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The purpose of this paper is to derive or determine the Credit Derivative, especially, the Credit Default Swap which is under the hazard rate (or default intensity) distributed as a multi-factor of the Cox, Ingersoll and Ross (CIR, 1985) models. It is crucial to know how default should be modelled for the valuation of credit derivatives. We are motivated by the idea that CIR term structure model, for example, must be effective for modelling hazard rate, and has some significant properties: mean-reversion and affine. We use South Africa (SA) credit spread market data on Defaultable bonds to estimate parameters associated with the stochastic single-factor hazard rate type CIR.

The major credit problems and significant failures faced by banks during Global Financial Crises, for example the recent financial crisis or credit crisis of 2007-2008 [

The designation or development of new products, such as Credit Derivatives (CDs), by all investors and financial institutions to reduce or remove any Credit Risk arises from lenders or bondholders (example banks), and to allow banks to deliver more loans seemed to want a share in it. The most widely used product of CDs is the Credit Default Swap (CDS). The Credit Default Swap is a contract entered between two parties that provide a protection against losses occurring due to a default event of a certain entity. Since its introduction in the mid-1990s, the growth of the global market has been overwhelming: for example, the market size for CDS almost doubled biannually from 1996 to 2004, and even quadrupled to over a peak notional outstanding amount of US $20 trillion during 2004-2006 [

The measurement or modeling of credit risk, however, provides its own set of challenges. There exist many ways of modeling credit risk [

The main point of this paper is to consider how the default is modeled for estimating the value of Credit Default Swaps. There are different ways for modeling the Credit Derivatives, typically characterized by how they characterize the default event. Duffie and Signeton [

The paper is organized as follows. In Section 2, we discuss how to estimate hazard rate which follows the dynamic of the multi-factor CIR-type model using the relationship between credit spread and hazard rate, that, is necessary for switching the market credit spread data (collecting directly from the market data) into the multi- factor CIR-type Hazard rate data. We analyze the hazard rate function of single-factor and two-factor CIR-type models and give some results. In Section 3, we follow the framework proposed by David and Mavroidis [

In this section, we discuss how to estimate hazard rate type Multi-factor CIR model. Using the relationship between credit spread and hazard rate, it is possible to convert the market credit spread data (collected directly from the market data) into CIR (or Vasicek) multi-factor type Hazard rate data. Our first analysis, thus, assumes the risk-free interest rate r to be independent of all the hazard rates. Therefore the occurrence of default is not correlated with bond prices. This assumption implies that the level of default is cause by some factors affecting the issuer, not the level of risk-free interest rate.

Multi-factor affine models of the term structure represent the yield of securities as affine function of a vector of n unobservable state variable

A model is CIR affine if all state variables

Therefore, for

where the parameters ?_{i}, b_{i} and σ_{i} are viewed as the speed of mean reversion, the long-term mean or the mean reversion level and the volatility respectively, and

Due to the independence of factors it is subsequently to derive the below formula. The price

where

At time t the credit spread, viewed as the difference between the default adjusted interest rate and the risk- free interest rate is given by

We are now in a position to define the relationship between the credit spread and hazard rate process. This is useful in converting the credit spread data given from market data into the hazard rate.

Lemma 2.1. Let

Proof 2.2. Note that it is impossible to estimate the recovery rate δ and the hazard rate γ separately from the credit spread: knowing δ (given by another technique), we may determine the parameters of the hazard rateγ.

We therefore need to determine the distribution of the random variable

This is contained in the formula of the bond price (for δ = 0), which is the expectation of the exponential of minus the integral of the short term process. Analogously to the work of the CIR model and Bond price under CIR, the CIR formula for the price of a zero-coupon bond is

where

This new process immediately yields

Given the relation of credit spread (8), we obtain

Equivalently, the hazard rate in a CIR-type model is given by

where

For the Single-factor model, the hazard rate CIR-type model is given by

where

For the two-factor model, the hazard rate CIR-type model is given by

where

two factors CIR-type model; given the market price of credit spread. We took the volatility to be σ = σ1 = σ2 = 0.9. The figure shows that the hazard rate function, is slightly variant (slightly increasing and decreasing from time t = 0 to t = 18) and increasing from t = 18 to t = 20. This two factors CIR-type mode (

Given the filtered space

be a two dimensional Brownian filtration and denote by τ a non-negative random default time. We assume the 2-dimensional

pose that we are given an auxiliary reference filtration

The following important lemmas are introduced for further evaluation of credit default swaps

Lemma 3.1. Assume that the hazard rate process γtis a non-negative

is a Martingale on

Lemma 3.2. Assume that, for

Corollary 3.3. For any bounded

Proof [

We follow the framework of David and Mavroidis [

A credit default swap is an agreement designed between two parties that provide a protection or assurance against losses occurring due to a default event of a certain entity (

One party agreed to buy protection called protection buyer B (e.g., a firm) and provides a regular payment^{2}.

