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The time evolution of the value of a firm is commonly modeled by a linear, scalar stochastic differential equation (SDE) of the type where the coefficient in the drift term denotes the (exogenous) stochastic short term interest rate and is the given volatility of the value process. In turn, the dynamics of the short term interest rate are modeled by a scalar SDE. It is shown that exhibits a lognormal distribution when is a normal/Gaussian process defined by a common variety of narrow sense linear SDEs. The results can be applied to different financial situations where modeling value of the firm is critical. For example, with the context of the structural models, using this result one can readily compute the probability of default of a firm.

Modeling the value of the firm is one of the more important research topics in finance. The value of an unlevered firm is the value of expected future cash flows discounted at a rate appropriate for an all-equity firm whereas the value of a levered firm is commonly expressed as the value of an unlevered firm plus the gain from leverage due to a tax shield provided by the debt. Including business disruption costs, the optimal capital structure can then be characterized as a trade-off between the interest tax shield and disruption costs. Recent analysis by Hackbarth, Hennessy and Leland [

Improved models for value of the firm are potentially useful in several contexts. For one example, consider models of credit spreads. Leland and Toft [

Our purpose is to derive distributions of whose evolution critically depends on the models for the short term interest rate process,. Models for can be broadly classified as (a) general linear and (b) non linear models. General linear models are also popularly known as affine models. In this context, we refer to Duffie, Filipovic and Schachermayer [

The next section describes the processes for value of the firm and short term interest rates. Next, we discuss a general framework for the solution of the distribution of. Then, we describe solutions in the cases where processes are narrow sense linear. Such processes are quite popular for models of credit risk. The shapes of the distributions are shown to be sensitive to such parameters as the correlation between the and processes. For example, a positive correlation displays a distribution with fatter tails than one with negative correlation.

The time evolution of the value, , of a firm is routinely modeled under the risk neutral measure by a linear, scalar, stochastic differential equation (SDE)

where the instantaneous drift denotes the (exogenous) stochastic variable known as the short-term interest rate process and is a deterministic function representing the instantaneous volatility as in Acharya and Carpenter [

This general form of value process has been used in numerous important structural models of credit risk. For example, see Merton [_{t} drift term. Many firms do not pay dividends and our model is one of zero coupon debt so that a γ of zero is reasonable. We note that Longstaff and Schwartz [

The drift of indicates our model is risk neutral. We could assume different firms have different drifts due to such things as different expected returns in their industry as well as different riskiness of assets and future projects. However, such an assumption is arbitrary and yields a model that is not arbitrage free. We believe it is much more theoretically credible to posit a risk neutral, arbitrage free model.

The dynamics of the short term interest rate are modeled, under the same risk neutral measure, by a (scalar) SDE of the type

where the instantaneous drift, and the volatility, are smooth functions. It is further assumed that the Wiener increment processes and are correlated; that is,

with. It is worth noting that in this set up the flow of information is only one way –affects and not vice versa. By combining several well known results from the literature, in this paper we characterize the distribution of the value process for different choices of the processes.

All the known stochastic interest rate models can be broadly classified into two classes—single factor models (SFMs) and multi-factor models (MFMs). We refer to Cairns [

and

A general linear model, on the other hand, has in the form (1.4) and

where and are smooth functions of time. The general linear models are also known as affine models as in Duffie, Filipovic, and Schachermayer [

Refer to Tables 1(a)-(c) for examples of these models. The narrow sense linear models of Merton [

We first solve the scalar SDE (1.2) for, and using it in (1.1), we then recover. It is well known that is a lognormal process when a constant. See Kloeden and Platen [

This problem of quantifying the probability distribution of is critical to credit risk analysis. For a review of various approaches to credit risk refer to the books by Duffie and Singleton [

In this section we develop a framework for solving (1.1)- (1.2). Setting and applying Ito’s lemma, equation (1.1) becomes.

See Kloeden and Platen [

Setting and

(Shreve [

and

where and are two independent Wiener increment processes and

Integrating (2.3), we obtain

where

and

From the properties of the Ito integral, Mikosch [

where

Since and are independent Wiener processes, it readily follows that . Thus, the distribution of and hence of critically depend on the properties of the process in (2.2).

In closing this section consider the special case when, a constant. Then,

Further, when, we obtain

4. Narrow Sense Linear Models for r_{t}

Setting

in (2.2), we get a narrow sense (time varying) linear model known as the generalized Hull and White [

Since all the other narrow sense linear models in

we get

This is known as the fundamental solution of (3.2). Hence the solution of (3.2) is given by (Arnold [

where

and

Hence,

where

Now combining (3.5)-(3.10), it follows that

Applying integration by parts to the second integral on the right hand side of (3.11) and using (3.7), it follows that

Is also a Gaussian process with mean zero and variance given by

Substituting (3.11)-(3.12) in (2.5), we get

where

and

Substituting (2.6), (3.12) and (2.7) in (3.16), the latter becomes

Since and are independent, it follows that where

where

and

.

Combining (3.15)-(3.18) with (3.14), we finally obtain

where given by (3.16) and given by (3.19).

We summarize the above developments in the following:

Theorem 3.1: Let the interest rate evolve according to a narrow sense linear, scalar, SDE of the type (3.2). Then, is a Gaussian process and consequently in (2.5) is also a Gaussian process given by (3.20).

Since, from (3.14)-(3.20), we get

where

and

The following corollary is immediate.

Corollary 3.2: Since,

is a lognormal process whose probability density function, as a function of time, is given by

It can be verified (Johnson et al. [

are given by

and

We now enlist a number of nested corollaries by considering special cases of interest rate models.

Case 1: Let, a constant. Then

, and is given by

(3.8). From (3.15) and (3.20), the mean is

where is given in (3.6). From (3.19)-(3.20), the variance is

Case 2: Hull and White [

Hence the mean is

and the variance is

Case 3: Ho-Lee [

The mean is

and the variance is

Case 4: Vasicek [

Hence, the mean is

and the variance is

Case 5: Merton [

Hence, the mean is

and the variance is

Case 6: Let, a constant and. Then, and. The mean

and the variance

We now provide sample plots of the distribution, when follows the Vasicek [

The primary motivation for characterizing the distribution of is to compute the probability of default. Within the framework of structural models, there has been an evolution of the definition of default. In the now classic paper, Merton [

However, Longstaff and Schwartz [

Recall that the probability of these later events can be readily calculated using the “reflection principle” if is a standard Wiener process or by using the Girsanov theorem if is a Wiener process with a drift. (See Elliott and Kopp [

To this end recall that every Ito integral is equivalent to a time changed Wiener process. (See Shiryaev [

where and are two independent Wiener process with

,

and and are given in (3.19). Since and are independent, there exists a Wiener process such that

where

as given by (3.18)-(3.19).

Combining (3.43) with (3.14), it follows that

where as in (3.20).

5. Conclusions

We have analyzed the impact of on when evolves according to a narrow sense linear model in

where the process is given by

with. Hence,

involves a process

which is an integral of the exponential functionals of the Wiener process. Processes of the type routinely arise in the evaluation of Asian type options (Vorst [

6. Acknowledgements

We are grateful to Robert J. Elliott (University of Calgary) and to Luciano Campi (Universite Paris Dauphine) for their interest and comments that improved the presentation.

REFERENCES