Bayesian Inference for a Structural Credit Risk Model ...

Journal of Financial Econometrics Advance Access published June 24, 2014 Journal of Financial Econometrics, 2014, Vol. 0, No. 0, 1--29

Bayesian Inference for a Structural Credit Risk Model with Stochastic Volatility and Stochastic Interest Rates ABEL R ODRÍGUEZ University of California Downloaded from http://jfec.oxfordjournals.org/ at University of Chicago on December 1, 2014

ENRIQUE TER HORST CESA and IESA SAMUEL MALONE University of the Andes School of Management

ABSTRACT We develop a novel structural credit risk model that extends the original Merton model by allowing for stochastic interest rates and stochastic volatility. The model is estimated using Bayesian methods implemented via a Markov chain Monte Carlo algorithm, in light of the demonstrable advantages of likelihood approaches and the importance of taking into account parameter uncertainty documented in the literature. We solve the nontrivial computational problem of contingent claim valuation in our set-up by using a Taylor series approximation to the expectation of the claim payoffs under the risk-neutral measure. Finally, we illustrate our model and compare it against the Merton model with real data on a nonfinancial firm (Ford Motor Company) and three financial firms (Citigroup, Goldman Sachs, and Lehman Brothers) during the recent financial crisis. ( JEL: G24, C01) KEYWORDS: Bayesian statistics, credit default swaps, structural credit risk models

Structural credit risk (SCR) models attempt to assess the credit-worthiness of a corporation by explicitly modeling the balance sheet of the firm as a portfolio consisting of the asset, implicit options, and risk-free debt, in which the market value of assets must add up to the market value of liabilities. Therefore, SCR models view the default process as endogenous and related to the capital structure of the

Address correspondence to Enrique ter Horst, CESA, Bogota, Colombia

doi:10.1093/jjfinec/nbu018 © The Author, 2014. Published by Oxford University Press. All rights reserved. For Permissions, please email: [email protected]

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firm, so that default probabilities and credit spreads can be computed as functions of the parameters governing the dynamics of the underlying asset prices. SCR models go back to Merton (1974), who applied the Black–Scholes–Merton option pricing model (Black and Scholes, 1973; Merton, 1973) to the valuation of corporate securities. In this framework, the asset price is assumed to follow a geometric Brownian motion and the firm’s debt is modeled as an implicit call option on the asset, with strike price equal to the face value of a zero-coupon bond. Merton’s model is clearly too simplistic; among other assumptions, it takes the parameters governing the evolution of the asset prices to be constant, assumes that all of the firm’s debt has been reduced to a single zero-coupon bond, allows bankruptcy to occur only at the time the debt matures, and assumes that the risk-free interest rate is constant. Since then, a number of authors have contributed to creating more realistic credit risk models; some noteworthy references include Geske (1974), who shows how to deal with multiple bonds; Longstaff and Schwartz (1995), who include stochastic interest rates; Leland and Toft (1996), who incorporate the effect of taxes and bankruptcy costs; Collin-Dufresne and Goldstein (2001), who account for claim dilution and other changes in capital structure; Black and Cox (1976) and Brockman and Turtle (2003), who study the effect of loan covenants and other barriers; and Fouque, Sircar, and Sølna (2006) and Zhang, Hao, and Zhu (2009), who consider the effect of changes in asset volatility on credit spreads. All of these models share a structure similar to Merton’s; they consist of two components: (i) a stochastic model for the dynamics of the firm’s asset price, and (ii) a pricing formula for the implied option that provides a link between the observed equity prices and the unobserved asset prices. In every case, the two components are linked to each other as the pricing formula is derived using the risk-neutral measure associated with the firm’s asset dynamics and the bankruptcy rules in the model. An important challenge associated with the practical application of SCR models is parameter estimation. Since the firm’s asset values and the parameters governing their dynamics cannot be observed directly, these must be inferred from the time series of equity prices, promised future repayments, and interest rates. A popular approach among practitioners is to estimate the parameters controlling the dynamics of asset prices (such as volatilities, jump rates, etc.) from proxy values (e.g., see Barclay and Smith Jr., 1995a, 1995b; Brockman and Turtle, 2003, and Zhang, Hao, and Zhu, 2009). This approach is simple to implement because it does not depend on the particular features of the SCR model at hand, but tends to underestimate the volatility of the asset (Li and Wong, 2008). In addition, the proxy approach is not self-consistent, as the market price of equity obtained by substituting the proxy-firm values and the estimated parameters into the bond pricing formula might result in equity prices that are different from those used to derive parameter values in the first place. An alternative to the proxy approach is the volatility restriction method (Merton, 1974; Ronn and Verma, 1986), which uses a system of equations at each time point (one of them being given by the equity pricing formula and another being obtained by restricting the equity volatility to match the volatility generated by applying Ito’s lemma to the equity pricing

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formula) to provide estimates of both volatilities and asset prices. Due to its relative simplicity, this approach has been used by a number of authors including Ogden (1987), Duan, Moreau, and Sealey (1995), Lyden and Saraniti (2000), and Huang and Huang (2012). However, volatility restriction has important drawbacks; for example, it might lead to systems of equations that have no solution (Duan, 1994; Li and Wong, 2008) and, in the case of models with constant volatility, to estimates that are inconsistent with the model. An additional drawback of ad hoc procedures such as the proxy and the volatility restriction approaches is that the properties of the estimators obtained are unknown, and it is therefore impossible to go beyond providing point estimates for the model parameters. A natural alternative that deals with all the issues discussed before is to use maximum likelihood estimates (MLEs). The use of MLEs in the context of SCR models was pioneered by Duan (1994), who discusses the derivation of the transformed likelihood for the model parameters as a function of the equity prices (as opposed to the asset values). This approach has been employed, among others, by Li and Wong (2008) to compare multiple SCR models, and by Duan and Fulop (2009) in the context of the original Merton model with microstructure noise (i.e., where the equity prices are assumed to be observed with noise rather than exactly). In this article, we present an SCR model that accounts for uncertainty in the path of interest rates while allowing for the volatility of the asset to change over time. Empirical evidence suggests that both interest rate risk (Longstaff and Schwartz, 1995) and volatility risk (Fouque, Sircar, and Sølna, 2006; Zhang, Hao, and Zhu, 2009) play an important role in determining the market behavior of risky debt and credit default swap prices. To the best of our knowledge, this is the first time a model of this type has been discussed in the literature. Furthermore, we develop Bayesian procedures for parameter estimation, in the belief that proper calibration is every bit as important as incorporating realistic features into SCR models. Indeed, the use of ad hoc procedures for parameter estimation makes assessment across competing SCR models extremely difficult, as it confuses model inadequacy with improper calibration. In addition, empirical evidence suggests that formal estimation procedures such as maximum likelihood generate better predictions for bond and credit default swap (CDS) spreads than proxy and volatility restriction procedures (Ericsson and Reneby, 2005; Tarashev, 2005; Li and Wong, 2008), and that information risk (i.e., parameter uncertainty) might explain an important percentage of the CDS spreads observed in the market (Korteweg and Polson, 2009). The use of established estimation procedures allows us to go beyond point estimation to generate interval estimates and perform formal model comparisons. Finally, we benchmark our SCR model against a Bayesian version of the Merton SCR model. We compare both models on the ground of their marginal log-likelihoods (Kass and Raftery, 2007). The remaining of the article is organized as follows: our SCR model with stochastic volatility and stochastic interest rates is described in Section 1. Section 2 introduces a Markov chain Monte Carlo (MCMC) algorithm for Bayesian inference.

