NUMERICAL COMPUTATION OF FIRST-CROSSING BOUNDARY PROBLEM. 1. Introduction

Transcription

1 NUMERICAL COMPUTATION OF FIRST-CROSSING BOUNDARY PROBLEM LAN CHENG, XINFU CHEN JOHN CHADAM & DAVID SAUNDERS Abstract.. Introduction In this paper, we study the inverse first-crossing problem. This problem originates from the Merton s structural model [7] for credit risk management, derived as follows. Consider a company whose asset value and debt at time t are denoted by A and D respectively. Assume the following: () D() A() and the company is in default at a time t > if A < D. () {A} is a log-normal process before the first time at which the company defaults. In mathematical finance, it is convenient to use the default index X and the barrier function b defined by X := log V V (), Then the assumption () is translated as D b := log V (). (.) dx = µ(x, t)dt + σ(x, t)db t t < τ, where τ = inf{s X(s) b(s)} is the default time, B t the standard Brownion Motion, µ the growth rate, and σ the volatility. Of importance are the following two problems: () The first-crossing problem: Given a barrier function b, find the survival probability p that a company does not default before t. (.) p := Prob{τ > t}. () The inverse first-crossing problem: Given a survival probability function p, find a barrier function b, such that (.) holds. In our paper Inverse First-Crossing Boundary Problem, we have formulating the inverse first-crossing problem into a free boundary PDE problem and we proved that

2 LAN CHENG, XINFU CHEN JOHN CHADAM & DAVID SAUNDERS there exists a unique viscosity weak solution to this PDE. Furthermore we estimate the free boundary when t is small and derived the integral equations for the purpose of numerical computation. In this paper we will apply those integral equations to compute the default boundary. To compute the boundary, Avellaneda and Zhu [4] applied the finite difference scheme to the PDE, Zucca, Sacerdote and Peskir [] applied the secant to a integral equation and I.Iscoe and A.Kreinin [6] use the Monte-Carlo approach. We will introduce their work in this paper compare all the schemes.. A Numerical Scheme For the simplicity when calculating the free boundary, we set µ = and σ =... Our Numerical Scheme. We use the integral equations (.) (.) Γ(b, t) = q = Γ x (b, t) Γ(b b(τ), t τ) dq(τ), Γ x (b b(τ), t τ) dq(τ), derived in 5 of Inverse First-Crossing Boundary Problem to compute the free boundary. Suppose b(τ) is known for all τ [, t). Set Q(x, t) = We want to solve b = b from the equation Note from (.) that Γ(x b(τ), t τ) dq(τ). Γ(b, t) Q(b, t) =. { } Γ(x, t) Q(x, t) = q x x=b. Thus, if we use Newton s method, a new approximation b new can be obtained from the old approximation b old by (.3) b new = ˆb old + Q(bold, t) Γ(b old, t). q From this, we can implement a numerical scheme as follows. Let {t i } N i= be the mesh points where = t < t < t <. Denote by b i the approximate value of b(t i ). For

3 NUMERICAL COMPUTATION OF FIRST-CROSSING BOUNDARY PROBLEM 3 n, suppose b,, b n have been calculated. We define b n by iteration b () = tn t n b n, b (k+) = b (k) + Q(b(k), t n ) tn e (b(k) ) /(t n), k =,,, K. q (t n ) b n = b (K), where the initial guess b () is obtained by the fact that b t scale. The iteration ends until b (k+) b (k) Q(b (k), t n ) = tn e (b(k) ) /(t n) q (t n ) changes slowly in the log t < ɛ and ε is tolerance, say 5. Since Q is an integral, a quadrature rule is needed for its numerical estimation. To take care of the singularity, we write it as Q(x, t n ) = n = π n e (x b(τ)) /(t n τ) q (τ)dτ tn τ q (τ)e (x b(τ)) /[(t n τ)] d ( t n τ ), when q is bounded. The left-point rule can be written as: Q(x, t n ) = n ( q (t i ) exp (x b i) ){ tn t i } t n t i+ π (t n t i ) i= Similarly the trapezoid rule is ( Q(x, t n ) = q (t π n ) ( t n t n + q () exp + n i= q (t i ) exp ) tn x t n t t n ) tn t ( (x b i) i ) t n t i+ (t n t i ) Nevertheless, due to the singularity of the integral, higher order quadrature rules (e.g., the Simpson s rule) are not recommended. Indeed, as we shall see, the trapezoid rule is quite satisfied at least for small t. The complexity of the scheme is O(N ) where N is the total number of mesh points. There are some other numerical treatments for the calculation of default boundary b. Avellaneda and Zhu ([4] ) used the finite difference method to (??). Zucca, Sacerdote and Peskir ([9])used the secant method to (??). Iscoe and Kreinin treated the problem as a conditional probability problem. In readers s convenience, we provide a little bit more of their contributions.

