Uncertainty Aware Trader-Company Method Interpretable Stock Price Prediction Capturing Uncertainty

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Uncertainty Aware Trader-Company Method:

Interpretable Stock Price Prediction Capturing

Uncertainty

1st Yugo Fujimoto

Innovation Lab

Nomura Asset Management Co, Ltd.

Tokyo, Japan

yu5fujimoto@gmail.com

2nd Kei Nakagawa

Innovation Lab

Nomura Asset Management Co, Ltd.

Tokyo, Japan

kei.nak.0315@gmail.com

3rd Kentaro Imajo

Preferred Networks, Inc.

Tokyo, Japan

imos@preferred.jp

4th Kentaro Minami

Preferred Networks, Inc.

Tokyo, Japan

minami@preferred.jp

Abstract—Machine learning is an increasingly popular tool

with some success in predicting stock prices. One promising

method is the Trader-Company (TC) method, which takes into

account the dynamism of the stock market and has both high

predictive power and interpretability. Machine learning-based

stock prediction methods including the TC method have been

concentrating on point prediction. However, point prediction in

the absence of uncertainty estimates lacks credibility quantiﬁ-

cation and raises concerns about safety. The challenge in this

paper is to make an investment strategy that combines high

predictive power and the ability to quantify uncertainty. We

propose a novel approach called Uncertainty Aware Trader-

Company Method (UTC) method. The core idea of this approach

is to combine the strengths of both frameworks by merging

the TC method with the probabilistic modeling, which provides

probabilistic predictions and uncertainty estimations. We expect

this to retain the predictive power and interpretability of the

TC method while capturing the uncertainty. We theoretically

prove that the proposed method estimates the posterior variance

and does not introduce additional biases from the original TC

method. We conduct a comprehensive evaluation of our approach

based on the synthetic and real market datasets. We conﬁrm with

synthetic data that the UTC method can detect situations where

the uncertainty increases and the prediction is difﬁcult. We also

conﬁrmed that the UTC method can detect abrupt changes in

data generating distributions. We demonstrate with real market

data that the UTC method can achieve higher returns and lower

risks than baselines.

Index Terms—Finance, Metaheuristics, Stock Price Prediction,

Uncertainty.

I. INTRODUCTION

Stock price predictability has been an important research

topic in both academia and industry since it reﬂects our

economic and social organization and the stock market plays

an important role in the world economy. Although the dynamic

nature of our economic activity makes it harder to predict

future stock prices, signiﬁcant efforts are made to explain

the dynamism. From this perspective, stock markets have

often been modeled as a complex, evolutionary, and nonlinear

dynamical system [1]–[3].

Due to the dynamic nature of our economic activity, ma-

chine learning is an increasingly popular tool with some

success in predicting stock prices [4]–[6]. This is because

many machine learning methods can automatically capture

nonlinear relationships between relevant factors from the input

data [7], [8]. One promising method among them is the Trader-

Company (TC) method, a metaheuristic stock prediction model

that mimics the roles of an actual ﬁnancial institute and traders

within it [9]. The TC method consists of two components,

the predictor called a Trader and the aggregation algorithm

called a Company. The TC method considers the dynamism

of the stock market and has both high predictive power and

interpretability.

Stock prediction methods based on machine learning, in-

cluding the TC method, have been concentrating on estimating

and improving point predictions. However, point predictions

in the absence of uncertainty estimates lack credibility quan-

tiﬁcation and raise concerns about safety. Considering the sig-

niﬁcant consequences of decision-making in ﬁnancial practice,

quantifying the uncertainty of predictions is proving to be a

key step in putting machine learning models into practice [10],

[11]. For example, most trades (80%) are automated [12] and

the algorithmic tradings based on the machine learning method

have played a crucial role in ﬁnancial markets. The algorithmic

tradings focused on the large investment universe of stocks

and sampled data at very high frequencies (intraday or tick

by tick). In such an environment with a large amount of data,

it is important for practitioners to quantify the uncertainty of

predictions. Therefore, the challenge in this paper is to make

an investment strategy with high predictive power and can

quantify the uncertainty of predictions.

To formalize our discussion of the uncertainty of predic-

tions, we will rely on probabilistic modeling. Probabilistic

modeling, which can provide probabilistic predictions and un-

certainty estimations simultaneously, has been a fundamental

tool in machine learning and related ﬁelds [13]. Most of these

studies rely on a Bayesian framework, and their applications

to complex models such as neural networks and decision tree

models have been actively studied. Among them, one standard

approach is directly estimating the distribution of predictions.

However, it has been pointed out that these methods tend to

arXiv:2210.17030v2 [q-fin.CP] 2 Nov 2022

make predictions biased toward one speciﬁc mode [14], [15].

Another approach is ensemble-based uncertainty estimation,

which focuses on the dispersion of predictions [15]–[17].

These methods are experimentally conﬁrmed to be more robust

to dataset shift than methods that explicitly learn distribu-

tions [15], [17]. That is effective in predicting a dynamic

environment, such as ﬁnancial markets.

Based on the above studies, we propose a novel approach

called the Uncertainty Aware Trader-Company (UTC) method.