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Section 3 presents empirical illustrations using data from Ford motor company and Lehman Brothers. Finally, we conclude in Section 4 with a brief summary and a discussion of shortcomings and future directions.

1 STOCHASTIC VOLATILITY AND STOCHASTIC INTEREST RATES IN STRUCTURAL CREDIT RISK MODELS 1.1 Modeling the System Dynamics

logVt = logVt−1 +μ+σt 1t ,

logσt = α +β(logσt−1 −α)+τ 2t ,

(1)

for t = 2,...,n, 1t and 2t are uncorrelated errors distributed according to standard Gaussian distributions, and initial conditions V1 ∼ N ξv ,v2 ,

logσ1 ∼ N ξσ ,σ2 .

(2)

Put succinctly, the previous model assumes that the physical measure for the asset returns is a geometric Brownian motion with constant drift μ and timevarying volatility, whose logarithm in turn follows a mean-reverting autoregressive process. The autocorrelation parameter β controls the speed of mean reversion toward its stationary distribution (which simply is a Gaussian distribution with mean α and variance τ 2 /(1−β 2 )). The previous specification, which acknowledges that prices are typically observed at discrete times, can be seen as a first-order Euler approximation of a continuous time stochastic volatility model (e.g., see Ghysels and Renault, 1996 or Hull and White, 1987). Our model for the behavior of the interest rates is similar to the volatility model. More specifically, the logarithm of the rate follows a first-order autoregressive process, logrt = γ +ρ(logrt−1 −γ )+κ3t ,


Our framework is an extension of the original SCR model (Merton, 1974). We assume that the firm has traded debt, with market value Dt , and traded equity, with market value Et at time t. The value of equity is observable in the market, and at all times we must have that Vt = Dt +Et , where Vt represents the value of the firm’s assets at time t. We model the dynamics of the firm’s assets using a discret-time stochastic volatility autoregressive model (Jaqcquier, Polson, and Rossi, 1994; Kim, Shephard, and Chib, 1998) where

(3)

where 3t is uncorrelated with 1t and 2t and follows a standard Gaussian distribution. As before, ρ controls the rate of mean reversion of the logarithmic of the risk-free rate to the stationary distribution, which is Gaussian with mean γ

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and variance κ 2 /(1−ρ 2 ). Similarly to our model for the asset prices, this scheme can be justified as a first-order Euler approximation to a geometric Vasicek model (Vasicek, 1977). As a preview of our results, it is worth mentioning that even without taking non-zero correlations into account, our model still does a respectable job in comparisons of historical versus model CDS spreads.

1.2 Equity Pricing Model

⎛

⎧ ⎨

Et = EQt ⎝exp − ⎩

t+m s=t+1

⎫ ⎬

⎞

rs max{VT −Kt (m),0}⎠ , ⎭

(4)

where Qt denotes the risk-neutral measure associated with the process described in Equations (1) and (3). Since a closed-form solution for equation (4) is not available, we instead price the firm’s equity using a slight modification of the standard Black– Scholes–Merton formula,

logVt −logKt (m)+ r˜t (m)+ σ˜ t2 (m)/2 Et = grt ,Kt (m),θt (Vt ) = Vt σ˜ t (m) logVt −logKt (m)+ r˜t (m)− σ˜ t2 (m)/2 −Kt (m)exp −˜rt (m) σ˜ t (m)

(5)


As with other SCR models, the equity represents a contingent claim on the firm’s assets at a future time (e.g., after 1 year), whose value depends on the ability of the firm to make future promised payments. More specifically, when the debt reaches maturity at time T = t+m, the firm will find itself in one of two situations. If the value of its assets at T is greater than the value of the promised payments that are currently promised for T, that is, VT > Kt (m), it will pay off the promised value of its debt to debtholders and will pay the residual, VT −Kt (m), to equityholders. Conversely, if VT < Kt (m), the firm will default, leaving equityholders with nothing, and will turn over its terminal asset value VT to debtholders, who will experience a partial default on what they are owed. Under these assumptions, the firm’s equity corresponds to an implicit European call option on the firm’s assets with strike price equal to the face value of the debt. The fair price for the equity is, therefore, given by the expected payoff of such an option,

where θt = (σt ,α,β,τ,γ ,ρ,κ) , denotes the cumulative distribution function of the standard normal distribution, ⎧ ⎫ m ⎨ t+m 2u ⎬ u 2 1−ρ u rs It = exp γ (1−ρ )+κ +ρ logrt r˜t (m) = E ⎭ ⎩ 1−ρ 2 s=t+1

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and σ˜ t2 (m) = E

⎧ ⎨ t+m ⎩

s=t+1

⎫ m ⎬ 2u 2 u 2 1−β u 2 σs It = exp 2α(1−β )+2τ +β logσt , ⎭ 1−β 2 u=1

1.3 Model-Induced Default Probabilities and Spreads Since a default is triggered if the value of asset at time T = t+m is lower than the corresponding face value of the firm debt expiring at that point, the probability of default is given by logKt (m)−logVt −μm (6) Pr(ST < Kt (m)) = σ˜ t (m)


are the expected risk-free return and expected total volatility over the next m periods, and It represents the information available up to time t. This pricing formula can be justified as a first-order Taylor approximation to the expectation in equation (4), where Qt is taken as normal distribution with mean r˜t − σ˜ t2 /2 and variance σ˜ t2 . Using this modification of the standard Black–Scholes–Merton formula is not new and has even been coined as the “plug-in” method as described in Bauwens and Lubrano (1997). For example, Noh, Engle, and Kane (1994) forecast the future mean volatility using a Generalized Autoregressive Conditional Heteroscedastic (GARCH) on the past returns and plug it in the Black– Scholes–Merton formula even though this formula is not the correct one for a GARCH specification (Heston and Nandi, 2000). Hull and White (1987) show that the Black–Scholes–Merton formula remains valid if the increments of the volatility process are independent of the increments of the process governing the asset returns. Even if Noh, Engle, and Kane (1994) violate the independence assumption required by Hull and White (1987), this plug-in method gives reasonable and easy-to-compute approximations of observed option prices. In this article, we assume that the increments of the price returns are not only uncorrelated with the increments of the volatility process, but also independent of the increments of the interest rates as well. Assuming non-zero correlations for the three processes would require a new option pricing formula in the same spirit as in Heston and Nandi (2000).