4 4 LAN CHENG, XINFU CHEN JOHN CHADAM & DAVID SAUNDERS.. Integral Equation by Peskir. Peskir derived a sequence of integral equations ( ) b t ( ) b b(s) (.4) t n/ H n (t s) n/ H n q(s)ds =, n =,,... t t s where H (x) = ϕ(x) = e x / and H n (x) = H n (z)dz for n. In particular, when n = and n =, they become (.5) (.6) t e b /t b t e z / dz x (t s) e (b b(s)) /(t s) q(s)ds =, n =, ( ) e z / dz q(s)ds =, n =. b b(s) t s One observe that (.5) is what we used in calculating the boundary. Instead of using (.5), Peskir and Zucca used (.6) to calculate the boundary. They discretize (.5) by the scheme: (.7) b(t i )/t i e z / dz = i b(t j= i ) b(t j ) ti t j e z / dz q(t j ) h i =,...n, e z / dz = q(t ) h i =. (.7) yields a non-linear system of n equations with n unknowns b(t ),...b(t n ). The secant method can used to solve it. We point the difference of this scheme with our scheme. In our approximation, we use the Newton iteration since we derived the derivative of (.5). Whereas for Peskir and Zucca s scheme, they used the secant method..3. Avellaneda-Zhu s Scheme. The spatial translated density function f(y, t) = u(y+ b, t) satisfies f t (y, t) = b f y (, y, t) + f(y, t) yy for y >, t >, f(, t) = for y =, t >, (.8) f(y, ) = δ (y b()) for y >, t =, f y(, t) = q fory =, t >. Here δ ( ) is a Dirac Measure concentrated at. It must be regularized consistently with the boundary condition f(, t) = for all t. Indeed, to take care of the singularity, Avellaneda and Zhu used the idea of initial layer. For a chosen small t, they replace the solution to [, t ] by an explicit solution to the first passage problem. A numerical simulation carries on after t. Based on Avellaneda and Zhu s idea of using initial layer and finite difference scheme, we implement their scheme as the following.. Analytic Small Time Approximation.

5 NUMERICAL COMPUTATION OF FIRST-CROSSING BOUNDARY PROBLEM 5 For any α > and β R, (.8) admits an exact solution. { f = t e ( x /t e α(x+α+βt)/t), (.9) b = α βt. Then the corresponding default probability and its derivative are given by: ( ) ( ) α βt α + βt q(t; α, β) = f(y, t)dy = N + e αβ N, t t q(t; α, β) = b α t t e (α+βt) /t. For a given t, the parameter α and β are chosen such that { q(t ) = q(t ; α, β), (.) q(t ) = q(t ; α, β). For example, when q =.t and t =., solution to (.) is α =.467 and β = Numerical Simulation Apply the finite difference scheme to the first equation of (.8) with the initial condition at t = t given by (.9). Define y i = ih ( i=,,...,m), t n = t +n t and let f n i represent the numerical approximation to f(y i, t n ) and b to b (t n+ n+ ). Their initial values are taken to be fi = f(y i, ) and b = β. The boundary conditions are f n = and fm+ n =. A Crank-Nicholson scheme reads as (.) f n+ i fi n t = b n+ f n+ i+ f n+ i h + f i+ n fi n + fi n + f n+ n+ i+ fi + f n+ i. 4h 4h f n+ is the numerical approximation of f(y + y, t + t ) by both Taylor expansion and i+ upwind scheme: f n+ i+ f n+ i+ = f n i+ + (λn t y)(fy ) n i+ + 4 (f yy) n i+ t, ifλ n, = f n i + (λn t + y)(fy ) n i + 4 (f yy) n i t, ifλ n <, where (f y ) n i and (f yy ) n i, are estimated by standard central difference and second order one-sided difference approximations for the boundary points (f y ) n i = f n i+ f n i h (f y ) n = 3f n + 4f n f n h (f y ) n M = 3f M n + 4f M n f 3 M h, (f yy ) n i = f n i+ f n i + f n i h, i =,..., M., (f yy ) n = f n 5f n + 4f n f n 3 h, (f yy ) n M = f n M 5f n M + 4f n M f n M 3 h.