The risky bond that the buyer B holds permits a fixed coupon

A credit default swap agreement includes a fixed premium leg or fixed side and a recovery side (or contingent default leg).

・ The fixed side corresponds to the series of payments made by the buyer B of the CDS-contract to the protection seller S of the contract up to the maturity time, unless a bankruptcy event or other credit event perturbs the contingent payment on a CDS.

・ The recovery side corresponds to the net payment delivered by the counter party protection seller S to the protection buyer in case of such default event happens.

The main goal of valuation of CDS is to obtain equilibrium premium (or regular payment) c_{i}’s paid periodically by the reference holder, which is followed from the equality of the value between fixed premium leg and contingent default leg. Consider the risk-free interest rate r be independent of all factors related to credit risk, alike default time and the hazard rates. This assumption implies that we can value default swap in term of the default-free zero coupon bond’s prices as follows.

Since all payments are evaluated at t = 0 and no payment is made after any default event occurs, the actual value of the fixed side

where r is the short rate interest and

is an explicit value of the fixed side, when the hazard rate function γ(t) is distributed as a multi-factor CIR model.

where

Let

where _{u}the hazard rate process. It is supposed that the fixed premium leg to the contract can recover the specific amount

This equation can be evaluated separately as

By Corollary 3.3, Lemma 3.2 and using hazard rate mean reversion. And for second term in (19) with application of Corollary 3.3

Therefore,

where

This is the price of the recovery side or contingent default leg for the hazard rate multi-factor CIR type. Given the value of the default free zero coupons bond

We shall estimate parameters associated with hazard rate models using the Moment Method. We investigate 20 South African firm’s debt terms, with different rating from AAA to BBB and different market credit spread for maturity one year, three years and five years as shown in

Company | Rating | 1 Year Credit Spread (bp) | 3 Year Credit Spread (bp) | 5 Year Credit Spread (bp) | Volatility of σ (%) | Mean Reversion ? (%) |
---|---|---|---|---|---|---|

1 | AAA | 0 | 7 | 28 | 32.1 | 3.29 |

2^{*} | AA^{+} | 0 | 1 | 6 | 19.03 | 1.09 |

3 | AA | 0 | 2 | 11 | 26.98 | 0.1018 |

4^{*} | AA | 0.2 | 7 | 18 | 5.31 | 10.3 |

5^{*} | AA | 0.2 | 7 | 18 | 4.212 | 4.48 |

6^{*} | AA | 0.4 | 12 | 29 | 7.153 | 0.0349 |

7^{*} | AA | 0.2 | 6 | 15 | 3.31 | 1.2945 |

8 | AA^{−} | 0 | 0 | 3 | 17.39 | 10.24 |

9 | AA^{−} | 0 | 5 | 19 | 20.15 | 4.25 |

10 | A^{+} | 0 | 2 | 10 | 21.18 | 5.85 |

11 | A^{+} | 0.6 | 33 | 85 | 29.02 | 0.0205 |

12^{*} | A^{+} | 0.4 | 7 | 17 | 3.13 | 4.61 |

13^{*} | A^{+} | 0.7 | 15 | 36 | 7.33 | 1.5287 |

14^{*} | A^{−} | 0 | 8 | 21 | 4.13 | 0.0412 |

15 | A^{−} | 0 | 6 | 24 | 23.6 | 3.77 |

16 | A^{−} | 16 | 3 | 18 | 41.37 | 4.97 |

17^{*} | BBB^{+} | 1.6 | 160 | 297 | 30.02 | 1.11 |

18^{*} | BBB | 1.6 | 34 | 77 | 15.07 | 2.18 |

19^{*} | BBB | 7.4 | 84 | 169 | 18 | 1.49 |

20^{*} | BBB | 9.8 | 108 | 211 | 22.02 | 1.33 |

An asterisk ^{*} indicates that the firm is in the banking sector.