(note that this probability of default is computed under the physical rather than the risk-neutral measure). The spread given in formula (7) simply follows from the same logic laid out in the original Merton model. We compute the spread associated with the rms debt (in basis points), following the logic of the original Merton model, as Pt (7) st = −10000ln 1− Kt (m)exp{−˜rt (m)} where Pt = Et +Vt −Kt (m)exp{−˜rt (m)} is the price of the implied put option.

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2 MODEL CALIBRATION This section discusses our approach to calibrate the model on the basis of the observed equity prices e1 ,...,en , the path of observed interest rates r1 ,...,rn , and the time series of promised payments K1 (m),...,Kn (m). Our approach relies on Bayesian statistical techniques implemented through MCMC algorithms as in Robert and Casella (2005), so we discuss both the structure of the likelihood function as well as prior elicitation for the unknown parameters.

Using the system dynamics described in Equations (1) and (2), the joint distribution of the asset prices given the volatilities σ1 ,...,σn and the mean expected return μ is given by n n n 1 1 1 1 p(v1 ,...,vn | μ,σ1 ,...,σn ) = √ v σt vt 2π t=2 t=1 n 2 logvt −logvt−1 −μ 2 logv1 −ξv exp − exp − , 2v2 2σt2

(8)

t=2

while the joint distribution for the interest rates given the median long-term interest rate γ , the mean-reversion parameter ρ, and the volatility of the interest rate κ is n−1 1 1 n p(r2 ,...,rn | r1 ,γ ,ρ,κ) = √ 2π κ t=2 rt n logrt −γ −ρ(logrt−1 −γ ) 2 . exp − 2κ 2

(9)

t=2

If the asset prices were directly observable in the market, Equations (8) and (9) together with the joint distribution of the log volatilities

1 p(σ1 ,...,σn | α,β,τ ) = √ 2π

n

2 logσ1 −ξσ 1 1 exp − 2σ2 σ τ n−1 nt=1 σt n logσt −α −β(logσt−1 −α) 2 exp − , 2τ 2


2.1 Likelihood Function

(10)

t=2

could be directly used to estimate the unknown parameters of the model. However, the asset prices are not directly observable and need to be inferred from the equity

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prices. To achieve this, we exploit the fact that the function grt ,Kt (m),θt (vt ) in Equation (5) defines a one-to-one correspondence between asset and equity prices, and use the transformed likelihood approach of Duan (1994) to obtain p(e1 ,...,en | μ,σ1 ,...,σn ,r1 ,...,rn ,K1 (m),...,Kn (m),α,β,τ,γ ,ρ,ω) = ⎧ 2 ⎫ −1 −1 ⎪ ⎪ n n ⎬ ⎨ (e )−logg (e )−μ logg t t−1 1 1 1 rt ,Kt (m),θt rt−1 ,Kt−1 (m),θt−1 n exp − √ 2 ⎪ ⎪ v t=2 σt 2σt 2π ⎭ ⎩ t=2

−1

, (11)

where gr−1 is the inverse function associated with grt ,Kt (m),θt (which can be t ,Kt (m),θt evaluated numerically using, for example, the Newton–Raphson method), and (et )−logKt (m)+ r˜t (m)+ σ˜ t2 (m)/2 loggr−1 ∂grt ,Kt (m),θt ,K (m),θ t t t = ∂vt σ˜ t (m) vt =g−1 (et ) θ

is the delta associated with the call option in Equation (5). (Note that, although Equation (8) is not a function of α, β, τ , γ , ρ or ω, the likelihood in Equation (8) is a function of these parameters because the pricing Equation (5) depends on them). Because of the conditional independence assumptions implied by our model, the final likelihood function for our model is obtained by multiplying Equations (9), (10), and (12), so that (μ,σ1 ,...,σn ,α,β,τ,γ ,ρ,κ;e1 ,...,en ,r1 ,...,rn ,K1 (m),...,Kn (m)) ∝ p(e1 ,...,en | μ,σ1 ,...,σn ,r1 ,...,rn ,K1 (m),...,Kn (m),α,β,τ,γ ,ρ,κ) p(r2 ,...,rn | r1 ,γ ,ρ,κ)p(σ1 ,...,σn | α,β,τ ) By setting β = τ = ρ = κ = 0, and σt = σ = exp(α) in Equations (1) and (3), we obtain the likelihood for the Merton SCR model (Merton, 1973):


⎧ 2 ⎫ n ⎪ n ⎨ loggr−1,K (m),θ (e1 )−ξv ⎪ ⎬ ∂gr ,K (m),θ 1 1 1 1 t t t exp − −1 ⎪ ⎪ ∂vt 2v2 ⎩ ⎭ t=1 grt ,Kt (m),θt (et ) t=1 vt =gθ−1 (et )

p(e1 ,...,en | μ,σ,...,r1 ,...,rn ,K1 (m),...,Kn (m)) = ⎧ 2 ⎫ −1 −1 ⎪ ⎪ n n ⎨ ⎬ (e )−logg (e )−μ logg t t−1 1 1 rt ,Kt (m),θt rt−1 ,Kt−1 (m),θt−1 exp − √ ⎪ ⎪ v 2σ 2 2π σ 2 ⎩ ⎭ t=2

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⎧ 2 ⎫ n ⎪ n ⎨ loggr−1,K (m),θ (e1 )−ξv ⎪ ⎬ ∂gr ,K (m),θ 1 1 1 1 t t t exp − −1 ⎪ ⎪ ∂vt 2v2 ⎩ ⎭ t=1 grt ,Kt (m),θt (et ) t=1 vt =gθ−1 (et )

−1

, (12)

where (et )−logKt (m)+(rt +σ 2 /2)(t+m) loggr−1 ∂grt ,Kt (m),θt t ,Kt (m),θt = √ ∂vt σ t+m vt =g−1 (et ) θ

(μ,σ ;e1 ,...,en ,r1 ,...,rn ,K1 (m),...,Kn (m)) ∝ p(e1 ,...,en | μ,σ,r1 ,...,rn ,K1 (m),...,Kn (m))

