Efficient solution of structural default models with ... - CiteSeerX

Efficient solution of structural default models with correlated jumps. A fractional PDE approach. Andrey Itkin1,2 and Alex Lipton1,3

arXiv:1408.6513v1 [q-fin.CP] 27 Aug 2014

1

Bank of America Merrill Lynch York University, School of Engineering 3 Oxford-Man Institute of Quantitative Finance, University of Oxford 2 New

August 28, 2014

Abstract Using a structural default model considered in Lipton and Sepp (2009) we propose a new approach to introducing correlated jumps into this framework. As the result we extend a set of the tractable L´evy models which represent the jumps as a sum of the idiosyncratic and common parts. So far in the literature only the discrete and exponential jumps were considered using Marshall and Olkin (1967) method, and only in one and two-dimensional cases. We present realization of our approach in two and three dimensional cases, i.e. compute joint survival probabilities Q of two or three counterparties. We also extend the model by taking into account mutual liabilities of the counterparties and demonstrate their influence on Q. The latter, however, requires a more detailed analysis which will be published elsewhere. Keywords: structural default model, join survival probability, mutual liabilities, ies jump-diffusion, PIDE, splitting, matrix exponential

1

Introduction

Multi-dimensional jump-diffusion models find various applications in mathematical finance. They are used in modeling baskets of stocks and their derivatives, computing prices and risk in various credit models which are based on a Merton (1974) structural model, etc. Various approaches and methods, which include semi-analytics, finite-difference (FD), Monte-Carlo The views represented herein are the author’s own views and do not necessarily represent the views of BAML or its affiliates and are not a product of BAML Research.

1

methods and combinations of all of them has been used for solving the corresponding problems, see, e.g., survey in Lipton and Sepp (2011) and references therein. Nevertheless, to the best of our knowledge the existing approaches suffer from the curse of dimensionality. Indeed, even very sophisticated analytical methods, e.g., Lipton and Savescu (2014), are compelled to eliminate jumps in order to make the problem tractable. However, it is wellknown that despite accounting for the jumps significantly increases the complexity of the model they need to be involved into simulation in order to describe the observed market behavior at short time scales. On the other hand, Monte Carlo methods are either slow or inaccurate. Therefore, finite difference approach seems to be an alternative for the 2D and 3D entire problems, despite even in the 3D case it could be relatively slow. So far the authors are aware of just few papers that used the finite difference approach to numerically solve 2D PIDE describing an evolution of the option price, Clift and Forsyth (2008), or the CDS prices, Lipton and Sepp (2009, 2011) written on two correlated assets. For instance, in Lipton and Sepp (2009) the assets were correlated twofold. First, the diffusion drivers of the asset represented as a Geometric Brownian motion were correlated with the correlation coefficient ρ. Second the jumps were also correlated by representing the jumps in each asset as a sum of i) the systemic jumps common for all assets, and ii) idiosyncratic jumps specific for this particular asset. To the best of our knowledge Vasicek (1987, 2002) was the first who proposed to factor asset moves into the idiosyncratic and common parts. Later this became a standard approach for multivariate L´evy models as well1 . In Clift and Forsyth (2008) the authors used a bivariate distribution proposed in Marshall and Olkin (1967) and considered normal and exponentially distributed multivariate jumps. However, other L´evy models could be even of greater interest as it is known that, e.g., Generalized Hyperbolic models, Eberlein and Keller (1995) fit the market data pretty well. Therefore, an extended framework which allows various L´evy models to be used when modeling jumps would be highly desirable. In the next sections we provide a short survey of various approaches to introduce multivariate correlated jumps via L´evy’s copula, multivariate subordinators of the Brownian motion, etc. as well as discuss their advantages and pitfalls. Another observation is that even in the one-dimensional case traditional methods of solving the PIDE experience some problems, see survey in Itkin (2014a), and references therein. In the multidimensional case these problems become even harder. Therefore, among the above approaches for setting multivariate correlated jumps, we choose just one and combine it with the matrix exponential method proposed first in Itkin and Carr (2012) and later further elaborated on in Itkin (2014a,c). Thus, we extend the approach of Itkin (2014a) to the multidimensional case. The presented construction allows various types of jumps to be used for modeling the idiosyncratic and common factors. For instance, in the 2D case one can represent idiosyncratic jumps of component 1 by using the Meixner model of Schoutens (2001), idiosyncratic jumps of component 2 by using the Merton model, and model the common jumps by using the Kou model. Certainly, we don’t consider every possible 1

Vasicek used a single period approach whereas for the L´evy models consideration is done in continuous time. Also he uses a Gaussian structure whereas this framework generalizes it to infinitely divisible distributions.

2

combination in this paper, since this could be done based on general principles described in Itkin (2014a,c). However, as an example we do consider a model where idiosyncratic jump components are represented using the Merton model, and the common component - using the Kou model. In this sense, our examples should be ideologically similar to that in Clift and Forsyth (2008). However, our method differs from Clift and Forsyth (2008) by: 1. Merton and Kou jumps are used just as an example. Other L´evy jumps could be used as well, see Itkin (2014a,c). 2. We use the matrix exponential method, rather than the traditional method of solving PIDE presented in Clift and Forsyth (2008). 3. We present a splitting method to provide solutions of the 2D and 3D problems with second order of accuracy in both space and time, and prove convergence and approximation of the method. The complexity of our method is O(N1 × N2 ) in the 2D case and O(N1 × N2 × N3 ) in the 3D case, which is better than that of the FFT method used in Clift and Forsyth (2008). Application-wise we are concentrated on a structural default model, while pricing options written on the basket of stocks also fits well to this framework. The above construction and results of the experiments are the main new results of this paper. Another innovation of this paper is as follows. We extend the approach of Lipton and Sepp (2009) by computing the joint survival probability Q of the firms (two firms in the 2D case, and three firms in the 3D case) and simultaneously taking into account that these firms might have some mutual liabilities. This problem was discussed, e.g., in Webber and Willison (2011) who mentioned that systemic capital requirements for individual banks, determined as the solution to the policymakers optimisation problem, depend on the structure of banks balance sheets (including their obligations to other banks) and the extent to which banks’ asset values tend to move together. And generally, banks’ systemic capital requirements are found to be increasing in balance sheet size relative to other banks in the system; interconnectedness; and, materially, contagious bankruptcy costs. From this prospective, Merton-type structural credit risk models were used to quantify risks to the solvency of an interconnected banking systems. For instance, Elsinger et al. (2006), Gauthier et al. (2010) consider this systemic risk to be attributed either to correlations between the asset values of the banks, so the latter could simultaneously default, or to interlinkages of the banks balance sheets which could result into contagious defaults. Thus, such a model could be constructed as a combination of the correlated Merton balance sheet models, calibrated by using the observed bank equity returns, and a network of interbank exposures cleared as in Eisenberg and Noe (2001). To that extent in this paper we provide some numerical results to demonstrate that accounting for the presence of the mutual debts could essentially affects Q, especially in the 3D case. We, however, don’t provide a detailed analysis of this effect, as well as we don’t consider how this consideration could affect prices of CDS and CDO, assuming to publish it elsewhere. 3

The rest of the paper is organized as follows. In section 2 we describe our multidimensional structural model inspired by the familiar model of Merton (1974), which is an extension of Lipton and Sepp (2009). Next section provides a short survey of the existing approaches to introducing multivariate correlated jumps, and argues our choice of the approach in use. In section 4 we shortly describe the method of Itkin (2014a,c), Itkin and Carr (2012) and extend it to the multidimensional case. Next section describes our splitting algorithm on financial processes that is adopted for solving the corresponding multidimensional PIDE. In section 6 we provide a detailed numerical scheme to solve the fractional jump equations and formulate and prove all necessary to confirm the unconditional stability, second order accuracy and convergence of the scheme. We also underline that our schemes preserve positivity of the solution. The results of the numerical experiments are provided in sections 7 (the 2D case) and 8 (the 3D case). Also in section 8 we describe all the details of the numerical scheme used in the 3D case. The final section concludes.

