Extreme Events and the Copula Pricing of Commercial Mortgage-Backed Securities

Abstract

Commercial mortgage-backed securities (CMBS), as a portfolio-based financial product, have gained great popularity in financial markets. This paper extends Childs, Ott and Riddiough’s (J Financ Quant Anal, 31(4), 581–603, 1996) model by proposing a copula-based methodology for pricing CMBS bonds. Default on underlying commercial mortgages within a pool is a crucial risk associated with CMBS transactions. Two important issues associated with such default—extreme events and default dependencies among the mortgages—have been identified to play crucial roles in determining credit risk in the pooled commercial mortgage portfolios. This article pays particular attention to these two issues in pricing CMBS bonds. Our results show the usefulness and potential of copula-based models in pricing CMBS bonds, and the ability of such models to correctly price CMBS tranches of different seniorities. It is also important to sufficiently consider complex default dependence structure and the likelihood of extreme events occurring in pricing various CMBS bonds.

Appendix

Basic Definitions and Properties

The theory of copula functions investigates and describes the dependence structure of multiple random variables. On one hand, copulas are functions that connect the marginal (individual) distribution functions of individual random variables to their multivariate distribution function. On the other hand, the copula function provides an analytical tractable way of characterizing the dependence structure of joint random variables.¹⁹

A copula can be defined as follows:

Definition: Let $C:\left[ {0,1} \right]^n \to \left[ {0,1} \right]$ be an n-dimensional distribution function on [0,1]ⁿ. Then C is called a copula if it has uniform marginal distributions on the interval [0, 1].

Based on the above definition, we have the following fundamental theorem and corollary for copulas.

Theorem (Sklar 1959): Let F be an n-dimensional joint distribution function with marginal distributions $F_1 \left( {x_1 } \right), \ldots ,F_n \left( {x_n } \right)$. Then there exists a copula function C, such that for all $\left( {x_1 ,...,x_n } \right) \in R^n$,

$$F{\left( {x_{1} , \ldots ,x_{n} } \right)} = C{\left( {F_{1} {\left( {x_{1} } \right)}, \ldots ,F_{n} {\left( {x_{n} } \right)}} \right)}.$$

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Also, C is unique if $F_1 \left( {x_1 } \right), \ldots ,F_n \left( {x_n } \right)$ are all continuous; if not, C is uniquely determined on $RanF \times \cdot \cdot \cdot \times RanF_n$, where RanF _i denotes the range of F _i (x _i ) for $i = 1, \ldots ,n$. Conversely, if C is an n-copula function and $F_1 \left( {x_1 } \right), \ldots ,F_n \left( {x_n } \right)$ are marginal distribution functions, then the function F defined above is an n-dimensional joint distribution function with margins $F_1 \left( {x_1 } \right), \ldots ,F_n \left( {x_n } \right)$. Sklar’s theorem shows that the copula function can partition a multivariate distribution into two components, i.e., the marginal distributions of the individual random variables and their dependence structure.

The following corollary shows how to obtain the copula of a multi-dimensional distribution function.

Corollary: Let F be an n-dimensional continuous distribution function with marginal distributions $F_1 \left( {x_1 } \right),...,F_n \left( {x_n } \right)$ . Then the corresponding copula C has representation

$$C{\left( {u_{1} , \ldots ,u_{n} } \right)} = F{\left( {F^{{ - 1}}_{1} {\left( {u_{1} } \right)}, \ldots ,F^{{ - 1}}_{n} {\left( {u_{n} } \right)}} \right)}$$

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where$F_1^{ - 1} ,...,F_n^{ - 1}$denote the generalized inverses of the distribution functions$F_{1} {\left( {x_{1} } \right)}, \ldots ,F_{n} {\left( {x_{n} } \right)}$, i.e. for all$u_1 , \ldots ,u_n \in \left( {0,1} \right)$: $F_i^{ - 1} \left( {u_i } \right) = \inf \left\{ {x \in R\left| {F_i \left( {x_i } \right) \geqslant u_i } \right.} \right\},\,\,i = 1, \ldots n.$