2.2 Hyperpriors To calibrate the models, we specify prior distributions for all unknown model parameters. For the mean of the log-returns " μ we assign a normal distribution ! with mean ξμ and variance μ2 , μ ∼ N ξμ ,μ2 , where ξμ is chosen so that the mean annualized returns are equal to 2% (roughly, the long-term average of the S&P500 index) and μ2 is chosen to be moderately flat. On the other hand, the prior for 2 the " asset price is given a log-normal prior with mean ξv and 2variance v , ! initial 2 N ξv ,v , with ξv chosen as the average value of of Kt (m) and v moderately large. On the other hand, to ensure the stationarity of the!volatility process we let " ξσ = α, σ2 = τ 2 /(1−β 2 ), and β ∼ N[0,1] (ξβ ,β2 ), where N a,b denotes the normal distribution with mean a and variance b truncated to the set . We select ξβ = 1 and β2 = 1, which is relatively flat on the [0,1] interval but slightly favors values close to 1. On the! other "hand, the average long-term log-volatility α is given a normal prior α ∼ N ξα ,α2 , where ξα is chosen to reflect a median annualized volatility of 20% and α2 is chosen so that the prior is relatively flat. !Finally, " the variance 2 is given an inverse gamma prior, τ 2 ∼ IG a ,b , where a = 2 of the volatility τ τ τ τ (so that E τ 2 is finite but Var τ 2 is infinite) and bτ is chosen so that E τ 2 = 0.2. Finally, we follow a similar approach to elicit priors on the parameters for the stochastic process associated with the interest rates. In particular, we let ρ ∼ ! " 2 2 2 N[−1,1] ξρ ,ρ with ξρ = 1 and ρ = 1, γ ∼ N ξγ ,γ with ξγ selected to represent


and the likelihood for the Merton SCR model is reduced to:

an annualized 5% risk-free rate and γ2 moderately large, and κ 2 ∼ IG(aκ ,bκ ) with aκ = 2 and bκ = 0.2.

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2.3 Computation Through MCMC Algorithms The posterior distributions associated with our model and the Merton model reduce, respectively, to: p(μ,σ1 ,...,σn ,α,β,τ,γ ,ρ,κ | e1 ,...,en ,r1 ,...,rn ,K1 (m),...,Kn (m)) ∝ p(e1 ,...,en | μ,σ1 ,...,σn ,r1 ,...,rn ,K1 (m),...,Kn (m),α,β,τ,γ ,ρ,κ) p(r2 ,...,rn | r1 ,γ ,ρ,κ)p(σ1 ,...,σn | α,β,τ )p(μ)p(α)p(β)p(τ )p(γ )p(ρ)p(κ).

(13)

p(μ,σ | e1 ,...,en ,r1 ,...,rn ,K1 (m),...,Kn (m)) ∝ p(e1 ,...,en | μ,σ,r1 ,...,rn ,K1 (m),...,Kn (m))

(14)

The posteriors in Equations (13) and (14) are too complex to be treated analytically. Therefore, we resort to MCMC algorithms (Robert and Casella, 2005), which allow us to simulate a dependent sequence of random draws from the target distribution. Given initial values for the parameters, the MCMC algorithm successively updates their values using the full conditional distributions derived from Equations (13) and (14). Standard Markov chain theory ensures that, after an appropriate burn-in, the samples generated by the algorithm are approximately distributed according to the posterior distribution. These posterior samples can be used to compute empirical counterparts for interesting summaries of the posterior distribution, such as posterior means and posterior credible intervals. To develop the MCMC algorithm for our SCR model, consider first sampling the parameters γ , ρ, and κ associated with the interest rate process from their full conditional distributions. If the assets were observed, our choice of conditionally conjugate prior distributions would imply that sampling from the full conditional distributions for all these three parameters would be straightforward. However, since Equation (12) depends on γ , ρ and κ through the expected value of future interest rates, direct samplers are not available for the full conditionals. Instead, we implement independent Metropolis–Hastings steps for each of these parameters. In each of these steps the full conditional posteriors from the model that assumes that asset prices are known are used as our proposal distributions. Our empirical experiments indicate that using these “partial” full conditionals as proposals leads to higher acceptance rates and lower autocorrelations than algorithms based on random walk proposals. Consider now sampling the volatilities σ1 ,...,σn and its associated parameters α, β, and τ . Given the volatilities σ1 ,...,σn , we can sample α, β, and τ using Metropolis–Hasting steps with “partial” full conditional proposals similar to the ones we used for the parameters of the interest rate process. On the other hand, to sample the volatilities given α, β, and τ we use Gaussian random-walk proposals

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to update each logσt separately. The variance of the proposal is tuned to obtain an average acceptance rate of 40%. Finally, the mean return μ is updated by directly sampling from its Gaussian full conditional distribution. Details of the MCMC algorithm for both our SCR and Merton models can be seen in Appendix A.

3 ILLUSTRATIONS

3.1 A Simulation Study of the Stochastic Volatility and Bayesian Merton Models We now benchmark the stochastic volatility and stochastic interest rate SCR model against the Bayesian Merton (1974) model using simulated data in the same spirit as in Ericsson and Reneby (2005). Our experimental design is as follows. We consider four scenarios, each of which represents a different, 2 × 2 combination of asset volatility (business risk) and leverage ratio (financial risk) for the firm. The two values considered for the annualized asset volatility are 20% and 40%, and the


To illustrate the features of our model, we discuss four brief case studies. One of them (Ford Motor Company) corresponds to a company in the industrial/manufacturing sector, whereas the other three (Citigroup, Golman Sachs, and Lehman Brothers) belong to the financial sector. In all cases, we work with weekly data and predict spreads and default probabilities associated with 1-year debt. The face value of the firm’s debt was derived from market data by adding half the value of long-term debt to the value of short-term debt (Gray and Malone, 2008; Loeffler and Posch, 2011). The risk-free rates1 correspond to the value of the 1day London Interbank Offered Rate (LIBOR) dollar contracts. All results presented below are based on 60,000 samples generated from MCMC algorithm after a burn-in period of 10,000 iterations. Convergence was assessed through visual inspection of trace plots from the posterior distribution and using the multiple-chain approach described in Gelman and Rubin (1992). No evidence of lack of convergence was found using these methods. Acceptance rates for all Metropolis–Hastings steps based on “partial likelihood” proposals were around 40–80% depending on the parameter and the data set under consideration for both our SCR model and the Merton model. The estimation of our model requires ∼24 hours per data set to run on a 2.6 Ghz Intel Core i5, making it very time consuming, whereas the Bayesian Merton structural model takes roughly 3 minutes to run for any data set. We performed sensitivity analysis for both model parameters and the estimation results where quite robust.2

1 The use of LIBOR has the risk-free rate has been critiqued widely during and since the crisis. Bloomberg,

for example, now uses swap and Treasury rates to build the risk-free rate yield curve. R codes implementing both algorithms used for the illustrations in this article can be obtained from the authors. For the sake of space, the numerous trace-plots are not presented here in the manuscript.