6 6 LAN CHENG, XINFU CHEN JOHN CHADAM & DAVID SAUNDERS Suppose f,..., f n, b n and λ n have been calculated, then so is f n+. (.) gives a i+ system of M linear equations with M unknown, f n+ i ( i M) and b. To n+ determine these M unknowns, another equation is required. Then the last equation in (.8) when t = t n+ : f y(, t n+ ) = q (t n+ ) is used, which can be rewritten by difference approximation as: (.) f n+ f n+ h = q (t n+ ). (.) and (.) gives a system of M linear equations with M unknowns. Solve the system of equations, we get the solution f n+ to the PDE and then update the boundary by b n+ = b n + b n+ t b = α βt..3.. An Alternative Explicit Scheme. In light of Avellanda and Zhu s scheme, where they impose an explicit finite difference scheme, Crank-Nicholson scheme, we discretize the PDE in (.8)as (.3) f n+ i fi n t = b f n+ i f n+ i n+ + f i+ n fi n + fi n. h h We corporate (.)with (.3) as i =. Namely, solving f n+ from (.3) and substituting the result into (.) gives an equation involving b n+. where solution is (.4) b n+ = hq (t n+ ) f n + ( t/h )(f n f n + f n ). tq (t n+ ) Using (.4) we can estimate the solution and update the boundary by f n+ j = f j n t h λn+ f n+ j + ( t/h )(fj n fj n + fj+) n, t h λn+ b n+ = b n + λ n+ t. The CFL stability condition requires that to be stable, we take t = h..4. Conditional default probability. Iscoe and Kreinin use a different approach to calculate the boundary. Instead of using the partial differential equations or integral equations to estimate the boundary, they used the theory of probability. They reduced the problem of estimating the default boundary to a sequential estimation of the quantities of the conditional default distributions. Instead of considering the continuous process,

7 NUMERICAL COMPUTATION OF FIRST-CROSSING BOUNDARY PROBLEM 7 they consider a discrete-time, mean zero process, S n, n =,,,..., S =, having a finite variance σ n = t n at time t n. It can be normalized by taking the value η n at time t n by η n = S n σ n, n =,,... ; η =, which satisfies the relation Eη n =, Eη n = for n. Then the default time τ can be formulated as τ = min n {n : η n < b n /σ n }. Denote Q(n) := P rob{τ t n }, π n = P rob{τ = t n } and ˆQ n = P rob{τ = t n τ t n }. Iscoe and Kreinin proved that the boundary {b k } N k=, the probability π n and ˆQ n satisfy the following equations when n =,,...N π n = P rob { n } {η k b k }, η n < b n, σ n k= π n = Q(n) Q(n ), ˆQ n = π n Q(n ). Based on this result, they estimated the boundary b n as follow.. Estimation of b. Based on Q() = P rob{η b }, one can calculate that b = F (Q()), where F denoting the the cdf of the random variable η.. Compute the conditional probabilities ˆQ n based on (??) and (??). 3. Suppose that the default boundary b k has already been computed for k =,..., n. To compute b n, generate a large number, M >>, of i.i.d sample paths η(m) = (η (m), η (m),..., η n (m)), (m =,,..., M), and retain only those vectors η(m) that satisfy the inequality (.5) η k (m) b k σ k, k =,,..., n. 4. Let F n (x) denote the conditional empirical cdf of the random variable η n under the condition (.5), b n is then the quantile of the distribution F n corresponding to the probability. 3. Numerical Simulation 3.. Linear boundary. If default probability function and its density function are given by (.) and (.) then the boundary is a straight line b = α βt, α >, t >. Here is the picture of the linear boundary for all four schemes with T = and t =.5. To have a better view, we plot four schemes separately. σ n

