A structural approach to default modelling with pure jump processes
AA structural approach to default modelling with pure jump processes
Jean-Philippe Aguilar ∗ Nicolas Pesci Victor JamesFebruary 11, 2021
Abstract
We present a general framework for the estimation of corporate default based on a firm’s capitalstructure, when its assets are assumed to follow a pure jump Lévy processes; this setup providesa natural extension to usual default metrics defined in diffusion (log-normal) models, and allows tocapture extreme market events such as sudden drops in asset prices, which are closely linked to defaultoccurrence. Within this framework, we introduce several pure jump processes featuring negative jumpsonly and derive practical closed formulas for equity prices, which enable us to use a moment-basedalgorithm to calibrate the parameters from real market data and to estimate the associated defaultmetrics. A notable feature of these models is the redistribution of credit risk towards shorter maturity:this constitutes an interesting improvement to diffusion models, which are known to underestimateshort term default probabilities. We also provide extensions to a model featuring both positive andnegative jumps and discuss qualitative and quantitative features of the results. For readers convenience,practical tools for model implementation and R code are also included.
Keywords:
Lévy process; Gamma process; Inverse Gaussian process; One-sided process; VarianceGamma process; Credit risk; Distance to default; Default probability.
AMS subject classifications (MSC 2020):
JEL Classifications:
C02, G12, G32. ∗ Corresponding author. Covéa Finance, Quantitative Research Team, 8-12 rue Boissy d’Anglas, FR-75008 Paris.Email: jean-philippe.aguilar@covea-finance.fr, nicolas.pesci@covea-finance.fr, victor.james@covea-finance.fr a r X i v : . [ q -f i n . P R ] F e b Introduction
We start the paper by providing a general introduction to the so-called structural approach to credit risk,recalling the main limitations of log-normal models and introducing the more realistic class of pure jumpmodels. We also detail the main contributions of the paper in this context, as well as its overall structure.
A popular approach to credit risk and corporate default modelling is to assume that the value of a firm’sassets is driven by a certain stochastic dynamics, and that a default occurs when the realization of thisprocess is lower than the facial value of the firm’s debt at its maturity. This approach, that has beencalled the structural approach by Duffie & Singleton (1999), was pioneered by the works of Black &Scholes (1973) and Merton (1974), the authors introducing a diffusion process to model the underlyingasset log returns; this setup is now universally known as the Merton model, and has subsequently beenextended to take into account various features, such as the existence of multiple maturities for the firm’sdebt (Geske, 1977) or the possibility for the default to occur prior to the debt’s maturity (Black & Cox,1976; Longstaff & Schwartz, 1995; Leland & Toft, 1996); these latter models are often referred to as firstpassage or barrier models. Extensions to sovereign issuers have also been considered (Gray & al., 2007).For a complete overview of default modelling and related topics within the structural approach (as wellas in the alternative class of reduced-form models), we refer to Lipton & Rennie (2013).A major difficulty in the structural approach lies in its calibration, because most of the corporate debtis not traded and, therefore, it is not possible to obtain directly the value of a firm’s assets by summing itsequity and debt values. In this context, assuming that assets log returns are driven by a diffusion process(i.e., a Wiener process, or Brownian motion) is particularly useful, because it allows for a closed-formrelationship between a firm’s value and its equity, via the celebrated Black-Scholes formula. This closedformula opens the way to simple historical calibrations for the asset’s volatility, by solving a system ofnonlinear equations with classical solvers such as the Newton-Raphson method in the algorithm developedby Vassalou and Xing (2004); other calibrations techniques for diffusion models have also been introduced,for instance based on the book value of the debt in Eom & al. (2004) or on maximum likelihood estimations(see details in Duan & al. (2004)).The Brownian hypothesis, however, has been severely criticized for being unrealistic and, notably,for producing an almost zero default probability for short maturities: if this was true, then short term1onds should have zero credit spread, which is typically not the case, as first noted by Jones & al. (1984)(see also many subsequent discussions such as Lyden & Saraniti (2000) or Demchuk & Gibson (2009) forinstance). This underestimation of short term default probabilities and of theoretical credit spreads isknown as the credit spread puzzle and has been evidenced many times and for every ranking, from highyield to investment grade issuers (see a recent overview in Huang & al. (2019)) and even sovereign issuers(see Duyvesteyn & Martens (2015) and references therein). Among the plausible explanations for thisdiscrepancy are the fact that other factors can influence credit spreads, such as taxes, liquidity premia(D’Amato & Remolona, 2003) and jump risk (Bai et al., 2020).This phenomenon is also evidenced when examining 1 year historical default rates; in 2019, the S&Pglobal default rate (all issuers) was of 1.03%, and, on the non investment grade or speculative sectorcorresponding to BB ratings and lower, it reached 2.10% (S&P, 2020b). With the dramatic COVID19events, the situation has worsened, and the default rate is expected to reach 10% in 2021 on the noninvestment grade sector, which now represents 30 % of all issuers, at a all time high (S&P, 2020a). Suchlevels are similar and even higher to those following the global financial crisis (GFC) of 2007-2008 (1 yearspeculative default rate was of 9.94% in 2009), and cannot be obtained from the classical Merton model,in particular because the normal assumption fails to reproduce extreme market events. A natural idea istherefore to introduce jumps in the assets dynamics, in order to better capture short term defaults occurringafter a brutal drop in the value of a company’s assets following, for instance, earning announcements,central bank meetings or major political events.Jump processes can be of two kinds: jump-diffusion processes, or pure jumps processes, and both areconveniently described by the formalism of Lévy processes. Jump diffusion processes were introduced instructural credit risk modelling by Zhou (1997), by adding a Poisson process whose discrete jumps arenormally distributed to the usual diffusion process, in order to materialize sudden changes in the firm’sassets value; other distributions for the magnitude of the Poisson jumps have subsequently been considered,such as negative exponential distributions in Lipton (2002), leading to higher short term probabilities andmore realistic credit spread curves. Self exciting Hawkes process have also been introduced in Ma & Xu(2016), a model for which an analytical formula for the equity value has been recently derived in Pasricha& al. (2021). Pure jump processes allow for an even richer dynamics and, notably, an interpretation interms of business time (differing from the operational time ), or the possibility for jumps to occur arbitrarilyoften on any time interval; such processes were introduced in the construction of credit risk models in thelate 2000s and early 2010s, notably in Madan & Schoutens (2008) for one-sided processes (i.e., featuring2ownward jumps only) and from the point of view of first passage models, and in Foriani & al. (2006);Fiorani & al. (2007); Luciano (2009) for double-sided processes, with CDS-based calibrations. Extensionsto multivariate processes have also been studied (Marfè, 2012).In this paper, we would like to demonstrate that pure jump processes are well suited to the structuralapproach, thus confirming the initial works cited above, but also that practical formulas can be derived forthe equity value, thus allowing for a precise issuer-based calibration. We will also show that incorporatingrealistic features such as non-normality of returns, jumps or asymmetry, allows to better capture theprobability of a default occurrence in particular during turbulent times, and provides a better fit tohistorical default rates notably for speculative issuers. This approach appears to be particularly relevantin the current COVID19 period: as already mentioned, short term defaults are expected to increase, asthey did during former major crisis (GFC, dot com crisis . . . ). Moreover, while former peaks of defaultevents lasted typically one or two years before going back to "usual" default rates, it is likely that, thistime, the situation may last far longer, given the enormous amounts of debt that have been accumulatedand the feeble revenues that most firms were able to collect between lockdown periods. We will focus on:(a) Extending the structural approach to the case where the stochastic process is no longer a Brownianmotion but a more general Lévy process, defining credit risk metrics in that context and showinghow model parameters can be calibrated using observable issuers data;(b) Deriving closed pricing formulas for the equity value of a firm when the Lévy process is assumed tobe spectrally negative, and implementing the calibration algorithm using these formulas;(c) Showing that obtained default probabilities are higher than the Merton default probabilities for shortmaturities, and comparing with historical default rates;(d) Using recent pricing formulas in the context of double-sided processes and, like in (b) and (c),discussing calibration and obtained probabilities.