2 The

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initial ratios of the default barrier to assets (K/V) considered are 0.5 and 1.0. These ratios are generated by setting the initial asset value equal to 100 and setting the value of K equal to 50 or 100, respectively. Second, within the aforementioned setup, we evaluate the models’ performances under two different data-generated processes (DGPs) for the underlying asset returns and return volatility. The first DGP is the discrete time version of the Black and Scholes (1973) model, with constant volatility, found by setting β = τ = ρ = κ = 0, and σt = σ = exp(α) in the stochastic volatility model. This case is of interest because it allows us to measure the performance of the stochastic volatility SCR model estimation algorithm against the Bayesian SCR Merton method when the true model is actually that assumed by the latter models. The second DGP is the stochastic volatility model with β = 0, τ = 0, ρ = 0, κ = 0, and σt = σ = exp(α). This second DGP is of interest because it allows us to measure the difference in performance between the Merton method and our method in a setting of stochastic volatility with stochastic interest rates. The first DGP allows us to measure as well the marginal contribution of our model without taking into account stochastic interes rates, but just the stochastic volatility part. The consideration of two DGPs, together with the four combinations of business risk and financial risk mentioned above, generates eight distinct simulation experiments. In each experiment, we proceed as follows. The time horizon is assumed to be 3 years throughout, and the risk-free rate is constant and set equal to 5% when the DGP is a Geometric Brownian Motion. In the case that the DGP of the firm’s assets is a Geometric Brownian Motion, we set the asset value, the default barrier, and the asset volatility according to the desired combination of business and financial risk, simulate 156 weeks of asset values, and then simulate 156 weeks of equity values by pricing equity as a call option on the firm’s assets using the Black–Scholes formula. We then take that 156-week time series of equity values as an input to our three models under consideration, the Bayesian Merton model and the stochastic volatility stochastic interest rates SCR model proposed in the current article. We run the MCMC algorithm for the two models and compute their marginal log-likelihood as summarized in tables (2) and (3). Overall the stochastic volatility model with stochastic interest rates is a global winner in terms of marginal log-likelihood, except for Lehman, where the score of −2730.04 for the Merton model is much better than −2784.63. When the DGP is a regular Geometric Brownian Motion,3 the stochastic volatility model still outperforms the Merton model but the gains are models. Those differences are more noticeable when the DGP4 is the one generated by stochastic volatility and stochastic interest rates. 3 The

stochastic volatility (SV) and the Merton model were estimated using equity data simulated from a Merton model with constant volatility and interest rates, and their respective marginal log-likelihoods were computed for the different four scenarios. 4 The SV and the Merton model were estimated using equity data simulated from a SV model with nonconstant volatility and non-interest rates, and their respective marginal log-likelihoods were computed for the different four scenarios.

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Table 1 Firm data. Models Ford Goldman Lehman Citibank

Merton

Stochastic volatility

−1920.01 −2852.40 −2730.04 −3036.99

−1830.06 −2486.12 −2784.63 −2850.83

Table 2 DGP with constant volatility and interest rates. Merton


σ = 20%, K = 50 σ = 40%, K = 50 σ = 20%, K = 100 σ = 40%, K = 100

−356.73 −745.02 −370.05 −518.31

−356.13 −409.22 −361.60 −496.04

Table 3 DGP with stochastic volatility and interest rates. Models

Merton


σ = 20%, K = 50 σ = 40%, K = 50 σ = 20%, K = 100 σ = 40%, K = 100

−397.87 −508.18 −507.78 −444.80

−365.92 −401.33 −394.97 −366.78

3.2 An Case Study Application of the Models to Three Banks and One Firm We start by comparing the results from Ford with those from Citigroup. Figure 1 shows the equity prices, barrier levels, and interest rates associated with these two data sets. Note that, while the value of Ford’s equity drops substantially in early 2005 and again toward the end of 2008, the value of the equity for Citigroup is quite stable until early November 2007, when it drops significantly. There are also important differences in the value of the default barriers. Although the path of the default barriers for both firms are characterized by sudden jumps (which make them resemble step functions), Ford’s default barrier does not seem to follow an overall increasing or decreasing trend overall, whereas the default barrier for Citigroup grows steadily until November 2007, when it starts to drop. Figure 1 also shows market prices of CDS for Ford and Citigroup. Although our model does not use the CDS data to calibrate the model, we use it as an informal benchmark against which we can evaluate model predictions. However, such comparisons need to be taken with a grain of salt, as market spreads reflect not only the circumstances of a

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Figure 1 Equity prices, promised payments, and 1-year market CDS spreads for Ford (left column) and Citigroup (right column).

firm’s balance sheet, but can also be affected by exogenous factors such as contagion and liquidity risks, without omitting the too-big-to-fail effect of implied (or explicit) bailout guarantees from the government to large companies, and in particular large banks, such as Citigroup (Gray and Malone, 2008; Schweikhard and Tsesmelidakis, 2011). Note that, although the average level of the CDS spreads is quite different for both companies, the overall behavior is very similar, with spreads increasing dramatically around March 2008. Figure 2 shows prior and posterior distributions of the mean expected returns μ and the parameters for the volatility process α, β, and τ 2 . Not surprisingly, the expected return for Ford is negative over this time period, whereas the expected


