CreditGrades Framework within Stochastic Covariance Models

In this paper we study a multivariate extension of a structural credit risk model, the CreditGrades model, under the assumption of stochastic volatility and correlation between the assets of the companies. The covariance of the assets follows two popular models which are non-overlapping extensions of the CIR model to dimensions greater than one, the Wishart process and the Principal component process. Under CreditGrades, we find quasi closed-form solutions for equity options, marginal probabilities of defaults, and some other major financial derivatives.


Introduction
We present a structural credit risk model which considers stochastic correlation between the assets of the companies.We allow the covariance of the assets to follow two popular stochastic covariance models.First we assume it follows a Wishart process [1] then we assume a Principal component Process (see [2]), both are non overlapping extensions of the CIR model to dimensions greater than one.
Modeling stochastic correlation has difficulties from the analytical as well as the estimation point of view.One of the attempts to x this gap began with a paper on Wishart processes by [1], which followed by a series of papers by Gourieroux, see [3].Several authors have recently brought the finance community's attention to the Wishart process and showed that the Wishart process is a good candidate for modeling the covariance of assets.A Wishart process is an affine symmetric positive definite process.The stream of papers [3][4][5][6][7] brought the finance community's attention to this process as a natural extension of Heston's stochastic volatility model, which has been a very successful univariate model for option pricing and reconstruction of volatility smiles and skews.The popularity of the Heston model as well as the empirical evidence of stochastic correlation and volatility has contributed to the recent popularity of the Wishart process.Risk is usually measured by the covariance matrix.Therefore Wishart process can be seen as a tool to model dynamic behavior of multivariate risk.Via the Laplace transform and the distribution of the Wishart process [3], prices derivatives with a generalized Wishart stochastic covariance matrix.This approach can be used to model risk in the structural credit risk framework.DaFonseca extends his approach in [7] to model the multivariate risk by a Wishart process.In this paper several risky stocks are considered and the pricing problem for one dimensional vanilla options and multidimensional geometric basket options on the stocks is presented.We adapt existing results about the Wishart process to the structural CreditGrades framework.We give quasi closed-form solutions for equity options, marginal probabilities of defaults, and some other major financial derivatives.For calculation of our pricing formulas we make a bridge between two recent trends in pricing theory; from one side, pricing of barrier options by [8] and [9] and from other side the development of Wishart process by [1].
In the second part of the paper we develop a new model for credit risk based on a model with stochastic eigenvalues called principal component stochastic covariance.To induce the stochasticity into the structure of volatilities and correlation, we assume that the eigenvectors of the covariance matrix are constant but the eigenvalues are driven by independent Cox-Ingersoll-Ross processes.To price equity options on this framework we first transform the calculations from the pricing domain to the frequency domain.Then we derive a closed formula for the Fourier transform of the Green's function of the pricing PDE.Finally we use the method of images to find the price of the equity options.Same method is used to find closed formulas for marginal probabilities of defaults and CDS prices.Inspired by the standard stochastic volatility models starting from Heston's paper [10] and the work on [2] the applications of the principal component model in credit risk is studied.The main idea is to identify the covariance matrix by its eigenvectors and eigenvalues.In the above papers, authors have assumed that the eigenvector of the covariance matrix are constant but the eigenvalues follow a CIR process.This implies a stochastic structure for the correlation between the assets.[2] prices the collateralized debt obligations under the Merton's model using a tree approach and the principal component model.
Merton's model [11] is the first structural credit risk model proposed which considers the company's equity as an option on the firm's asset.There has been numerous extensions for the Merton's model in the literature including incorporating early defaults, stochastic interest rates, stochastic default barriers and jumps in the asset's price process.A simpler approach was jointly developed by CreditMetrics, JP Morgan, Goldman Sachs and Deutsche Bank, called the CreditGrades model, this can be seen as a particular case of Merton with zero time to maturity.The simplicity allows for closed for expression on some derivatives as shown in our paper which can not be found closed form under Merton or Black Cox structural frameworks.We extend the CreditGrades model using stochastic covariance Wishart process focusing on the role of stochastic correlation.The performance of a company is usually monitored by observing its equity's volatility or the CDS spread.CreditGrades model can be considered as a down-and-out barrier credit risk model.This means that default is triggered if the value of the asset reaches a certain level identified by the recovery part of the debt.[12] has extended the CreditGrades model to price equity options by introducing the equity as a shifted log-normal process.[9] has extended [12] idea by embedding the Heston's volatility into the model and pricing equity derivative.Both [9] and [12] models are univariate credit risk models.We extend the Credit-Grades model by use of the Wishart and PC processes to dimensions greater than one implementing stochastic correlation into the dynamics of the assets.We give quasi closed formulas for equity derivatives based on these models.
This paper is organized as follows: in Section 2 we use Wishart process as a candidate to model the covariance matrix of the assets' prices within a CreditGrades model.The pricing problem for some derivatives on the equities is derived in Section 2.2.Section 3 presents and uses the Principal component process for the covariance matrix of the assets' prices within the same structural framework.The pricing problem is derived in Section 3.2.Section 4 concludes.The proofs are given in the appendix.

