Simulation of an SPDE Model for a Credit Basket

Transcription

1 Simulation of an SPDE Model for a Credit Basket Christoph Reisinger Joint work with N. Bush, B. Hambly, H. Haworth christoph.reisinger@maths.ox.ac.uk. MCFG, Mathematical Institute, Oxford University C. Reisinger p.1

2 Outline Introduction, structural models of credit Example: Credit default swap spreads A general multi-factor model Large basket limit and CDO pricing Numerical simulation of the SPDE Calibration and pricing examples Improvements, extensions C. Reisinger p.2

3 Structural setup As in Merton (1974), and Black and Cox (1976), A t the company s asset value, governed by da t A t = µdt + σ dw t µ the mean return, σ the volatility, W a standard Brownian motion denoting the default threshold barrier by b, say constant, define the distance to default, X t, as X t = 1 σ ( loga t logb ). Merton: company defaults if X T < 0 Black/Cox: default time τ is given by the first time X t hits 0 C. Reisinger p.3

4 Default probabilities By first exit time theory, the probability of survival to T is where Q(T τ X t ) = Q(X s 0, t s T X t ) H(x, s) = Φ ( = H(X t, T t), ) ( x + ms e 2mx x + ms Φ ). s s Φ the standard Gaussian CDF, m the risk-neutral drift of X s C. Reisinger p.4

5 More companies Consider firm values, for i = 1,...,N (risk-neutral measure), da i t = (r f q i )A i t dt + σ i A i t dw i (t) where r f risk-free rate q i dividend yields σ i volatilities W i (t) Brownian motions and cov(w i (t), W j (t)) = ρ ij t. C. Reisinger p.5

6 More companies Consider firm values, for i = 1,...,N (risk-neutral measure), da i t = (r f q i )A i t dt + σ i A i t dw i (t) where r f risk-free rate q i dividend yields σ i volatilities W i (t) Brownian motions and cov(w i (t), W j (t)) = ρ ij t. We assume that each company has an exponential default barrier, b i (t) = K i e γ i(t t) for constants K i, γ i. T represents the maturity of the product. C. Reisinger p.5

7 Transformation Setting ( A Xt i i = ln t A i 0 ) e γ it = α i t + σ i W i (t) with α i = r f q i γ i 1 2 σ2 i, leads to a Brownian motion with ( ) drift and constant barrier B i = ln bi (0) 0, X0 i = 0. A i 0 C. Reisinger p.6

8 Transformation Setting ( A Xt i i = ln t A i 0 ) e γ it = α i t + σ i W i (t) with α i = r f q i γ i 1 2 σ2 i, leads to a Brownian motion with ( ) drift and constant barrier B i = ln bi (0) 0, X0 i = 0. Defining the running minimum A i 0 X i t = min 0 s t Xi s, and default time, τ i, as the first hitting time of the default barrier, τ i = inf{t : X i t = B i }, the survival probability is then Q(τ i > s) = Q(X i s B i). C. Reisinger p.6

9 2D case For two firms, this problem is analytically tractable, e. g. Q(t) = Q(X 1 t B 1, X 2 t B 2) = 2 ( ) β βt ea 1B 1 +a 2 B 2 +bt e r2 0 /2t nπθ0 sin β where g n (θ) = 0 n=1 a 1 = α 1σ 2 ρα 2 σ 1 (1 ρ 2 )σ 2 1 σ 2 ( re r2 /2t e A(θ)r rr0 ) I ( nπ β ) t dr, a 2 = α 2σ 1 ρα 1 σ 2 (1 ρ 2 )σ 1 σ ( ) nπθ sin g n (θ) dθ β b = α 1 a 1 α 2 a σ2 1a ρσ 1 σ 2 a 1 a σ2 2a ρ 2 tan β =, β [0, π] ρ etc, and I ( nπ β ) ( rr0 t ) is a modified Bessel s function. C. Reisinger p.7

10 CDS spread calculation The buyer of a k th -to-default credit default swap (CDS) on a basket of n companies pays a premium, the CDS spread, for the life of the CDS until maturity or the k th default. In the event of default by the k th underlying reference company, the buyer receives a default payment and the contract terminates. C. Reisinger p.8

11 CDS spread calculation The buyer of a k th -to-default credit default swap (CDS) on a basket of n companies pays a premium, the CDS spread, for the life of the CDS until maturity or the k th default. In the event of default by the k th underlying reference company, the buyer receives a default payment and the contract terminates. Equating discounted spread payment and discounted default payment, T DSP = ck e rfs Q(τ k > s) ds DDP = (1 R)K 0 T 0 e r fs Q(s τ k s + ds) gives the market k th -to-default CDS spread, c k, as { (1 R) 1 e rft Q(τ k > T) } T 0 r fe rfs Q(τ k > s) ds c k = T. 0 e rfs Q(τ k > s) ds C. Reisinger p.8

12 Varying T First-to-default CDS, varying T Second-to-default CDS, varying T year 2 years 3 years 4 years 5 years year 2 years 3 years 4 years 5 years Spread Correlation Spread Correlation σ 1 = σ 2 = 0.2, K 1 = 100, r f = 0.05, q 1 = q 2 = 0, γ 1 = γ 2 = 0.03, initial distance-to-default = 2, R = 0.5 C. Reisinger p.9