In section 2, we recall basic facts on Lévy processes, we define credit risk metrics (distance to default,default probability) in this context, and we present the calibration algorithm that extends the algorithm3f Vassalou and Xing (2004) to the case of pure jump processes. Then, in section 3, we introduce twospectrally negative processes which are similar to the Gamma and inverse Gaussian subordinators, butwith a Lévy measure translated to the negative real axis, allowing us to define the NegGamma and NegIGcredit risk models; we derive explicit formulas for equity values in these models and, using the calibrationalgorithm, we determine model parameters, compute default probabilities and compare them with Mertonprobabilities as well as with historical default rates. We proceed to a similar discussion in section 4, butfor a process featuring a double-sided Lévy measure which, for simplicity, is assumed to be symmetric;equity value in this case is based on series expansions whose terms are powers of the distance to default.Section 5 is dedicated to concluding remarks and future works. For reader’s convenience, we have alsoequipped the paper with an appendix, which summarizes the papers notations, provides details on thedata set of test issuers used in the study, and contains an example of code that can be used to implementthe models in practice.
In this section, we start by recalling some fundamental concepts on Lévy processes and asset pricing (werefer to the classical references Bertoin (1996) and Schoutens (2003) for all technical details), that extendthe classical log-normal Merton model. Then, we introduce the generalization of distance to default anddefault probability in this context, as well as the general algorithm that will be used to calibrate thevarious models in the rest of the paper.
Following the classical setup of structural credit risk modelling, we consider a company possessing a totalasset V A ( t ) at time t ∈ [0 , T ] , financed by an equity V E ( t ) and a zero coupon debt of maturity T and facevalue K : V A ( t ) = V E ( t ) + K. (1)At t = T , two situations can occur:- V A ( T ) ≥ K : the issuer has enough financial resource to pay off for the full amount of its debt K ,and in that case its equity still has a positive value equal to V A ( T ) − K ;- V A ( T ) < K : the company is in default, and in that case its equity value falls down to .4n short, we see the equity value of a firm as a European call option written on its assets, whose strikeprice (resp. maturity) equals the face value (resp. the maturity) of the company’s debt; following theusual financial notation, we will therefore write that V E ( T ) = [ V A ( T ) − K ] + . (2)We will furthermore assume that the instantaneous variations of the firm’s assets V A ( t ) can be writtendown in local form as d V A ( t ) V A ( t ) = r d t + d X t , t ∈ [0 , T ] , (3)under the risk-neutral measure. In (3), r is the (continuously compounded) risk-free interest rate and X t is a Lévy process, that is, a càdlag process satisfying X = 0 almost surely, and whose increments areindependent and stationary. Note that when X t = W t where W t ∼ N (0 , σ t ) is a centered Wiener process,then V A ( t ) follows a geometric Brownian motion, and in that case (3) corresponds to the traditional Mertonmodel.The solution to the stochastic differential equation (3) is the exponential process V A ( T ) = V A e ( r + ω X ) T + X T (4)where V A := V A (0) and where ω X is a convexity adjustment computed in a way that the discounted assetprices are a martingale, i.e. E [ V A ( T ) | V A ] = e rT V A . (5)As X t is a Lévy process, its probability distribution is infinitely divisible and, consequently, its char-acteristic function admits a semigroup structure. In other words, there exists a function ψ X ( u ) calledcharacteristic exponent, or Lévy symbol, such that E (cid:2) e iuX t (cid:3) = e tψ X ( u ) , and which is entirely character-ized by the Lévy-Khintchine formula ψ ( u ) = a iu − b u + + ∞ (cid:90) −∞ (cid:0) e iux − − iux {| x | < } (cid:1) Π X (d x ) , (6)where a is the drift and is b the Brownian (or diffusion) component. The measure Π X (d x ) , assumed to be5oncentrated on R \{ } and satisfy + ∞ (cid:90) −∞ min(1 , x ) Π X (d x ) < ∞ , (7)is called the Lévy measure of the process, and determines its tail behaviour and the distribution of jumps.If the Lévy measure is discrete (i.e., is supported by a countable subset of R ), one speaks of a jump-diffusion process and in that case only a finite number of jumps can occur on any finite time interval;among the most popular distributions for the random size of jumps are Gaussian distributions (Zhou,1997) or exponential distributions (Lipton, 2002). Non-random discrete (Dirac) jumps have also beenintroduced in Lipton & Sepp (2009), and shown to provide remarkably good results when implementedin a model featuring multi-name joint dynamics, notably because they allow to achieve higher levels ofcorrelations than exponential jumps. Among other jump-diffusion models, let us also cite the works byHilberink & Rogers (2002) or Le Courtois & Quittard-Pinon (2006) (which is the credit risk counterpart ofthe geometric Kou model (Kou, 2002) for option pricing). When the Lévy measure is absolutely continuouswith respect to the Lebesgue measure, then an infinite number of jumps can occur on any time intervaland one speaks of a pure jump process; in this case the diffusion component b can be chosen equal to ;among the most prominent pure jump Lévy processes are the so-called subordinators (i.e. models for which Π X ( R − ) = 0 ) such as the Gamma or Inverse Gaussian processes, or models featuring both positive andnegative jumps such as the Variance Gamma (VG) process (Madan et al. , 1998) or the Normal InverseGaussian (NIG) process (Barndorff-Nielsen, 1995, 1997). Let us also mention that the drift a actuallydepends on the choice of the so-called truncation function (in (6) we choose iux {| x | < } ) , but the choice isnot unique), and therefore can be eliminated by a suitable choice of truncation. Last, let us remark that,as a direct consequence of the definition of the Lévy symbol, the condition (5) can be reformulated as: ω X = − ψ X ( − i ) . (8) Remark 1. If X t = W t where W t ∼ N (0 , σ t ) is a centered Wiener process, it is well known that E (cid:2) e iuW t (cid:3) = e − σ u t . From the point of view of the representation (6) , this means that the Lévy measureis identically null and that the Lévy symbol resumes to its diffusion component i.e. ψ W ( u ) = − σ u ; it esults that the convexity adjustment in this case is ω W = − ψ W ( − i ) = − σ (9) which is the usual Gaussian (Black-Scholes-Merton) adjustment. Let us now introduce the quantity k X := log V A K + ( r + ω X ) T (10)which, in the context of option pricing, is generally called the log forward moneyness. Then, we canformulate a proposition: Proposition 2.1 (Equity value) . Under the dynamics (3) , the equity value of a firm can be written as V E = Ke − rT ∞ (cid:90) − k X (cid:16) e k X + x − (cid:17) f X ( x, T ) d x. (11) Proof.