The CreditGrades Wishart Process
In this section, we introduce our Wishart CreditGrades model.CreditGrades model is a version of the Merton model jointly developed by CreditMetrics, JP Morgan, Goldman Sachs and Deutsche Bank.The original version of the CreditGrades model assumes that volatility is deterministic.We extend CreditGrades model, by means of stochastic covariance Wishart process.Our model allows correlation and volatility be stochastic.By considering the stochastic covariance Wishart process, we have more flexibility and degree of freedom in the marginal, while analytic tractability is preserved when extending CIR process to Wishart process.We first present Wishart process of integer degree of freedom and derive their matrix stochastic differential equation which later on will give a natural representation of Wishart process with fractional degree of freedom.A Wishart process with integer degree of freedom K is a sum of K independent n-dimensional Ornstein-Uhlenbeck process.We first remind the formal definition below: where A and are Q

 
, n n matrices with invertible.Then a Wishart process of degree is defined as  is the transpose of the vector .

 
k t U Ito's lemma can be used to find a diffusion SDE for the process t  As it can be seen, the drift term of the SDE above contains t  , but the diffusion part contains the terms  and  separately.[1,3] show that t  also satisfies the following matrix SDE   where is an t W n n  standard Brownian motion matrix.

The Dynamics of the Assets
The assets are defined on a probability space   where 0 t t  is the information up to time and Q is the risk-neutral measure equivalent to the real-world measure P. Let's assume that the firm's asset price per share is given by   Here we review the results regarding the dynamics of the assets with stochastic covariance Wishart process.As before, assume the assets' prices follow the multivariate real-world model Here the vector .Up to now we have described the dynami as-f the ts' prices with stochastic covariance structure coming from the Wishart process.The asset's price for a firm is not directly observed from the market.This leads us to structural credit risk models which introduce the equity as a form of a derivative on the asset of the company.Then the role of the model is in connecting the equity market to the default event.For example [3] proposes a dynamics for the assets and liabilities in the Merton's model where the equity is defined as a call option on the asset with liability as the strike price.This is a direct extension of the Merton model with multivariate stochastic volatility.In that case the price of a bond has quasi closed-form formulas based on the closed formulas for the conditional Laplace transform of the joint price-covariance process.We prefer not to use this model for several reasons.First, we found the CreditGrades model more popular in financial markets because of its ability to link the structural framework to equity derivatives.On the other hand, the model proposed by [3] has one common Wishart process and one distinct Wishart processes for each asset.Therefore, the number of parameters for the model is high and the calibration is extremely illposed because of the high degree of freedom imposed by the number of parameters inside the model.We found that in the case of two companies, assuming only one Wishart process driving the covariance matrix gives a fairly flexible model to capture market's behavior, while at the same time provides a fewer number of parameters.
Next, we introduce the equity process based on the reditGrades' perspective rather than Merton's perspective taking advantage of the flexibility of the Credit-Grades model.This will enrich the credit risk modeling with possibility of early default and also a straightforward link between credit risk and the equity market.We will derive a formula for the infinitesimal generator of the joint equity-covariance process below.This operator will play an important role in the partial differential equation of the equity option's price.Now that we have identified the dyna e explain the mechanism of the CreditGrades model.As before, we assume that the th i firm's value where for some , is the default barrier for the asset.There time based on CreditGrades model is given by iden fore, the default In the framework of CreditGrades model, the equity's va inf 0 In terms of the equity, the default time can be written as Zero is an absorbing state for ich makes the pricing of the equity option similar to pricing of down-and-out options studied by [11].By using

   
the equity process wh , the dynamics of   i D t and eq a shifted l rmal SDE.We will use the notation u o ation (1), the equity follows g-no   diag x ,   vec x and I denoting diagonal matrix and vector with elements x and t pec-he matrix of ones res tively 1 .
Note that the solution of the dynamics above can reach negative values but not before the stopping time i  .We force sufficient conditions on the Wishart process to make t  mean reverting.For our purposes, we assume M is negative definite and , Tr is the trace of a matrix, and used the notatio we've ve Pricing; Analytical Results

our rm and
The price of a European Call option on the equity is calsk-neutral expectation of the In this section, we tackle the pricing problem of credit risk model.We will use the fourier transfo method of images to solve the pricing problem for European calls and puts on the equity