2

Statement of the problem

Similar to Lipton and Sepp (2009, 2011) we consider a multi-dimensional structural model inspired by the familiar model of Merton (1974). For survey of the existing literature on this problem and proposed approaches to its solution also see Lipton and Sepp (2009, 2011) and references therein. First, for simplicity, assume that we have just two firms with the firm asset values a1,t and a2,t . Also assume that the default barrier of the firm is a deterministic function of time l(t): l(t) = l(0)G(t) where G(t) is the deterministic growth factor Z t  0 0 0 (r(t ) − q(t )) dt . G(t) = exp

(1)

0

Here r(t) is the forward rate, q(t) is the continuous dividend rate, l(0) = RL(0), R is an average recovery of the firm’s liabilities, and L(0) is the total debt per share2 . Usually, R could found from CDS quotes and has a typical value R = 0.4. We define firm’s default time τd assuming continuous monitoring τd = min [at < l(t)]. 0T ] The join survival probability solves the following partial Integro-differential equation (PIDE), see Lipton and Sepp (2011) and also Clift and Forsyth (2008) Qτ = [L + J ]Q

(3)

where a backward time τ = T − t is introduced. Here L is the two-dimensional linear convection-diffusion operator which reads  2 2  X 1X ∂ ∂ 1 2 ∂ + ρi,j σi2 σi2 L= r − σi 2 ∂xi 2 i,j=1 ∂xi ∂xj i=1 and J is the jump operator Z ∞h ∂Q(x1 , x2 , τ, 0) JQ = Q(x1 + y1 , x2 + y2 , τ, 0) − Q(x1 , x2 , τ, 0) − (ey1 − 1) ∂x1 −∞ i ∂Q(x1 , x2 , τ, 0) − (ey2 − 1) ν(dJ)dy1 dy2 . ∂x2 where ν(dJ) is the two-dimensional L´evy measure. 6

(4)

(5)

This PIDE has to be solved subject to the boundary and terminal conditions. The terminal condition reads Q(x1 , x2 , 0, 0) = 1x1 >log ˆl1 , x2 >log ˆl2 . The boundary conditions could be set as the Dirichlet conditions at ±∞. Obviously, Q(x1 , x2 , τ, 0) → 0, at xi → −∞. As xi → ∞, i = 1, 2, Q(x1 , x2 , τ, 0) should replicate the marginal survival probability Q(x3−i , τ, 0). This condition, however, must be supplemented with the boundary condition when both x1 → ∞ and x2 → ∞. A natural choice is Q(x1 → ∞, x2 → ∞, τ, 0) = 1. Various choices of the L´evy measures that could be used for this model as well as an approach to introduce the correlated jumps are discussed in the next section.

3

Correlated jumps and structured default models

There exist at least three known ways of introducing the correlated jumps, see Cont and Tankov (2004), Deelstra and Petkovic (2010) and references therein. The first one is L´evy copula. In application to the structural credit models this approach was used by Baxter (2007), Moosbrucker (2006). However, Copula-based constructions issue some restrictive constraints on the jump parameters to preserve marginal distributions, which make it difficult to model arbitrary (positive and negative) correlations between jumps. In other words, due to restrictions on the parameters controlling marginal distributions the correlation coefficient is not able to take all values in the continuous range [−1, 1]. Same problem is inherent to the popular Marshall and Olkin (1967) construction used, e.g., in Clift and Forsyth (2008), as this model doesn’t allow negative correlations between jumps. Some other types of L´evy copulas were also discussed in the literature but not in the context of the problem under consideration. Another construction in Lipton and Sepp (2009) is also partly inspired by the work of Marshall and Olkin (1967). However, in contrast to Marshall and Olkin (1967) it doesn’t make use of copula. Therefore, both positively and negatively correlated jumps can be represented using their model. The second approach suggests to use multivariate subordinated Brownian motions (or multivariate subordinators of Brownian motions), where the L´evy subordinator could consist of the common and idiosyncratic parts. This approach was advocated by Guillaume (2013), Luciano and Semeraro (2010), Sun et al. (2011), see also survey in Ballotta and Bonglioli (2014) and references therein. As applied to our problem this approach provides analytical tractability if the local volatility is ignored. In this case the characteristic function of the entire jump-diffusion model is known in closed form, and transform methods, like FFT or cosine transform could be used. With allowance for the local volatility this approach becomes inefficient, as the jump integral must now be computed at every point in time and space. Also both constructions considered in above can only accommodate strictly positive correlation values due to restrictions on the parameter controlling the correlation coefficients, 7

which are required to ensure the existence of the characteristic function of the processes involved, see Ballotta and Bonglioli (2014). Therefore, we introduce the correlated jump following the third approach, see Ballotta and Bonglioli (2014). Again in the spirit of Vasicek (1987, 2002) this method constructs a jump process as a linear combination of two independent L´evy processes representing respectively the systematic factor and the idiosyncratic shock. Note, that such an approach was also previously mentioned in Cont and Tankov (2004). It has an intuitive economic interpretation and retains nice tractability, as the multivariate characteristic function in this model is available in closed form. The main result of Ballotta and Bonglioli (2014) that immediately follows from Theorem 4.1 of Cont and Tankov (2004) (see also Deelstra and Petkovic (2010), Garcia et al. (2009) is given by this Proposition Proposition 3.1 Let Zt , Yj,t , j = 1, ..., n be independent L´evy processes on a probability space (Q, F, P ), with characteristic functions φZ (u; t) and φYj (u; t), for j = 1, ..., n respectively. Then, for bj ∈ R, j = 1, ..., n Xt = (X1,t , ..., Xn,t )> = (Y1,t + b1 Zt , ..., Yn,t + bn Zt )> is a L´evy process on Rn . The resulting characteristic function is ! n n X Y φX (u; t) = φZ bi u i ; t φYj (uj ; t), u ∈ Rn . i=1

i=1

By construction every factor Xi,t , i = 1, ..., n includes a common factor Zt . Therefore, all components Xi,t , i = 1, ..., n could jump together, and loading factors bi determine the magnitude (intensity) of the jump in Xi,t due to the jump in Zt . Thus, all components of the multivariate L´evy process Xt are dependent with pairwise correlation (again see Ballotta and Bonglioli (2014) and references therein) ρj,i = Corr(Xj,t , Xi,t ) = q

bj bi Var(Z1 ) q Var(Xj,1 ) Var(Xj,1 )

Such a construction has multiple advantages, namely: 1. As sign(ρi,j ) = sign(bi bj ), both positive and negative correlations can be accommodated 2. In the limiting case bi → 0 or bj → 0 or Var(Z1 ) = 0 the margins become independent, and ρi,j = 0. The other limit bi → ∞ or bj → ∞ represents a full positive correlation case, so ρi,j = 1. Accordingly, bi → ∞, b3−i → ∞, i = 1, 2 represents a full negative correlation case as in this limit ρi,j = −1. One more advantage of this approach comes out if we want the margin distribution Xi,t to be fixed. Then a set of conditions to convolution coefficients could be imposed to preserve the margin. This is reasonable from the practitioners point of view as the entire credit 8

product could be illiquid, and, therefore, the market quotes necessary to calibrate the full correlation matrix could not be available. Hence, alternatively, the margin distributions could be first calibrated to a more liquid market of the components Xi,t , and the entire correlation structure should preserve these marginals. This first of all defines parameters of the idiosyncratic factors. The remaining parameters of the entire correlation structure could be found at the second step given another consideration. According to this setup, the instantaneous correlation between the drivers a1 and a2 reads ρσ1 σ2 + b1 b2 Var(Z(1)) q ρ12 = q σ12 + Var(X1,1 ) σ22 + Var(X2,1 )

(6)

As far as the structural default model is concerned, positive jumps might not be necessary for this problem. However, below we keep this generality, as the proposed approach to modeling the correlated jumps amounts to some other asset classes as well without any modification, where both positive and negative jumps are important.

