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In the framework of CreditGrades model, the equity’s value is given by
In terms of the equity, the default time can be written as. Zero is an absorbing state for the equity process which makes the pricing of the equity option similar to pricing of down-and-out options studied by . By using, the dynamics of and equation (1), the equity follows a shifted log-normal SDE. We will use the notation, and I denoting diagonal matrix and vector with elements and the matrix of ones respectively1.
Note that the solution of the dynamics above can reach negative values but not before the stopping time. We force sufficient conditions on the Wishart process to make mean reverting. For our purposes, we assume is negative definite and for some. Moreover, without loss of generality, we assume. We first derive the infinitesimal generator of the joint process. This operator will appear in the pricing PDE for equity options and the probabilities of default in the next section Proposition 1: The infinitesimal generator of the joint process is given by
where, is the trace of a matrix, and we’ve used the notation and.
2.2. Derivative Pricing; Analytical Results
In this section, we tackle the pricing problem of our credit risk model. We will use the fourier transform and method of images to solve the pricing problem for European calls and puts on the equity
2.2.1. Equity Call Options
The price of a European Call option on the equity is calculated by discounting the risk-neutral expectation of the payoff at maturity. Since
the price of the call option could be rewritten as
The price of a single name derivative on one of the equities satisfies the partial differential equation . Specially, the price of an equity call option is given by the PDE
where is the infinitesimal generator of the joint process given by the proposition 1. We first change the variables by,
to transform the PDE to
To use the method of images, we need to eliminate the drift term first, hence we change the variables by
Then, PDE (3) transforms to
The PDE (5) is our reference PDE to solve the pricing problem for equity options on. We have the following proposition for the Fourier transform of the Green’s function of PDE Proposition 2: The Fourier transform of the Green’s function of PDE is given by
Now that 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.
Proposition 3: The price of a call option on with maturity date and strike price is given by
and the function is defined by
with and given as in proposition 2.
For large values of k, the integrand in (7) is exponentially decreasing which makes it easy to evaluate the integral numerically.
Remark 1: As we have mentioned before, our result covers  as a special case. If in the dynamics of the asset (1), we assume and for the parameters we let, and, propositions 2 and 3 yield. Now to find andby proposition 2 we have
is a matrix with
The function can be found by integration from. This gives the price of equity call option in the presence of Heston stochastic volatility (as in  Equations (3.5)-(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
. Therefore, the price of the put option could be rewritten as
Equations (3) and (10) give the put-call parity for the equity options
2.2.2. Survival Probabilities and Credit Default Swaps
Suppose is the survival probability for the company
then using Feynman-Kac formula, satisfies the partial differential equation.
Proposition 4: The survival probability for the firm is given by
with functions and as in equation (7).
Credit default swaps are one of the most popular credit derivatives traded in the market. A CDS provides protection against the default of a firm, known as reference entity. The buyer of the contract pays periodic payments, called CDS spreads, until the default time or maturity date. In return, the seller of the CDS provides the buyer with the unrecovered part of the notional if default occurs. The valuation problem of a CDS is then to give the CDS spread a value such that the contract begins with a zero value. This means that the value of the floating leg and the 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, the time of default is denoted by τ and the recovery rate is the constant. The fixed leg of the CDS is the value at time 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, is
The CDS spread is chosen such that the contract has a fair value at. By setting the fixed leg equal to the floating leg, the equations and imply
3. The CreditGrades Principal Component Model
In this section, we first present an stochastic eigenvalue process which is used for the covariance of the assets process. The section then covers pricing of derivatives using the CreditGrades model. We first remind the formal definition below:
Definition 2: The instantaneous stochastic covariance follows a Principal Component Model if:
where is a diagonal matrix whose elements are real valued CIR process defined, for, by:
the’s are independent one-dimensional Brownian motion and is an orthogonal constant matrix. We also assume, for andwithout lost of generality,.
The main ingredient of this multivariate process is a family of one-dimensional stochastic processes for the eigenvalues. We assume for simplicity Heston-type processes but this approach works for other kind of processes.
The conditions, ensure stationarityergodicity and mixing conditions for the one-dimensional processes (see ). The constraints ensure that the eigenvalues process will keep, on average, the same order but their paths could eventually cross over. This ordering on average allows us to keep the eigenvalues with greatest mean reverting levels while dropping the less significant ones.