Equity Call Options
culated by discounting the ri payoff at maturity.Since .
The price of a single name derivative on one of the equities satisfies the partial differential equation . Specially, the price of an equity call tion is given by the PDE where is the infinitesimal generator of the process , S  given by the proposition 1.We first the v change ariables by To use the method of images, we need to e drift term first, hence we change the variables by liminate the
The PDE ( 5) is our reference PDE to solve th problem for equity options on We have the following proposition for the Fourier transform of the G ansf reen's function of PDE Proposition 2: The Fourier tr orm of the Green's function of PDE is given by is involved, it is followed by the truncating factor   ( as in Equations (3) and (10) for the payoffs of call and put options.) , , d , where at we have found the Fourier transform of the Green's function of the pricing PDE, we solve the pricing problem for an equity call option by the method of images.

Now th
Proposition 3: The price of a call option on   j S t with maturity date T and strike price K is given by given as in proposition 2. For u ntegrand in ( 7) is exponentially decreasing which makes it easy to evaluate the integral numerically. mark where The function

 
, A k  ves the pr can be found by integration from.This gi ice of equity call option in the presence of Heston stochastic volatility (as in [9] Equa-.5)-(3.7)).The price of a European put option on the equity is calculated by discounting the risk-neutral expectation of the payoff at maturity.Similarly to payoff of the call option, one can check that . price of the put option could be rewritten as 1 Therefore, the Equations ( 3) and (10) give the put-call parity for the eq uity options , , ,0 exp d .

Survival Probabilities and Credit Default Swaps
Suppose   i any , , P t T S is the survival probability for the comp
most popu DS provid ce ts ds, until the default time or maturity Credit default swaps are one of the lar credit derivatives traded in the market.A C es protection against the default of a firm, known as referen entity.The buyer of the contract pays periodic paymen , called CDS sprea date.In return, the seller of the CDS provides the buyer with the unrecovered part of the notional if default oc urs.The valuation problem of a CDS is then to give the CDS spread a value such that the contract begins with a zero ans that lue of t th , .
The CDS spread is chosen such that the contract has a fair value at . By setting the fixed leg equal to th c value.This me the va he floating leg and e fixed leg should coincide when the contract is written.Assume that the CDS spread is denoted by Sp the periodic payments occur at , the notional is N , the time of default is denoted by τ and the recovery rate is the constant R .The fixed leg of the CDS is the value at time 0 t  of the cash flow corresponding to the payments the buyer makes.With the above notation we have On the other hand, the floating leg, which is the value of the protection cash flow at 0 , is In this section, we f present an stochastic eigenvalue process which is used for the covariance of the assets process.The section then covers pricing o the i Z 's are independent one-dimensional Brownian m otion and E is an orthogonal constant matrix.We also assume for i p and, without lost of generality,

 
The in ingr t of this multivariate process is a fam ly ne-dimensional stochastic processes for the We assume for simplicity Heston-type procs approa ma edien i of o eigenvalues.esses but thi ch works for other kind of processes.

The conditions 0
ensure stationarity, ergodicity and mixing conditions for the one-dimensional processes i  (see [4]).The constraints , on av ensure that the valu ss will erage, eigen es proce keep ths could ev tually cross over.Th e irm's valu the same order but their pa en is ordering on average allows us to keep the eigenvalues with greatest mean reverting levels while dropping the less significant ones.

The Dynamics of the Assets
We assume that th f e ics f In the two assets case, the above dynam ollows where the eigenvalues of the covariance process follow And assuming  e as the angle that the first eigenvector makes with th real axis, the eigenvector matri is given by We assume that assets are driven by the Brownian motion , the covariance matrix of the assets is driven by the B wnian motion T dependence assu ption between stock and its volatility is that closed form formulas for the value of double-ba rier options and equity options are not available when the asset and its volatility are correlated as pointed out by [8,9].
The infinitesimal generator of the joint process   , S  ,   , S  , appears in the pricing PDE.Here we find a fomula for this operator to use it for our pricing purposes in the next section.Since A can be divided into three terms related to the stock's operator, the covariance operator and their joint operator Z and d t W are independent, the last term is zero.From the dynamics of the equity, we know that

And fr
the classical results regarding the infinitesimal generator of the CIR process is a derivative on the first underly ing asset only, we have In the next section, we derive closed formulas for the pr pt

Derivative Pricing; Analytical Results
In a model with two underlyings, the first asset follows the following process: ice of equity o ions and marginal probabilities of default.
es as ions Calculating equity option prices is essential to calibrate the stochastic correlation CreditGrades model since this model uses the information available from the equity options to estimate the parameters of the model.Later, we will use the evolutionary algorithm method to match the theoretical results of our extended CreditGrades model with the market data.One of the advantages of the C with the equity option markets.The price of the equity option can be calculated nction at the maturity.The We will show next the prices of several derivativ seen from a credit perspective.

