Analyzing Equilibrium in Incomplete Markets with Model Uncertainty

• Date: 01 June 2017
• DOI: http://doi.org/10.1111/irfi.12119
• Author: Daisuke Yoshikawa
• Published date: 01 June 2017

Analyzing Equilibrium in

Incomplete Markets with Model

Uncertainty*

DAISUKE YOSHIKAWA

Hokkai-Gakuen University, Sapporo, Japan

ABSTRACT

In this paper, we analyze equilibrium in incomplete markets of random

endowments by adopting utility indifference pricing and utility-based pricing.

Addressing model uncertainty, we also consider agents who adopt maxmin

expected utility and a risk management policy. Using this framework, we

demonstrate the existence of equilibrium. Moreover, we clarify the differences

in the features of equilibria derived using thesemethods. Further, we showthat

the traded amount of randomendowments in equilibrium by indifference pric-

ing depends onthe degree of risk aversion, initialcapital, and agents’risk limits.

JEL Codes: G13; C62; D81

I. INTRODUCTION

Over the past 30 years, various methods have been developed to derive an

optimal strategy for hedging risk in financial markets especially in cases

involving incomplete markets. However, a unique method to diminish risks in

incomplete markets does not exist.

Many authors have analyzed this subject to overcome this problem. Local-risk

minimization and mean–variance hedging are two major approaches [Schweizer

(1999) provided a good summary]. Various martingale measures, as pricing

standards, have also been developed: Minimal martingale measures (Föllmer and

Schweizer 1991),p-optimal martingale measures(Grandits 1999), and minimal en-

tropy martingale measures (Frittelli 2000; Goll and Rüschendorf 2001; Monoyios

2007). These methods offer inspiration not only to academics but also to market

practitioners especially when the principle based on the method has appropriate

features for practitioners to utilize. However, even if such a method is appropriate

for particular types of agents, it might not besuitable for other types of agents.

Therefore, these methods do not lead us to a unique price of a random

endowment in incomplete market appropriate for agents with different risk

aversions. Davis and Yoshikawa (2015a) analyzed the features of utility

* This research was supported by the JSPS KAKENHI (15K03546); Zengin Foundation for Studies on

Economics and Finance; and Hokkai-Gakuen Gakujutsu Kenkyu Josei (Sogo Kenkyu).

© 2017 International Review of Finance Ltd. 2017

International Review of Finance, 17:2, 2017: pp. 235–262

DOI: 10.1111/irfi.12119

indifference pricing and utility-based pricing, and the features of the partial

equilibrium

1

of random endowments in incomplete market.

Utility indifference price is the price of a random endowment that equates one

agent’s expected utility, comprising terminal wealth, including the random

endowment and the same agent’s expected utility excluding the random

endowment. By this definition, the utility indifference price will be the threshold

price. If the agent tries to sell the random endowment at a price below the

indifference price, the expected utility without the random endowment will be

larger than that with the random endowment. Therefore, the agent cannot

tolerate selling it at such a price. Similarly, if the agent tries to buy the random

endowment, then the utility indifference price is the maximum price. We can

enjoy various benefits from utility indifference pricing, such as those mentioned

by Becherer (2003), Monoyios (2004), Sircar and Zariphopoulou (2004), Davis

(2006), Monoyios (2008), and Biagini et al. (2011), while utility-based pricing is

a pricing method of random endowment in incomplete markets [e.g., Hugonnier

and Kramkov (2004) and Hugonnier et al. (2005)]. In principle, it is a hedging

method rather than a pricing method, because utility-based pricing considers

the optimal amount of random endowment to optimize an agent’s preference.

In sum, both indifference and utility-based pricings consider optimal

strategies based on the agents’preferences: Utility indifference pricing considers

the standard price for the random endowment, whereas utility-based pricing

considers the optimal amount for the random endowment. By these definitions,

if the traded price and the amount of random endowment is derived using

indifference pricing and utility-based pricing as an equilibrium, it will satisfy all

agents’preferences.

Indeed, Davis and Yoshikawa (2015a) showed the existence of equilibrium in

the case in which agents optimize their strategy using utility-based pricing and

the case in which agents optimize their strategy using utility indifference pricing.

However, even if the equilibrium is shown to exist, the result in Davis and

Yoshikawa (2015a) implies a market in which random endowment will not be

traded at all; this is called a “zero trade equilibrium.”This is counterintuitive

and results from two different factors. One is the assumption in Davis and

Yoshikawa (2015a) that an agent’s preference type is given by an exponential

utility function. The other reflects the structure of market incompleteness as

defined in Davis and Yoshikawa (2015a), where a random endowment does not

necessarily have perfect correlation with tradable assets.

In terms of exponential utility, the solution of utility maximization problem is

often independent of initial wealth, which is considered a beneficial feature of

exponential utility, but is effective for deriving zero trade equilibrium. However,

the fact that even such a well-utilized type of utility function cannot derive a

“non”-zero trade equilibrium seems to pose a problem.

1 Hereafter, we discuss partial equilibrium. As long as there is no confusion, we describe it with-

out denoting “partial.”

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