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

Published date01 September 2016
AuthorMing Shann Tsai,Shu Ling Chiang
Date01 September 2016
The Valuation Model for a Risky
Asset When Its Risky Factors Follow
Gamma Distributions*
Department of Finance, National University of Kaohsiung, Kaohsiung, Taiwan and
Department of Business Management, National Kaohsiung Normal University,
Kaohsiung, Taiwan
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 specications 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.
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 specic termination risks
(e.g., default and prepayment) associated with risky assets. Managing these is
an important but difcult assignment for nancial 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 -
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. 421444
DOI: 10.1111/ir.12088
efcient hedging analyses.
Therefore, it is important and even necessary for
market practitioners and nancial 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 species 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 specied. Nowadays, the termination hazard
rate for a risky asset is usually specied as following one of the following three
distributions: the normal distribution, the log-normal distribution, and the
non-central chi-square distribution. Each specication has its specic advantages
and problems when pricing a risky asset.
The termination hazard rate follows a normal distribution if it is specied 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 specication 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 specied 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 efciency; and (3) it provides basic building blocks that nancial institutions can
use to price complicated nancial products (Liao et al. 2008).
International Review of Finance
© 2016 International Review of Finance Ltd. 2016422

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