From (2) and the standard theory of martingale pricing, the value of a firm’s asset is V E = e − rT E (cid:2) [ V A ( T ) − K ] + | V A (cid:3) . (12)Using (4) and the notation (10), we can re-write (12) as V E = e − rT E (cid:104) [ V A e ( r + ω X ) T + X T − K ] + | V A (cid:105) = Ke − rT E (cid:104) [ e k X + X T − + | V A (cid:105) . (13)The last expectation can be carried out by integrating all possible realizations for the terminal payoff overthe process density, resulting in formula (11). The occurrence of a default at time T corresponds to the situation where the value of the firm’s assetis lower than the facial value of the debt; in this situation, the firm’s equity becomes equal to 0 and,7herefore, it follows from eq. (13) that the probability of such an event is P [ V A ( T ) < K | V A ] = P [ X T < − k X ] (14) = F X [ − k X ] (15)where F X ( . ) is the cumulative distribution function of the process X t evaluated at t = T : F X ( x ) = x (cid:90) −∞ f X ( y, T ) d y. (16)These observations motivate the following definitions: Definition 1 (Default metrics) .
1. Under the pure jump dynamics (3) , the distance to default of a firm is defined to be k X := log V A K + ( r + ω X ) T ; (17)
2. The corresponding default probability of the firm is defined to be P X := F X ( − k X ) . (18) Example 1 (Merton model) . If X t = W t where W t ∼ N (0 , σ t ) , then the process admits the density (heatkernel): f W ( x, t ) = 1 σ √ πt e − x σ t . (19) The cumulative distribution function is therefore F W ( x ) = x (cid:90) −∞ σ √ πt e − x σ t d x = N (cid:18) xσ √ t (cid:19) (20) where N ( . ) is the cumulative distribution function of the standard normal distribution N (0 , . It follows rom remark 1 and definition 1 that the default probability is F W (cid:18) − (cid:18) log V A K + ( r − σ T (cid:19)(cid:19) = N (cid:32) − log V A K + ( r − σ ) Tσ √ T (cid:33) := N ( − d ) (21) where d (following the usual Black-Scholes notation) is often interpreted as the distance to default in theMerton model. Let us assume that, using proposition 2.1, we were able to derive a closed formula for the equity value,that is, that there exists an analytic function P such that one can write V E = P ( V A , K, r, T, a (1) X , . . . , a ( n ) X ) (22)where a (1) X , . . . , a ( n ) X are the model’s parameters, i.e. the degrees of freedom of the process X t . Assumealso that there exists a function M X : R n → R n such that µ (1) X . . .µ ( n ) X = M X a (1) X . . .a ( n ) X , (23)where µ (1) X := E [ X ] , and, for j = 2 , . . . , n , µ ( j ) X := E [( X − E [ X ]) j ] (central moments). Last, let us denoteby A X the vector in the right hand side of (23); the purpose of algorithm 1 is to determine A X as thelimit of a sequence of vectors A k calibrated from observable market data, i.e., formally, A k −→ k →∞ A X . (24) Algorithm 1 (Calibration) . (1) Choose n real numbers a (1)0 , . . . , a ( n )0 , and introduce the vector A := t [ a (1)0 , . . . , a ( n )0 ] . For every datein a fixed period of N consecutive trading days { t j } j =1 ,...,N (e.g. 6 months, 1 year . . . ), determine he value of the firm’s assets V (0) A ( t j ) by numerically inverting the equity formula V E ( t j ) = P ( V (0) A ( t j ) , K, r, T − t j , a (1)0 , . . . , a ( n )0 ) . (25) (2) Compute the mean and central moments for the obtained time series (cid:26) log V (0) A ( t j +1 ) V (0) A ( t j ) (cid:27) j =0 ,...,N − up toorder n , group them under the form of a vector C , and solve C = M X ( A ) (26) where A := t [ a (1)1 , . . . , a ( n )1 ] is to be determined.(3) Repeat step (1) and (2) using the elements of A to determine a new series of log-returns and centralmoments C , and solve the system C = M X ( A ) (27) where A := t [ a (1)2 , . . . , a ( n )2 ] is to be determined.(4) Stop the algorithm once a desired level of precision is attained, e.g. || A k − A k − || ∞ < − . In this section, we will be particularly interested in the case where the Lévy process X t in (3) is spectrallynegative, that is, features negative (downward) jumps only; this will lead us to introduce two spectrallynegative pure jump processes, namely the NegGamma and NegIG processes. Let us note that one-sidedprocesses have already been applied to default modelling, e.g. in Madan & Schoutens (2008), but froma point of view of negative subordinators, and with a first passage time approach and a CDS-basedcalibration. Our approach is different: we will first establish closed formulas for the equity value of thefirms under these dynamics and show how they can be used for historical issuer-based calibrations andestimation of default probabilities. We define the NegGamma process to be the pure jump process whose Lévy measure reads Π G (d x ) = ρ e − λ | x | | x | { x< } d x (28)10or some positive real numbers λ and ρ ; within this parametrization λ is called the rate parameter, and ρ the shape parameter. It follows from the Lévy-Khintchine representation (6) that the Lévy symbol of theNegGamma process is, with an adapted choice of truncation function, ψ G ( u ) = + ∞ (cid:90) −∞ ( e iux −
1) Π G (d x ) = − ρ log (cid:16) i uλ (cid:17) . (29)The cumulant generating function is therefore κ G ( p ) = ψ G ( − ip ) = − ρ log (cid:16) pλ (cid:17) (30)from which we easily deduce the cumulants κ ( n ) G = ( − n ( n − ρ/λ n (see further details e.g. in Küchler& Tappe (2008)). In particular, we obtain, for the central moments: κ (1) G = − ρλ (Mean) κ (2) G = ρλ (Variance) κ (3) G / ( κ (2) G ) / = − √ ρ (Skewness) κ (4) G / ( κ (2) G ) = 6 ρ (Excess kurtosis) . (31) It follows from (29) that the martingale adjustment for the NegGamma process is ω G = ρ log (cid:18) λ (cid:19) (32)and therefore the distance to default is k G = log V A K + (cid:18) r + ρ log (cid:18) λ (cid:19)(cid:19) T. (33)The characteristic function also follows immediately from (29): Ψ G ( u, t ) = (cid:16) i uλ (cid:17) − ρt (34)11nd, inverting this Fourier transform, we get the density function f G ( x, t ) = λ Γ( ρt ) | λx | ρt − e − λ | x | { x< } , (35)from which we obtain the cumulative distribution function: F G ( x, T ) = Γ( ρT, − λx )Γ( ρT ) if x <
01 if x ≥ (36)where Γ( a, z ) stands for the upper incomplete Gamma function (see appendix A and Abramowitz & Stegun(1972)). As a consequence of definition 1, the default probability of a firm thus writes F G ( − k G , T ) = Γ( ρT, λk G )Γ( ρT ) if k G >
01 if k G ≤ . (37)As a consequence of the definition of the upper incomplete Gamma function, it is clear that F G ( − k G , T ) → when k G → , and that there is a 100% default probability as soon as the distance to default is nullor negative. In other words, there is no more chance for V A ( T ) to be greater than K as soon as k G ≤ ,which is a consequence of the occurrence of downward jumps only. Let γ ( a, z ) denote the lower incomplete gamma function, such that the equality γ ( a, z ) + Γ( a, z ) = Γ( a ) holds for all z ≥ . Proposition 3.1.
