dt}{\overline{\gamma}\coth\left(\overline{\gamma}\sqrt{\rho}\left(t-\mathbb{T}\right)\right)+\tanh\left(\sqrt{\rho}\left(T-t\right)\right)}$
   	$\displaystyle=\int_{0}^{(T-\mathbb{T})\wedge\Delta}\frac{dt}{\overline{\gamma}\coth\left(\overline{\gamma}\sqrt{\rho}t\right)+\tanh\left(\sqrt{\rho}\left(T-\mathbb{T}-t\right)\right)}$
   	$\displaystyle\leq\int_{0}^{T\wedge\Delta}\frac{dt}{\overline{\gamma}\coth\left(\overline{\gamma}\sqrt{\rho}t\right)+\tanh\left(\sqrt{\rho}\left(T-t\right)\right)}$

   We conclude that the optimal time is $\mathbb{T}=0$ and so one should peek ahead immediately.

2. (ii)

   How much of an advantage does continual probing have over periodic probing? For this we can compare the continual probing where we keep predicting what is going to happen over the next $\Delta$ time units with its discrete counter part where we only peek ahead every $\Delta$ units:

   $\displaystyle\tau^{c}(t)=(t+\Delta)\wedge T\quad\text{vs.}\quad\tau^{p}(t)=\left((\lfloor\frac{t}{\Delta}\rfloor+1)\Delta\right)\wedge T,$

   where $\lfloor 2.8\rfloor=2$ denotes the floor function. Clearly, the continual probing is superior to periodic probing, but its advantage depends on the probing period length $\Delta$. Indeed, Figure 1 below shows the difference in certainty equivalents for the choices $\Delta=T/n$ corresponding to $n=1,2,\dots$ updates in the periodic case. We see that, for the considered parameters, the periodically updating investor will be at the greatest disadvantage compared to her more diligent, continually probing counterpart when updating about 80 times for periods of length $\Delta=T/80$.

   

We start this section by outlining the main steps of the proof. The first step in Section 4.1 deals with simplifying the general setup to obtain a notationally and computationally more convenient special case of our problem. The second step (Section 4.2) consists of the main idea which is to apply duality theory. We show that the dual problem can be reduced to the solution of three separate deterministic variational problems. The third step (Section 4.3) is to solve these deterministic problems, which are quite involved and for which we therefore compute only their values and the essential properties of the corresponding optimal control. The fourth step (Section 4.4) combines the second and the third step to solve the dual problem. The last step (Section 4.5) uses duality and the Markovian structure of our problem to finally construct the solution to the primal problem from the dual solution.