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return for Citigroup is positive. On the other hand, whereas the expected value of the annualized median long-term volatility is very similar for both companies (6.83% annualized for Ford and 6.37 for Citigroup), the mean reversion parameter and the volatility of volatility are very different, with the posterior mean for β being 0.51 for Ford and 0.12 for Citigroup and the posterior mean of τ 2 being 0.61 for Citigroup and 0.21 for Ford. This fundamental difference in the behavior of both firm’s volatilities could be partially explained by the nature of the sectors to which the firms belong. Indeed, we could expect the volatility of industrial assets (which are mostly physical) to be more persistent than the volatility of financial assets. This interesting observation is supported by out last two case studies, which correspond to two financial institutions for which the persistence of the volatility is also very low. Figure 3 shows prior and posterior distributions for the parameters of the interest rate process. In both cases, the annualized expected long-term median interest rate is 3%, while the persistence is very close to 1. Figure 4 presents estimates of the volatility and firm’s asset prices, along with predictions of the 1-year spreads and of the probability of default. Interestingly, the path of the firm’s assets is similar to the path of the default barrier, and uncertainty associated with it very low. Also, as would be expected from our estimates of the persistence parameters, the volatility path of Citigroup has sharper and more frequent spikes than the volatility path of Ford. Furthermore, these spikes seem to coincide with the abrupt changes in promised payments. In terms of the spread, we see that the overall shape of our predicted path is very similar to the shape of the path of the market prices presented in Figure 1. However, while the overall level of the predicted spreads for Citigroup is close to that observed in the market, the overall level of our predictions for Ford is somewhat lower than that of the spreads observed in the market (which at one point runs above 120%). Finally, note that, unsurprisingly, the path of the default probabilities is similar to the path of spreads. Furthermore, whereas the probability of default for Ford reached close to 50% at the end of 2008, the probability of default for Citigroup does not exceed 30% over the period studies here. Next we analyze data from two investment banks which are considered examples of well (Goldman Sachs) and badly (Lehman Brothers) run institutions during the run-up to the financial crises. Figure 5 shows the raw data associated with these two firms, along with the market price of CDS. Note that the overall shape of the time series of equity prices, default barriers, and market spreads associated with these two investment banks closely resemble those of Citigroup, but that the levels are very different. In particular, while the spreads for Goldman Sachs peaked at around 150 basis points, those of Lehman peaked at 700 basis points a few days before its bankruptcy. Figure 6 shows prior and posterior distributions of the mean expected returns μ and the parameters for the volatility process α, β, and τ 2 . Note that both firms have positive expected returns over this period and that, surprisingly, the median longterm volatility of the returns associated with Goldman Sachs is somewhat larger

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than that of Lehman Brothers (expected posteriors are 5.5% and 3.3% in annualized terms, respectively), and both are lower than the median long-term volatility of Citigroup’s asset returns. As in the case of Citigroup, the persistence parameters associated with the volatilities of both firms are relatively low (the posterior means for β are 0.13 and 0.10 for Goldman and Lehman, respectively), while the volatility of volatility is quite high (and in particular, larger than that for Citigroup). Figure 7 shows prior and posterior distributions for the parameters of the interest rate process. The results are very similar to those presented in the previous subsection, with the interest rates exhibiting high persistence, low volatility of volatility, and a annualized median long-term rate of around 3%.

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Figure 5 Equity prices, promised payments, and and 1-year market CDS prices for Goldman Sachs (left column) and Lehman Brothers (right column).

Finally, Figure 8 presents estimates of the volatility and firm’s asset prices, along with predictions of the 1-year spreads and of the probability of default. As with Citigroup, the low values of β are associated with a volatility path that exhibits frequent large jumps (which, as before, tend to coincide with sudden changes in the default barrier), and we observe low levels of uncertainty associated with the asset prices. Another interesting observation is that, although the path of market spreads we predict for Lehman Brothers has an overall shape and level that is similar to that of the price of its CDS contracts, the predicted path for Goldman Sachs does not resemble that of its CDS contracts. In particular, we do not predict the large jump in spreads that is seen in the market, and that is successfully predicted for Citigroup


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Figure 7 Prior (continuous lines) and posterior distributions (histograms) for the parameters for the interest rate process γ , ρ, and κ 2 for Ford (left column) and Citigroup (right column).

and Lehman Brothers. This observation is consistent with the idea that Goldman was a better-run bank than Lehman or Citigroup, and that the peak in the price of its CDS contracts in 2008 is the consequence of contagion from Lehman Brothers’ increasing problems (and eventual bankruptcy) and is not due a sudden dramatic deterioration of Goldman Sach’s balance sheet.


γ

4 DISCUSSION We have described a novel SCR model that allows for stochastic interest rates and stochastic volatility. We have also devoted substantial effort to developing a Bayesian approach for model calibration. To obtain predictions that account for

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5/2/03 6/27/03 8/22/03 10/17/03 12/12/03 2/6/04 4/2/04 5/28/04 7/23/04 9/17/04 11/12/04 1/7/05 3/4/05 4/29/05 6/24/05 8/19/05 10/14/05 12/9/05 2/3/06 3/31/06 5/26/06 7/21/06 9/15/06 11/10/06 1/5/07 3/2/07 4/27/07 6/22/07 8/17/07 10/12/07 12/7/07 2/1/08 3/28/08 5/23/08 7/18/08

5/23/03 7/18/03 9/12/03 11/7/03 1/2/04 2/27/04 4/23/04 6/18/04 8/13/04 10/8/04 12/3/04 1/28/05 3/25/05 5/20/05 7/15/05 9/9/05 11/4/05 12/30/05 2/24/06 4/21/06 6/16/06 8/11/06 10/6/06 12/1/06 1/26/07 3/23/07 5/18/07 7/13/07 9/7/07 11/2/07 12/28/07 2/22/08 4/18/08 6/13/08 8/8/08

0.0

0.1

0.2

0.3 0.4

0.4

0.6


0.2


0.8

0.5

5/2/03 6/27/03 8/22/03 10/17/03 12/12/03 2/6/04 4/2/04 5/28/04 7/23/04 9/17/04 11/12/04 1/7/05 3/4/05 4/29/05 6/24/05 8/19/05 10/14/05 12/9/05 2/3/06 3/31/06 5/26/06 7/21/06 9/15/06 11/10/06 1/5/07 3/2/07 4/27/07 6/22/07 8/17/07 10/12/07 12/7/07 2/1/08 3/28/08 5/23/08 7/18/08

5/23/03 7/18/03 9/12/03 11/7/03 1/2/04 2/27/04 4/23/04 6/18/04 8/13/04 10/8/04 12/3/04 1/28/05 3/25/05 5/20/05 7/15/05 9/9/05 11/4/05 12/30/05 2/24/06 4/21/06 6/16/06 8/11/06 10/6/06 12/1/06 1/26/07 3/23/07 5/18/07 7/13/07 9/7/07 11/2/07 12/28/07 2/22/08 4/18/08 6/13/08 8/8/08

0

0

100

300

200 300

400

Predicted one−year spreads (basis points)

200

Predicted one−year spreads (basis points) 100

500

400

600 5/2/03 6/27/03 8/22/03 10/17/03 12/12/03 2/6/04 4/2/04 5/28/04 7/23/04 9/17/04 11/12/04 1/7/05 3/4/05 4/29/05 6/24/05 8/19/05 10/14/05 12/9/05 2/3/06 3/31/06 5/26/06 7/21/06 9/15/06 11/10/06 1/5/07 3/2/07 4/27/07 6/22/07 8/17/07 10/12/07 12/7/07 2/1/08 3/28/08 5/23/08 7/18/08