The equity value of a firm in the NegGamma credit risk model can be written as: V E = V A γ ( ρT, ( λ + 1) k G )Γ( ρT ) − Ke − rT γ ( ρT, λk G )Γ( ρT ) if k G > , k G ≤ . (38) Proof. If k G > , then using proposition 2.1 with the density (35) and changing the variable x → − x , wecan write the equity value as: V E = λ ρT Γ( ρT ) Ke − rT e k G k G (cid:90) e − ( λ +1) x x ρT − d x − k G (cid:90) e − λx x ρT − d x . (39)12sing the definition of the lower incomplete gamma function (see (73) in appendix A) and recalling that Ke − rT e k G = V A e ω G T , we have: V E = V A e ω G T (cid:18) λλ + 1 (cid:19) ρT γ ( ρT, ( λ + 1) k G )Γ( ρT ) − Ke − rT γ ( ρT, λk G )Γ( ρT ) (40)which simplifies into the first equation of (38) because e ω G T = (( λ + 1) /λ ) ρT .If k G ≤ , then ( − k G , ∞ ) ⊂ R + and therefore the integral (11) is equal to zero as the density (35) issupported by the real negative axis only. There are 2 parameters to calibrate, λ and ρ , and therefore it suffices to use two central moments inalgorithm 1, for instance the variance µ and excess kurtosis µ . Given the cumulants (31), we introducethe function M G : λρ −→ ρ/λ /ρ . (41)Let { V E } denote the series of observable equity prices during a one year (252 business days) period preced-ing the calculation date, and let us initiate the algorithm by choosing λ and ρ such that M G ( λ , ρ ) =(Variance( { V E } , ex . Kurtosis( { V E } )) . Following algorithm 1, we perform a classical Newton-Raphson al-gorithm for the pricing formula in proposition 3.1 with λ := λ and ρ := ρ for every day in { V E } , andgenerate a time series of log returns of V A ( t ) (step 1) for this period. Then we denote C := t [ µ (0)2 , µ (0)4 ] the variance and excess kurtosis of the obtained time series and we compute the vector A := M − G ( C ) (42)where the inverse of M G is straightforward to obtain from definition (41) (step 2). We repeat step 1using A instead of A , and repeat step 2 to compute the new vector A , and so on until obtaining twosufficiently close consecutive vectors || A k − A k − || ∞ < − . Calibration results are displayed in table 1for a data set constituted of investment grade and speculative issuers with various rankings (see details intable 5 in appendix B); practical details for implementing this calibration process can be found in the Rcode provided in appendix C. 13able 1: Parameter calibration for several issuers in the NegGamma credit risk model: asset value V A (inMM GBP or EUR), rate λ and shape ρ . We choose a maturity T =1 year and a risk free interest rate r = 0% . negGamma model Merton model K (in MM) V A (in MM) λ ρ V A (in MM) σ SAN FP 25 933 136 114 4.739 1.021 136 115 21.32%SAP GY 16 196 180 913 3.280 0.888 180 914 28.73%AI FP 14 730 78 928 3.194 0.559 78 931 23.40%SU FP 8 473 71 471 2.200 0.510 71 474 32.45%CRH LN 10 525 33 935 2.700 0.684 33 965 30.38%DAI GY 161 780 213 453 6.736 0.530 214 039 10.78%VIE FP 16 996 27 243 4.102 0.452 27 319 16.14%AMP IM 1 339 8 627 1.784 0.414 8 629 35.95%FR FP 4 879 11 379 2.746 0.774 11 415 31.98%EO FP 4 838 9 993 3.786 1.129 10 023 27.75%GET FP 4 498 11 658 3.230 0.612 11 675 23.98%LHA GY 10 106 14 635 4.074 0.784 14 730 21.61%PIA IM 609 1 491 4.138 1.050 1 492 24.54%CO FP 14 308 16 445 11.896 0.745 16 494 7.11%
We define the NegIG process to be the pure jump process whose Lévy measure reads Π I (d x ) = (cid:114) λ π e − λ µ | x | | x | { x< } d x (43)for some positive real numbers λ (shape) and µ (mean); from the Lévy-Khintchine representation (6), wededuce the Lévy symbol for the NegIG process: ψ I ( u ) = + ∞ (cid:90) −∞ ( e iux −
1) Π I (d x ) = λµ (cid:32) − (cid:114) iu µ λ (cid:33) . (44)The cumulant generating function is therefore κ I ( p ) = ψ I ( − ip ) = λµ (cid:32) − (cid:114) p µ λ (cid:33) (45)14rom which we easily deduce the cumulants κ ( n ) I and the corresponding central moments: κ (1) I = − µ (Mean) κ (2) I = µ λ (Variance) κ (3) I / ( κ (2) I ) / = − (cid:114) µλ (Skewness) κ (4) I / ( κ (2) I ) = 15 µλ (Excess kurtosis) . (46) It follows from (44) that the martingale adjustment for the NegIG process is ω I = λµ (cid:32)(cid:114) µ λ − (cid:33) (47)and therefore the distance to default is k I = log V A K + (cid:32) r + λµ (cid:32)(cid:114) µ λ − (cid:33)(cid:33) T. (48)The characteristic function also follows immediately from (44): Ψ I ( u, t ) = e λµ (cid:18) − (cid:113) iu µ λ (cid:19) t (49)and, inverting this Fourier transform, we get the density function f I ( x, t ) = (cid:114) λt π e − λ ( | x |− µt )22 µ | x | | x | { x< } . (50)From definition 1 and Shuster’s integrals (see (77) in appendix A), we therefore obtain the default proba-bility in the negIG model: F I ( − k I , T ) = N − (cid:115) λT k I (cid:18) k I µT − (cid:19) − e λTµ N − (cid:115) λT k I (cid:18) k I µT + 1 (cid:19) if k I >
01 if k I ≤ . (51)where N(.) stands for the standard normal cumulative distribution function (see apprendix A).15 .2.2 Equity value Let us define the function ϕ ( x, t, λ, µ ) := N (cid:32)(cid:114) λt x (cid:18) xµt − (cid:19)(cid:33) + e λtµ N (cid:32) − (cid:114) λt x (cid:18) xµt + 1 (cid:19)(cid:33) . (52) Proposition 3.2.
