Pricing Defaultable Securities under Actual Probability Measure

1 Introduction

Credit risk, one of the most pervasive threats in today’s financial markets, cannot be completely diversified away. How to manage the credit risk has become the focus in Basel II and Basel III. Researchers have developed many models to estimate default probability and price defaultable securities like [1, 4, 5, 13, 16], especially after financial crisis in 2008, most of scholars began to be concerned about correlated default^{[7, 8]}. Most of them discuss credit risk under risk neutral probability measure, which makes it is difficult to calibrate model parameters by market data and difficult to use these models to price the securities not traded. On the other hand, The models applied to manage credit risk by financial institution are still some simple models such as KMV based on Black-Scholes-Merton^[15] and CreditMetrics model developed by Morgan^[9] which can’t satisfy the needs of investors.

Here we will give a new approach extended from [3] to price defaultable securities (like contingent securities, defaultable corporate debt or securities) under real probability measure. In [3], the author mainly presented the value of life insurance contracts without continuous market risk. We consider how to price the defaultable securities which have both diffusion risk^[13] and default risk and the price framework is more generalized. If we match this approach to existing default models, it is a kind of intensity-based model, thus the default time is completely unpredictable. In addition, this approach frames the price process of defaultable securities by the backward stochastic differential equation (BSDE). It looks the defaultable security’s value as the discount value of the cash flow happened during its lifespan, here we call it Risk-Adjusted Value (RA-value).

In this paper, we reference two market price processes of risk, i.e. market price of continuous market risk and market price of default risk, to compensate the holder of defaultable securities appropriately. The two risk prices can be estimated from the financial market data and there have existed several literatures to calculate them (like Hull^[10], Giesecke^[13]). It is interesting that they also prescribe the mapping between probabilities under the physical measure and an equivalent martingale measure of it if we strengthen some conditions of the model. This conclusion is consistent with Giesecke^[13]. Given the two market prices of risk (deterministic or stochastic), the hazard rate (or intensity) of default time under real probability measure and the risk-free rate, we can estimate the price of any defaultable security under real probability measure. Further when the security is a tradable one, its price given in this study is no arbitrage. On the other hand, noticing it doesn’t need the exists of risk neutral probability, the conditions model parameters must satisfy will be certainly relaxed. Finally, for easy to apply the model in practice, we will give the security’s price framework in discrete time and continuous time respectively.

The price formulas given here are mainly the price of corporate securities, corporate debt, or Mortgage based securities, which is called defaultable security for simplicity. Those securities may have zero-coupon or zero recovery, which doesn’t affect our results. In addition, this model can also be used to price other credit derivatives, like CDS, CDO, and others.

The paper is made as follows: In section 2, the basic concepts and notations are given. In section 3, results for the price of defaultable securities in discrete time and their proofs are presented. And the price framework in continuous time and their proofs are given in section 4. In section 3 and section 4, we will prove the prices are no arbitrage by strengthening some conditions of model’s parameters. the last section we will give a summary of this research.

2 Modelling the value of defaultable securities

Model the uncertainty in the capital market with a completed filtered probability space (Ω, 𝒢, (𝒢_t)_0≤t≤T, ℙ), where T is a given positive number representing the largest possible life of defaultable securities considered. Ω denotes the state space, including all possible pathes the securities price moves as time goes on. 𝒢 is a σ-field, representing measurable events in the state space, ℙ is the actual probability measure and the information available at time t is captured by the σ-field 𝒢_t. 𝔾^T = {𝒢_t : t ∈ [0,T]} is a standard filtration on (Ω, 𝒢, ℙ). All filtration are assumed to satisfy the ‘usual conditions’ of right-continuity and completeness.