8 8 LAN CHENG, XINFU CHEN JOHN CHADAM & DAVID SAUNDERS.8 T=,N=4,chen* zucca + alex o,zhu aquare Figure. Linear Boundary (Our scheme, Peskir and Zucca +, Alex o, Zhu, original line ) Integral equation with newton iteration Figure. Linear Boundary with N=4 (Our scheme o, original line x) 3.. Our scheme vs Zhu and Avellanda s. Use Zhu and Avellanda s finite difference method, we can get not only the free boundary, but also the solution of the PDE. Meanwhile we get more information from this scheme than all the other three schemes. However they used the idea of initial layer so that the boundary estimated at least starting from t. Here is an example for q =., with T =, t =. (Table (), () and (3)).

9 NUMERICAL COMPUTATION OF FIRST-CROSSING BOUNDARY PROBLEM 9.6 Integral equation by Peskir and Zucca Figure 3. Linear Boundary with N=4 (Peskir and Zucca o, original line x).8 PDE Figure 4. Linear Boundary with N=4 t =.(Zhu o, original line x) 3.3. Our scheme vs Peskir and Zucca s. Use the scheme by Peskir and Zucca, the computation result is very closed to ours. In fact the integral equations used by Peskir, Zucca an us are both from the sequence of equations refpeskir3). The difference is that for the first point from initial point, we use the estimation from section 4, however Peskir and Zucca used (??). Both schemes need to solve the nonlinear equations. We used the newton iteration and they used the secant method. Here we list the results by both

10 LAN CHENG, XINFU CHEN JOHN CHADAM & DAVID SAUNDERS.6 conditional default probability Figure 5. Linear Boundary with N=4 m= (alex o, original line x) s N/4 s N/4 s N/ N/A N/A N/A E E Table. Default Boundary at T = with q =.t (Our scheme) schemes with the different default probability functions with both left-point rule and the trapezoid rule (inside the parenthesis).. q(x) = t and T =. (table (4) and (5)). q = t (table (6) and (7))

11 NUMERICAL COMPUTATION OF FIRST-CROSSING BOUNDARY PROBLEM s N/4 s N/4 s N/ N/A N/A N/A Table. Default Boundary at T = with q =.t t =.(explicit) s N/4 s N/4 s N/ N/A N/A N/A Table 3. Default Boundary at T = with q =.t t =.(Zhu) We make some adjustment for Q(x, t) and s (k+) when we do the calculation use our scheme. Indeed Q(x, t) = = = e (x s(τ)) (t τ) dq(τ) (t τ) e (x s(τ)) (t τ) (t τ) e (x s(τ)) (t τ) 8π ( d τ dτ arcsin( τ t ) t ),

12 LAN CHENG, XINFU CHEN JOHN CHADAM & DAVID SAUNDERS s N/4 s N/4 s N/ N/A N/A.7 ( ) N/A N/A (.88) N/A.38 ( ) (.66) N/A (.3) ( ) (.4897) ( 3.9) (.4) ( ) (.3439) (3.5) (.766) ( ) (.4575) (.94) (.375) ( ) (.543) (3.) (4.8) ( ) (.57) (.97) (6.5) ( ) (.8) (.54) (34.56) ( ) (.465) (4.98) (93.78) Table 4. Default Boundary at T =. with q = t s N/4 s N/4 s N/6.868 N/A N/A N/A Table 5. Default Boundary at T =. with q = t (by Peskir & Zucca s scheme) and s (k+) = s (k) + t n Q(s (k) ) π e (s(k) ) tn. When we use Peskir and Zucca s Scheme, we make some adjustment too. The reason is that q = t as t. Note that (.4) can be written as: t n/ H n ( b t ) ( ) b b(s) (t s) n/ H n dq(s) =, t s

