3.2 Fitting the Credit Models
3.2.6 Model-Specific Implementation
3.2.6.4 Constant Elasticity of Variance
The firm’s asset value is assumed to transition in continuous time by
dV(t) = (r−δv)V(t)dt+σv¯V(t)ρdW(v,t)Q, (3.37)
whereris the risk-free rate,δvis the firm asset payout rate, and ¯σvis a volatility scalar. The instantaneous variance of the firm is ¯σv2V(t)2ρ, and the variance of the firm’s return is ¯σv2V(t)2(ρ−1).
The default boundary is assumed constant. Let the latent state variable beX(t) = (V(t)−K)/K. Default occurs on the first passage ofX(t)to zero. For comparison with the other structural models, the equivalent log-solvency ratio ofx(t) =ln(V(t)/K), is recovered from X(t), by the relationship x(t) =ln(X(t) +1). From Ito’s lemma and equation (3.37), it follows thatx(t)is also a CEV process
dX(t) = (r−δv)X(t)dt+σv¯ X(t)ρdW(v,t)Q. (3.38)
Whenρapproaches one, the CEV model approaches the single-factor LS1 model, how- ever the two models are not strictly nested because, as discussed further below, ρ is restricted to be less than one.
The CEV model has the convenient property that a closed-form solution for the crossing-time to a zero-value boundary is known whenρ<1. Cox (1975) shows that
Q(t,T) =Γ(νQ,HQ) Γ(νQ) (3.39) νQ= 1 2−β HQ=kX(t)(2−β)exp µ(t)(2−β)(T−t) k= 2µ(t) σ2 v(2−β) exp(µ(t)(2−β)(T−t))−1
whereβ =2ρ,µ(t) = (r−δv),νQis a shape parameter andHQis the evaluation point in the standard complementary gamma function. A necessary restriction is thatρ<1 in order for the boundary to be an absorbing state and therefore for the closed-form crossing-time solution to be equivalent to the cumulative default probability. This is because, whenρ<1, the local volatility of the proportional change in solvency becomes infinite as the firm approaches insolvency. Consequently, the CEV model is not strictly nested within the LS model.
The time-t value of a one-dollar face value, default-risky, zero-coupon bond is given by
p(t,T) =e(−r(t,T)(T−t)) 1−ωQ(t,T)
where the risk-free rate is allowed to vary deterministically with time and is maturity matched with the timing of the promised payment,Q(t,T)is the cumulative risk-neutral probability of default at any time between times t and T, and ω is the exogenously determined writedown rate.
The valuation of a coupon bond follows the method used for the LS1 model, and is solved for by the same sum of zeros approach, as shown in equation (3.29). The CEV and LS1 models share common valuation assumptions conditional on the firm’s level of solvency, but the potential paths are different. The CEV model’s firm volatility increases as solvency reduces, whereas firm asset return volatility is independent of solvency level in the LS1 model.
The elements of the transition equation are found by firstly expressing the continuous state process shown in equation (3.38) into discrete time using the Eueler approximation and time dependency on the risk-free rate
X(t) =X(t−1) + (r(t−1)−δv)∆t (3.41)
+σv¯ X(t−1)ρ√∆tη(t), η(t)∼N(0,q(t)).
It follows that the transition equation for the CEV model is the generic transition equa- tion (equation (3.12)) with the elements:
α(t) =X(t), c¯(t) =0,
¯
T =1+ (r(t−1)−δv)∆t, R=σv¯ X(t−1)ρ√∆t. (3.42)
The initial value of the state variable, for each firm, is obtained from the observed market capitalisation and book debt by
X(0) = V(0)−K K ≡exp(x(0))−1, (3.43)
where,x(0)is the initial sample value of the firm’s observed log-solvency ratio, shown in Table 3.10.
No constraint is placed on the path that X(t) may take other than zero being an absorbing boundary. Due to the discretisation, it is possible that X(t) may be projected below zero during the filtering procedure. Any prediction of X(t) below zero in the EKF is trimmed to zero.
To initialise the variance of the state vector, the volatility scalar is first estimated by equation (3.44) using the observed sample asset return volatility,σv(0), and an assumed initial elasticity parameter,ρ(0), taken from a prior result by Albanese & Chen (2005). They find that equity default swap prices can be explained, on average, by a CEV equity diffusion model withρ equal to -0.65. Since our state variable is defined as the scaled net worth of the firm, it is reasonable to consider a similar result may hold in our sample.
The initial estimate of the volatility scalar is then given by ¯
σv(0) =σv(0)X(0)(1−ρ(0))=σv(0)X(0)1.65. (3.44) Having estimated ¯σv(0), the initial diffuse variance of firm solvency is calculated as
P(0) =σv¯ (0)2X(0)2ρ(0)(t(1)−t(0))·1000, (3.45) where(t(1)−t(0))is the length of the first observation time interval.
There are three hyperparameter sets conditional on the treatment of the non-credit component of the credit spread. Suppressing dependence on the firm for notational clar- ity, we have: No liquidity: ψh(1) ={ln ¯σ2 v,lnσm2,ln(δv/(1−δv)),ln(1−ρ)}, Constant liquidity: ψh(2) ={ln ¯σ2 v,lnσm2,ln(δv/(1−δv)),ln(1−ρ),lndi}, Time-varying liquidity: ψh(3) ={ln ¯σ2 v,lnσm2,ln(δv/(1−δv)),ln(1−ρ),lnβR,lndi}. (3.46)
As shown in equation (3.46), the volatility scalar and measurement errors are trans- formed to be non-zero, and the elasticity parameter is transformed to constrain the op- timal estimate to be less than one. The writedown rate is assumed to be exogenously known, applying the same industry-specific values as applied to the LS1 and LS2 mod- els, as shown in Table 3.10.