4

Fractional PDE and jump integrals

Assuming that some particular L´evy models are chosen to construct processes Yj,t , j = 1, ..., n and Zt , let us look more closely at Eq.(5). In doing that we follow the method proposed in Itkin and Carr (2012) (first presented at Global Derivatives and Risk Conference, Roma 2009) and then further elaborated on in Itkin (2014a,c). The key idea is to represent the jump integral in the form of a pseudo-differential operator and then formally solve the obtained evolutionary partial pseudo-differential equation via a matrix exponential. For the sake of clearness let us start with a one-dimensional case. It is well known from quantum mechanics (de Lange and Raab (1992)) that a translation (shift) operator in L2 space could be represented as   ∂ Tb = exp b , (7) ∂x with b = const, so Tb f (x) = f (x + b). Therefore, the one-dimensional integral corresponding to Eq.(5) can be formally rewritten as  ∂Q(x, t) ν(dy) = J Q(x, t), [Q(x + y, t) −Q(x, t) − (e − 1) ∂x R    Z  ∂ ∂ y J ≡ exp y − 1 − (e − 1) ν(dy). ∂x ∂x R

Z

y

(8)

In the definition of the operator J (which is actually an infinitesimal generator of the jump process), the integral can be formally computed under some mild assumptions about existence and convergence if one treats the term ∂/∂x as a constant. Therefore, operator J 9

can be considered as some generalized function of the differential operator ∂x . We can also treat J as a pseudo-differential operator. It is important to recognize that J = ψ(−i∂x ) − [log ψ(−i)]∂x = MGF(∂x ) − [log MGF(1)]∂x ,

(9)

where ψ(u) is the characteristic exponent of the jump process, and MGF(u) is the moment generation function corresponding to the characteristic exponent. This directly follows from the L´evy-Khinchine theorem. Note, that the last terms in the right hands part of Eq.(9) is a compensator as the characteristic exponent is computed using the expectation under a risk-neutral measure Q. In other words, the last term is added to make the forward price to be a true martingale under this measure. The advantage of such a representation consists in the fact that we transformed a linear non-local integro-differential operator (jump operator) into a liner local pseudo-differential (fractional) operator. The operator J can be analytically computed for various popular L´evy models, hence J admits an explicit representation in the form of the pseudo-differential operator. Then a pure jump evolutionary equation Qτ = J Q under some mild existence conditions could be formally integrated to provide Q(x, τ + ∆τ ) = e∆τ J Q(z, τ ). Operator A = e∆τ J is the matrix exponential and is understood as a Taylor series expansion of the exponent in ∆τ J . In Itkin (2014a,c), Itkin and Carr (2012) it was shown that the matrix exponential can be efficiently computed on a finite difference grid for various jump models, namely Merton, Kou, CGMY, NIG, General Hyperbolic and Meixner models. Efficiency of this method in general is not worse than that of the FFT, and in many cases is linear in the number of the grid points N . The proposed method is almost universal, i.e., it allows solving PIDEs for various jump-diffusion models in a unified form. Second order finite difference schemes in both space and time were constructed that i) are unconditionally stable, and ii) preserve positivity of the solution. Now let us use the same idea for getting fractional representation of the jump integral in the two-dimensional case. The translational two-dimensional operator in L2 × L2 space could be similarly represented as     ∂ ∂ exp y2 , (10) Ty1 ,y2 = exp y1 ∂x1 ∂x2 with y1 , y2 = const, so Q(x1 + y1 , x2 + y2 , τ ) = Ty1 ,y2 Q(x1 , x2 , τ )

10

Therefore, the whole integral in Eq.(5) could be re-written in the form Z ∞h i J = ey1 ∂x1 ey2 ∂x2 − 1 − (ey1 − 1)∂x1 − (ey2 − 1)∂x2 ν(dJ)dy1 dy2 .

(11)

−∞

With allowance for Proposition 3.1 and using the L´evy-Khinchine theorem in the way similar to how the Eq.(9) was derived, we can show that " 2 ! # 2 2 X X X J =λ ψXj (−i∂xj ) + ψZ −i bj ∂xj + 1 − [log ψXj (−i)]∂xj . (12) j=1

j=1

j=1

where λ is the intensity of the jumps which here is same for all idiosyncratic and systemic jumps. Based on Itkin (2014a,c), Itkin and Carr (2012) we know how to deal with all the terms in this expression except the new term ψZ which represents a two-dimensional characteristic exponent of the common jump process Zt . We shall discuss this in the next sections.

5

Splitting on financial processes

To solve Eq.(3) we use a finite-difference approach with splitting in financial processes. We refer the reader to Itkin (2014a) to the detailed description of the splitting algorithm. Splitting (a.k.a. the method of fractional steps) reduces the solution of the original kdimensional unsteady problem to the solution of k one-dimensional equations per time step. For example, consider a two-dimensional diffusion equation with a solution obtained by using some finite-difference method. At every time step, a standard discretization in space variables is applied, such that the finite-difference grid contains N1 nodes in the first dimension and N2 nodes in the second dimension. Then the problem is solving a system of N1 × N2 linear equations, and the matrix of this system is block-diagonal. In contrast, utilization of splitting results in, e.g., N1 systems of N2 linear equations, where the matrix of each system is banded (tridiagonal). The latter approach is easy to implement and, more importantly, provides significantly better performance. A natural choice for the first step of splitting would be to split operators L and J in Eq.(3) due to their different mathematical nature. So a special scheme could be applied at every step of the splitting procedure. As operators L and J are non-commuting, we use Strang’s splitting scheme, Strang (1968), which provides second order approximation in time τ assuming that at every step of splitting the corresponding equations are solve also with the second order accuracy in time. For more details on how to apply Strang’s splitting to nonlinear equations see Itkin (2014a) and references therein. The entire numerical scheme reads Q(1) (x1 , x2 , τ ) = e (2)

Q (x1 , x2 , τ ) = e Q(x1 , x2 , τ + ∆τ ) = e 11

∆τ 2

D

Q(x1 , x2 , τ ),

∆τ J

Q (x1 , x2 , τ ),

∆τ 2

Q(2) (x1 , x2 , τ ).

D

(1)

(13)

Thus, instead of an unsteady PIDE, we obtain one PIDE with no drift and diffusion (the second equation in Eq.(13)) and two unsteady PDEs (the first and third ones in Eq.(13)). Proceeding in a similar way the second step is to apply splitting to the second equation in Eq.(13). Taking into account Eq.(12) for the brevity of notation represent it as J = J1 + J2 + J12

(14)

where   Jj = λ ψXj (−i∂xj ) − [log ψXj (−i)]∂xj , " !# 2 X J12 = λ 1 + ψZ −i bj ∂xj

j = 1, 2

j=1

Obviously, operators J1 and J2 commute, i.e. et(J1 +J2 ) = etJ1 etJ2 Therefore, replacing the second step in Eq.(13) with another Strang’s splitting with allowance for Eq.(14) we finally obtain Q(1) (x1 , x2 , τ ) = e

∆τ 2

D

Q(2) (x1 , x2 , τ ) = e

∆τ 2

J1

Q(1) (x1 , x2 , τ ),

Q(3) (x1 , x2 , τ ) = e

∆τ 2

J2

Q(2) (x1 , x2 , τ ),

Q(x1 , x2 , τ ),

(15)

Q(4) (x1 , x2 , τ ) = e∆τ J12 Q(3) (x1 , x2 , τ ), Q(5) (x1 , x2 , τ ) = e

∆τ 2

J2

Q(1) (x1 , x2 , τ ),

Q(6) (x1 , x2 , τ ) = e

∆τ 2

J1

Q(1) (x1 , x2 , τ ),

∆τ 2

D

Q(x1 , x2 , τ + ∆τ ) = e

6

Q(6) (x1 , x2 , τ ).

Numerical procedure

Due to the splitting nature of our entire algorithm represented by Eq.(15), each step of splitting is computed using a separate numerical scheme. All schemes provide second order approximation in both space and time, are unconditionally stable and preserve positivity of the solution. For the first and the last step where a pure convection-diffusion two-dimensional problem has to be solved we use a Hundsdorfer-Verwer scheme, see In’t Hout and Foulon (2010), In’t Hout and Welfert (2007), Itkin (2014b). A non-uniform finite-difference grid is constructed similar to Itkin and Carr (2011). For the steps 2,3,5,6 we choose a Merton jump model. In other words, the idiosyncratic jump part of each component Xj,t , j = 1, 2 is represented as Normal jumps. Computation ∆τ of the matrix exponential Aj Q(x1 , x2 , τ ) = e 2 Jj Q(x1 , x2 , τ ), j = 1, 2 could be done with 12

complexity O(N1 N2 ) at every time step. That is because when computing A1 the second variable x2 is a dummy variable, while computation of A1 Q(x1 , x2 = const, τ ) is O(N1 ), see Itkin (2014a). Construction of the jump grid, which is a superset of the finite-difference grid used at the first (diffusion) step is also described in detail in Itkin (2014a). For step 4 (common or systemic jumps) we choose Kou double exponential jumps model proposed in Kou and Wang (2004). Its L´evy density is   ν(dx) = λ pθ1 e−θ1 x 1x≥0 + (1 − p)θ2 eθ2 x 1x 1, θ2 > 0, 1 > p > 0; the first condition was imposed to ensure that the underlying price S(t) has finite expectation. Using this model a one-dimensional representation for J is given in Itkin (2014a). Similarly, in a two dimensional case we obtain   (17) J12 = λ pθ1 (θ1 − b1 Ox1 − b2 Ox2 )−1 + (1 − p)θ2 (θ2 + b1 Ox1 + b2 Ox2 )−1 , Ox1 ≡ ∂x1 , Ox2 ≡ ∂x2 , −θ2 < Re(b1 Ox1 + b2 Ox2 ) < θ1 . The inequality −θ2 < Re(O) < θ1 is an existence condition for the integral defining J and should be treated as follows: the discretization of the operator O should be such that all eigenvalues of matrix A, a discrete analog of O, obey this condition. We proceed in a way similar to the one-dimensional case. To this end we can use the (1,1) P´ade approximation of e∆τ J12 which provides O((∆τ )2 ) approximation 1 1 e∆τ J12 ≈ [1 − ∆τ J12 ]−1 [1 + ∆τ J12 ] + O(∆τ 3 ). 2 2

