Endogeneity of Return Parameters and Portfolio Selection: An Analysis on Implied Covariances

• Published date: 01 October 2017
• Author: Koohyun Park,Thomas Rhee
• DOI: http://doi.org/10.1111/ajfs.12187
• Date: 01 October 2017

Endogeneity of Return Parameters and

Portfolio Selection: An Analysis on Implied

Covariances*

Koohyun Park

School of Information and Computer Engineering, Hongik University, Republic of Korea

Thomas Rhee**

Department of Finance, College of Business Administration, California State University, Long Beach, CA,

United States

Received 2 December 2016; Accepted 7 May 2017

Abstract

The paper presents a method to measure forward-looking covariance risk for any two assets

even when the explicit market for barter trades does not exist. We argue that the terms of trade

in any barter exchanges also follow a martingale process with no arbitrage. We then compute

various bivariate martingale probabilities for different assets to value all possible pseudo

exchange options. This makes it possible for one to compute implied covariances embedded in

the value of any exchange options as in Margrabe (1978). The paper also discusses how these

“recoverable” implied return distribution parameters can impact portfolio choice.

Keywords Endogeneity of return parameters; Option implied covariance; Option implied

volatility; Forward-looking volatility; Forward-looking covariance; Risk-neutral probability;

Portfolio selection; Quadratic programming

JEL Classification: G11, G12

1. Introduction

Typical portfolio selection models utilize historical mean and variance/covariance

returns. Despite the fact that these return parameters themselves are random vari-

ables, as they are subject to their inherent sampling distribution, we often use them

in constructing portfolios without question. In addition, we make no specific refer-

ences as to why the use of certain values of such parameters is particularly correct

*The authors thank Hongik University for financial support (2014 Hongik University

Research Fund) and California State University for the research opportunities while Park was

a visiting scholar in the 2014–15 academic year.

**Corresponding author: Thomas Rhee, Department of Finance, College of Business Admin-

istration, California State University, Long Beach, CA 90840-8501, USA. Tel: +1-212-729-

0074, Fax: +1-562-985-1754, email: thomas.rhee@csulb.edu.

Asia-Pacific Journal of Financial Studies (2017) 46, 760–789 doi:10.1111/ajfs.12187

760 ©2017 Korean Securities Association

in portfolio analysis as we often fail to explain how the historical mean and vari-

ance/covariance returns are consistent with any market equilibrium.

Market prices signal useful information, and, for example, European option

prices give us valuable information about the variance of stock returns with no bias.

We say “without bias,” as we believe that the price as a market statistic will always

carry truthful information. Obviously, option-embedded volatility does not rely

upon particular sample observations and/or statistics. Since the implied volatility is

drawn from real-time market prices, they must be forward-looking statistics to the

extent that the market price always looks forward. Why then do we not use these

forward-looking probability parameters when constructing a portfolio?

Option-implied volatility does not appear to be unique, however, which may be

one of the reasons for not using implied volatility in portfolio analysis. Numerous

empirical studies point to the fact that implied volatility differs for different exercise

prices of options, that is, seeming volatility smiles and frowns, even for differing

maturities, that is, volatility term structures and/or surfaces. Which volatility num-

ber should we use then in our portfolio?

Consequently, we often question the validity of our original assumptions about

the stochastic process of the underlying securities. Perhaps the underlying asset

prices may not be log-normally distributed, as has been assumed typically in deriv-

ing option pricing formulas. Consequently, many researchers conclude that the

underlying securities returns themselves may be severely skewed or leptokurtic.

Alternatively, we often maintain the log-normality but conjecture that the volatility

itself may not be constant or that the underlying returns are mixed primarily with

frequent jumps and discontinuities.

1

What about implied covariance? Unfortunately, the idea of measuring forward-

looking implied covariance poses even more difficult problems as neither the mar-

ket price of exchange barter trades nor the market price for accompanying exchange

options are available. Exchange options are those options that allow a barter trade

between two non-money assets. In reality, they are not common except perha ps in

the case of foreign currencies and stock-for-stock tender offers in a mergers and

acquisition trade. The objective of this paper is to show a numerical method to

compute the unique value of not only the implied volatilities of a non-money asset

but also, in particular, the covariances between it and other non-money assets, even

when the market for such exchange options does not exist.

Section 2 presents a possibility that under some circumstances one can actually

compute not only the implied volatility but also the implied covariances embedded

in the market price of various existing options. The idea is simple: if one can some-

how impute a unique value of martingale probabilities

2

from option prices, one

1

There has been much research done in this area and discussing it in detail here would be

beyond the scope of this paper.

2

{X

t

,t≥0}is said to be a martingale under the measure ~

Pif X0¼E~

P½XTand the measure

~

Pis said to be the martingale probability measure.

Implied Covariance

©2017 Korean Securities Association 761