Plan de Salud Ocupacional y Seguridad industrial Contenido del Plan

Note that this expectation can be obtained via

∂

∂sQ(τ^1st > s) = ∂

∂sIE

exp(− Zs

t

λ^1st_u du)

= IE exp(−

Zs

t

λ^1st_u du)λ^1st(s) .

Further on, Schmidt and Ward (2002) derive interesting results on spread widening, once a default occurred. For example, if one of two strongly related companies defaults, it might be likely that the remaining one gets into difficulties, and therefore credit spreads increase. It seems interesting that traders have a good intuition on this amount of spread widening, which also could be used as an input parameter to the model, which determines the copulas.

1.6 Hybrid models

Hybrid models incorporate both preceding models, for example the firm value is modeled, and a hazard rate framework is derived within this model.

1.6.1 Madan and Unal (1998)

The approach of Madan and Unal (1998) mimics the behavior of the Merton model in a hazard rate framework. They assume the following structure for the default intensity:

λ(t) = c

ln_{F ·B(t)}^{V (t)} 2.

Here V (t) denotes the firm value which as in Merton’s model is assumed to follow a geometric Brownian motion. B(t) is the discounting factor exp(−Rt

0 rudu) and F is the amount of outstanding liabilities. If the firm value approaches F the default intensity increases sharply and it is very likely that the bond defaults. As defaults can happen at any time this model is much more flexible than the Merton model. Unlike in Longstaff and Schwartz’s model, the default can even happen when the firm value is far above F , though with low probability.

The authors also consider parameter estimation in their model. A closed form solution for the bond price is not available and for calculating the prices of derivatives numerical methods need to be used.

Further hybrid models of this type can be found in Ammann (1999) or Bielecki and Rutkowski (2002).

1.6.2 Duffie and Lando (2000)

The model of Duffie and Lando (2001) accounts for the fact that bond holders receive only imperfect information on the issuer’s assets. The approach starts with a structural model for the firm value and assumes that the bond holder obtains observations on the firm value disturbed and only at discrete time points, which leads to a hazard rate model. After presenting the framework proposed by the authors we derive the hazard rate explicitly.

Suppose the firm value can be modeled by a geometric Brownian motion, as in the Merton framework, i.e.,

Vt = V0exp((µ− σ²

2 )t + σWt) =: V0exp(mt + σWt).

The firm is operated by equity owners, which have complete information on the firm’s assets, represented by F^t = σ(Vs : 0 ≤ s ≤ t). The first step is to determine the optimal liquidation policy.

Assume that the drift of the firm value is smaller than the risk-free interest rate, µ < r, and further on, the firm generates cash flow at the rate δVt for some constant δ > 0. Then the present value of the firm’s future cash flow is finite, respectively

IEhZ^∞

t

e^−r(s−t)δVsds Ft

i

= δVt

Z∞

t

e^{(µ−r)(s−t)}ds = δVt

r− µ.

If µ≥ r the present value of the firm’s future cash flow is infinite. This case poses several problems and an optimal exercise policy like the one determined in equation (1.11) below is not available. Nevertheless, one could assume that equity owners liquidate the firm at the first time when the firm value falls below a certain boundary, thus, assuming directly that (1.11) holds.

If the equity holders choose to liquidate the firm, a fraction α∈ [0, 1] of the assets is lost because of liquidation costs. The outstanding debt D has to be paid to the debt-holders, if possible, and the remaining value goes to the equity holders, that is

min(D, (1− α) δVt

r− µ) −→ debtholders

max(0, (1− α) δVt

r− µ − D) −→ equity.

If the debt takes the form of a consol bond, meaning that the coupons are paid continu-ously at rate C > 0 and the tax benefit therefore yields the constant rate θC, we conclude for the initial value of equity, according to a certain liquidation policy represented by a (Ft)_t≥0-stopping time τ , that

F (V₀, C, τ ) = IEhZ^τ

0

e^−rt δV_t+ (θ− 1)C

dt + e^−rτmax(0,(1− α)δV^τ

r− µ − D)i .

1.6 Hybrid models 31

As the equity owners will choose the liquidation policy maximizing the initial value of equity, this leads to the optimization problem

S0 = sup

τ ∈T

F (V0, C, τ ),

whereT is the set of all (F^t)_t≥0-stopping times. The optimal strategy, as shown by Leland and Toft (1996), is given by

τ (VB) = inf{t : Vt ≤ VB}, (1.11) with a certain level VB which can be determined by solving a Hamilton-Jacobi-Bellman differential equation¹⁶. For “conventional parameters”, the authors are able to show that

VB = VB(C) = (1− θ)Cγ(r − µ)

r(1 + γ)δ , γ = m +√

m²+ 2rσ²

σ² .

Turning to the bond holder’s perspective, we notice that they receive information on the firm value just at selected times t1, t2, . . . . This is modeled by a noisy observation of Vti, i.e., instead of observing Vti the market participants observe¹⁷

V˜ti := Vti · exp(Zⁱ− σ_Z² 2 ).

The Zi are assumed to be independent normally distributed random variables with vari-ance σ²_Z and being independent of (Ws)_s≥0.

If we assume for simplicity that equity is not traded on the public market, the information available to the bond holder is

H^t = σ( ˜Vt1, . . . , ˜Vtn, 1_{{τ ≤s}} : t1, . . . , tn ≤ t and 0 ≤ s ≤ t).

In this framework the probability for no default until T equals 1_{{τ >t}}IP(τ > T|H^t) = 1_{{τ >t}}IP( inf

s∈(t,T ]Vs > VB|H^t).

Fix t and denote by tk the last tn which is smaller than or equal to t. Because (Wt)_t≥0 is a Markov process, it is sufficient to condition on a smaller σ−algebra, and therefore

1_{{τ >t}}IP(τ < T|H^t)

= 1_{{τ >t}}IP Vt · inf

s∈(t,T ]exp(m(s− t) + σ(Bs− Bt)) > VB

 ˜Vtk, 1_{{τ ≤t}}

= 1_{{τ >t}}IP inf

s∈(t,T ]m(s− t) + σ(Bs− Bt) > lnV_B Vt

 ˜Vtk, 1_{{τ ≤t}}

, (1.12)

16For a detailed treatment of optimization problems in the financial context, see Korn and Korn (1999, Chapter V).

17Duffie and Lando (2001) use ˜Vti := Vti· exp(Zⁱ) instead. This is equivalent in terms of information, but seems counterintuitive as in that case the expectation of ˜Vti is not Vti.

where

where ξ has a standard normal distribution and is independent of ˜V_tk. We obtain the decomposition

This decomposition of Vt into independent random variables will enable us to calculate the desired probability. Consider

(1.12) = 1_{{τ >t}}IP inf of B, we can apply equation (B.2). Recall that equation (B.2) yields for c < 0

IP inf