5/23/03 7/18/03 9/12/03 11/7/03 1/2/04 2/27/04 4/23/04 6/18/04 8/13/04 10/8/04 12/3/04 1/28/05 3/25/05 5/20/05 7/15/05 9/9/05 11/4/05 12/30/05 2/24/06 4/21/06 6/16/06 8/11/06 10/6/06 12/1/06 1/26/07 3/23/07 5/18/07 7/13/07 9/7/07 11/2/07 12/28/07 2/22/08 4/18/08 6/13/08 8/8/08

3e+05

3e+05

4e+05

4e+05

Assets

5e+05

Assets

5e+05

6e+05

6e+05

7e+05

5/2/03 6/27/03 8/22/03 10/17/03 12/12/03 2/6/04 4/2/04 5/28/04 7/23/04 9/17/04 11/12/04 1/7/05 3/4/05 4/29/05 6/24/05 8/19/05 10/14/05 12/9/05 2/3/06 3/31/06 5/26/06 7/21/06 9/15/06 11/10/06 1/5/07 3/2/07 4/27/07 6/22/07 8/17/07 10/12/07 12/7/07 2/1/08 3/28/08 5/23/08 7/18/08

5/23/03 7/18/03 9/12/03 11/7/03 1/2/04 2/27/04 4/23/04 6/18/04 8/13/04 10/8/04 12/3/04 1/28/05 3/25/05 5/20/05 7/15/05 9/9/05 11/4/05 12/30/05 2/24/06 4/21/06 6/16/06 8/11/06 10/6/06 12/1/06 1/26/07 3/23/07 5/18/07 7/13/07 9/7/07 11/2/07 12/28/07 2/22/08 4/18/08 6/13/08 8/8/08


0.0

0

0

20

20

60

40

60


40


80

80

100

100


Figure 8 Estimates of the volatility and asset price paths along with predictions of the 1-year spreads and of the probability of default for Ford (left column) and Citigroup (right column).

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estimation risk, as well as benchmarking it against a Bayesian version of the Merton SCR model. Three variations of our approach are natural and have the potential to greatly improve the quality of the model predictions. First, introducing correlations among asset price, volatility and interest rate innovations 1,t , 2,t , and 3,t would allow us to capture leverage effects, as well as have a more flexible model for high-quality short-term bonds. Introducing these correlations is conceptually straightforward, but substantially increases the computational complexity of our calibration procedure. Second, we could treat the equity prices as being observed under noise. A number of models with microstructure noise have been recently presented in the literature (e.g., see Huang and Yu, 2008, Korteweg and Polson, 2009 or Duan and Fulop, 2009), but in most cases they assume that both the volatility of the asset prices and the interest rates are constant in time. Again, the extension is conceptually straightforward and the main challenges are associated with devising an appropriate MCMC algorithm for calibration. It would be useful to replace our approximate pricing function with the true pricing function associated with our model for the dynamics of the process. Besides the complexities associated with deriving such pricing formula, using such formula would be computationally difficult because it is likely to be unavailable in closed form, or available but up to numerical inversion as in Heston and Nandi (2000). As this model is a discrete-time model, computing the risk-neutral measure in order to get the exact pricing formula would be crucial to assess any Jensen effect or bias that is introduced in the whole estimation process when using only a first-order Taylor series approximation of the risk-neutral expectation as the option pricing formula, as it is the case in our article. To solve a similar problem for option pricing models in continuous time, Durham (2013) uses a polynomial approximation of degree three which could prove to be superior to formula (5). However, the approach taken in Durham (2013) “uses option prices to back out implied volatility states with an explicitly specified risk-neutral measure,” making the need to compute the exact option pricing formula for our model a first priority, and therefore its exact risk-neutral measure. Furthermore, Hull and White (1987) show that the Black– Scholes–Merton formula remains valid if the increments of the volatility process are independent of the increments of the process governing the asset returns. In this article, we assume that the price return is uncorrelated with the volatility process as well as the interest rate process. Finally, the calibration approaches discussed in this article ignore the information about the firm’s assets contained in the market prices of CDS. An alternative model would use both the equity prices (which provide information about the implied call option on the firm’s assets) and the CDS prices (which provide information about the implied put option), along with the call-put parity equation, to derive the implied asset prices, which could then be modeled using a stochastic volatility model. This type of approach has been recently used by Rodriguez and ter Horst (2010) to measure expectations in option markets or to calibrate SCR models as in Forte (2011), and can be extended to the context of credit

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risk models to construct predictions that can account for features such as as, for example, liquidity risk or measuring the size of the sovereign guarantee for toobig-to-fail banks as described in Gray and Malone (2012). These approaches will be explored elsewhere.

A DETAILS OF THE MCMC ALGORITHM A.1 The General SCR model

(1)

Update μ from a Gaussian distribution, ⎛

n 1 1 + 2 μ | ··· ∼ N ⎝ 2 σ μ t=2 t

(2)

−1 n

n 1 ξμ 1 , + + 2 2 2 2 σ σ μ μ t=2 t t=2 t xt

−1

⎞ ⎠,

where xt = logVt /Vt−1 is the return at time t. Update αby first making a proposal α ∗ from a Gaussian distribution α ∗ | ··· ∼ ˜ α2 , where N ξ˜α ,

ξ˜α =

(1−β)2 z1 τ2

+

#n

t=2

(1−β)2 τ2

(1−β)2 (zt −βzt−1 ) + ξα2 τ2 α

2 + (n−1)(1−β) τ2

+ 12 α

,

˜ α2 =

1 (1−β)2 τ2

2 + (n−1)(1−β) τ2

+ 12

,

α

and zt = logσt . This proposal is accepted with probability min{1,α } where α =

p(e1 ,...,en | μ,σ1 ,...,σn ,r1 ,...,rn ,K1 (m),...,Kn (m),α ∗ ,β,τ,γ ,ρ,ω) , p(e1 ,...,en | μ,σ1 ,...,σn ,r1 ,...,rn ,K1 (m),...,Kn (m),α,β,τ,γ ,ρ,ω)

and the likelihood p(e1 ,...,en | μ,σ1 ,...,σn ,r1 ,...,rn ,K1 (m),...,Kn (m),α,β,τ,γ ,ρ,ω)

(3)


In this section, we describe the sampling steps associated with our MCMC algorithm.

is given in Equation (12). Update first making a proposal β ∗ from a Gaussian distribution β ∗ | ··· ∼ β by ˜ 2 , where N[0,1] ξ˜β , β

#n ξ˜β =

t=2

ξ (zt −α)(zt−1 −α) + β2 τ2 β

#n−1 (zt −α)2 t=1

[14:41 12/6/2014 nbu018.tex]

τ2

+ 12 β

,

˜ β2 = #

1

n−1 (zt −α)2 t=1 τ2


+

1 β2

.