An important property of copula is the invariance property. That is, if one carries out strictly increasing transformations for the underlying random variables, the transformed variables have the same copula as the original variables. When the random variables are independent, their copula can be simply written as

$$C{\left( {u_{1} , \ldots ,u_{n} } \right)} = F{\left( {x_{1} , \ldots ,x_{n} } \right)} = {\prod\limits_{i = 1}^n {F{\left( {x_{i} } \right)}} } = {\prod\limits_{i = 1}^n {u_{i} } }.$$

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On the other hand, the density of the multi-dimensional distribution function F can be expressed as follows

$$f{\left( {x_{1} , \ldots ,x_{n} } \right)} = c{\left( {F_{1} {\left( {x_{1} } \right)}, \ldots ,F_{n} {\left( {x_{n} } \right)}} \right)}{\prod\limits_{i = 1}^n {f_{i} {\left( {x_{i} } \right)}} }$$

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where $c\left( {.,...,.} \right)$ is the density of the copula C

$$c{\left( {F_{1} {\left( {x_{1} } \right)}, \ldots ,F_{n} {\left( {x_{n} } \right)}} \right)} = \frac{{\partial ^{n} {\left[ {C{\left( {F_{1} {\left( {x_{1} } \right)}, \ldots ,F_{n} {\left( {x_{n} } \right)}} \right)}} \right]}}}{{\partial F_{1} {\left( {x_{1} } \right)} \ldots \partial F_{n} {\left( {x_{n} } \right)}}}$$

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and $f_i \left( \cdot \right)$ are the densities of the marginal distributions.

Any copula $C\left( {u_1 , \ldots ,u_n } \right)$ also satisfies the following bounds

$$max{\left( {u_{1} + \ldots + u_{n} - 1,0} \right)} \leqslant C{\left( {u_{1} , \ldots ,u_{n} } \right)} \leqslant min{\left( {u_{1} , \ldots ,u_{n} } \right)}.$$

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This inequality is known as Fréchet–Hoeffding Bounds, which represent the largest possible positive and negative dependence of the underlying random variables.

An n-copula $C\left( {u_1 , \ldots ,u_n } \right)$ is non-decreasing in each argument. In particular, its partial derivative with regard to u _i exists almost everywhere and satisfies

$$0 \leqslant \frac{{\partial C}}{{\partial u_{i} }}{\left( {u_{1} , \ldots ,u_{n} } \right)} \leqslant 1;$$

it also has mixed kth-order partial derivatives almost surely, which for $1 \leqslant l \leqslant n$, satisfies

$$0 \leqslant \frac{{\partial ^{l} C{\left( {u_{1} , \ldots ,u_{n} } \right)}}}{{\partial u_{1} , \ldots ,\partial u_{l} }} \leqslant 1.$$

The properties imply that copulas have nice smoothness conditions.

Tail Dependence

Tail dependence is a powerful measure of the dependency between the occurrences of extreme observations of the underlying random variables, and can therefore be used to model probabilities of highly correlated defaults.

Definition (Tail dependence)

Let X = (X ₁,X ₂) be a two-dimensional random vector. Then the upper tail dependence of X is defined as

$$\lambda _{U} = {\mathop {lim}\limits_{u \to 1} }Pr{\left[ {X_{1} \geqslant F^{{ - 1}}_{1} {\left( u \right)}\left| {X_{2} \geqslant F^{{ - 1}}_{2} {\left( u \right)}} \right.} \right]},$$

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while its lower tail dependence is

$$\lambda _{L} = {\mathop {lim}\limits_{u \to 0} }P{\left[ {X_{1} \leqslant F^{{ - 1}}_{1} {\left( u \right)}\left| {X_{2} \leqslant F^{{ - 1}}_{2} {\left( u \right)}} \right.} \right]},$$

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where F₁ and F ₂ are the marginal distribution functions of X ₁ and X ₂, respectively.