3.1. The Dynamics of the Assets
We assume that the firm’s value is driven by the dynamics
where, , with
Each follows a CIR process of the type
In the two assets case, the above dynamics follows
where the eigenvalues of the covariance process follow
And assuming as the angle that the first eigenvector makes with the real axis, the eigenvector matrix is given by
We assume that assets are driven by the Brownian motion, the covariance matrix of the assets is driven by the Brownian motion and two Brownian motions and are uncorrelated. The reason we make the independence assumption between stock and its volatility is that closed form formulas for the value of double-barrier 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, , 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, the equity satisfies the stochastic differential equation
can be divided into three terms related to the stock’s operator, the covariance operator and their joint operator
Since and are independent, the last term is zero. From the dynamics of the equity, we know that
And from the classical results regarding the infinitesimal generator of the CIR process
Therefore, if is a derivative on the first underlying asset only, we have
In the next section, we derive closed formulas for the price of equity options and marginal probabilities of default.
3.2. Derivative Pricing; Analytical Results
In a model with two underlyings, the first asset follows the following process:
We will show next the prices of several derivatives as seen from a credit perspective.
3.2.1. Equity Call Options
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 CreditGrades model compared to Merton’s model, is the straight forward link it makes with the equity option markets. The price of the equity option can be calculated by discounting the payoff function at the maturity. The 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 with maturity date and strike price is given by:
3.2.2. Survival Probabilities and Credit Default Swaps
Similar techniques can be used to find the marginal probabilities of default. Suppose is the survival probability for the company
Using the Feynman-Kac formula, satisfies the partial differential equation with boundary conditions and. We have the following proposition for the survival probabilities Proposition 6: The survival probability for the firm is given by
Knowing the probability of the default, one can find the CDS spread for the underlying company. Assume that the CDS spread is denoted by, the periodic payments occur at, the notional is, the time of default is denoted by and the recovery rate is the constant. The fixed leg of the CDS is the value at time 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, is
The CDS spread is chosen such that the contract has a fair value at. By setting the fixed leg equal to the floating leg, the Equation (18) imply
We presented a structural credit risk model which considers stochastic correlation between the assets of the companies. The stochasticity of the volatility and correlation comes from first a Wishart process and then a principal component stochastic covariance process which drives the covariance matrix of the assets. To model credit risk, we use the so called CreditGrades model. Using the affine properties of the joint log-price and volatility process, we solved the pricing problem of the equity options. We used our analytical techniques to derive quasi closed-form solution for equity options, probabilities of defaults and prices of CDSs issued by the companies.
Proof proposition 1:
can be divided into
Since and are independent, the last term is zero. By :
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. and) do not depend on the variable. So we set to get
and then by substituting (20) into (19) leads to
To solve the above ODE, we rearrange the equation as
Since the function 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 ) and in Affine term structure models (see ) 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 is
By Duhamel’s formula
With the consequent changes of variables
one can conclude that.
Proof Proposition 4:
The PDE for survival probability is:
Using the change of variables,
and, the PDE transforms to
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 absorbed aggregated Green’s function as
Now by Duhamel’s formula
Using the change of variable, one can find the survival probability from the above formula for as:
Proof proposition 5:
By risk neutral valuation, W satisfies
where is the infinitesimal generator of the SDE driving the equity. By substitution
From now, we drop the index. We change the variables as
We perform the second change of variables as
And finally we perform the third change of variables
We claim that the Fourier transform of the Green’s function for the above PDE is of the form
We know that satisfies the corresponding PDE. Plugging into the PDE, one gets Ricatti ODE’s for and’s, which finally gives the function as
The representation for the function comes from equation
Note that has a structure invariant with respect to the change of variables
Therefore, the Fourier transform absorbed at is
the above expression and Duhamel’s formula leads to:
Since the result follows.
Proof proposition 6:
Substituting for the infinitesimal generator from Equation (17), solves
Using the change of variables,
and, the PDE (26) transforms to
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
and the functions and are given. In order to find a bounded solution reflected at, we use the method of images to write the absorbed aggregated Green’s function as
Now by the Duhamel’s formula
Therefore, the survival probability is given by
1Note that with the above dynamics is allowed to gain negative values but not prior to the stopping time. Even though it might seem unreasonable to allow have negative values, this doesn’t affect any of the pricing formulas since whenever the process is involved, it is followed by the truncating factor ( as in Equations (3) and (10) for the payoffs of call and put options.)