More specifically, using a debt term of 3 years, the credit spreads of AA-rated companies vary between 2 and 12 bps whereas observed spreads in the South African market at the time of writing and in general, vary between 30 and 60 bps [

We recall that the parameter of hazard rate process can be estimated from historical market data of credit spread. In the special cases, such as: -Vasicek model, and CIR model, (except for the recovery rate δ), we have to secure the parameters estimator for hazard rate ?, b and σ. Meaning that we need to have at least four different credit spread

There exist different methods of estimating the parameters, including the implied volatility which is used in option valuation. Typically they are characterized or evaluated from the historical data. Though we consider the CIR processes in particular, we note that a similar procedure is possible for other models (Vasicek, et). We follow [

・ Characteristic of the long-term mean hazard rate type CIR process b.

Following the assumption that the long-term mean hazard rate process is similar for the same category of industry and the rating of the same class, we consider the estimation of b as the mean value of the probability of default on every category of industry and rating collected from rating agencies such as Standard & Poor, Moody’s and Fitch. We assume the long-term mean b is 10%.

・ Rule of estimating recovery rate δ.

Moody Agency’s database includes detailed bond prices information after default, the historical market price of the bond for 30 days (one month) after the firm experienced the default event. This is viewed as the recovery rate from the default bond. Basically, the average of debt differs from issuers to issuers (or from firm to firm), in this discussion we assume that the recovery rate of the same class of financial rating and the same class of industry are shared and regard as the recovery rate δ of the firm. That is the mean value computed from Moody’s data of recovery rates every class of industry and rating [

Due to the difficulty of estimating the volatility σ and mean-reverting speed ? separately from market data of credit spread using only one bond issuer (or reference bond), we restrict ourselves to the limit distribution of the single factor CIR hazard rate-type model, and attempt to use the moment method to find those parameters. The moment method is a generic method or the most preferred numerical technique of estimating parameters in statistical model due to its less requirements of information.

Consider a set of observations of hazard rate

From the discussion about hazard rate models above, we recall that the elements of γ are the speed of mean reversion ?, the long mean rate b and the volatility σ.

In the case of the CIR model, CIR hazard rate

where

Obtained by historical market data calculated from the formula (13). Using the limit of expectation value of the hazard and substituting Equation (13) for the CIR model, we have

Using the limit of variance and the definition formula of variance, and substituting Equation (13) for the CIR model, we have

We have, therefore,

Having the estimator parameters δ and b, we have only to obtain the parameters values σ and ? which obey Equations (21) and (22) simultaneously. This may be done by solving, for example, the following system of two equations with two unknowns, ? and σ:

Note that this system of equations in (23) is nonlinear equation. The solution of this system with ? and σ unknowns can be obtained by solving the systems of nonlinear simultaneous equations. (Typically, this is very difficult). Assume that

This system of two nonlinear simultaneous equations can be solved on Matlab. This problem can also be formulated as the optimization problem i.e. we will seek to minimize

This optimization problem is the problem of making the best possible choice of σ and ? that can minimize the objective function

Because of the rarity of data, we use 20 South Africa firm debts for maturity one year, three years and five years [

The results shown in

In this paper, we have been concerned how the default is modeled for estimating the value of credit default swaps. We have considered the default model which takes the hazard rate as principal factor and distributed as the multi-factor Cox, Ingersoll and Ross (CIR) model, since it has some significant properties: mean reversion and affine. The explicit value of fixed side and recovery side of credit default swaps was determined in quite general form that contained counter party risk, under the multi-factor affine hazard rate. We suggested one implementation procedure when the available market data were not sufficiently rich and provided some simulation results. However, the work done in this paper has mainly been on the study of credit default swap under multi- factor hazard rate model, and the estimation of parameters associated with the single factor hazard rate model. These are done under the martingale measure or the original risk neutral measures. Moreover, as in real world the market is often incomplete, that is the existence of many martingale measures (or the risk neutral probability is not unique, refers to asset pricing theorem).

A natural extension of our estimation approach is to consider the calculation under the objective probability (or real world probability), investigate a case of correlation between a defaultable bond and the risk free interest rate (or correlation between the risk-free interest rate and all the hazard rates), and consider when defaultable bonds are issued by different firms.

The authors would like to acknowledge support of this work from Centre for Business Mathematics and Informatics and AIMS (African Institute for Mathematical Sciences). The work was done while Hopolang P. Mashele was a visiting researcher at AIMS.

Alma P. BimbabouMaboulou,Hopolang P.Mashele, (2015) Credit Derivative Valuation and Parameter Estimation for Multi-Factor Affine CIR-Type Hazard Rate Model. Journal of Mathematical Finance,05,273-285. doi: 10.4236/jmf.2015.53024