The core idea of this approach is to combine the strengths of

both frameworks by merging the TC method with the proba-

bilistic modeling framework. We expect to retain the predictive

power and interpretability of the TC method while capturing

the uncertainty. To be more concrete, we propose the method

of estimating the prediction’s uncertainty from the Traders’

output. We estimate the variance by the two-stage algorithm:

estimation by the Trader and estimation by the Company.

The Trader estimates the uncertainty on weighting given a

trader’s strategy while the Company estimates the uncertainty

about the effectiveness of the whole traders’ strategies. We

theoretically show that our uncertainty estimation reﬂects the

posterior variance of predictive return given the past return.

Also, we prove that the predictive return of the UTC method

is identical to that of the original TC method under some

assumptions in the prior distribution of traders. That means

our method does not introduce additional biases.

We conduct a comprehensive evaluation of our approach

based on the synthetic and real market datasets. Our evaluation

of the synthetic datasets demonstrates that the UTC method

can detect situations where the uncertainty increases and the

prediction is difﬁcult. We also conﬁrmed that our method can

detect abrupt changes of data generating distributions. Further-

more, experiments using actual data show that the investment

strategy based on our UTC method gains stable returns while

suppressing risks compared to existing investment strategies.

The remainder of this paper is organized as follows. Section

2 describes our problem formulation and TC method brieﬂy.

Section 3 presents our UTC method and theoretical properties.

Section 4 performs experiments. Section 5 reviews the related

work, and Section 6 is the conclusion.

II. PRELIMINARY

In this section, we formulate our problem and then provide

the overview of the TC method [9].

A. Problem Formulation

Our problem is to forecast future returns of stocks based on

their historical observations.

Let Xi[t]be the price of stock iat time twhere 1≤i≤S

denotes the index of stocks and 0≤t≤Tdenotes the time

index. We use the logarithmic returns of stock prices as input

features of models; we denote the one period ahead return of

stock iby

ri[t] := log(Xi[t]/Xi[t−1]) ≈Xi[t]−Xi[t−1]

Xi[t−1] .(1)

TABLE I

NOTATION.

Notation Meaning Def.

Xi[t]stock price of stock iat time t

where 1≤i≤S, 0≤t≤T§II-A

ri[t]logarithmic return of iat t(1)

ri[u:v] (ri[u],··· , ri[v]) (2)

ri:j[u:v] (ri[u:v],··· , rj[u:v]) (2)

ˆri[t+ 1],ˆσi[t+ 1] predicted value and standard deviation of ri[t](4)

M, Pj, Qj, Dj

Fj, Aj, Ojhyper-parameters of Traders (5)

Then we deﬁne the returns of stock iand returns of multiple

stocks 1≤i≤j≤Sfrom time period uto v(u≤v)by

ri[u:v] := (ri[u],· · · , ri[v]),(2)

ri:j[u:v] := (ri[u:v],· · · , rj[u:v]) (3)

We can formulate our main problem as follows.

Problem 1 (one-period-ahead prediction).We sequentially ob-

serve the returns ri[t](1≤i≤S) at every time 0≤t≤T−1.

We predict the one-period-ahead return ri[t+ 1] and estimate

its predictive uncertainty σi[t+ 1] based on the past treturns

r1:S[0 : t]. That is, the one-period-ahead return and uncertainty

prediction can be written as

ˆri[t+ 1],ˆσi[t+ 1] = ft(r1:S[0 : t]) (4)

for some function ft. The purpose of this study is to ﬁnd ft

whose output ˆri[t+ 1] approximates the true return ri[t+ 1]

and predictive standard deviation ˆσi[t+ 1] approximates the

estimation error between ˆri[t+ 1] and ri[t+ 1] well.

B. Trader Company Method

This section introduces the Trader-Company method brieﬂy.

The TC method consists of two main components, Traders and

Companies. A Trader predicts the returns using a simple model

expressing realistic trading strategies, while a Company com-

bines strategies from multiple Traders into a single prediction.

A Company applies an evolutionary algorithm that mimics

the role of ﬁnancial institutes as employers of traders. During

training, a Company generates promising new candidates for

Traders and deletes poorly performing ones. We provide more

detailed deﬁnitions and training algorithms for the TC method.

1) Traders - Simple Prediction Module:

Deﬁnition 1. A Trader is a predictor of one period ahead

returns deﬁned as follows. Let Mbe the number of terms

in the prediction formula. For each 1≤j≤M, we deﬁne

Pj, Qjas the indices of the stock to use, Dj, Fjas the delay

parameters, Ojas the binary operator, Ajas the activation

function, and wjas the weight of the j-th term. Then, the

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UncertaintyAwareTrader-CompanyMethod:InterpretableStockPricePredictionCapturingUncertainty1stYugoFujimotoInnovationLabNomuraAssetManagementCo,Ltd.Tokyo,Japanyu5fujimoto@gmail.com2ndKeiNakagawaInnovationLabNomuraAssetManagementCo,Ltd.Tokyo,Japankei.nak.0315@gmail.com3rdKentaroImajoPreferredNetworks,I...

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Uncertainty Aware Trader-Company Method Interpretable Stock Price Prediction Capturing Uncertainty

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