A positive probability of positive or negative outliers jointly occurring implies the presence of upper or lower tail dependence, respectively. Eqs. 41 and 42 can be rewritten as

$$\lambda _{U} = {\mathop {lim}\limits_{u \to 1} }\frac{{1 - 2u + C{\left( {u,u} \right)}}}{{1 - u}}$$

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and

$$\lambda _{L} = {\mathop {lim}\limits_{u \to 0} }\frac{{C{\left( {u,u} \right)}}}{u}.$$

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If λ _U or λ _L  > 0, the two random variables (X ₁,X ₂) are asymptotically dependent in the upper or lower tail and their extreme observations tend to occur simultaneously with probability λ _U or λ _L . On the other hand, if λ _U or λ _L  = 0, the two random variables are asymptotically independent in the upper or lower tail. That is, the copula has no upper or lower tail dependence [(see, e.g., Meneguzzo and Vecchiato (2004)].

If the two random variables are independent in the upper and lower tails, then $C\left( {u,u} \right) = u^2$

$$\lambda _{U} = {\mathop {lim}\limits_{u \to 1} }\frac{{1 - 2u + C{\left( {u,u} \right)}}}{{1 - u}} = {\mathop {lim}\limits_{u \to 1} }{\left( {1 - u} \right)} = 0$$

and

$$\lambda _{L} = {\mathop {lim}\limits_{u \to 0} }\frac{{C{\left( {u,u} \right)}}}{u} = {\mathop {lim}\limits_{u \to 0} }u = 0.$$

On the other hand, if

$${\mathop {lim}\limits_{u \to 1} }C{\left( {u,u} \right)} = {\mathop {lim}\limits_{u \to 1} }{\left( {1 - 2{\left( {1 - u} \right)} + o{\left( {1 - u} \right)}} \right)},$$

then

$$\lambda _U = \mathop {\lim }\limits_{u \to 1} \frac{{1 - 2u + C\left( {u,u} \right)}}{{1 - u}} = \mathop {\lim }\limits_{u \to 1} o\left( {1 - u} \right) = 0.$$

If

$${\mathop {lim}\limits_{u \to 0} }C{\left( {u,u} \right)} = o{\left( u \right)}.$$

then

$$\lambda _{L} = {\mathop {lim}\limits_{u \to 0} }\frac{{C{\left( {u,u} \right)}}}{u} = {\mathop {lim}\limits_{u \to 0} }\frac{{o{\left( u \right)}}}{u} = 0.$$

The two cases have no tail dependence of the underlying random variables. To analyze their tail dependence structure, the copula functions are usually chosen with these two limits not equal to zero.

If copulas have no closed-form expressions, we can use the approach of Embrechts et al. (1999, 2002) in calculating tail dependence. It is shown that the upper tail dependence λ _U can be expressed using conditional probabilities if the following limit exists:

$$\lambda _{U} = {\mathop {lim}\limits_{u \to 1} }Pr{\left[ {{\left( {U_{1} > u\left| {U_{2} = u} \right.} \right)} + {\left( {U_{2} > u\left| {U_{1} = u} \right.} \right)}} \right]}.$$

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If (U ₁,U ₂) have the same marginal distribution of normality or Student-t and the copula is exchangeable, then:

$$\lambda _{U} = {\mathop {lim}\limits_{u \to 1} }Pr{\left[ {{\left( {U_{1} > u\left| {U_{2} = u} \right.} \right)} + {\left( {U_{2} > u\left| {U_{1} = u} \right.} \right)}} \right]} = 2{\mathop {lim}\limits_{u \to 1} }Pr{\left( {U_{1} > u\left| {U_{2} = u} \right.} \right)}$$

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However, the tail dependent coefficient for the bi-normal distribution is zero, implying that extreme events occur independently in each margin. Thus, the Gaussian or normal copula does not have a useful tail dependence structure for mortgage default risk management.