13 Varying σ First-to-default CDS, varying σ Spread σ i =0.1 σ i =0.15 σ i =0.2 σ i =0.25 σ i = Correlation Second-to-default CDS, varying σ Spread σ i =0.1 σ i =0.15 σ i =0.2 σ i =0.25 σ i = Correlation K 1 = 100, r f = 0.05, q 1 = q 2 = 0, R = 0.5 γ 1 = γ 2 = 0.03, initial distance to default = 2, T = 5 C. Reisinger p.10

14 Portfolio model Consider a portfolio of N different companies. Denoting the companies asset values at time t by A i t, we assume that under the risk neutral measure they follow a diffusion process given by da i t = µ(t, A i t) dt + σ(t, A i t) dw i t + m σ ij (t, A i t) dm j t, j=1 W i t and M j t are Brownian motions satisfying d W i t, M j t = 0 i, j and d W i t, W j t = d M i t, M j t = δ ij dt. C. Reisinger p.11

15 Numerical PDE solution Solve Kolmogorov equation numerically? Isotropic grid for N = 2 directions (firms): C. Reisinger p.12

16 Numerical PDE solution Solve Kolmogorov equation numerically? Isotropic grid for N = 2 directions (firms): C. Reisinger p.12

17 Numerical PDE solution Solve Kolmogorov equation numerically? Isotropic grid for N = 2 directions (firms): C. Reisinger p.12

18 Numerical PDE solution Solve Kolmogorov equation numerically? Isotropic grid for N = 2 directions (firms): Curse of dimensionality : After n refinements, unknowns h N 2 nn Accuracy ǫ has complexity C(ǫ) ǫ N/2 (for an order 2 method) exponential growth of required computational resources C. Reisinger p.12

19 What is high dimensional? Example, 30 firms: If each asset is represented by only two states, the total number of variables is already 2 30 = C. Reisinger p.13

20 What is high dimensional? Example, 30 firms: If each asset is represented by only two states, the total number of variables is already 2 30 = If we choose a reasonable number of points in each direction, say 32 = 2 5, the same total number is already obtained for dim = 6. C. Reisinger p.13

21 Sparse grids Full grid, h N points Sparse, h 1 log h N 1 But: require too much smoothness and have large constants. C. Reisinger p.14

22 Correlation data (typical) C. Reisinger 0.18 p.15 0

23 Correlation data l σ(σ) λ Spectral gap after λ Exponential decay from λ Asymptotic expansion? Not enough regularity C. Reisinger 0.18 p.16 0 k

24 A different viewpoint It is irrelevant which of the firms default. Assume interchangeable. Let ν N,t the empirical measure for the entire portfolio, ν N,t = 1 N N i=1 δ A i t Let φ Cc (R) and for measure ν t write φ, ν t = φ(x)ν t (dx). Define a family of processes F N,φ t by F N,φ t = φ, ν N,t = 1 N N φ(a i t). i=1 C. Reisinger p.17

25 Evolution of measure Applying Ito s lemma to F N,φ t F N,φ t = 1 N N i=1 t N we have φ (A i s) µ(s, A i s) dt + σ(s, A i s) dws i + N i=1 t φ (A i s) m σij 2 + σ 2 ds. j=1 m σ ij (s, A i s) dms j j=1 We now assume that σ ij = σ j for all i = 1, 2..., then t t m F N,φ t = F N,φ 0 + Aφ, ν N,s ds + σ j φ, ν N,s dms j t 0 1 N 0 j=1 N φ (A i s) σ(s, A i s) dws. i i=1 A = µ x σ2 2 x 2, σ2 = m σj 2 + σ 2, j=1 C. Reisinger p.18

26 ing equation The idiosyncratic component becomes deterministic in the infinite dimensional limit, see also [Kurtz & Xiong, 1999]. As the market factors affect each company to the same degree, we can write F N,φ t F φ t = φ, ν t as N, with df φ t = Aφ, ν t dt + m σ j φ, ν t dm j t. j=1 Alternatively, we can write this in integrated form as φ, ν t = φ, ν 0 + t 0 Aφ, ν s ds + m t j=1 0 σ j φ, ν s dm j s C. Reisinger p.19

27 Density Assume the measure ν t to be absolutely continuous with respect to the Lebesgue measure, to write ν t (dx) = v(t, x)dx In differential form, dv = µ v x dt σ2 2 v x 2 dt m j=1 x (σ j(t, x)v)dm j t. This is a stochastic PDE that describes the evolution of an infinite portfolio of assets whose dynamics were given before. For solubility see [Krylov, 1994]. Can now use this to approximate the loss distribution for a portfolio of fixed size N. C. Reisinger p.20

28 Simplification Assume asset processes are correlated via a single factor. Specifically, let the SDEs for the asset processes be given by da i t A i t = r dt + 1 ρσ dw i t + ρσ dm t, A i 0 = a i, where ρ [0, 1), d W i t, M t = 0. The distance-to-default X i t = 1 σ ( log A i t log B i t ) evolves according to dx i t = µdt + 1 ρdw i t + ρdm t, X i 0 = x i with µ = 1 σ ( r 1 2 σ2) and x i = 1 σ ( log a i log B i 0 ). C. Reisinger p.21

