Real‐time waiting‐price trading interval in a heterogeneous options market: a Bernoulli distribution

• Author: Avi Herbon,Yossi Shvimer
• Date: 01 November 2020
• Published date: 01 November 2020
• DOI: http://doi.org/10.1111/itor.12778

Intl. Trans. in Op. Res. 27 (2020) 2817–2840

DOI: 10.1111/itor.12778

INTERNATIONAL

TRANSACTIONS

IN OPERATIONAL

RESEARCH

Real-time waiting-price trading interval in a heterogeneous

options market: a Bernoulli distribution

Yossi Shvimer and Avi Herbon∗

Department of Management, Bar-Ilan University, Ramat-Gan, Israel

E-mail: yosis@MSH.CO.IL [Shvimer]; avher@bezeqint.net [Herbon]

Received 31 March 2019; receivedin revised form 27 October 2019; accepted 1 January 2020

Abstract

Options pricing remains an open research question thatis challenging for both theoreticians and practitioners.

Unlike many classical binomial models that assume a “representative agent,” the model suggested herein

considers two players who are heterogeneous with respect to their estimations of the distribution of the

underlying asset price on expiration day, and with respect to their levels of willingness to make a transaction

(eagerness level). A two-player binomial model is developed to find the real-time optimal option price in

two stages. First, we determine a primary feasible pricing domain. We then find a narrower feasible domain,

termed the “waiting-price trading interval,” meaning the region within which the players may either wait for

better offers (due to a change in market conditions or player beliefs), or make an immediate transaction.

The suggested model is formulated by a nonlinear optimization problem and the optimal price is shown to

be unique. We demonstrate that the counter player’s eagerness level has a significant effect on the proposed

optimal option price. Using empirical analysis, several known lattice-based models for option pricing, such

as CRR and Tian, are compared with the current model (herein, S-H) in which the price offered by the

model player takes into account the subjective beliefs of the opposing market player. The comparison shows

significant advantages to the S-H model in terms of the expected profit on expiration day.

Keywords:option pricing; optimization; waiting-price trading interval; heterogeneous players

1. Introduction

1.1. Options overpricing

Option contracts havebeen widely used in supply chain management to provide the retailer with the

flexibility to respond to unanticipatedcustomer demand. Under an option contract, the retailer gains

the right, but not the obligation, to engage in the transaction. Wang and Chen (2018) investigated a

∗Corresponding author.

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2818 Y. Shvimer and A. Herbon / Intl. Trans. in Op. Res.27 (2020) 2817–2840

newsvendor problem for Put options pertaining to fresh produce in which the stochastic demand is

price sensitive. The newsvendorcan obtain products by ordering from a firm, and can return unsold

products by purchasing and exercising Put options. Yuan et al. (2020) analyzed a system in which a

newsvendor-like retailer orders Call options from a supplier with an emergency order opportunity.

Wan and Chen (2019) considered a supply chain consisting of a supplier who manufactures one

type of perishable product, characterized by a long lead time, and a retailer who purchases these

products from the supplier and sells them to end consumers. In order to hedge against the risks of

price and demand caused by inflation, the retailer obtains the products from the supplier by means

of both portfolio contracts and Put options (i.e., a combination of wholesale price contracts and

Put option contracts).

Empirical evidence for “overpriced” options has been provided in a number of previous studies

dating back several decades (see, e.g., Bakshi et al., 1997; Jackwerth, 2000; Broadie and Detemple,

2004). An important study in terms of understanding the causes of overpricing is that of Pena et al.

(1999), who examined the overpricing of Call and Put options in the case where the theoretical

prices were predicted by the Black and Scholes’ (1973) model (hereafter the B-S model). They found

that the main explanation foroptions overpricing is associated with high transaction costs and large

Bid–Ask spreads, especially in the case of out-of-the-money options. Due to options mispricing, the

use of common models might lead buyers to avoid a transaction (in the case of an underestimation)

or to agree to pay a higher price (in the case of overestimation). Similarly, sellers may expect higher

prices (in the case of overestimation), thus reducing tradability or generating losses for numerous

players (buyers) who based their decisions on classical models.

Little attention has been paid to understanding the effect of heterogeneous beliefs on overpric-

ing, or on measures such as the expected profit gained by speculators. The mismatch between

theoretical and market option prices is strongly related to the general problem of evaluating op-

tions’ prices. The solution to this research problem seems challenging, as options pricing is also

related to the underlying market activity. Detemple and Selden (1991) demonstrated that the un-

derlying asset price cannot be “exogenic” to option pricing because it is itself affected by the

option price. Thus, when new options are issued, there is an influence on the underlying asset—its

price tends to increase because of risk-taking investors who increase the demand for Call op-

tions. Friesen et al. (2012) showed empirically, using options traded on the CBOE between 2003

and 2006, that heterogeneous beliefs cause overpricing of options beyond the effect of demand

(which is also an important determinant of overpricing, as argued by Bollen and Whaley, 2004).

Bondarenko (2014) found that historical prices of S&P 500 Put options are excessive and in-

compatible with canonical asset-pricing models such as the CAPM model (Sharpe, 1964). They

found that none of the proposed models can possibly explain the anomaly, and postulated that

the subjective beliefs of investors might provide the key. Buraschi et al. (2014) examined the het-

erogeneous beliefs of players regarding specific stocks on the S&P 1000 index in 1996–2007. They

found that heterogeneous beliefs partly explained the overpricing of options both at the index

level and with respect to specific stocks. More specifically, they found that overpricing arises due

to disparate estimations of different players regarding the future growth of a particular firm and

of the index being traded. Kang and Luo (2016) argued that for heterogeneous players, Call op-

tions are overpriced because model prices are calculated from the perspective of the “representative

agent” (meaning that all players are assumed to have the same distribution for the underlying as-

set price on expiration day). However, option overpricing in general remains a puzzle. Despite

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2020 The Authors.

International Transactionsin Operational Research C

2020 International Federation ofOperational Research Societies