Pricing Credit Default Swaps Under Multifactor Reduced-Form Models: A Differential Quadrature Approach

Abstract

We present a new numerical method for pricing credit default swaps under fully correlated multifactor reduced-form models. In particular, the proposed approach combines an implicit/explicit operator splitting procedure with the harmonic differential quadrature scheme, and is so efficient that it can be applied to models with up to six stochastic factors. This is a remarkable advantage, as we can use two factors to describe the interest rate, other two factors to describe the default probability, and other two factors to take into account, for example, the so-called counterparty risk. The performances of the novel method are demonstrated by extensive simulation, in which various kinds of models with four and six fully correlated factors are considered.

Appendix

In this section we show how to compute the matrix $M_1$ appearing in equation (54).

Let $\epsilon _{j,l}$, $\omega _{j,l}$ be defined as in (47)–(51), and let $a_1$, $b_1$ be the coefficients of Eq. (6). Let us define:

$$\begin{aligned} h_{j,l}=\epsilon _{j,l}\, a_1\big (x^l_1\big ) +\frac{1}{2} b_1^2\big (x^l_1\big ) \omega _{j,l}, \quad j,l=1,2,\ldots , N_1, \end{aligned}$$

(82)

and

$$\begin{aligned} z_{j,j}= & {} 1+f_1(x^j_1)\,\Delta t, \quad j=1,2,\ldots , N_1, \end{aligned}$$

(83)

$$\begin{aligned} z_{j,l}= & {} 0,\;\; j=1,2,\ldots , N_1, \quad l=1,2,\ldots , N_1,\;\;j\ne l. \end{aligned}$$

(84)

Moreover, let us define the matrices (of size $N_1 \times N_1$):

$$\begin{aligned} H= & {} \left[ \begin{array}{cccc} h_{1,1} &{}h_{1,2}&{} \dots &{}h_{1,N_1}\\ h_{2,1} &{}h_{2,2}&{} \dots &{}h_{2,N_1}\\ \vdots &{}\vdots &{} &{}\vdots \\ h_{N_1,1} &{}h_{N_1,2}&{} \dots &{}h_{N_1,N_1}\\ \end{array}\right] , \end{aligned}$$

(85)

$$\begin{aligned} Z= & {} \left[ \begin{array}{cccc} z_{1,1} &{}z_{1,2}&{} \dots &{}z_{1,N_1}\\ z_{2,1} &{}z_{2,2}&{} \dots &{}z_{2,N_1}\\ \vdots &{}\vdots &{} &{}\vdots \\ z_{N_1,1} &{}z_{N_1,2}&{} \dots &{}z_{N_1,N_1}\\ \end{array}\right] . \end{aligned}$$

(86)

The matrix $M_1$ is obtained as follows:

$$\begin{aligned} M_1 = Z - H\,\Delta t. \end{aligned}$$

(87)