Let H = {H_t}_0≤t≤T be a right-continuous nondecreasing 𝒢_t-adapted process valued in {0,1}, and denote by τ a non-negative random variable on (Ω, 𝒢, ℙ),

satisfying: ℙ(τ = 0) = 0, ℙ(τ > t) > 0 for any t ∈ [0,T]. Then H_t = 1_(τ≤t), and H is a one-jump process with the form

Where (A_t) is the compensator of (H_t) and (M_t) is a 𝔾-martingale under measure ℙ. In fact, H describe whether the security defaults in future and M denotes the default risk. Let ℍ be the associated filtration: ℋ_t = σ(H_u : u ≤ t), 0 t ≤ T and assume we are given an auxiliary filtration 𝔽 such that 𝔾 = ℍ ∨ 𝔽, i.e. 𝒢_t = ℋ_t ∨ ≤ ℱ_t for any t ∈ [0, T].

Let a standard Brownian motion on (Ω, 𝒢, ℙ) W = {W_t : 0 ≤ t ≤ T} denote continuous market risk (diffusion risk^[13]). We are common to suppose 𝔽 is the natural filtration of W.

Let S¹, S², ⋯ ,Sⁿ be traded risk securities, and the risks include the continuous market risk and default event risk, both of which can’t be diversified away. It is a standard economic principle that those risks undiversified commands premiums. Generally, premiums are proportionability to the price volatility induced by the corresponding risk. We call the proportion market price of risk. Here, we denote by process η^={ηt}0≤t≤T,η~={η~t}0≤t≤T the relative market prices with respect to the risks M and W, respectively. It is reasonable to assume η̂ is a 𝔽-predictable process irrelevant to default information and ῆ is a 𝔾-predictable process, which is affected by both of common market information and default information. Let S={St=(Sti)n×1:0≤t≤T} are given by the following stochastic differential equation (SDE):

for i = 1, 2,⋯, n, where (a_i)_n×1 represents a n-dimensional vector. B^si is an adapted finite variation process, which is cash flow associated with Sⁱ. r = {r_t : 0 ≤ t ≤ T}, is a non-negative 𝔽-adapted integrable process representing default-free spot rate. β={βt=(βti)n×1:0≤t≤T},α={αt=(αti)n×1:0≤t≤T} are all (𝒢_t)-adapted measurable processes such that (1) has a unique strong solution. An element of S is called a market index if it is the price of the traded security in the sense of “securitization” (see [17]). Suppose there exists a saving account at the default-free spot rate r = {r_t : 0 ≤ t ≤ T}. Using those market indexes and saving account, we can calculate η̂, ῆ. Therefore we suppose η̂, ῆ are given.

They give the fluctuation intensity of the value V_t of B and thus measure the risks of B. Let RB=(RtB:0≤t≤T) be a Itò process given by the following SDE,

Then, formula (5) can be verified as for any t₁≤ t₂≤ T,

That is, R^B is the instantaneous growth (return) process in the value of B. Consider two securities B¹and B², suppose that they are valued by their RA-discounts (Vti,σ^ti,σ~ti),i = 1,2 under the same market price of risk η̂, ῆ and default-free spot rate r. If δ^1=δ^2,δ~1=δ~2, we have RB1=RB2. In other words, under the RA-valuation, two securities with the same risks will have the same growth (return) rate in their value. Therefore the RA-valuation gives a fair valuation or pricing system without ambiguity.

3 Pricing discrete defaultable securities

In this section, we consider securities those pay coupons or recoveries only at the moments 0 = t₀ < t₁ < ⋯ < t_N = T, where t_i = ih, i = 0,1, ⋯ , N, N is a positive integer and Nh = T. Assume all time variables in this subsection are valued on {t₀, t₁,⋯, t_N}. Our pricing model can be restated as follows:

ω1:={1,−1}N=(ω11,ω21,⋯,ωN1)whereωi1=1or−1;

Ω1:={ω1}={(ω11,ω21,⋯,ωN1)};

F:=2Ω1; 𝔽: a filtration defined as ℱ₀ is a trivial σ-field on Ω¹,

for any t > 0, thus ℱ = ℱ_N.

