The Valuation Model for a Risky Asset When Its Risky Factors Follow Gamma Distributions

• Published date: 01 September 2016
• Author: Ming Shann Tsai,Shu Ling Chiang
• Date: 01 September 2016
• DOI: http://doi.org/10.1111/irfi.12088

The Valuation Model for a Risky

Asset When Its Risky Factors Follow

Gamma Distributions*

MING SHANN TSAI

†

AND SHU LING CHIANG

‡

†

Department of Finance, National University of Kaohsiung, Kaohsiung, Taiwan and

‡

Department of Business Management, National Kaohsiung Normal University,

Kaohsiung, Taiwan

ABSTRACT

This paper constructed a pricing model for the asset with multi-risks by specifying

the risky factors (i.e., interest rate and termination hazard rates) to follow gamma

distributions. The model not only avoids the possibility of the termination hazard

rate takingan irrational (i.e.,negative) value, but it also makes it easier to derive a

valuation formula for a risky asset. Our model can also effortless apply because the

parametersof the gamma distribution can easilybe estimated from market data.

An example using Taiwanese bond data illustrates how the model can be utilized

for practical applications. To facilitate understanding of how accurately the

different models price risky bonds, we compare their out-of-sample pricing

errors for different hazard rate specifications assuming normal and gamma

distributions. The results show that our pricing formula is realistic and accurate

in its applications. Therefore, it should help market participants to a ccurately

price riskyassets and to effectively manage complicated portfolios.

I. INTRODUCTION

Because of the recent crises involving subprime mortgages in the US and the gov-

ernment bond markets in Europe, researchers and market practitioners have

been paying a great deal of attention to studies of the specific termination risks

(e.g., default and prepayment) associated with risky assets. Managing these is

an important but difficult assignment for financial institutions. Because a pricing

formula for the risky asset can adequately appraise asset values from correlated

multiple risk sources, it can greatly reduce the hard works of managing risks. By

utilizing a suitable pricing formula for the risky asset, portfolio managers and fi-

nancial institutions cannot only accurately appraise risky assets but also perform

* We are grateful for valuable comments and suggestions on earlier drafts by International Review of

Finance Editor Huining Cao, an anonymous Associate Editors and anonymous referees. We also thank

the Ministry of Science and Technology for providing fund in support of this study.

© 2016 International Review of Finance Ltd. 2016

International Review of Finance, 16:3, 2016: pp. 421–444

DOI: 10.1111/irfi.12088

efficient hedging analyses.

1

Therefore, it is important and even necessary for

market practitioners and financial institutions to derive a pricing formula for

the risky assets. The main goal of this paper is to provide a general and accurate

valuation model to accomplish this task.

To assess these probabilities of risky events (e.g., default, prepayment, and li-

quidity risk) occurring prior to contract maturity, researchers usually utilize two

models: the structural-form model (Merton 1974; Kau et al. 1993; Yang et al.

1998; Ambrose and Buttimer 2000; Azevedo-Pereira et al. 2003) and the

reduced-form model (Jarrow and Turnbull 1995; Jarrow 2001). Because it is easier

to derive a pricing formula for the risky asset by using the reduced-form model

than the structural-form model, researchers have increasingly chosen the

reduced-form model for pricing risky securities and determining the probability

of termination (Liao et al. 2008; Tsai et al. 2009; Tsai and Chiang 2012, 2015).

For our study, we also chose the reduced-form model for this purpose.

The reduced-form model usually specifies the unpredictability of risky events

as exogenous random variables that follow a Poisson distribution (Bielecki and

Rutkowski 2001). The analyses of default risk depend mainly on how the termi-

nation hazard rate is appropriately specified. Nowadays, the termination hazard

rate for a risky asset is usually specified as following one of the following three

distributions: the normal distribution, the log-normal distribution, and the

non-central chi-square distribution. Each specification has its specific advantages

and problems when pricing a risky asset.

The termination hazard rate follows a normal distribution if it is specified as a

Vasciek-form (Vasicek 1977). As shown by Jarrow (2001) and by Liao et al. (2008),

it is easiest to derive a pricing formula for a risky asset with multi-risks under this

construction. To depict reasonably the risks that an asset incurs, the hazard rates

associated with the various risks have to be modeled as the function of state var-

iables (i.e., interest rate and stock return). It is also necessary to consider the cor-

relations among the hazard rates in the pricing model. For example, the decisions

of prepayment and default for mortgage obligations are viewed as the correlated

competing risks and thus are jointly estimated when investigating mortgages and

mortgage-based securities (Han and Hausman 1990; Sueyoshi 1992; McCall

1996; Deng et al. 2000; Kuznetsovski and Hwang 2010). Nowadays, among the

aforementioned three distributions, this is only possible to more easily deal with

this situation if hazard rates are assumed to follow a normal distribution. How-

ever, some scholars have argued that such a specification yields hazard rates with

unrealistic values (e.g., negative values), which in turn leads to inaccurate pricing

of the securities. To avoid such problems, additional constraints are needed that

restrict the range of values the hazard rate can assume (Liao et al. 2008).

In what is usually called the Cox hazard rate model for analyzing termination

probabilities for risky securities, the termination hazard rates are specified as

1 Some of the other advantages of a pricing formula for the risky asset are as follows: (1) it helps

us better appreciate how sensitive asset values are to changes in relevant factors; (2) it improves

calculation efficiency; and (3) it provides basic building blocks that financial institutions can

use to price complicated financial products (Liao et al. 2008).

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