The equity value of a firm in the NegIG credit risk model can be written as: V E = V A ϕ (cid:32) k I (cid:114) µ λ , T, λ (cid:114) µ λ , µ (cid:33) − Ke − rT ϕ ( k I , T, λ, µ ) if k I > , k I ≤ . (53) Proof. If k I > , then using proposition 2.1 with the density (50), and changing the variable x → − x , wecan write the equity value as: V E = (cid:114) λT π Ke − rT e k I k I (cid:90) e − x − λ ( x − µT )22 µ x x / d x − k I (cid:90) e − λ ( x − µT )22 µ x x / d x (54)The second integral can be performed directly from formula (77) and the definition of the ϕ ( . ) function;the first integral can be carried out in a similar way, after completing the square − x − λ ( x − µT ) µ x = − λ (cid:16)(cid:112) µ /λx − µT (cid:17) µ x + λTµ (cid:32) − (cid:114) µ λ (cid:33) (55)and recalling that Ke − rT e k I = V A e ω I T = V A e λTµ (cid:16) √ µ /λ − (cid:17) . (56)Last, if k I ≤ , then ( − k I , ∞ ) ⊂ R + and therefore the integral (11) is equal to zero as the density (50) issupported by the real negative axis only. Calibration process for the NegIG parameters λ and µ goes like the NegGamma calibration in sub-subsection 3.1.3, by using variance and excess kurtosis, and by introducing the function M I : λµ −→ µ /λ µ/λ (57)16hose inverse M − I is straightforward to compute. In table 3, we compare the default probabilities obtained for the set of issuers (see details in appendixB) with our NegGamma and NegIG models, to the one obtained via the classical Merton model, for twodistinct horizons T (1 year and 5 years). We also compare with historical default rates of the investmentgrade and speculative categories displayed in table 2. Several observations can be made:Table 2: Historical default rates for the investment grade and speculative categories (source: S&P (2020b)).Category 1 year default rate 5 years default rate 10 years default rate2010-2019 2009 2010-2015 2009 2000-2008 2009Investment Grade 0.01% 0.33% 0.17% 0.63% 2.15% 0.92%Speculative 2.53% 9.95% 10.01% 16.47% 21.3% 20.74%All ratings 1.19% 4.19% 4.61% 6.99% 9.43% 8.88%Table 3: Comparison of NegGamma, NegIG and Merton default probabilities, for time horizons T=1 and5 years and for several categories/ratings.Rating Category NegGamma NegIG Merton T = 1 y. T =5 y. T = 1 y. T =5 y. T =1 y. T =5 y.SAN FP AA Inv 0.02% 0.61% 0.02% 0.60% 0.00% 0.06%SAP GY A Inv 0.01% 0.47% 0.01% 0.46% 0.00% 0.03%AI FP A- Inv 0.08% 1.26% 0.08% 1.22% 0.00% 0.16%SU FP A- Inv 0.14% 2.02% 0.14% 1.95% 0.00% 0.51%CRH LN BBB+ Inv 1.10% 10.11% 1.02% 9.97% 0.01% 10.85%DAI GY BBB+ Inv 3.26% 20.94% 3.00% 21.06% 0.55% 34.13%VIE FP BBB Inv 2.62% 14.96% 2.39% 14.77% 0.21% 17.14%AMP IM BB+ Spec 0.50% 4.82% 0.47% 4.66% 0.00% 3.16%FR FP BB+ Spec 3.12% 21.63% 2.90% 21.84% 0.62% 33.48%EO FP BB Spec 3.06% 22.02% 2.87% 22.13% 0.65% 29.00%GET FP BB- Spec 1.50% 11.43% 1.38% 11.26% 0.03% 12.11%LHA GY BB- Spec 7.29% 34.11% 6.93% 34.67% 5.09% 49.57%PIA IM B+ Spec 1.08% 11.22% 1.02% 11.10% 0.02% 11.66%CO FP B Spec 5.62% 24.97% 5.29% 24.94% 2.48% 29.28%- We can clearly observe that, both in the NegGamma and NegIG models, 1 year default probabilitiesare significantly higher than the Merton ones, except for very well rated issuers (A and beyond),which is coherent given the high level of quality of their signature (but, even for these issuers,NegGamma and NegIG probabilities remain strictly bigger than 0, which is not the case with theMerton model); it is also interesting to note that, for longer horizons, most NegGamma and NegIGdefault probabilities grow slower than Merton default probabilities, notably for speculative issuers.17his risk redistribution towards shorter maturities is a clear improvement of classical diffusion mod-els, which underestimate (resp. overestimate) short (resp. long) term default probabilities.- NegGamma probabilities are generally higher than NegIG probabilities: this is coherent with thefact that the Lévy measure decreases in / | x | in the NegGamma model and in / | x | / in theNegIG model, making for a fatter left tail in the NegGamma model and therefore for more frequentoccurrences of downward jumps.- On the speculative segment, 1 year default probabilities are typically around 1-3% in the NegIG andNegGamma models, which provides a very good agreement with the 2010-2019 average for 1 yeardefault rates in this category (2.53%). This is again a clear improvement when compared to theMerton model, whose default probabilities lie around 0-0.5%, except for very indebted firms with a V A /K ratio close to 1 such as LHA GY (which, as an airline company, has particularly suffered fromthe pandemics and saw its stock price V E drop by 35% in 2020) or CO FP. It is also interesting to notethat, for these 2 particular issuer, NegGamma and NegIG probabilities (5-7%) remain significantlyhigher than the Merton probabilities (2-5%), and are closer to the 1 year default rate that could beobserved during periods of intense crisis (reaching more than 9% in 2009, consecutively to the globalfinancial crisis).- On the same segment but for a time horizon T=5 years, the Merton model returns very high defaultprobabilities (up to 50% for LHA GY), while the NegIG model returns far more realistic figures(10-20%, except for the two very endebted issuers cited above, but figures remain far lower than theMerton probabilities), which are coherent with historical default rates for this category ( (cid:39)
10% forthe 2010-2019 average rate of 5 years default, peaking at 16% during the GFC).- It is interesting to note that, for the AA/A issuers retained in our study, 1 and 5 years default ratesprovide a remarkably good fit to historical default rates. For the other issuers of the investmentgrade segment, NegGamma and NegIG default probabilities appear to slightly overestimate shortterm historical default rates; we may note, however, that these probabilities can easily be diminishedby usual data re-processings of the equity or debt values for such issuers. For instance, an importantpart of a firm’s debt can be concentrated by one of its financial captive, as frequently the case inthe automobile industry (DAI GY); it is therefore common to lower the face value of the total debt K by the captive debt amount. Other issuers, for instance operating in the energy sector (like VIEFP), require expensive equipment with long term financing, and in that case a common practice is18o integrate operational cashflows to the market capitalization ( V E ) of the firm when evaluating itsdefault risk.Figure 1: Term structure for the default probabilities given by several models, for the issuer GET FP.NegGamma and NegIG models provide better fit to historical default rates in the speculative categorythan the Merton model.In Fig. 1, we plot the term structure of the default probabilities for the GET FP issuer (0 to 10years), in the NegGamma, NegIG and Merton models. We can recognize the characteristic shape ofthe Merton default probabilities: short term probabilities are very low, with a sudden increase and aninflection around 5 years and a flattening for longer horizons (which is a direct consequence of the shapeof the normal cumulative distribution function N(.)). On the contrary, NegGamma and NegIG defaultprobabilities display a more regular structure across maturities; note that both NegGamma and NegIGprobabilities are particularly close to the average historical rates of table 2 in the speculative segment,with approximately 2% for T=1, 11% for T=5 and 24% for T=10, while Merton default probabilities arenull for T ≤
1, and reach 35% on the 10 years horizon, which is completely incoherent with the historicaldefault rates of this category.
