Modeling eBay price using stochastic differential equations

• Published date: 01 January 2019
• Date: 01 January 2019
• DOI: http://doi.org/10.1002/for.2551

Received: 23 August 2017 Revised: 30 May 2018 Accepted: 3 August 2018

DOI: 10.1002/for.2551

RESEARCH ARTICLE

Modeling eBay price using stochastic differential equations

Wei Wei Liu1Yan Liu2Ngai Hang Chan3

1School of Mathematics and Statistics,

Lanzhou University, Lanzhou, China

2Department of Finance, Ocean

University of China, Qingdao, China

3School of Statistics, Southwestern

University of Finance and Economics,

China and Department of Statistics,

Chinese University of Hong Kong, Shatin,

NT, Hong Kong

Correspondence

Ngai Hang Chan, Department of Statistics,

Chinese University of Hong Kong, Shatin,

N.T.,Hong Kong.

Email: nhchan@sta.cuhk.edu.hk

Abstract

Online auctions have become increasingly popular in recent years. There is a

growing body of research on this topic, whereas modeling online auction price

curves constitutes one of the most interesting problems. Most research treats

price curves as deterministic functions, which ignores the random effects of

external and internal factors. To account for the randomness, a more realis-

tic model using stochastic differential equations is proposed in this paper. The

online auction price is modeled by a stochastic differential equation in which

the deterministic part is equivalent to the second-order differential equation

model proposed in Wang et al. (Journal of the American Statistical Association,

2008, 103, 1100–1118). The model also includes a component representing the

measurement errors. Explicit expressions for the likelihood function are also

obtained, from which statistical inference can be conducted. Forecast accu-

racy of the proposed model is compared with the ODE (ordinary differential

equation) approach. Simulation results show that the proposed model performs

better.

KEYWORDS

online auction, price dynamics, stochastic differential equation

1INTRODUCTION

eBay Inc., which celebrated its 20th anniversary in 2015 as

an Internet pioneer, is today'slargest global online auction

marketplace. It attractedmore than 162 million active buy-

ers and more than 800 million listings globally by the end

of 2015. Complete auction information of eBay is available

to the public, which has led to a surge of empirical studies

of online auction phenomena.

Unlike offline auctions, online auction bid data have

their own characteristics such as the low barriers of entry

for bidders and sellers, the globalism and bid revising;1

among which the most interesting issues are bid sniping2

1An individual bidder places multiple bids.

2The time spacing of the bids from each auction is not evenly distributed.

It is very sparse for a long time from the auction start and becomes very

dense at the end of the auction, since bidders place their bids at the last

moment.

and bid jumping3Modeling these statistical features is an

important topic in an online auction study. Further, price

prediction constitutes another interesting topic. Most stud-

ies are conducted using the framework of the functional

data analysis (FDA) approach. Since the auction price is

monotonically nondecreasing, the FDA approach recovers

the price curve from discrete measurements using non-

parametric smoothing, such as monotone splines (see, e.g.,

Hyde, Jank, & Shmueli, 2006; Jank, Shmueli, & Zhang,

2010; Shmueli et al., 2007; Wang, Jank, & Shmueli, 2008)

and parametric models such as the four-member para-

metric growth model (Hyde, Jank, & Shmueli, 2008) and

the beta model (Jank et al., 2010). Alternatively, the point

3Wedefine bid jumping for an auction if the bid increment is significantly

higher than the previous one during the bidding process (see Easley &

Tenorio, 2004). Herein, bid jumping refers to a bid that is at least 30%

higher than the previous bid.

Journal of Forecasting. 2019;38:63–72. wileyonlinelibrary.com/journal/for © 2018 John Wiley & Sons, Ltd. 63