(18)

This scheme can be also re-written as 1 Q(x1 , x2 , τ + ∆τ ) − Q(x1 , x2 , τ ) = ∆τ J12 [Q(x1 , x2 , τ + ∆τ ) + Q(x1 , x2 , τ )] , 2

(19)

and this equation could be solved using the Picard fixed-point iterations. In doing so observe that the entire product J12 Q(x1 , x2 , τ ) with J12 given in Eq.(17) can be calculated as follows. First term. Observe that vector z(x1 , x2 , τ ) = pθ1 (θ1 − b1 Ox1 − b2 Ox2 )−1 Q(x1 , x2 , τ ) solves the equation (θ1 − b1 Ox1 − b2 Ox2 )z(x1 , x2 , τ ) = pθ1 Q(x1 , x2 , τ ). (20) This is a two-dimensional linear PDE of the first order. It could be solved numerically with the second order approximation in x1 , x2 using the Peaceman-Rachford ADI method, see McDonough (2008)       1 1 ∗ s − θ1 + b2 Ox2 z k (x1 , x2 , τ ) + b (21) s + θ1 − b1 Ox1 z (x1 , x2 , τ ) = 2 2       1 1 k+1 s + θ1 − b2 Ox2 z (x1 , x2 , τ ) = s − θ1 + b1 Ox1 z ∗ (x1 , x2 , τ ) + b 2 2 b ≡ pθ1 Q(x1 , x2 , τ ). 13

Here s > 0 is some parameter that could be chosen in a special way to provide convergence of the method, see Appendices. The number k denotes the iteration number. So the whole process starts with k = 1. Before constructing a finite difference scheme to solve this equation we need to introduce some definitions. Define a one-sided forward discretization of O, which we denote as AF : AF C(x) = [C(x + h, t) − C(x, t)]/h. Also define a one-sided backward discretization of O, denoted as AB : AB C(x) = [C(x, t) − C(x − h, t)]/h. These discretizations provide first order approximation in h, e.g., OC(x) = AF C(x) + O(h). To provide the second order F ˙B approximations, use the following definitions. Define AC - the central difference 2 = A A 2 C F approximation of the second derivative O , and A = (A + AB )/2 - the central difference approximation of the first derivative O. Also define a one-sided second order approximations B to the first derivatives: backward approximation AB 2 : A2 C(x) = [3C(x)−4C(x−h)+C(x− F F 2h)]/(2h), and forward approximation A2 : A2 C(x) = [−3C(x)+4C(x+h)−C(x−2h)]/(2h). Also I denotes a unit matrix. All these definitions assume that we work on a uniform grid, however this could be easily generalized for the non-uniform grid as well, see, e.g., In’t Hout and Foulon (2010). The following Proposition now solves the problem Eq.(21) Proposition 6.1 Consider the following discrete approximation of the ADI scheme Eq.(21): i h i 1  1  ∗ s + θ1 Ix1 − b1 A(x1 ) z (x1 , x2 , τ ) = s − θ1 Ix2 + b2 A(x2 ) z k (x1 , x2 , τ ) + b (22) 2 2 i h i h 1  1  k+1 s + θ1 Ix2 − b2 A(x2 ) z (x1 , x2 , τ ) = s − θ1 Ix1 + b1 A(x1 ) z ∗ (x1 , x2 , τ ) + b 2 2 ( F A2 (xi ), bi > 0 b ≡ pθ1 Q(x1 , x2 , τ ), A(xi ) = AB bi < 0, i = 1, 2 2 (xi ),

h

Then this scheme is unconditionally stable, approximates the original PDE Eq.(21) with O((∆x1 )2 + (∆x2 )2 + (∆x1 )(∆x2 )) and preserves positivity of the solution. Proof See Appendix A. We can start iterations in Eq.(22) by choosing z (1) (x1 , x2 , τ ) = Q(x1 , x2 , τ ). In our experiments the scheme converges to the solution after 5-6 iterations. Second term. Observe that vector z(x1 , x2 , τ ) = (1−p)θ2 (θ2 +b1 Ox1 +b2 Ox2 )−1 Q(x1 , x2 , τ ) solves the equation (θ2 + b1 Ox1 + b2 Ox2 )z(x1 , x2 , τ ) = (1 − p)θ2 Q(x1 , x2 , τ ).

14

(23)

This is also a two-dimensional linear PDE of the first order, so again we apply the PeacemanRachford method       1 1 ∗ s + θ2 + b1 Ox1 z (x1 , x2 , τ ) = s − θ2 − b2 Ox2 (1 − p)z k (x1 , x2 , τ ) + b (24) 2 2       1 1 k+1 s − θ2 − b1 Ox1 z ∗ (x1 , x2 , τ ) + b s + θ2 + b2 Ox2 z (x1 , x2 , τ ) = 2 2 b ≡ (1 − p)θ2 Q(x1 , x2 , τ ) Next Proposition provides a construction of the finite difference scheme to solve the problem Eq.(24) Proposition 6.2 Consider the following discrete approximation of the ADI scheme Eq.(24): i h i 1  1  ∗ s + θ2 Ix1 + b1 A(x1 ) z (x1 , x2 , τ ) = s − θ2 Ix2 − b2 A(x2 ) z k (x1 , x2 , τ ) + b (25) 2 2 h i h i 1  1  k+1 s + θ2 Ix2 + b2 A(x2 ) z (x1 , x2 , τ ) = s − θ2 Ix1 − b1 A(x1 ) z ∗ (x1 , x2 , τ ) + b 2 2 ( B A2 (xi ), bi > 0 b ≡ (1 − p)θ2 Q(x1 , x2 , τ ), A(xi ) = AF2 (xi ), bi < 0, i = 1, 2 h

Then this scheme is unconditionally stable, approximates the original PDE Eq.(24) with O((∆x1 )2 + (∆x2 )2 + (∆x1 )(∆x2 )) and preserves positivity of the solution. Proof See Appendix B. Overall, our experiments show that the first Picard scheme Eq.(19) converges after 2-3 iterations to the absolute accuracy 10−3 . To summarize, the total complexity of the proposed splitting algorithm at every time step is O(αN1 N2 ), where α is some constant coefficient. To estimate it, observe that the solution of the convection-diffusion equation requires five sweeps, where at every sweep either N1 systems of linear equations with the tridiagonal matrix of size N2 , or N2 systems of size N1 have to be solved, see In’t Hout and Foulon (2010). The idiosyncratic jump parts modeled by the Merton jump model are solved with the complexity O(N1 N2 ) (i.e., at this step α = 1) using the Improved Fast Gauss Transform (IFGT), see Itkin (2014a). As we need to provide two steps of splitting in the x1 dimension, and two other steps in the x2 , the total number of sweeps is four. Finally the above algorithm for computing common jumps using the Kou model requires 2-3 Picard iterations for the matrix exponential, and at every iteration we solve 2 ADI systems of linear equations using 5-6 iterations, so totally about 30 sweeps. Thus, overall α is about 44. This is still better then a straightforward application of the FFT which usually requires the number of FFT nodes to be a power of 2 with a typical value 211 . It is also better than the traditional approach which considers approximation of the linear non-local jump integral J on some grid and then makes use of the FFT to compute a matrix by vector product. Indeed, when using FFT for this purpose we need two sweeps 15

per dimension using a slightly extended grid (with, say, the tension coefficient ξ) to avoid wrap-around effects, d’Halluin et al. (2005). Therefore the total complexity per time step could be at least O(4ξ1 ξ2 N1 N2 log2 (ξ1 N1 ) log2 (ξ2 N2 )) which even for a relatively small FFT grid with N1 = N2 = 512, and ξ1 = ξ2 = 1.1 is about 9 times slower than our method. Also traditional approach experiences some other problems for jumps with infinite activity and infinite variation, see survey in Itkin (2014a) and references therein. If instead of the Kou model one wants to apply the Merton jump model for systemic jumps, it becomes a bit more computationally expansive. Indeed, at every time step the multidimensional diffusion equation with constant coefficients could be effectively solved by using the IFGT. Suppose in doing so we want to achieve the accuracy 10−3 . Then, roughly, we need to keep p = 9 terms in the Taylor series expansion of IFGT, and the total complexity for the two-dimensional case d = 2 is O(90N1 N2 ), see Yang et al. (2003).

