Data-driven prediction for volatile processes based on real option theories

https://doi.org/10.1016/j.ijpe.2019.107605Get rights and content

Highlights

  • We propose a new approach for predicting future observations.

  • The underlying dynamics follows the inhomogeneous Geometric Brownian Motion (GBM).

  • GBM parameters change over time adaptively to characterize time varying dynamics.

  • The prediction problem is solved using real option theories.

  • Our approach allows overestimation to be handled differently from underestimation.

Abstract

This paper presents a new prediction model for time series data by integrating a time-varying Geometric Brownian Motion model with a pricing mechanism used in financial engineering. Typical time series models such as Auto-Regressive Integrated Moving Average assumes a linear correlation structure in time series data. When a stochastic process is highly volatile, such an assumption can be easily violated, leading to inaccurate predictions. We develop a new prediction model that can flexibly characterize a time-varying volatile process without assuming linearity. We formulate the prediction problem as an optimization problem with unequal overestimation and underestimation costs. Based on real option theories developed in finance, we solve the optimization problem and obtain a predicted value, which can minimize the expected prediction cost. We evaluate the proposed approach using multiple datasets obtained from real-life applications including manufacturing, and finance. The numerical results demonstrate that the proposed model shows competitive prediction capability, compared with alternative approaches.

Introduction

In many applications including manufacturing, energy, and finance, accurate prediction is required to support strategic, tactical and/or operational decisions of organization (Chatfield, 2000). When physical information about the underlying mechanism that generates the time series data is limited, data-driven methods can be useful for predicting future observations (Zhang, 2003). In general, data-driven forecasting methods predict future observations based on past observations (Box et al., 2015). Several data-driven methods have been proposed in the literature for modeling time series data, among which Auto-Regressive Integrated Moving Average (ARIMA) and its variants such as the ARIMA-General Auto Regressive Conditional Heteroskedasticity (ARIMA-GARCH) have been widely used in many applications due to their flexibility and statistical properties (Sohn and Lim, 2007, Hahn et al., 2009, Lu et al., 2014, Ruppert, 2015). ARIMA assumes a constant standard deviation of stochastic noises, whereas ARIMA-GARCH extends it by allowing the standard deviation to vary over time. Some studies modify the original ARIMA model to update the parameters using new observations (Ledolter, 1981, Tran and Reed, 2004). The basic idea of these ARIMA-based models is that the future observation can be predicted by using a linear combination of past observations (and estimated noises). Therefore they assume a linear correlation structure between consecutive observations (Brooks, 2002). However, when the underlying dynamics exhibits a highly volatile process, such a simple linear structure may provide poor prediction performance (Kantz and Schreiber, 2004).

This study aims to provide accurate predictions for a highly volatile and time-varying stochastic process whose underlying dynamics is complicated and possibly nonlinear. As an example, let us consider a prediction problem faced by a contract manufacturer (CM) located in Michigan in the U.S, which motivates this study. The CM is a manufacturing company that produces various automotive parts, such as front and rear bumper beams, for several large automotive companies worldwide. The CM deals with a large number of orders for bumper beams from several automotive companies and the order sizes are time-varying. The CM should plan its production capacity carefully so that it can deliver products promptly when it gets orders. When an actual order size is greater than expected (i.e., when an order size is underestimated), overtime wages must be paid to workers to meet demands. On the other hand, when an order size is smaller than predicted (i.e., when an order size is overestimated), workers and equipment become idle.

As such, CM wants to predict future order sizes accurately, so that it can reduce its operating costs resulting from the discrepancy between its predicted value and actual sizes. Currently, CM uses its own proprietary prediction model, but its prediction performance is not satisfactory. The details of CM’s proprietary model are confidential, so we cannot find reasons for its unsatisfactory performance. When we apply the ARIMA and ARIMA-GARCH models to CM’s datasets, we also do not obtain significantly better prediction results (detailed results will be provided in Section 3). We believe such poor performance of ARIMA-based approaches is because they cannot fully characterize the underlying volatile dynamics. In addition to historical data, the future order size may depend on other factors which possibly make the order process behave nonlinearly. A new prediction approach that can adapt to such time-varying, and possibly nonlinear, dynamics is needed for providing better forecasts.

To this end we develop a new method for predicting future values in highly volatile processes, based on real option pricing theories typically used in financial engineering. In valuing investment projects, standard Discount Cash Flow (DCF) methods may fail to capture future opportunities (Benninga and Tolkowsky, 2002). The research on valuing investments under uncertainty was motivated by the pioneering work of Black and Scholes (Black and Scholes, 1973) and Marton (Merton et al., 1973) in financial options theory. As a result, options pricing theory has emerged in capital budgeting, and valuing projects or real options. The real options approach has an advantage over traditional methods because it includes the value of various managerial options (Hahn and Dyer, 2008, Bengtsson, 2001). Unlike standard financial options, real options reflect “real” and not traded assets or investments. Other investment options include deferring building a product, abandoning a project upon completion, or expanding a product to a new market (Trigeorgis et al., 1996). Consider a project with two stages: a pilot stage for research and development (R&D) and a second stage for commercialization. At the end of the first stage, project managers can decide whether to proceed to the second stage or to terminate the project and avoid potential poor outcomes. Real options can be used to evaluate such decision (Datar and Mathews, 2004), in which a call option represents the pilot stage. To justify the project, the investments in the first stage should not exceed the option’s value

One of the popularly used stochastic process models for pricing real options is the Geometric Brownian Motion (GBM) model. Brownian motion is a continuous-time stochastic process, describing random movements in time series variables. The GBM, which is a stochastic differential equation, incorporates the idea of Brownian motion and consists of two terms: a deterministic term to characterize the main trend over time and a stochastic term to account for random variations. In GBM the random variations are represented by Brownian Motion (Björk, 2009). GBM is useful to model a positive quantity whose changes over equal and non-overlapping time intervals are identically distributed and independent.

