The Distribution of the Value of the Firm and Stochastic Interest Rates

The time evolution of the value of a firm is commonly modeled by a linear, scalar stochastic differential equation (SDE) of the type   d d d t t t v t V rV t t V W      v t  d t r v is 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. t r


Introduction
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 [1] extends this line of research by examining an optimal mixture of debt; that is, the optimal mixture of bank debt and market debt (bonds).
Improved models for value of the firm are potentially useful in several contexts.For one example, consider models of credit spreads.Leland and Toft [2] develop an ambitious model of firm value that addresses optimal capital structure, optimal debt maturity, and the term structure of credit spreads.Recently, Qi [3] has modified the Leland and Toft [2] model by setting the lower bankruptcy boundary to be a fraction of bond face value.The importance of good structural models for credit spreads has been enhanced with the growth of credit derivatives and the credit crisis of 2007 and 2008.More specifically, notional amounts of credit derivatives grew by over 100% for every year from 2004 through 2006.At the end of 2006, there was 34.5 trillion outstanding (see Saha-Bubna and Barrett [4]).The weakened credit quality of many financial firms in 2007 and 2008 caused high volatility in equity markets and, also, large changes in the value of credit spreads and credit default swaps.
Our purpose is to derive distributions of t whose evolution critically depends on the models for the short term interest rate process, t .Models for t 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 [5]; Duffie and Singleton [6]; and Lamberton and Lapeyre [7].In this paper we are particularly interested in a special class of the general linear models called narrow sense linear models described by Arnold [8].

V r r
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 t .Then, we describe solutions in the cases where t r processes are narrow sense linear.Such t r processes are quite popular for models of credit risk.The shapes of the t V d tributions are shown to be sensitive to such parameters as the correlation between the V and processes.For example, a positive correlation displays a distribution with fatter tails than one with negative

The Processes for Value of the Firm and Interest Rates
The time evolution of the value, t , of a firm is routinely modeled under the risk neutral measure by a linear, scalar, stochastic differential equation (SDE) where the instantaneous drift t 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 [9].


This general form of value process has been used in numerous important structural models of credit risk.For example, see Merton [10] and Acharya and Carpenter [9] where any dividends and coupon payments, outflows of γ from the firm to investors, are subtracted from the r 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 [11] similarly have a zero γ.
The t drift of t 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.

V r
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, It is worth noting that in this set up the flow of information is only one wayt affects t V 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 [12] and Privault [13] for details.In this paper, we are primarily interested in the SFMs.These SFMs can be divided into linear and nonlinear models.Following Arnold [8], linear models can be further subdivided into two subclasses.The SFM in ( where 2 and are smooth functions of time .The general linear models are also known as affine models as in Duffie, Filipovic, and Schachermayer [5]; Duffie and Singleton [6]; and Lamberton and Lapeyre [7]  The narrow sense linear models of Merton [14], Vasicek [15], Ho and Lee [16], and Hull and White [17] are special cases of the Heath, Jarrow and Morton [18] model and define normal/Gaussian processes.r We first solve the scalar SDE (1.2) for t , and using it in (1.1), we then recover .It is well known that t is a lognormal process when t a constant.See Kloeden and Platen [19].We extend this result by first showing that t also inherits this lognormal distribution where t is a normal process defined by the narrow sense linear models in Table 1.
This problem of quantifying the probability distribution of t is critical to credit risk analysis.For a review of various approaches to credit risk refer to the books by Duffie and Singleton [6], Bielecki and Rutkowski [20], and Jarrow et al. [21].Clearly computation of the default probability in structural models requires knowledge of the probability distribution of contingent on the chosen model for the interest rate.

A framework for the Solution
In this section we develop a framework for solving (1.1)-(1.2).Setting ln and applying Ito's lemma, Equation (1.1) becomes.
Black, Derman and Toy [28]  where ln where and are two independent and From the properties of the Ito in and Shreve [22], it follows that we get This is known as the fundamental solution of (3.2).Hence the solution of (3.2) is given [31], Kuo [32], Lamberton and Lap by (Arnold [8], Gard eyre [7]) where and where where ry is immediate.

Corollary 3.2: Since
The following corolla bability density function, as a function of time, is given by It can be verified (Johnson et al. [33]) that the time evolution of the mean and variance of the value process and e now enlist a number of nested corollaries by considering special cases of interest rate models.
Case 1: Case 3: Ho-Lee [16] model: In this model, and and the variance is Case 4: Vasicek [15] model: Hence, the mean is and the variance is Hence, the mean is and the variance is   .T e primary motivation for characterizing the distribution of t V is to compute the p o bility of default.Within the framework of structural models, there has been an evolution of the definition of default.In the now classic paper, M n [10] defines default as the event where K is the face value of the discount bond with aturity T .Using the results described above, we could readily compute the probability default according to this classical definition 1   However, Longstaff and Schwartz [11] define default by the event   . Recently, Giesecke [34] has expanded on this theme and has defined the default by the compound event   0 min or 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 [35] and Giesecke [34]).To enable co putation of default probability according to Giesecke [34], in the following, we seek to express To this end recall that every to integral is equiv e W pr I alent to a tim changed iener ocess.(See Shiryaev [36], Oksendal [37], Karatzas and Shreve [25]).Accordingly, om (3.17) we obtain fr  which is an integral of the exponential functionals of the Wiener process.Processes of the type t A routinely arise in the evaluation of Asian type options (Vorst [39]).By relating to a Bessel process, Yor [40] and Geman and Yor [41] have provided a complete characterization of the distribution of the t A process.Combining these results with (2.5)

t
Refer to Tables 1(a)-(c) for examples of these models.

Figure 1
Figure 1.Probability density of for Vasicek with

2: Hull and White [17] model: In this
s Case to derive the distribution of t g is an interesting open problem.Similar , computing the dis-