ℙ¹: a probability measure on the space Ω¹, such that

i.e. ωi1andωj1 are independent each other when i ≠ j, and for any 1 ≤ i ≤ N

The above probability space (Ω¹, ℱ, {ℱ_t}_0≤t≤T, ℙ¹) is a completed filtered probability space, and we denote by it the market information captured. ω¹ can be looked on as a path of a binomial tree with N steps, and Ω¹ is the set of all possible pathes of the binomial tree. The binomial tree represents the path where the price of a security possibly moves by as time goes on before the security defaults. For a path ω1=(ω11,ω21,⋯,ωN1) and for any i = 1, 2, ⋯ ,N,

ωi1=1 implies the security price will grow up between t_i−1 and t_i, otherwise ωi1=−1 implies the security price will drop down between t_i−1 and t_i.

For any ω1=(ω11,⋯,ωN1)∈Ω1, denote ω1(t)=ωt/h1, then E¹ω¹(t)= 0, E¹(ω¹(t))² = 1. Define a process W as

and W₀ ≡ 0, then W is a square integrable 𝔽-martingale under ℙ¹, and the sharp bracket process < W, W > (t) is the unique predictable increasing process such that Wt2−<W,W>(t) is a martingale, where <W,W>(t)=[W,W](t)=E1(Wt2)=t. That is to say, t is the compensator of [W, W]_t. It is easy to prove when h tends to zero from positive direction, W is just a standard Brownian motion under ℙ¹.

In addition, let {ωt2(⋅),0≤t≤T}: a non-decreasing function family, such that ωt2(t−h)=0,ωt2(t)=1,0<t≤Tandω02(t)=0,∀t∈{t0,t1,⋯,tN};

Ω2:={ω2}={ωt2,t=t0,t1,⋯,tN};

H:=2Ω2;

H: a nondecreasing process which is defined as: Ht(ω2):=ω2(t)=ωs2(t)ifω2=ωs2, then we have H0(ω)≡0,Ht(ω02)=0,∀0≤t≤T; ℍ: the associated filtration of H, i.e. ℋ_t = σ(ℋ_u, u ≤ t). Then ℋ = ℋ_T.

Given a sequence number {qt1,qt2,⋯,qtN}, where 0<qti<1h,i=1,2,⋯,N, we can define a probability measure ℙ² such that for any 0 < t ≤ T, ℙ²(H_t = 1|H_t–h = 0)/h = q_t and ℙ²(H₀=0) = 1.

The above probability space (Ω², ℋ,{ℋ_t}_0≤t≤T, ℙ²) is another completed filtered probability space, we denote by it the default information captured.

Let

then 1_(τ(ω)≤t) = H_t (ω). If we denote by M a process : Mt=1(τ≤t)−∑(s≤t)qs∧τh,∀t>0 and M₀ = 0, it is easy to see that M is a ℍ-martingale under ℙ² on the probability space (Ω², ℋ, {ℋ_t}_{0 ≤t≤T}, ℙ²).

Now, let Ω = Ω¹ × Ω², 𝒢 = ℱ ⊗ ℋ, 𝒢_t = ℱ_t ⊗ ℋ_t, ℙ = ℙ¹ ⊗ ℙ², where ℙ satisfies

and for any ω ∈ Ω, it has the form ω = (ω¹, ω²). Obviously, W and M are also 𝔾-martingales under ℙ and <W,W>(t)=t,<M,M>(t)=∑(s≤t)qs∧τhon(Ω,G,{Gt}0≤t≤T,P). So we have