Equity Call Opt
reditGrades model compared to Merton's model, is the straight forward link it makes by discounting the payoff fu only subtle point here in pricing these options lies in the specific dynamics of the equity itself and the possibility of default for the company.In Black-Scholes model, the stock follows geometric Brownian motion which is a strictly positive process with a log-normal distribution and never hits zero.In the CreditGrades model, equity is modeled as a process satisfying a shifted log-normal distribution which hits the state zero when the company defaults.Because of the absorbing property of the state zero for the equity process, there is a resemblance in pricing the equity options and the pricing of the downand-out options.By considering the barrier condition for equity, the payoff of an equity call option is given by . Therefore the price of an equity call option can be written as: Similarly, the payoff of an equity put option is . Therefore, the price of an equity put option is given by: Equation (18) give the put-call parity for the equity options: The following proposition gives a closed form solution for the price of an equity call option on the first asset.Proposition C5 and Equation (18) give the price of an equity put option.This result is an essential tool to calibrate the model in the next section.
Proposition 5: The price of a call option on   1 S t n by: with maturity date and strike price is give

Survival Probabilities and Credit Default Swaps
Similar techniques can be used to find the marg al probabilities of default.Suppose is the survival probability for the com in  

, , i P t T S pany
Using the Feynman-Kac formula, satisfies the partial differential equation with boundary conditions   We have the fo probabilities Proposition 6: The survival probability for the firm is given by llowing proposition e su for th Knowing the probability of the defa ne can find the CDS spread for the underlying co e s i n 2e , , ult, o mpany.Assume that the CDS spread is denoted by , the perio ic payments occur at , the notional is N , the time of default is denoted by  and the of the CDS is recovery rate is the constant R .The fixed leg the value at time 0 t  of the cash ding ts the buyer makes.With the above notation we have f correspon e paymen .
e at low   On the other hand the floating leg, which is the value of the protection cash flow at 0 t  , is The CDS spread S is chosen such that the contract has a fair valu

Conclusion
We presented a structural credit risk m dit risk, we use the o called CreditGrades model.Using the affine properties of the joint log-price and volatility process, we solved the pricing problem a

Appendix
Proof proposition 1: Since d t Z and are independent, the last term is zero.By [1]: To find , by the dynamics of Proof proposition 2: Define and substituting into (5) yields Note that the functions satisfying the ODE above (i.e.

 ,
A k  and ) do not depend on the variable  , and then by substituting (20) into (19) leads to To solve the above ODE, we rearrange the equation as Therefore, H  satisfies: Since the function ij H is independent of  , assuming  to be a zero matrix except for the entry.Therefore This matrix Ricatti equation has been studied in the literature (see [13]) and in Affine term structure models (see [14]) leading to:  can be found by integration.Proof proposition 3: The previous proposition gives the Fourier transform of the Green's function of the pricing PDE.Now note is invariant with respect to the change of variables and , therefore is an even function with respect to .This implies that the Fourier transform of the Green's function absorbed at b q y y q y y q y y With the consequent chan s of variables ge

P S D t P r t d t S P P P t T P T T S
,0 e , 0, 0.
This PDE is the same as (5).In Proposition 2, we have proved that the aggregated Green's function for this PDE is of the form (28). To find a bounded solution reflected at we use the method of images to write the abso 0 x  , rbed aggregated Green's function as y q y y q y y   is the infinitesimal generator of the SDE driving the equity.By substitution From now, we drop the index We change the variables as We claim that the Fourier transform of the Green function for the above PDE is of the form , , e d 4 e e , 2 l n ( ) 2 1 , 4 We know that In the proof of the proposition 5 we showed that the Fourier transform of the Green's function for the above PDE is of the form

2 :
floating leg, the equations and imply model.We first remind the formal definition below:Definition The instantaneous stochastic covariance follows a Principal Component Model if:

Z
and two Brownian otions m t W and t Z are unco he reason we make the in m r-rrelated.
above for-y one can find the survival probab mula for as:


perform the second change of variables as And finally we perform the third change of vari Moreover, without loss of generality, we as- , S  is given by se ,