7 7.1

Numerical experiments One-dimensional problem

We start with the one-dimensional model for two reasons. First, the solution of this model is used as the boundary condition for the two-dimensional problem. Second, in some cases, e.g., for the exponential jumps, this model could be solved in closed form, and, therefore, can be utilized for verification of the method. In the first test we consider a one-dimensional pure diffusion problem. We solve it as a limiting case of the two-dimensional problem when the volatility and drift of the second asset vanish. This solution for the survival probability was compared with the analytical solution which in this case coincides with the price of a digital down-and-out call option, see Howison (1995). Thus, in this test the robustness of our convection-diffusion FD scheme is validated. Parameters of the model used in this test are given in Table 1, and the results are presented in Fig. 2 where a relative difference (the absolute value) between the analytical price and that computed by our finite-difference method is depicted as a function of a0,1 . As it could be seen it is better than 1% everywhere except at the point very close to the barrier. That is because the value of Q is small there, so the relative difference is not a good measure. a0,2 l0,1 100 40

l0,2 0

l1,2 0

l2,1 0

R1,2 0

R2,1 0

rf q T 0.05 0 1

σ1 σ2 0.2 0.

Table 1: Parameters of the structural 1D default model.

In the second test we extend the previous case by adding the exponential jumps to the first component. Again, this problem admits an analytical solution which could be expressed via the inverse Laplace transform, see Lipton (2002) where a similar problem was considered in the presence of the upper barrier. As we didn’t find a reference which could explicitly provide the value of Q in this case, in Appendix C we give a short derivation 16

Figure 2: Relative difference (the absolute value) between the analytical price and the FD solution for the convection-diffusion problem % as a function of the initial asset value a0,1 .

of the corresponding expression. Also as far as our numerical approach is concerned, the exponential jumps were never considered within this framework. Therefore, in Appendix D in a spirit of Itkin (2014c) we construct a finite-difference algorithm for the exponential jumps. In Fig. 3 a relative difference (the absolute value) between the analytical price and that computed by this method is depicted as a function of a0,1 . In this experiment we set the intensity of the jumps λ = 0.7, and the parameter of the exponential distribution φ = 2. The difference is less than 1% except close to the barrier. That is because the values of Q are small in this region. To see this we collect the results obtained numerically (FD) and analytically in Table 2.

7.2

Two-dimensional problem

In the first test we solve the problem Eq.(3) with parameters of the model given in Table 3. We also chose jump parameters of the idiosyncratic jumps (the Merton model with parameters µM , σM ) and for systemic jumps (the Kou model with parameters p, θ1 , θ2 ) as shown in Table 4. We use upper script (i) to mark the ith component of the model. We computed all test using a 100 × 100 spatial grid for the convection-diffusion problem. Also we use a constant step in time ∆τ = 0.01, so the total number of time steps for the given maturity is also 100. The non-uniform grid for jumps in each direction is a superset of the convection-diffusion grid up to ai = 105 . It is built using a geometric progression and contains 80 nodes. In Fig. 4 the joint survival probability Q(x1 , x2 , t, T ) computed in such the experiment 17

Figure 3: Relative difference (the absolute value) between the analytical price and the FD solution for the jump-diffusion problem % as a function of the initial asset value a0,1 .

is presented at t = 0.

Figure 4: Joint survival probability Q(x1 , x2 , t, T ) at t = 0.

To better see the behavior of the graph close to the initial values of a1 , a2 we zoom-in this picture in the vicinity of these values, and thus obtained results are shown in Fig. 5. 18

a0,1 40.85 41.69 42.53 43.36 44.18 44.99 45.79 46.59 47.38 48.16 48.94 49.70 50.46 51.22 51.96 52.70 53.43 54.16 54.88 55.60

Qanal 0.008805 0.017710 0.026710 0.035802 0.044982 0.054251 0.063610 0.073058 0.082601 0.092241 0.101984 0.111833 0.121795 0.131876 0.142080 0.152413 0.162881 0.173486 0.184233 0.195123

QF D 0.008909 0.017874 0.026972 0.036188 0.045523 0.054981 0.064563 0.074273 0.084116 0.094094 0.104212 0.114472 0.124878 0.135431 0.146134 0.156985 0.167985 0.179132 0.190423 0.201854

∆Q -0.000103 -0.000164 -0.000262 -0.000387 -0.000541 -0.000729 -0.000953 -0.001215 -0.001515 -0.001853 -0.002228 -0.002639 -0.003083 -0.003556 -0.004054 -0.004571 -0.005104 -0.005646 -0.006190 -0.006732

Table 2: Results of the 1D jump-diffusion test: Qanal - the survival probability computed analytically, QF D - same obtained by using our FD scheme, ∆Q = Qanal − QF D . a0,1 a0,2 110 100

l0,1 40

l0,2 50

l1,2 10

l2,1 15

R1,2 0.4

R2,1 rf q T σ1 σ2 ρ 0.35 0.05 0 1 0.2 0.3 0.5

Table 3: Parameters of the structural default model. (1)

λ µM 3 0.5

(2)

µM 0.3

(1)

σM 0.3

(2)

σM 0.4

p θ1 θ2 b1 b2 0.3445 3.0465 3.0775 0.2 0.3

Table 4: Parameters of the jump models.

To compare this result with the situation when two banks don’t have mutual liabilities, we first reduce l1 and l2 by the previous amounts l1,2 and l2,1 , and then put l1,2 = l2,1 = 0. The difference in two solutions (the first one presented in Fig. 4 and the second one is shown in Fig. 6. As expected the maximal difference occurs at the default boundaries and at large x1 or x2 where the difference could reach even 1. To see how pronounced is this effect at low x1 19

Figure 5: Joint survival probability Q(x1 , x2 , t, T ) at t = 0, a zoomed-in picture.

Figure 6: Difference in the joint survival probability ∆Q = Q1 − Q2 , Q1 (x1 , x2 , 0, T ) - two banks have mutual liabilities, Q2 (x1 , x2 , 0, T ) - no mutual liabilities.

and x2 we redraw the above plot using a finer scale, and the results are shown in Fig. 7. In this region the effect is significant, e.g. the difference could achieve 0.4. Obviously, the magnitude depends on the values of the jump parameters used in the test as well as on the parameters of the deterministic model of the default boundaries. Also, the effect should be 20

Figure 7: Difference in the joint survival probability ∆Q = Q1 − Q2 , Q1 (x1 , x2 , 0, T ) - two banks have mutual liabilities, Q2 (x1 , x2 , 0, T ) - no mutual liabilities. A larger scale.

Figure 8: Difference in the joint survival probability ∆Q = Q1 − Q2 , Q1 (x1 , x2 , 0, T ) - two banks have mutual liabilities, Q2 (x1 , x2 , 0, T ) - no mutual liabilities. No jumps, only the diffusion components (the picture is rotated by 180◦ ).

21

more pronounced when mutual liabilities of two banks increase, so the ratio of the mutual liabilities to the other liabilities increases. To underline the influence of jumps the same test was computed with no jumps, i.e. only the diffusion component was taken into account. The results are given in Fig. 8. Clearly, presence of jumps significantly changes the picture, while still the effect of the mutual liabilities exists even in this case. In the second set of tests we setup a local volatility function for components 1 and 2, that is given in Tables 5, 6 t, yrs 0.1 0.2 0.4 0.6 0.8

70 0.447 0.500 0.548 0.592 0.632

80 0.455 0.507 0.554 0.597 0.638

90 0.459 0.511 0.558 0.601 0.641

100 0.462 0.514 0.560 0.603 0.643

a0,1 110 0.465 0.516 0.562 0.605 0.645

120 0.467 0.518 0.564 0.607 0.646

130 0.468 0.519 0.565 0.608 0.648

140 0.470 0.520 0.566 0.609 0.649

150 0.471 0.522 0.567 0.610 0.650

Table 5: Local volatility function of the component X1,t .

t, yrs 0.1 0.2 0.4 0.6 0.8

a0,1 50 0.548 0.592 0.632 0.671 0.707

60 0.554 0.597 0.638 0.676 0.712

70 0.558 0.601 0.641 0.679 0.715

80 0.560 0.603 0.643 0.681 0.719

90 0.562 0.605 0.645 0.683 0.718

100 0.564 0.607 0.646 0.684 0.720

110 0.565 0.608 0.648 0.685 0.721

120 0.566 0.609 0.649 0.686 0.722

130 0.567 0.610 0.650 0.687 0.722

140 0.568 0.611 0.650 0.688 0.723

Table 6: Local volatility function of the component X2,t .