The GBM and its variants have represented various real processes in finance, physics, etc. (Gardiner, 1986, Wu and Wu, 2015, Zhai and Ye, 2017). In particular, it is fundamental to many asset pricing models (Björk, 2009), and recently it has been applied to facilitate the use of a rich area of options theory to solve various pricing problems (see, for example, Whitt, 1981, Thorsen, 1999, Benninga and Tolkowsky, 2002, Nembhard et al., 2002, Boomsma et al., 2012, Chiu et al., 2017). However, most of the current studies in real options have been limited to solving pricing problems and have not been used to make forecasts (Xiao et al., 2015).

In this study, by utilizing the full power of real options theory, we present a new approach for predicting future observations when the system’s underlying dynamics follows the GBM process. In particular, we allow the GBM parameters to adaptively change over time in order to characterize time-varying dynamics. The inhomogeneous GBM offers significantly more flexibility in characterizing the non-stationary nature of the stochastic process. Moreover, GBM, which includes the volatility term that changes dynamically, can capture highly volatile and nonlinear processes.

We formulate the prediction problem as an optimization problem and provide a solution using real option theories. Our approach provides extra flexibility by allowing overestimation (or over-prediction) to be handled differently from underestimation (or under-prediction). The overestimation and underestimation costs are determined in real life applications, depending on a decision-maker’s (or organization’s) preference. For example, in the aforementioned CM case, overestimation and underestimation of order sizes could cause different costs. The CM may want to put a larger penalty on the demand underestimation than on the overestimation, so that it can avoid extra overtime wages. We incorporate unequal overestimation and underestimation costs into the optimization problem and find the optimal forecast that minimizes the expected prediction cost. To the best of our knowledge, our study is the first attempt to incorporate options theory in the prediction problem.

The proposed approach with unequal over and underestimation penalties is closely related to the bad news principle (Bernanke, 1983). This principle states that an investment decision is only affected by the severity of future expected bad news. The choice of the parameter ω introduced in our approach, which denotes the ratio of overestimation penalty to the underestimation penalty, is highly dependent on the expected severity of future outcomes. In the CM example, ω reflects the expected severity of overtime and inventory costs.

Another way to see the role of ω is through good and bad economic uncertainties. The changes in economic activities depend on the type of uncertainty (Segal et al., 2015). Good uncertainty represents positive shocks that translate into higher asset prices and investments. On the other hand, bad uncertainty is associated with the volatility that impacts asset prices and investments negatively. The two types of volatility can be illustrated in two well-known financial incidents, namely, the high tech revolution in the 1990s and the collapse of Lehman Brothers. The former incident has introduced positive volatility and hence growth to the economy. The latter collapse has sent negative shocks through the markets. The variations of good and bad uncertainty have been demonstrated in [25] where each has significant but opposing impacts on the economy. The key finding in [25] is that good uncertainty predicts future growth in the economy while bad uncertainty predicts falling in asset prices. In our model, one may think the introduced parameter ω is a proxy to risk preference or type of uncertainty. In situations where positive shocks are expected, overestimation (ω>1) is preferred. In this case, the predicted value will be biased in that direction. Conversely, bad uncertainty would be associated with underestimation (ω<1). The predicted value, therefore, will be biased towards that preference.

To evaluate the prediction performance, we use two datasets collected from different applications, including the demand for bumper beams in CM (manufacturing), and stock prices (finance). We compare the performance of our model with ARIMA and ARIMA-GARCH models (and the proprietary prediction model in the CM case study) with different combinations of overestimation and underestimation costs. In most cases, our model outperforms those alternative models. In particular, we find that when the process is highly time-varying such as stock prices, the proposed approach provides much stronger prediction capability than ARMA and ARIMA-GARCH.

The remainder of the paper is organized as follows. The mathematical formulation and solution procedure are discussed in Section 2. Section 3 provides numerical results in two different applications. Section 4 concludes the paper.

Section snippets

Problem formulation

Consider a real-valued variable S(t) which represents a system state at time t. For example, the state variable can be a stock market index price, or a manufacturer’s order size. This state variable is assumed to follow an inhomogeneous GBM with time-varying parameters.

Let us consider a filtered probability space (Ω,F,P,Ft), where the filtration Ft is generated by the Brownian motion W, i.e. Ft=FtW so that Ft contains all information generated by W(t), up to and including time t. With GBM, the

Case studies

This section implements the proposed prediction model using multiple datasets obtained from real-life applications. Specifically we examine the performance of the predictive model in predicting the size of a manufacturer’s order, and a stock market index price.

Conclusion

In this study, we present a new prediction methodology for the time series data, based on option theories in finance when the underlying dynamics is assumed to follow the GBM process. To characterize time-varying patterns, we allow the GBM model parameters to vary over time and update the parameter values using recent observations. We formulate the prediction problem with unequal overestimation and underestimation penalties as the stochastic optimization problem and provide its solution

CRediT authorship contribution statement

Abdullah AlShelahi: Conceptualization, Methodology, Software, Formal analysis, Writing - original draft, Writing - review & editing. Jingxing Wang: Conceptualization, Methodology, Formal analysis. Mingdi You: Data curation, Validation. Eunshin Byon: Conceptualization, Methodology, Writing - review & editing, Funding acquisition, Supervision. Romesh Saigal: Conceptualization, Methodology, Writing - review & editing, Supervision, Resources.

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  • This work was supported by the National Science Foundation, United States under Grant IIS-1741166.

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