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25


This proposal is accepted with probability min 1,β where

1/2

1−β ∗2 β = 1−β 2

(z1 −α)2 (β ∗2 −β 2 ) exp 2τ 2

p(e1 ,...,en | μ,σ1 ,...,σn ,r1 ,...,rn ,K1 (m),...,Kn (m),α,β ∗ ,τ,γ ,ρ,ω) . p(e1 ,...,en | μ,σ1 ,...,σn ,r1 ,...,rn ,K1 (m),...,Kn (m),α,β,τ,γ ,ρ,ω) (4)

n 1 2 2 2 b˜ τ = bτ + (1−β )z1 + [zt −βzt−1 −α(1−β)] . 2

n a˜ τ = aτ + , 2

t=2

This proposal is accepted with probability min{1,τ } where τ = (5)

p(e1 ,...,en | μ,σ1 ,...,σn ,r1 ,...,rn ,K1 (m),...,Kn (m),α,β,τ ∗ ,γ ,ρ,ω) , p(e1 ,...,en | μ,σ1 ,...,σn ,r1 ,...,rn ,K1 (m),...,Kn (m),α,β,τ,γ ,ρ,ω)

Update γ by first making a proposal γ ∗ from a Gaussian distribution γ ∗ | ··· ∼ ˜ γ2 , where N ξ˜γ ,

ξ˜γ =

(1−ρ)2 x1 κ2

+

#n

t=2

(1−ρ)2 κ2

ξ (1−ρ)2 (xt −ρxt−1 ) + γ2 κ2 γ

2 + (n−1)(1−ρ) κ2

+ 12 γ

,

˜ γ2 =

1 (1−ρ)2 κ2

2 + (n−1)(1−ρ) κ2

+ 12

,

γ

and xt = logrt . This proposal is accepted with probability min 1,γ where γ = (6)

p(e1 ,...,en | μ,σ1 ,...,σn ,r1 ,...,rn ,K1 (m),...,Kn (m),γ ∗ ,ρ,κ,γ ,ρ,ω) . p(e1 ,...,en | μ,σ1 ,...,σn ,r1 ,...,rn ,K1 (m),...,Kn (m),γ ,ρ,κ,γ ,ρ,ω)

∗ ∗ Update ρ by first making a proposal ρ from a Gaussian distribution ρ | ··· ∼ ˜ ρ2 , where N[0,1] ξ˜ρ ,

#n ξ˜ρ =

t=2

ξ (xt −γ )(xt−1 −γ ) + ρ2 κ2 ρ

#n−1 (xt −γ )2 t=1

κ2

+ 12 ρ

,

1

˜ ρ2 = #

n−1 (xt −γ )2 t=1 κ2

+ 12

.

ρ


∗2 Update τ 2 by first making a proposal τ from an inverse Gamma distribution, τ ∗2 | ··· ∼ IG a˜ τ , b˜ τ where

This proposal is accepted with probability min 1,ρ where

1−ρ ∗2 ρ = 1−ρ 2

1/2

(x1 −γ )2 (ρ ∗2 −ρ 2 ) exp 2κ 2

p(e1 ,...,en | μ,σ1 ,...,σn ,r1 ,...,rn ,K1 (m),...,Kn (m),γ ,ρ ∗ ,κ,γ ,ρ,ω) . p(e1 ,...,en | μ,σ1 ,...,σn ,r1 ,...,rn ,K1 (m),...,Kn (m),γ ,ρ,κ,γ ,ρ,ω)

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(7)

Journal of Financial Econometrics ∗2 Update κ 2 by first making a proposal κ from an inverse Gamma distribution, κ ∗2 | ··· ∼ IG a˜ κ , b˜ κ where

n a˜ κ = aκ + , 2

n 1 2 2 2 b˜ κ = bκ + (1−ρ )x1 + [xt −ρxt−1 −γ (1−ρ)] . 2 t=2

This proposal is accepted with probability min{1,κ } where κ =

Update each σt one at a time by proposing a new value of zt∗ = logσt∗ using a Gaussian random walk where zt∗ ∼ N(zt ,ζz2 ) This proposal is accepted with probability min{1,z } where z =

p(σ1 ,...,σt∗ ,...,σn | α,β,τ ) p(σ1 ,...,σt∗ ,...,σn | α,β,τ )

p(e1 ,...,en | μ,σ1 ,...,σt∗ ,...σn ,r1 ,...,rn ,K1 (m),...,Kn (m),γ ,ρ,κ ∗ ,γ ,ρ,ω) p(e1 ,...,en | μ,σ1 ,...,σt ,...,σn ,r1 ,...,rn ,K1 (m),...,Kn (m),γ ,ρ,κ,γ ,ρ,ω)

A.2 The Merton SCR Model We describe how to sample the two parameters σ"2 ≡ exp(2α), assuming a ! μ and 2 Normal Inverse Gamma (NIG(μ0 ,λ0 ,α0 ,β0 ) = N μ0 ,σ /λ0 IG(α0 ,β0 )) as joint prior distribution for both parameters (Robert and Casella, 2005). (1)

Update μ from a Gaussian distribution, μ | ··· ∼ N μn ,σ 2 /λn

where xt = logVt /Vt−1 is the return at time t # from the proposed asset levels by # inverting the option pricing formula, x¯ = 1/n nt=1 xt , s2 = nt=1 (xt − x¯ )2 /(n−1), λn = λ0 +n−1, and μn = (λ0 μ0 +(n−1)s2 )/(λ0 +(n−1)). (2) Update σ 2 by first making a proposal σ ∗2 from an inverse Gamma distribution, σ ∗2 | ··· ∼ IG(αn ,βn ) where αn = α0 +(n−1), and βn = β0 +0.5 (n−1)s2 +(λ0 (n−1)(s2 −μ0 )2 )/(λ0 +n−1)


(8)

p(e1 ,...,en | μ,σ1 ,...,σn ,r1 ,...,rn ,K1 (m),...,Kn (m),γ ,ρ,κ ∗ ,γ ,ρ,ω) , p(e1 ,...,en | μ,σ1 ,...,σn ,r1 ,...,rn ,K1 (m),...,Kn (m),γ ,ρ,κ,γ ,ρ,ω)

.

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$ % This proposal is accepted with probability min 1,(μ,σ 2 ) where (μ,σ 2 ) =

p(e1 ,...,en | μ∗ ,σ ∗ ,r1 ,...,rn ,K1 (m),...,Kn (m)) . p(e1 ,...,en | μ,σ,r1 ,...,rn ,K1 (m),...,Kn (m))

Received February 19, 2010; revised March 24, 2014; accepted May 21, 2014.

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