The results of this test are given in Fig. 9. It is seen that larger volatilities make the effect of mutual liabilities significantly more pronounced, as well as the shape of the plot in Fig. 9 becomes highly asymmetric. We also recomputed the results of the first test now setting T = 10 years to see how the time horizon affects the contribution of the mutual financial obligations to the join survival probability Q(x1 , x2 , 0, T ). This is shown in Fig. 10, and Fig. 11. It is seen that the effects of mutual obligations significantly decreases when T increases. That is because Q(x1 , x2 , 0, T ) itself decreases in absolute value with larger T , and therefore the absolute value of the effect also drops down, despite the relative contribution remains almost the same. The next tests show the influence of correlations on the mutual debts effect. In Fig. 12 same results as in Test 1 are presented when ρ12 = 0, while in Fig. 13 the loading factors b1 , b2 vanish. 22

Figure 9: Difference in the joint survival probability ∆Q = Q1 −Q2 , when the model includes a local volatility function: Q1 (x1 , x2 , 0, T ) - two banks have mutual liabilities, Q2 (x1 , x2 , 0, T ) - no mutual liabilities. A larger scale.

Figure 10: Joint survival probability Q(a1 , a2 , 0, T ) at T = 10 years

Based on these results we can conclude that both contributions of the Brownian and jump correlations are important. 23

Figure 11: Difference in the joint survival probability ∆Q = Q1 − Q2 at T = 10 years, when the model includes a local volatility function: Q1 (x1 , x2 , 0, T ) - two banks have mutual liabilities, Q2 (x1 , x2 , 0, T ) - no mutual liabilities. A larger scale.

Figure 12: Difference in the joint survival probability ∆Q = Q1 − Q2 at ρ12 = 0, when the model includes a local volatility function: Q1 (x1 , x2 , 0, T ) - two banks have mutual liabilities, Q2 (x1 , x2 , 0, T ) - no mutual liabilities.

24

Figure 13: Difference in the joint survival probability ∆Q = Q1 − Q2 at b1 = b2 = 0, when the model includes a local volatility function: Q1 (x1 , x2 , 0, T ) - two banks have mutual liabilities, Q2 (x1 , x2 , 0, T ) - no mutual liabilities.

8

Three-dimensional case

A more realistic situation would be if we consider three firms, ai , i = 1, 2, 3 using the same structural default model as above. Also assume that all three firms have mutual liabilities to each other, as well as liabilities with respect to the other world. An advantage of our approach lies in the fact that just minor changes in the pricing algorithm need to be done to include the third component to the whole picture. Indeed, as now Q = Q(x1 , x2 , x3 , t, T ), we need to replace two-dimensional matrices with three-dimensional matrices. Therefore, the expected complexity of the method becomes O(N1 N2 N3 ). As the idiosyncratic jumps are still independent, our splitting algorithm remains the same, although we need to add two more steps in dimension x3 to Eq.(15). Hence,

25

the 3D splitting algorithm reads Q(1) (x1 , x2 , τ ) = e

∆τ 2

D

Q(2) (x1 , x2 , τ ) = e

∆τ 2

J1

Q(1) (x1 , x2 , τ ),

Q(3) (x1 , x2 , τ ) = e

∆τ 2

J2

Q(2) (x1 , x2 , τ ),

Q(4) (x1 , x2 , τ ) = e

∆τ 2

J3

Q(3) (x1 , x2 , τ ),

Q(x1 , x2 , τ ),

(26)

Q(5) (x1 , x2 , τ ) = e∆τ J12 Q(3) (x1 , x2 , τ ), Q(6) (x1 , x2 , τ ) = e

∆τ 2

J3

Q(3) (x1 , x2 , τ ),

Q(7) (x1 , x2 , τ ) = e

∆τ 2

J2

Q(1) (x1 , x2 , τ ),

Q(8) (x1 , x2 , τ ) = e

∆τ 2

J1

Q(1) (x1 , x2 , τ ),

∆τ 2

D

Q(x1 , x2 , τ + ∆τ ) = e

Q(7) (x1 , x2 , τ ).

Now at step 5 let us still use the Kou model for the systemic jumps. Then we need to solve the corresponding 3D linear equations of the first order similar to Eq.(20) and Eq.(23). This could be done using a 3D version of the ADI scheme which could be derived in the similar to the 2D case way, see McDonough (2008). For the sake of brevity we formulate two propositions with no proof (which could be obtained in the exactly same manner as in Appendices) Proposition 8.1 Consider the following PIDE (θ1 − b1 Ox1 − b2 Ox2 − b3 Ox3 )z(x1 , x2 , τ ) = pθ1 Q(x1 , x2 , τ ).

(27)

and solve it using the following ADI scheme i h i h 1  1  ∗ s + θ1 − b1 Ox1 z (x, τ ) = s − θ1 + b2 Ox2 + b3 Ox3 z k (x, τ ) + b 2 2 h i h i 1  1  ∗∗ s + θ1 − b2 Ox2 z (x, τ ) = s − θ1 + b1 Ox1 + b3 Ox3 z ∗ (x, τ ) + b 2 2 i h h i 1  1  k+1 s + θ1 − b3 Ox3 z (x, τ ) = s − θ1 + b1 Ox1 + b2 Ox2 z ∗∗ (x, τ ) + b 2 2 b ≡ pθ1 Q(x1 , x2 , τ ) Then the discrete approximation of this ADI scheme h i h i 1  1  s + θ1 Ix1 − b1 A(x1 ) z ∗ (x, τ ) = s − θ1 Ix2 + b2 A(x2 ) + b3 A(x3 ) z k (x, τ ) + b 2 2 i h i h 1  1  s + θ1 Ix2 − b2 A(x2 ) z ∗∗ (x, τ ) = s − θ1 Ix1 + b1 A(x1 ) + b3 A(x3 ) z ∗ (x, τ ) + b 2 2 h i h i 1  1  s + θ1 Ix3 − b3 A(x3 ) z k+1 (x, τ ) = s − θ1 Ix1 + b1 A(x1 ) + b2 A(x2 ) z ∗∗ (x, τ ) + b 2 2 ( F A2 (xi ), bi > 0 b ≡ pθ1 Q(x1 , x2 , τ ), A(xi ) = AB bi < 0, i = 1, 2 2 (xi ), 26

is unconditionally stable, approximates Eq.(27) with O itivity of the solution.

P

3 i,j=1

∆xi ∆xj



and preserves pos-

The second proposition is similar. Proposition 8.2 Consider the following PIDE (θ2 + b1 Ox1 + b2 Ox2 + b3 Ox3 )z(x1 , x2 , τ ) = (1 − p)θ2 Q(x1 , x2 , τ ).

(28)

and solve it using the following ADI scheme h i h i 1  1  ∗ s + θ2 + b1 Ox1 z (x1 , x2 , τ ) = s − θ2 − b2 Ox2 − b3 Ox3 z k (x1 , x2 , τ ) + b 2 2 h i h i 1  1  ∗∗ s + θ2 + b2 Ox2 z (x1 , x2 , τ ) = s − θ2 − b1 Ox2 − b3 Ox3 z ∗ (x1 , x2 , τ ) + b 2 2 h i h i 1  1  k+1 s + θ2 + b3 Ox3 z (x1 , x2 , τ ) = s − θ2 − b1 Ox1 − b2 Ox2 z ∗∗ (x1 , x2 , τ ) + b 2 2 b ≡ (1 − p)θ2 Q(x1 , x2 , τ ) Then the discrete approximation of this ADI scheme h i h i 1  1  ∗ s + θ2 Ix1 + b1 A(x1 ) z (x1 , x2 , τ ) = s − θ2 Ix2 − b2 A(x2 ) − b3 A(x3 ) z k (x1 , x2 , τ ) + b 2 2 h i h i 1  1  ∗∗ s + θ2 Ix2 + b2 A(x2 ) z (x1 , x2 , τ ) = s − θ2 Ix1 − b1 A(x1 ) − b3 A(x3 ) z ∗ (x1 , x2 , τ ) + b 2 2 h i h i 1  1  k+1 s + θ2 Ix3 + b3 A(x3 ) z (x1 , x2 , τ ) = s − θ2 Ix1 − b1 A(x1 ) − b2 A(x2 ) z ∗∗ (x1 , x2 , τ ) + b 2 2 ( B A2 (xi ), bi > 0 b ≡ (1 − p)θ2 Q(x1 , x2 , τ ), A(xi ) = AF2 (xi ), bi < 0, i = 1, 2 is unconditionally stable, approximates Eq.(28) with O itivity of the solution.