13 NUMERICAL COMPUTATION OF FIRST-CROSSING BOUNDARY PROBLEM 3 s N/4 s N/4 s N/ N/A N/A (-.85) N/A N/A (.5) N/A (-.9954) (.5678) N/A () ( ) (.65989) ( 3.7) () (-.9963) (.7337) (.3) (.63) ( ) (.33488) (.7) (.94) ( ) (.683) (.6) (.38) ( -.995) (.8847) (.96) (.687) (-.998) ( ) (.45) (.75) ( -.993) (.68437) (.) (.93) ( ) (.8355) (.) (46.735) Table 6. Default Boundary at T =. with q = t (.7) becomes b(t i )/t i e z / dz = i j= b(t i ) b(t j ) ti t j e z / dz (q(t j ) q(t j )), and (??) becomes b(t )/t e z / dz = q(t ). 3. q = e t (table (8) and (9)) 4. q = e /t (table () and ()) We make some adjustment for Q(x, t) and s (k+) when we do the calculation. Since q = e /t, t Q(x, t) = = s(τ)) τ exp( (x (t τ) (t τ) τ )dτ (x s(τ)) exp( ( t τ 8π (t τ) τ )d tτ + ) t t ln t τ t, t τ + t

14 4 LAN CHENG, XINFU CHEN JOHN CHADAM & DAVID SAUNDERS s N/4 s N/4 s N/ N/A N/A E-5 N/A E Table 7. Default Boundary at T =. with q = t (by Peskir & Zucca s scheme) s N/4 s N/4 s N/ N/A N/A.4 ( ) N/A N/A (.9) N/A.344 ( ) (.58) N/A (.35) ( ) (4.4763E-5) (3.9) (.88) ( ) (.3773E-5) (3.5) (.3) ( ) (4.5935E-6) (.94) (7.7) E ( ) (.53E-6) (3.) (.99) E ( ) (5.9558E-7) (.5) (77.4) E ( ) (.68E-7) (5.6) (36.9) E ( ) (4.839E-8) (.6) (956.8) Table 8. Default Boundary at T =. with q = e t s N/4 s N/4 s N/6.87 N/A N/A N/A Table 9. Default Boundary at T =. with q = e t (by Peskir & Zucca s scheme)

15 NUMERICAL COMPUTATION OF FIRST-CROSSING BOUNDARY PROBLEM 5 s N/4 s N/4 s N/ N/A N/A N/A Table. Default Boundary at T =. with q = e /t s N/4 s N/4 s N/6.76 N/A N/A N/A Table. Default Boundary at T = with q = e /t (by Peskir & Zucca s scheme) s N/4 s N/4 s N/ N/A N/A N/A Table. Default Boundary at T =. with q = e t / and s (k+) = s (k) + t Q(s (k) )/e /t t 3 n π e /tn e (s (k) ) tn For the scheme by Peskir and Zucca, due to the same reason as in the second case when q = t, we use the same adjustment. 5. q = e t / (table () and (3))

16 6 LAN CHENG, XINFU CHEN JOHN CHADAM & DAVID SAUNDERS s N/4 s N/4 s N/6.76 N/A N/A N/A.9 Table 3. Default Boundary at T = with q = e t / (by Peskir & Zucca s scheme) Iscoe and Kreinin s scheme is different from all the other three. The other scheme used either the PDE or the integral equations. However this used neither. It just based on the theory of probability and used the simulation to calculate the boundary. The advantage of this scheme is that it is not like all the other schemes that we have discuss the existence of solutions to the equations. However it takes so much time to do the simulations and the result is a little less accuracy. References [] B.Økesendal, Stochastic differential equations, fourth edition, Springer, New York, 998. [] Lawrence C. Evans, Particle Differential Equations, American Mathematical Society, Rhode Isiand, 998. [3] A.Friedman, Variational Principles and Free-Boundary Problems, John Wiley& Sons, New York, 98 [4] M.Avellaneda & J.Zhu, Modeling the distance-to-default process of a firm, Risk 4() (), 5-9. [5] J.Hull & A.White, Valuing credit defaultswaps II: Modeling default correlations, Journal of Derivatives 8(3) (), -. [6] I.Iscoe & A.Kreinin, Default Boundary problem () [7] R.C.Merton, On the pricing of corporate debt: The risk struture of interest rates,, Jounal of Finance (974), [8] G.Peskir, On integral equations arising in the first-passage problem for Brownian motion, Integral equations Appl. 4(4) (), [9] G.Peskir, Limit at zero of the Brownian first-passage density, Probab. Theory Related Fields 4 (), -. [] C.Zucca & L.Sacerdote & G.Peskir On the inverse first-passage problem for a wiener process, Preprints. [] M.G.Crandall & H.Ishii & P.Lions, User s Guide to viscosity solutions of second order partial differential equations, Bulletin of the American Mathematical Society7() (99), -67.