Let us now consider the case where the stochastic process X t in (3) is a symmetric Lévy process, thatis, a process with equally probable downward and upward jumps. We will consider a Variance Gamma19VG) process, which, for simplicity, will be assumed to be symmetric. VG processes have already beenintroduced in credit risk models in Foriani & al. (2006), but using a calibration from CDS indices (MarkitCDX.NA.HY and CDX.NA.IG). Again, we adopt a different approach, by taking advantage of recentlyderived pricing formulas for the European call to perform algorithm 1 on each firm’s own equity, anddeduce its default probability. The Variance Gamma (VG) process, which has been popularized in financial modelling in Madan et al. (1998), can be defined by a time changed drifted Brownian motion: V t := θG t + σW G t , (58)where G t is a Gamma process of mean µ := 1 and variance ν > (note that this parametrization isequivalent to the rate/shape parametrization of subsection 3.1, for ρ = µ /ν and λ = µ/ν ). σ > is thescale parameter and θ ∈ R the asymmetry, or sknewness parameter; in the case θ = 0 the distribution of theVG process is symmetric around the origin (this model was introduced earlier, in Madan & Seneta (1990)).The definition (58) shows that the VG process is actually a so-called normal mean mixture (Barndorff-Nielsen, 1982), where the mixing distribution is given by the Gamma distribution and materializes thepassage of the business time, jumps in the process materializing periods of intense trading activity. When ν grows high then jumps are more frequent and the business time admits staircase-like realizations; on thecontrary, the ν → configuration corresponds to a linear passage of time.The Lévy measure of the VG process is known to be Π V (d x ) = e θxσ ν | x | e − (cid:114) θ σ νσ | x | d x (59)which shows that the VG process is a particular case of a CGMY process (Carr & al., 2002). As the measure(59) is defined on the whole real axis, realizations of the VG process feature both upward and downwardjumps; while downward jumps still materialize a brutal drop in a firm’s assets, we may think of upwardjumps as cash injections from governmental institutions or central banks in order to help recapizalizingseverely indebted issuers. Let us also remark that when θ = 0 then the Lévy measure is symmetric around0 (and in that case positive and negative jumps occur with equal probability). The Lévy symbol can be20omputed from (59) and the Lévy-Khintchine formula (6), resulting in ψ V ( u ) = − ν log (cid:18) − iθνu + σ ν u (cid:19) (60)from which we easily deduce the cumulant generating function as well as the first cumulants and centralmoments: κ (1) V = θ (Mean) κ (2) V = θ ν + σ (Variance) κ (3) V / ( κ (2) V ) / = θν (2 θ ν + 3 σ )( θ ν + σ ) / (Skewness) κ (4) V / ( κ (2) V ) = 3 ν (2 θ ν + 4 θ νσ + σ )( θ ν + σ ) (Excess kurtosis) . (61)If we assume θ = 0 (we will call this configuration the symVG model), only moments of even order differfrom zero, the variance being equal to σ and the excess kurtosis to ν . When the VG process V t is symmetric, its density is known to be f V ( x, t ) = 2 √ π Γ( tν )( σ ν ) tν | x | σ (cid:113) ν tν − K tν − (cid:32) σ (cid:114) ν | x | (cid:33) (62)and, taking θ = 0 in (60), we easily obtain the convexity adjustment ω V = 1 ν log (cid:18) − σ ν (cid:19) . (63)From definition 1, the distance to default in the symVG model is therefore k V = log V A K + (cid:18) r + 1 ν log (cid:18) − σ ν (cid:19)(cid:19) T (64) Remark 2.
In the low variance regime ν → , Taylor expanding (63) yields ω V ∼ − σ / , thus recoveringthe Gaussian adjustment; in other words, the symVG model recovers the Merton model in its low variancelimit. The cumulative distribution function of the Variance Gamma distribution is not known in exact formbut easily accessible from any programming language. For instance in R one can use the CDF function21pplied to the Variance Gamma density (denoted by "dvg" in the Variance Gamma package), and thedefault probability in the symVG model can be written formally as: F V ( − k V ) = CDF (cid:18) dvg (cid:18) − k V √ T , µ = 0 , σ √ T , θ = 0 , νt (cid:19)(cid:19) . (65) An exact formula for the price of the European call under the VG process has been established in Madan et al. (1998), but it involves complicated products of Bessel and hypergeometric functions; in practice,Fourier techniques are favored (Lewis, 2001), notably because of the relative simplicity of the characteristicexponent (60). This technology, however, is not adapted to the implementation of algorithm 1 becauseif would call for solving an equation involving an integral in the Fourier space. Instead, we propose touse a recently derived pricing formula for the European call in the symmetric VG model, which takes theform of quickly convergent series of powers of the log forward moneyness (or, in the context of defaultmodelling, of the distance to default). To that extent, let us introduce the notations T ν := Tν − σ ν := σ (cid:114) ν , (66)and assume T ν / ∈ Q . For X, Y ∈ R , let us also define the quantity a n ,n ( X, Y ) := ( − n n ! (cid:34) Γ( − n + n +12 + T ν )Γ( − n + n + 1) (cid:18) − XY (cid:19) n Y n + 2 Γ( − n − n − − T ν )Γ( − n + − T ν ) (cid:18) − XY (cid:19) n +1+2 T ν ( − X ) n (cid:35) . (67) Proposition 4.1.
The equity value of a firm in the symVG credit risk model can be written as:- If k V < , V E = Ke − rT Tν ) ∞ (cid:88) n =0 n =1 a n ,n ( k V , σ ν ) . (68) - If k V > , V E = V A − Ke − rT − Ke − rT Tν ) ∞ (cid:88) n =0 n =1 a n ,n ( k V , − σ ν ) . (69)22 If k V G = 0 , V E = Ke − rT Tν ) ∞ (cid:88) n =1 Γ( n +12 + T ν )Γ( n + 1) σ nν . (70) Proof.