P

3 i,j=1

∆xi ∆xj



and preserves pos-

The solution of the 3D convection-diffusion problem at steps 1 and 9 is a more challenging problem. So far the unconditional stability of some schemes (Craig-Sneid, Modified Craig-Sneid (MCS), Hundsdorfer-Verwer (HV), etc.) was proven only in the case when no drift terms exist in the corresponding diffusion equation, see In’t Hout and Mishra (2013). Therefore, this problem requires further attention. Nevertheless, these schemes were successfully used in the 3D setup by Haentjens and In’t Hout (2012) where the MCS and HV schemes demonstrated good stability when the scheme parameter θ was chosen similar to In’t Hout and Mishra (2013).

27

a0,1 a0,2 a0,2 l0,1 110 100 120 40 l1,2 l2,1 l1,3 l3,1 20 20 15 20

l0,2 50 l2,3 10

l0,3 60 l3,2 15

rf q T 0.05 0 1 R1,2 R2,1 R1,3 0.4 0.35 0.4

ρxz ρyz 0.5 0.3 R3,1 R2,3 0.35 0.4

φxy π/4 R3,2 0.35

Table 7: Parameters of the 3D structural default model. (1)

λ µM 3 0.5

(2)

µM 0.3

(3)

µM 0.4

(1)

σM 0.3

(2)

σM 0.4

(3)

σM 0.5

p θ1 θ2 b1 b2 b3 0.3445 3.0465 3.0775 0.2 0.3 0.25

Table 8: Parameters of the 3D jump models.

8.1

Numerical experiments

In our test we chose parameters of the model similar to the 2D case, see Tables 7, 8 We recall that a correlation matrix Σ of N assets can be represented as a Gram matrix with matrix elements Σij = hxi , xj i where xi , xj are unit vectors on a N − 1 dimensional hyper-sphere SN −1 . Using the 3D geometry, it is easy to establish the following cosine law for correlations between three assets: q ρxy = ρyz ρxz + (1 − ρ2yz )(1 − ρ2xz )cos(φxy ) with φxy being an angle between x and its projection on the plane spanned by y, z. As discussed, e.g., by Dash (2004) three variables ρyz , ρzz , φxy are independent, but ρxy , ρxz , ρyz are not. Based on the values given in Tab. 7 we find ρxy = 0.7342. We computed the test using a 50 × 50 × 50 spatial grid for the convection-diffusion problem. Also we use a constant step in time ∆τ = 0.04, so the total number of time steps for the given maturity is 25. The jump non-uniform grid in each direction is a superset of the convection-diffusion grid up to ai = 104 build using a geometric progression. So the jumps were computed on the grid 62 × 64 × 63 nodes. We again compare the result of this test, Q1 (a1 , a2 , a3 ), with the situation when three banks don’t have mutual liabilities, Q2 (a1 , a2 , a3 ). To obtain that we first reduce l1 , l2 , l3 by the amounts li,j , i ∈ [1, 3], j ∈ [1, 3], lii = 0, and then put li,j = 0. The difference in two solutions ∆Q = Q1 (a1 , a2 , a3 ) − Q2 (a1 , a2 , a3 ) is presented in Fig. 14,15,16. Since the whole picture in this case is 4D, we represent it as a series of 3D projections, namely: Fig. 14 represents the X − Y plane at various values of the Z coordinate which are indicated in the corresponding labels; Fig. 15 does same in the X − Z plane, and Fig. 16 - in the YZ plane. Analyzing these results two observations could be made. First, in the case of three banks with the mutual liabilities, the effect of the latter on the joint survival probability is more profound. Second, this effect has an irregular shape as a function of 3 coordinates. For instance, in YZ plane it has two local maximums (in absolute value) while the 2D case doesn’t demonstrate such a behavior. Also this effect disappears if the model doesn’t take jumps 28

into account. This is similar to the effect observed in Itkin and Carr (2011) when asymmetric positive and negative jumps each modeled by the CGMY model, but with different α produce a qualitatively new effect. It consists in the appearance of a big dome close to ATM at the moderate values of instantaneous variance v in addition to a standard arc of the barrier options which is also close to ATM, but at the small values of v. In general, the whole picture is rather complicated. Moreover, as it is affect by the number of the model parameters, which could be difficult to replicate from the liquid market data, it could be a problem to calibrate such a model. A standard recipe is to first c calibrate marginals of the distribution to the corresponding market, and then use some other data for calibration of the remaining parameters.

9

Conclusion

In this paper we presented three main innovations that seems to be rather general, namely: 1. Consideration of the mutual obligations of the firms is introduced into the structural default model. We briefly consider how significant could be this effect when computing the join survival probability of the firms. Our numerical results justify that this can be important. A more detailed consideration of this phenomena will be given elsewhere. 2. We develop a scheme which introduces correlated jumps into the model. For every asset the jumps are represented as a weighted sum of the common and idiosyncratic parts. Both parts could be simulated by an arbitrary L´evy model which is an extension of the previous approaches where only the discrete or exponential jumps were considered. We provide an efficient numerical (splitting) algorithm to solve the corresponding 2D and 3D jump-diffusion equations, and prove it convergence and second order of accuracy in both space and time. 3. The join survival probability of three firms Q(x1 , x2 , x3 , 0, T ) has been computed using the above jump-diffusion framework. To the best of our knowledge there were no any similar results in the literature. We found that in the presence of jumps the difference ∆Q = Q1 − Q2 ( Q1 (x1 , x2 , x3 , 0, T ) - three banks have mutual liabilities, Q2 (x1 , x2 , x3 , 0, T ) - no mutual liabilities) demonstrates a bimodal profile in some projections, and this effect disappears when jumps are omitted. This is similar in the behavior to what was observed in Itkin and Carr (2011) where interaction of jumps also produced a bimodal distribution of the double barrier option prices. Despite the present approach seems to be efficient and attractive it is not clear, however, how to extend it to the case when the number of firms exceeds 3 unless some simplifications are introduced into the model. This is a standard limitation of the finite-difference approach which experience curse of dimensionality. A possible way to overcome this could be to combine the analytical and numerical methods, similar to how this was done, e.g., in Lipton and Savescu (2014).

29

30

Figure 14: Difference in the joint survival probability ∆Q = Q1 − Q2 , Q1 (x1 , x2 , x3 , 0, T ) - three banks have mutual liabilities, Q2 (x1 , x2 , x3 , 0, T ) - no mutual liabilities. X − Y plane.

31

Figure 15: Difference in the joint survival probability ∆Q = Q1 − Q2 , Q1 (x1 , x2 , x3 , 0, T ) - three banks have mutual liabilities, Q2 (x1 , x2 , x3 , 0, T ) - no mutual liabilities. X − Z plane.

32

Figure 16: Difference in the joint survival probability ∆Q = Q1 − Q2 , Q1 (x1 , x2 , x3 , 0, T ) - three banks have mutual liabilities, Q2 (x1 , x2 , x3 , 0, T ) - no mutual liabilities. Y − Z plane.

Acknowledgments We thank Peter Carr, Peter Forsyth, Igor Halperin for useful comments. We assume full responsibility for any remaining errors.