Now consider the following security on (Ω, 𝒢, {𝒢_t}_0≤t≤T,ℙ): at time t = t_i, i = 1,2, ≤ , N, the coupon of b_t is paid to a security holder in case of H_t = 0, otherwise the recovery of c_t is paid to the security holder at the moment t if H_t = 1, H_t–h = 0, and at maturity, if H_T = 0,

the face value X is received by the holder. Let B₀ = 0,

for any t ∈ {t₁, t₂, ⋯, t_N}, define π_t := b_t + X1_(t=T), then

The defaultable security is thus represented by the random cash flow B = {B_t, t = t₀, ⋯, t_N} and its RA-value will be calculated under a given default-free spot rate r = {r_t : t = t₀, t₁, ⋯, t_N–1} and two relative market price of risks η^={η^t:t=t0,t1,⋯,tN−1}andη~={η~t:t=t0,t1,⋯,tN−1}.r_t, η̂_t and ῆ_t represent annual risk free rate and market price of risks at t respectively. Suppose r, η̂ are 𝔽-adapted processes and ῆ is a ℍ-predictable process. We first look for a RA-discount (B~s,σs)={(B~ts,σ^ts,σ~ts):0≤t≤s} of B_s for every s = t₁,⋯, t_N, that is, a solution of the following backward difference equation

where ΔB~ts=B~t+hs−B~ts,ΔMt=Mt+h−Mt=Ht+h−Ht−1(τ>t)qt+hh. If we define process ξ as ξt(ω)=ξt(ω1)=ωt/h1,0<t≤T, then

Therefore BSDE(9) is verified as

where η~t′=qt+h(η~t−1),ΔHt=Ht+h−Ht,σ^tsandσ~t+hs represent the viability of discount value caused by continuous market risk and default risk in the period [t, t + h]. We assume σ^ts is determined by the information until t and σ~ts is determined by the market information until t and default information before time t. that is, σ^ts is 𝒢_t measurable, and σ~ts is ℱ_t∨ℋ_t–h measurable.

For any t ≤ s, set

and let R(t,s,h)=D^(t,s,h)=D~(t,s,h)=1 if s < t.

From the structure of ℱ_t, we can see it denotes all possible pathes ω¹walks until time t and for any ℱ_t measurable random variable, its value is only determined by the path ω¹ walks before time t. We denote by a sequence x(t) = {x₁, ⋯, x_t/h} a path before time t, where x_i = 1 or –1, i = 1, ⋯ , t/h and denote A(t,x)={ω1:(ω11,⋯,ωN1)=(x1,⋯,xt/h,ωt/h+11,⋯,ωN1)},

then Ft = σ{A(t, x): x ∈ {1, –1}^t/h}. For any ω1,ω^1∈A(t,x), the value of any ℱ_t measurable random variable y satisfies y(ω1)=y(ω^1), denoted by y(x). In addition, rewrite 𝒢_t as 𝒢_t = σ{A(t, x)× {ω²} : × ∈ {1, –1}^t/h, ω² ∈ Ω²}, and for any 𝒢_t measurable variable z, denote its value at any ω = (ω¹, ω²) ∈ {A(t, x)× {ω²}} by z(x, ω²).

If H_t(ω²) = 1, for t < s,

Obviously satisfies (10).

If H_t (ω) = 0, consider two cases.

1^o If H_t+h(ω) = 1, then ω2=ωt+h2, for t ≤ s – 2h,

Let A={ω^1:ω^1(u)=ω1(u),u≤t},A1={ω^1:ω^1(u)=ω1(u),u≤t+h},A2={ω^1:ω^1(u)=ω1(u),u≤t,ω^1(t+h)=−ω1(t+h)}, we have

Thus follows

For t = s – h, σ^ts=0 we have

So (10) follows in this case.

2° If H_t+h(ω) = 0, then ∃k > 1, such that ω2=ωt+kh2. At that time,

For t ≤ s –2h, by equation (11) and (14), we have

For t = s – h, σ^ts=0,

and the proof of the first assertion is completed.

We now show that ℙ^η is a probability measure on 𝔾, for any ω=(ω1,ω2)=((ω11,⋯,ωN1),ω2)∈Ω,

Then,

ℙ^η is a probability measure. In addition, we can easily gain

For any s ≤ T and each ω1=(ω11,⋯,ωN1)∈Ω1, note As(ω1)={ω^1:ω^1(t)=ω1(t),∀t≤s}, then