See Aguilar (2020).
The calibration procedure is, like for one-sided models, based on algorithm 1 and proposition 4.1 (it issufficient to truncate the series at n = m = 10 ); given the moments (61) for θ = 0 , we need to considerthe function M V : σν −→ σ ν (71)whose inversion is immediate. Calibration results and default probabilities are displayed in table 4.Table 4: Parameter calibration and 1 year default probabilities in the symVG model, compared withNegGamma and Merton default probabilitiesIssuer Calibrated parameters Default probabilities σ ν V A (in MM) SymVG Merton NegGammaSAN FP 0.2133 1.9575 136 121 0.00% 0.00% 0.02%SAP GY 0.2873 2.2526 180 904 0.00 % 0.00% 0.01%AI FP 0.2340 3.5774 78 923 0.03 % 0.00% 0.08%SU FP 0.3246 3.9171 71 472.2 0.07% 0.00% 0.14%CRH LN 0.3038 2.9005 33 948.2 0.57% 0.01% 1.10%DAI GY 0.1026 3.5876 213800 1.61% 0.55% 3.26%VIE FP 0.1618 4.4099 27 281.3 1.42% 0.21% 2.62%AMP IM 0.3598 4.809 8 627.48 0.26% 0.00% 0.50%FR FP 0.2741 3.1256 11 931.5 1.02% 0.62% 3.12%EO FP 0.2754 1.7378 10010.3 1.69% 0.65% 3.06%GET FP 0.2402 3.2453 11 666.7 0.79% 0.03% 1.50%LHA GY 0.2092 2.5558 14700 4.40% 5.09% 7.29%PIA IM 0.2513 1.8526 1 494.92 0.57% 0.02% 1.08%CO FP 0.0713 2.6652 16 427.4 3.30% 2.48% 5.62%It is interesting to note that 1 year symVG default probabilities are all comprised between 1 year Mertonand 1 year NegGamma default probabilities (except for LHA GY). The fact that symVG probabilities arelower than the probabilities computed in a one-sided model (the NegGamma model) is no surprise, sincethe symVG model features also positive jumps (i.e. sudden raises in asset prices), which can diminishthe probability for V A ( T ) to be lower than K ; the symVG model is therefore more tempered than theNegGamma model, but its default probabilities remain very significantly higher than in the Merton model,except for top quality issuers. 23 Concluding remarks
The main conclusions of our study are the following:(a) We have generalized the notions of distance to default and default probabilities to the class of purejump structural models, and studied two sub-classes: one-sided models (NegGamma, NegIG) andsymmetric models (symVG).(b) One-sided models are particularly tractable, thanks to the closed formulas for the equity value thatwe have established (propositions 3.1 and 3.2). Such closed formulas make the calibration algorithmeasy to implement.(c) One-sided models produce higher short term default probabilities than the Merton model. In par-ticular, these probabilities typically lie between 0.5% and 3% for speculative issuers, which is closeto the historical 1 year default rate of this category. On the investment grade segment, high qualityissuers (with a ranking higher or equal to A) have a small default probability (0.01-0.15%) whichis coherent with the quality of their signature; the figures, however, remain clearly higher than theMerton ones.(d) In double-sided models (symVG), the presence of upward jumps moderates the probability for V A ( T ) to be lower than K , and therefore default probabilities are lower than NegGamma and NegIG defaultprobabilities, while remaining higher than the Merton ones. The model is, consequently, well suitedto investment grade issuers for longer horizons, but as the equity value is only accessible via truncatedseries expansions, the calibration algorithm takes more time to converge.Future work should include the consideration of other dynamics for asset prices, such as a double-sidedprocess with asymmetric distribution of jumps; recent pricing formulas established for instance in the caseof an asymmetric NIG process could be used for computing the equity value, but their series nature willcall for several optimization in the calibration algorithm to maintain a satisfactory level of performance. Acknowledgments
We thank Alexander Lipton for his interest in this work and his many valuable comments.24 eferences
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Special functions
Standard normal cumulative distribution function: N ( x ) := 1 √ π x (cid:90) −∞ e − y d y (72)Lower incomplete Gamma function: γ ( a, z ) := z (cid:90) e − x x a − d x. (73)27pper incomplete Gamma function: Γ( a, z ) := ∞ (cid:90) z e − x x a − d x. (74)Bessel function of the second kind (MacDonald function): K ν ( z ) := 12 (cid:16) z (cid:17) ν ∞ (cid:90) e − x − z x x − ν − d x. (75)Shuster’s integrals (Shuster, 1968): x (cid:90) e − λ ( y − µt )22 µ y y d y = (cid:114) πλt (cid:34) N (cid:32)(cid:114) λt x (cid:18) xµt − (cid:19)(cid:33) + e λtµ N (cid:32) − (cid:114) λt x (cid:18) xµt + 1 (cid:19)(cid:33)(cid:35) ∞ (cid:90) x e − λ ( y − µt )22 µ y y d y = (cid:114) πλt (cid:34) N (cid:32) − (cid:114) λt x (cid:18) xµt − (cid:19)(cid:33) − e λtµ N (cid:32) − (cid:114) λt x (cid:18) xµt + 1 (cid:19)(cid:33)(cid:35) . (77) Model notations
Characteristic function and Lévy symbol: Ψ X ( u, t ) := E [ e iuX t ] := e tψ X ( u ) . (78)Moment generating function (double-sided Laplace transform): M X ( p, t ) := E [ e pX t ] = Ψ( − ip, t ) . (79)Cumulant generating function: κ X ( p ) := log M X ( p,
1) = ψ X ( − ip ) . (80) This is a consequence of the Cauchy - Schlömilch substitution (cid:90) R f ( x ) d x = (cid:90) R f (cid:18) x − x (cid:19) d x (76)which is itself a consequence of the more general Glasser’s master theorem (Glasser, 1983). th cumulant: κ ( n ) X := ∂ n κ X ( p ) ∂p n (cid:12)(cid:12) p =0 . (81)Cumulative distribution function (process with density f X ( x, t ) ): F X ( x ) := ∞ (cid:90) −∞ f X ( y, T ) d y. (82)Martingale adjustment: ω X := − ψ X ( − i ) = − log E (cid:2) e X (cid:3) . (83)Distance to default: k X := log V A K + ( r + ω X ) T. (84)Risk-neutral default probability: P X := F X ( − k X ) . (85) B Data set
To test the models, we have chosen a set of issuers from various European markets, which are representativeof several industrial sectors and S&P rating categories. In table 5 we provide details on the issuers andthe debt level chosen in our study.Table 5: Data set used in the paper; debt values are as of 13 oct. 2020. Source: Bloomberg.Ticker Name Industry group Total debt (in MM) CategorySAN FP FP Sanofi Pharmaceuticals 25 933 InvestmentSAP GY SAP SE Software 16 196 InvestmentAI FP Air Liquide SA Chemicals 14 730 InvestmentSU FP Schneider Electric SE Electrical Components 8 473 InvestmentCRH LN CRH PLC Building materials 10 525 InvestmentDAI GY Daimler AG Auto manufacturers 161 780 InvestmentVIE FP Veolia Environment Water 16 996 InvestmentAMP IP Amplifon SPA Pharmaceuticals 1 339 SpeculativeFR FP Valeo SA Auto parts & Equipment 4 879 SpeculativeEO FP Faurecia Auto parts & Equipment 4 838 SpeculativeGET FP Getlink SE Transportation 4 998 SpeculativeLHA GY Deutsche Lufthansa Airlines 10 106 SpeculativePIA IM Piaggio Motorcycles 609 SpeculativeCO FP Casino Foods 14 308 Speculative29
Code (in R)
We provide the code used to calibrate the NegGamma model using algorithm 1 on the whole data setdisplayed in table 5, and to compute the related default metrics (distance to default, default probabilities).The code has been written in R and implemented on a personal computer with Intel(R) Core(TM) i5-8265UCPU @1.60GHz. ptm <- proc.time()library(readxl)library(pracma)library(xlsx)library(scales)library(GeneralizedHyperbolic)library(moments) par_rho <- function(kurt){ par_rho = 6/kurt}par_lambda <- function(variance,rho){ ar_lambda = sqrt(rho/variance)}Theoretical_equity_value <- function(x){ (cid:44) → Gamma Process k_G <- log(x/K)+(r+rho*log(1+1/lambda))*tif (k_G>0){ _ G > 0 we apply equation (29)....