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36

Appendices A

Proof of Proposition 6.1

Introduce a definition of an EM-matrix, see Elhashash and Szyld (2008). Definition An N × N matrix A = [aij ] is called an EM-Matrix if it can be represented as A = sI − B with 0 < ρ(B) < s, s > 0 is some constant, ρ(B) is the spectral radius of B, and B is an eventually nonnegative matrix. Now suppose b1 < 0, b2 < 0. Then matrix h i 1  M1 = s + θ1 Ix1 − b1 AB (x ) 1 2 2 in the first row of Eq.(22) is an EM-matrix, see Lemma A.2 in Itkin (2014c). Therefore, the inverse of M1 is a non-negative matrix, see Lemma A.3 in Itkin (2014c). The matrix i h 1  (x ) M2 = s − θ1 Ix2 + b2 AB 2 2 2 is a eventually non-negative matrix3 if s is chosen to provide 1 3 s > θ1 − b2 2 h

(29)

  Therefore, the solution of the first row of Eq.(22) is z ∗ (x1 , x2 , τ ) = M1−1 M2 z k (x1 , x2 , τ ) + b which by construction is a non-negative vector. Also eigenvalues of M1−1 M2 are 1 s − θ1 + 3b2 /h 2 λi = < 1, 1 s + θ1 − 3b1 /h 2

i ∈ [1, N1 ]

Therefore, this scheme converges unconditionally provided Eq.(29) is satisfied. Also, by construction matrix AB 2 (x) approximates the operator Ox with the second order, i.e., with O(h2 ). Therefore, the whole scheme provides the second order approximation. The second row of Eq.(22) could be analyzed in the same way. In all other cases b1 < 0, b2 > 0, b1 > 0, b2 < 0 and b1 > 0, b2 > 0 the proof could be done by analogy.

B

Proof of Proposition 6.2

The proof is completely analogous to that in the Appendix A. 3

By definition of AB 2 matrix M2 is a lower triangular matrix with three non-zero diagonals. The main and the first lower diagonals are positive and the second lower diagonal is negative. However, first two diagonals dominate the second one.

37

C

Survival probabilities in a 1D jump-diffusion model

Similar to Lipton and Sepp (2009) we consider a one-dimensional structural model inspired by the familiar model of Merton (1974). Accordingly, we consider a firm with the firm asset value at . Also assume that the default lower barrier of the firm Ba is constant. We define firm’s default time τd assuming continuous monitoring τd = min [at < Ba ]. 0T ] where τ is the time of default. It is easy to see that Q is equal to the price of a Down-and-Out undiscounted digital call C(x, l, T ). To compute this price define Z ∞ e−izx g(x)dx gˆ(z) = −∞

Similar to Lewis (2000) represent the call option value as  Z iη+∞ e−rT izXT −rT e gˆ(z)dz EQ C = e EQ [g(XT )] = 2π iη−∞   Z iη+∞ Z iη+∞ e−rT e−rT izxT iz(XT −xT ) izxT −T ψYT (z) = e EQ e gˆ(z)dz = e e gˆ(z)dz 2π 2π iη−∞ iη−∞ where gˆ(z) is the Fourier transform of the payoff, ψXt (z) is the characteristic exponent of the process Xt , xT = (r − q − 21 σ 2 − κλ)T , the process YT = XT − xT is an exponential martingale, so EQ [eiuYT ] = 1. In our case the process XT has to be replaced with the process X t = inf 0 1 is the parameter of the exponential distribution. This process is meromorphic, and, therefore,  −  φq (z)/q EQ [e−T ψX T (z) ] = EQ [eizX t ] = L−1 q izX θ (q) where L is the inverse Laplace transform, φ− ] is the Wiener-Hopf factor, and q (z) = EQ [e θ(q) is the exponentially distributed random time with the parameter q > 0, see Kuznetsov et al. (2011). It is also shown there that φ− q (z) reads

iz Y 1 + ρˆ n . φ− q (z) = iz n≥1 1 + ξˆn

(35)

Here ξˆn > 0 are the negative roots with the minus sign of the characteristic equation ¯ q − ψ(z) = 0, ¯ ¯ with ψ(z) be the Laplace exponent of the process, ψ(iz) = ψ(z), and ρˆn > 0 are the negative poles with the minus sign of this equation. The roots and poles satisfy the following interlacing condition ... − ρˆ2 < −ξˆ2 < −ˆ ρ1 < −ξˆ1 < 0 < ξ1 < ρ1 < ξ2 < ρ2 < ... For a chosen model of the exponential jumps we have 1 z ¯ ψ(z) = σ 2 z 2 + µz + λ 2 φ−z

(36)

where µ = r − q − κλ − 21 σ 2 . As φ > 1 this equation doesn’t have negative poles, while by Vieta’s theorem it could have only one negative root. Denote a, b, c, d the coefficients of the characteristic polynomial, i.e. 1 a = − σ2, 2

b = −φa − µ,

c = q + µφ + λ, 39

d = −qφ.

Then the roots could be found via the celebrated Cardano formula B = b/a, C = c/a, D = d/a, (37) 1√ 2 1 1 R = (9BC − 27D − 2B 3 ), ρ = B − 3C, θ = arccos(R/ρ), 54 3 3 B √ B z1 = 2ρ cos(θ) − , z2 = −ρ cos(θ) − − 3 sin(θ), 3 3 B √ z3 = −ρ cos(θ) − + 3 sin(θ). 3 Thus, in Eq.(35) we have just one negative root (so ρˆ1 = −z2 ) and no negative poles. Combining the above expressions together we arrive at the following representation Z iη+∞ Z γ+i∞ qT −iβz e 1 1 dqdz, Im(z) ≥ 0. (38) Q=− 2 iz qz 4π iη−∞ γ−i∞ 1+ ρˆ1 (q) The integrand has two poles at z = 0 and at z = iˆ ρ1 (q). Accordingly using the residue calculus and Cauchy’s theorem we obtain ) (   Z γ+i∞ β ρˆ1 (q)     i 1 dq e Q=− eqT 1 − eβ ρˆ1 (q) = L−1 1 − eβ ρˆ1 (q) = 1 − L−1 . (39) q q 2π γ−i∞ q q q

D

Matrix exponential approach for the exponential jumps

In the 1D case we still want to use the splitting algorithm in Eq.(13). To proceed, let us define an explicit model for jumps, so the pseudo-differential operator J defined in Eq.(8) could be computed in the explicit form. For the exponentially distributed jumps with the L´evy measure ν(dy) given in Eq.(34) and intensity of jumps λ ≥ 0 we can plug ν(dy) into Eq.(8) and integrate. The result reads J =

λ (φ − Ox )−1 (O2x − Ox ), φ−1

Ox ≡ ∂x .

(40)

Since x = log a this could be re-written as J =

λ (φ − aO)−1 a2 O2 , φ−1

O ≡ ∂a .

(41)

Proposition D.1 Consider the following discrete approximation of Eq.(41): J=

λ (φI − aAF2 )−1 a2 AC 2. φ−1

(42)

Then this scheme is a) unconditionally stable; b) approximates the operator J in Eq.(41) on a certain non-uniform grid in variable a with O(max(hi )2 ), where hi , i = 1, ..., N are the steps of the grid; c) and preserves positivity of the solution. 40

Proof For the sake of clarity we give the proof for the uniform grid, as an extension to the non-uniform grid is straightforward. As shown in Itkin (2014c) matrix −AF2 is an EM matrix. Therefore, matrix φI −AF2 is also an EM-matrix. Therefore, it inverse is a non-negative matrix. Matrix AC 2 by construction is the Metzler matrix. A product of the non-negative and Metzler matrices is the negative of an EM-matrix4 . As φ > 1 matrix J is also the negative of an EM-matrix. Then unconditional stability and positivity of the solution follows from the main Theorem in Itkin (2014c). As matrix AF2 is the second order approximation in h to O, and AC 2 is the second order 2 approximation in h to O , the whole scheme approximates the operator J with the second order in h, . In practical applications the complexity of this scheme could be linear in the number of the grid nodes N . Indeed, suppose that somebody wants to compute Q with the second order of approximation in step in time ∆t, i.e. with the accuracy O((∆t)2 ). Represent e∆tλJ in the second step of the splitting algorithm Eq.(13) using a Pad´e rational approximation (1, 1): −1    1 1 ∆tλJ I + λ∆tJ e = I − λ∆tJ 2 2 With allowance for Eq.(42) after some algebra this could be re-written in the form     λ λ F 2 C 2 C F φI − aA2 − ∆ta A2 Q(a, t + ∆t) = φI − aA2 + ∆ta A2 Q(a, t). 2(φ − 1) 2(φ − 1) (43) Matrices in the square brackets are banded (three or five diagonal), therefore this system of linear equations could be solved with the complexity O(N ).

4

Some care should be taken regarding the boundary values of AC 2 to guarantee this. Usually, introduction of ghost points at the boundaries is helpful to increase the accuracy of the method. Alternatively, one could use another approximation of the term (φ − aO)−1 in Eq.(41) which is (φI − aAF )−1 . This reduces slightly the order of approximation from the exact second order to some order in between of 1 and 2, but, however, significantly improves the properties of the resulting matrix J.

41