Generalizing Merton’s approach of pricing risky debt: some closed form results

In this work, I generalize Merton’s approach of pricing risky debt to the case where the interest rate risk is modeled by the CIR term structure. Closed form result for pricing the debt is given for the case where the firm value has non-zero correlation with the interest rate. This extends previous closed form pricing formular of zero-correlation case to the generic one of non-zero correlation between the firm value and the interest rate. PACS number: 05.40.+j Fluctuation phenomena, random processes and Brownian motion; 01.90+g other topics of general interest Typeset using REVTEX 1 One well known approach of pricing risky debt was pioneered by Merton a long time ago. Within the framework, one assumes a stochastic process for the firm value and treats the risky debt as a option [1–5]. In Merton’s original work [1], he assumed that the interest rate is constant and that the default event of the debt can only occur at the time of maturity. Closed form results for pricing the risky debt were given explicitly. As he has pointed out, one can also study the case when stochastic interest rate is taken into account. This can easily be achieved by using Merton’s work that generalized Black-Scholes formular of the option pricing with stochastic interest rate [1]. However, one has to make the assumption that the bond process of the stochastic interest rate has a non-stochastic volatility which is allowed to be deterministic time-dependent [1]. Therefore, Shimko etc [2] applied Merton’s results to the case of stochastic interest rate described by the Vasicek model, the bond process of which has a non-stochastic volatility. However, they were unable to handle the case where the interest rate is modeled with the CIR term structure, as the CIR interest rate model will give rise to a bond process having stochastic volatility [7]. It remained open whether one can give a similar closed form results for the risky debt when the interest rate risk is modeled with CIR term structure. In one recent work, I gave the closed form pricing formular for the risky debt in the case where CIR term structure is used for the default-free interest rate. However, using the moment generating functional, one has to assume that the firm value is uncorrelated with the interest rate [8]. In this paper, I extend my closed form results of pricing risky debt to the generic case where the correlation between the firm value and the interest rate is nonzero. Let us first assume a probability space denoted by (Ω, P, {Ft}, F ), with the filtration {Ft}. Consider the value of the firm that is described by the following process dV V = μdt+ σdZ1, (0.1) where Z1 is a Brownian motion in the probability space. The interest rate process is assumed to be the one given by Cox-Ingersoll-Ross [7]: dr = (a− βr)dt+ ηdZ2, (0.2) 2 where η = σrr 1/2 with σr as a constant. The co-quadratic variational process is [Z1, Z2] = ρt. The correlation coefficient ρ is a non-zero constant during the following consideration. The assumptions in Merton’s paper are also made here [1]. The firm value is assumed to be independent of its capital structure by assuming that MM theorem is valid. The firm issues debt and equity. The total value of the firm is the sum of equity and debt. The PDE satisfied by the equity is given by Hτ = σ 2 V HV V + ρησV HV r + η 2 Hrr + rV HV + (α− βr)Hr − rH (0.3) where H = H(V, r, T − t) and τ = T − t is time to the maturity, and α is sum of a plus the constant representing the market price of the interest rate risk. At τ = 0, the equity should satisfy the boundary condition that H = max(0, V (T ) − B), where B is the face value of the debt issued by the firm maturing at time T . The risky debt price is therefore given by Y = V (t)−H(V, r, T − t). For simplicity, it is assumed here, as Merton did, that event of default of the risky debt can only occur at the time of maturity. Following the standard risk neutral approach, we write the equity price as below: H = E(e ∫ T t max(0, V (T )−B)|Ft), (0.4) where the expectation E means that in the risk-neutral-adjusted world. In this risk-neutral world, the firm value and the interest rate will follow the stochastic differential equations as dlnV = (r − 1 2 σ)dt+ σdẐ1 dr = (α− βr)dt+ ηdẐ2. (0.5) Here, both Ẑ1 and Ẑ2 are Wiener processes in the risk-neutral world, and the co-quadratic process is [Ẑ1, Ẑ2] = ρt. In principle, one can go on with Monte-Carlo simulation based on the above formular Eq.(0.4) and Eq.(0.5). However, we are interested in finding closed form result for the equity price here. In the risk-neutral world, the two Brownian motions Ẑ1 and Ẑ2 can be represented in the following way. Suppose that X and Y are two independent Brownian motions in the risk3 neutral world. We can find such independent Brownian motions that Ẑ1 = ρX+ √ (1− ρ2)Y , and Ẑ2 = X. The stochastic differential equation for the firm value is governed by dlnV = [rdt+ σρdX − 1 2 σρdt]− 1 2 σ(1− ρ)dt+ √