Computing default probabilities and pricing with de-

Z >

Z t 0

λ(s)ds

G∞



= e^−R

t 0λ(s)ds

.

Applying the tower property of the conditional expectation eventually gives

P[τ > t|G^t] = e⁻

Rt 0λ(s)ds

,

hence (A1) and (A2) are satisfied. Finally, define H_t = 1{τ ≤t}, H_t = σ(Hs|s ≤ t) and Ft:= Gt∨ Ht.

5.1.3 Computing default probabilities and pricing with default risk

Up till now, all results were stated w.r.t. to the physical measure P. As it is known, for pricing, a risk-neutral measure is needed. Since in general, assumptions (A1) and (A2) are not preserved under an equivalent change of measure (for details, see [12, Chapter 12.3.4]), one needs to explicitly assume that (A1) and (A2) are satisfied for a measure Q ∼ P. Furthermore it is supposed that there exists a process r and a non-negative process λ^Q, both being (G_t)-progressive, which represent the interest rate and intensity respectively, with

Z t 0

|r(s)| + λ^Q(s)ds < ∞ Q − a.s..

Default probabilities

Recall the formula for the conditional default probability

Q[τ > T |Ft] =1^{{τ >T }}E[e⁻

RT

t λ(s)ds|G_t].

One could decide to use one of the positive stochastic processes used for interest rate modelling, such as the CIR or CIR++ model, to describe the dynamics of the process λ^Q, where the process W (t), which drives the

ran-dom shocks, is assumed to be a Brownian motion w.r.t. to G. Then, the conditional expectation in the expression above is equal to the bond price, and is therefore explicitly given. In mathematical terms

Q[τ ≤ T |Ft] =

The goal is to determine the price of a defaultable bond under different recovery assumptions, namely zero recovery and recovery of market value. It is also possible to assume a partial recovery at default or at maturity of the bond, but these cases are not going to be dealt with in this thesis.

Pricing with zero recovery

Zero recovery means that, in the event of default, the bond becomes worthless and no payment at all is made. The payoff of this bond at maturity T is 1{τ >T }. Because of the results stated in Chapter 2.3, the no-arbitrage price at time t is given by

Using the theoretical framework established in the previous section, espe-cially Lemma 5.1.2, the above expression can be simplified to

P (t, T ) = E¯ Q

This means that in the zero recovery case, a defaultable bond can be priced by discounting by the sum of the risk-free rate r and a spread λ^Q. For certain dynamics of r and λ^Q, one can just apply the theory developed in the chapters on interest rate models, which can be seen in the following example:

Proposition 5.1.7. Assume the interest rate dynamics are given by a Cox-Ingersoll-Ross process, i.e.,

dr(t) = k(θ − r(t))dt + σ^qr(t)dW (t), r(0) = r₀,

with r₀, k, θ, σ > 0, where W (t) is a Brownian motion w.r.t. to (Q, Gt). Let λ^Q = c₀ + c₁r(t) be an affine function in r(t) with non-negative constants c₀, c₁. Then the zero-recovery defaultable bond price at time t with maturity T is given by

Proof. Inserting the model into the general bond price formula yields P (t, T ) =¯ 1{τ >t}EQ

Since

d(1 + c₁)r(t) = k((1 + c₁)θ − (1 + c₁)r(t))dt + (1 + c₁)σ^qr(t)dW (t)

⇔ d¯r(t) = k((1 + c₁)θ − ¯r(t))dt +^q(1 + c₁)σ^qr(t)dW (t),¯

¯

r(t) is also a CIR process, just with parameters ¯k = k, ¯θ = θ(1 + c₁), ¯σ =

q(1 + c₁)σ, which means that the expectation in (∗) is nothing else than the default-free bond price of the CIR process ¯r(t). Therefore, one can use the explicit formula (3.3.2), which immediately proofs the claim.

Pricing with recovery of market value

In a lot of cases, even if a company defaults, a claim does not become com-pletely worthless. Therefore, a pricing formula under the recovery of market value assumption will be derived in this section. The derivation is based on the works of Duffie & Singleton [9].

Consider a defaultable claim with terminal payoff X. Assume that upon default, the claim holder receives a payment X⁰. The price of that claim at time t, assuming default has not yet occurred, equals

V_t:= EQ

and is assumed to be continuous. Further, it will be postulated, that X⁰ = (1 − L)V_{τ −}, i.e., upon default the claim pays a fraction of its market value just before default occured. L ∈ [0, 1] represents the fractional loss given default and V_{τ −} = lim_s%τV_s. Under mild conditions (see [9]),

holds. This means, that the claim is priced as in the default-free case, except that discounting happens with respect to an additional credit spread λ^Q_sL to the interest rate r_t. Consequently, after choosing suitable models for r and λ^Q, the same methods of simulation can be performed as in the default-free case to price contingent claims.

To proof this formula in a somewhat informal way, recall Proposition 5.1.4, which states that the dynamics of the default indicator process H_t can be written as

dH_t= λ^Q_t(1 − H_t)dt + dN_t,

where N_t is a martingale with respect to F under the risk-neutral measure Q. Let further dV^t = α_tdt + dM_t be the Doob-Meyer decomposition of V_t, where M_t is another martingale under Q. A priori, αt does not need to be absolutely continuous, but in the course of our reasoning it will be shown that indeed it is. Let G_t be the discounted gain process of the claim, i.e.,

G_t= exp where the first expression represents the discounted price of the claim and the second stands for the discounted payoff if default occurs. Using Itˆo’s formula yields: since all the covariation/quadratic variation terms except [V ]_t are zero, be-cause 1 − Ht and exp^−^R₀^trsds^ are of finite variation. However, [V ]t is irrelevant, since _∂v^∂22f (t, v, h) = _∂v^∂22 exp^−^R₀^tr_sds^v(1 − h) = 0. Inserting the expressions for dH_t and dV_t leads to

dG_t= − exp

where V_t− = V_t holds because of the continuity assumption and M_t^∗ is a mar-tingale under Q, because it is the sum of stochastic integrals with respect to martingales, where the integrands are L¹-integrable. Since the gains process is a martingale and the Doob-Meyer decomposition is unique, the drift needs to be zero, i.e. (r_tV_t− α_t+ LV_tλ^Q_t)(1 − H_t) = 0. Since t < τ , the second term equals one, which implies

0 = r_tV_t− α_t+ Lλ^Q_t α_t= V_t(r_t+ Lλ^Q_t).

Since under the risk-neutral measure, drift and risk-free rate need to coincide, discounting for the defaultable claim has to be done by r_t + Lλ^Q_t, which mathematically speaking yields our conjecture (5.1.4).