Theoretical_equity_value = x*as.numeric(gammainc((lambda+1)*k_G,rho*t)[1])/gamma(rho (cid:44) → *t)-K*exp(-r*t)*as.numeric(gammainc(lambda*k_G,rho*t)[1])/gamma(rho*t)-equity_ (cid:44) → value}else{ _ E = 0
Theoretical_equity_value = -equity_value}}Distance_to_default <- function(K,actif_value,rho,lambda,r,t){ (cid:44) → Gamma Process framework
Distance_to_default = log(actif_value/K)+(r+rho*log(1+1/lambda))*t}Default_probability <- function(K,actif_value,rho,lambda,r,t){ (cid:44) → Negative Gamma Process framework. Input : distance to default of the issuer k_G <- log(actif_value/K)+(r+rho*log(1+1/lambda))*tif (k_G>0){Default_probability = as.numeric(gammainc(lambda*k_G,rho*t)[2])/gamma(rho*t) else{Default_probability = 1}} data_equity = read_excel("my_dataset.xlsx",sheet = 1)data_debt = read_excel("my_dataset.xlsx",sheet = 2) r <- 0 Time <- 1 convergence_threshold <- 0.0001 default_probabilities <- matrix(1,1,ncol(data_debt)) distances_to_default <- matrix(1,1,ncol(data_debt)) parameters_issuer <- matrix(1,3,ncol(data_debt)) (cid:44) → (asset value, rho and lambda) for (k in 1:length(data_debt)){K <- as.double(data_debt[1,k]) equity <- as.double(as.matrix(data_equity[(nrow(data_equity)-251):nrow(data_equity),k (cid:44) → +1])) rV_E <- diff(log(equity)) rho_0 <- par_rho(kurtosis(rV_E)) lambda_0 <- par_lambda(var(rV_E)*252,rho_0) ctif_value <- matrix(0,length(equity),1) (cid:44) → each time hist_rho_actif <- append(-100,rho_0) hist_lambda_actif <- append(-100,lambda_0) print(k)cpt <- 0 (cid:44) → corresponding asset value at each time, we stop when convergence is reached while ((abs(hist_rho_actif[cpt+2]-hist_rho_actif[cpt+1])>convergence_threshold)|(abs( (cid:44) → hist_lambda_actif[cpt+2]-hist_lambda_actif[cpt+1])>convergence_threshold)){rho <- tail(hist_rho_actif,1)lambda <- tail(hist_lambda_actif,1) for (i in 1:length(equity)){equity_value <- as.double(equity[i])t <- Time + (252-i)/252 (cid:44) → equity value to find the asset value of the issuer actif_value[i,1] <- newtonRaphson(Theoretical_equity_value,K)$root}rV_A <- diff(log(actif_value)) variance <- var(rV_A)*252 kurt <- kurtosis(rV_A) print(paste("It\’eration␣num\’ero",cpt+1,sep="␣"))print(paste("rho:",hist_rho_actif[cpt+2],sep="␣"))print(paste("lambda:",hist_lambda_actif[cpt+2],sep="␣"))print(paste("V_A:",tail(actif_value,1),sep="␣"))print(paste("Variance:",variance,sep="␣")) ist_rho_actif[cpt+3] <- par_rho(kurt) hist_lambda_actif[cpt+3] <- par_lambda(variance,hist_rho_actif[cpt+3]) (cid:44) → the new lambda cpt <- cpt+1} default_probabilities[1,k] <- Default_probability(K,tail(actif_value,1),tail(hist_rho_ (cid:44) → actif,1),tail(hist_lambda_actif,1),r,t)distances_to_default[1,k] <- Distance_to_default(K,tail(actif_value,1),tail(hist_rho_ (cid:44) → actif,1),tail(hist_lambda_actif,1),r,t)parameters_issuer[1,k] <- tail(hist_rho_actif,1)parameters_issuer[2,k] <- tail(hist_lambda_actif,1)parameters_issuer[3,k] <- tail(actif_value,1)print(paste("Number␣of␣iterations␣for␣Vassalou’s␣algorithm␣:",cpt,"iter",sep="␣"))print(paste("Default␣probability␣of␣the␣issuer␣=",default_probabilities[1,k],sep="␣"))print(paste("Distance␣to␣default␣of␣the␣issuer␣=",distances_to_default[1,k],sep="␣"))print(paste("Parameters␣=␣rho␣:",tail(hist_rho_actif,1),"lambda␣:",tail(hist_lambda_ (cid:44) → actif,1),"V_A:",tail(actif_value,1),sep="␣"))} data_to_export <- as.data.frame(rbind(format(default_probabilities,scientific=FALSE),as. (cid:44) → numeric(distances_to_default),parameters_issuer))colnames(data_to_export) <- colnames(data_debt)rownames(data_to_export) <- c(paste("Probabilit\’e␣de␣d\’efaut␣",Time,"Y",sep=""),paste( (cid:44) → "Distance␣to␣default␣",Time,"Y",sep=""),"Calibrated␣Rho","Calibrated␣Lambda"," (cid:44) → Calibrated␣Asset␣Value")setwd("G:\\my_folder") rite.xlsx(data_to_export,file=paste("DD_DP_ModMarketCap_Debt_",Time,"Y.xlsx",sep=""), (cid:44) → append=FALSE)print(proc.time()-ptm)append=FALSE)print(proc.time()-ptm)