29 Portfoilio loss Constant coefficient SPDE for v. The portfolio loss variable L N t = (1 R) 1 N measures the losses at time t. N i=1 1 {τi t} Assume that a fraction R of losses, 0 R 1, the recovery rate, is recovered after default. Absorbing barrier condition: v(t, 0) = 0 for t 0. The loss distribution is then ( L t = (1 R) 1 0 ) v(t, x) dx. C. Reisinger p.22

30 Properties Matching the initial conditions for both measures, we need v(0, x) = 1 N N δ A i 0. Given this initial condition, i=1 L 0 = N ( 1 1 N N δ A i 0 i=1 B t ) = 0, Also, 0 L t 1, for t 0 Q(L s K) Q(L t K), for s t, which ensure that there is no arbitrage in the loss distribution. C. Reisinger p.23

31 CDOs CDOs slice up the losses into tranches, defined by their attachment and detachment points. Typical numbers are 0-3%, 3%-6%, 6%-9%, 9%-12%, 12%-22%, 22%-100%. For an attachment point a and detachment point d > a, the outstanding tranche notional and tranche loss X t = max(d L t, 0) max(a L t, 0) Y t = (d a) X t = max(l t a,0) max(l t d, 0) determine the spread and default payments for that tranche. Spreads are quoted as an annual payment, as a ratio of the a notional, but assumed to be paid quarterly. a there is a variation for the equity tranche, but we do not go into details. C. Reisinger p.24

32 Spread calculation Assume the notional to be 1, c the spread payment; n the maximum number of payments up to expiry T ; T i the payment dates for 1 i n, δ = 0.25 the interval between payments. The expected sum of discounted fee payments, referred to as the fee leg, is given by cv fee = c The protection leg is n δe rt i E Q [X Ti ], i=1 V prot = n e rt i E Q [X Ti 1 X Ti ]. i=1 The fair spread payment is then given by s = V prot V fee. C. Reisinger p.25

33 Computation Recall the SPDE dv = 1 (r 12 ) σ σ2 v x dt v xx dt ρv x dm t, t > 0, x > 0 v(0, x) = v 0 (x), x > 0 v(t, 0) = 0 t 0 Monte Carlo on top of a PDE solver computationally costly. For each simulated path of the market factor, a PDE needs to be solved, e. g. by a finite difference method. From the PDE solution, quantities like tranche losses can be computed, then averaged over the simulated paths. C. Reisinger p.26

34 Discrete defaults First make the assumption that defaults are only observed at a discrete set of times, taken quarterly to coincide with the payment dates, i.e. if a firm s value is below the default barrier on one of the observation dates T i, removed from the basket. We therefore solve the modified SPDE problem dv = 1 σ (r 12 ) σ2 v x dt v xx dt ρv x dm t, t (T k, T k+1 ), v(0, x) = v 0 (x), v(t k, x) = 0 x 0, 0 < k n C. Reisinger p.27

35 Lagrangian coordinate The boundary condition is not active in intervals (T k, T k+1 ). The Brownian driver only introduces a random shift. The solution can therefore be written as v(t, x) = 8 < : v (i) (t T k, x ρ(m t M Tk )) 0 x 0 t {T k, 1 i n} else if t (T k, T k+1 ], 0 k < n where v (k) is the solution to the (deterministic) problem v (k) t = 1 (1 ρ)v(k) xx 1 2 σ (r 12 σ2 ) v (k) x, t (0, τ) = (0, T k+1 T k ) v (k) (0, x) = v(t k, x) assuming payment dates are equally spaced, τ = T k+1 T k. C. Reisinger p.28

36 Algorithm This suggests the following inductive strategy for k = 0,...,n 1: 1. Start with v (0) (0, x) = v 0 (x). 2. Solve the PDE numerically in the interval (0, T 1 ), to obtain v (0) (T 1, x). 3. Simulate M T1, evaluate v(t 1, x). 4. For k > 0, having computed v(t k, x) in the previous step, use this as initial condition for v (k), and repeat until k = n. C. Reisinger p.29

37 Finite differences The initial distribution is assumed localised. Approximate the distribution by one with support [x min, x max ] with x min < 0 and x max > 0. Can ensure that the expected error of this approximation is much smaller than the standard error of the Monte Carlo estimates. Grid x 0 = x min, x 1 = x min + x,...,x min + j x,..., x J = x min + J x = x max, where x = (x max x min )/J, timesteps t 0 = 0, t 1 = t,...,t I = I t = τ, where t = τ/i. Define an approximation v i j to v(t i, x j ) as solution to a FD/FE scheme with θ timestepping. C. Reisinger p.30

38 Stability Standard central differencing is second order accurate in x. The backward Euler scheme θ = 1 is of first order accurate (in t) and strongly stable. The Crank-Nicolson scheme θ = 1 2 is of second order accurate, and is unconditionally stable in the l 2 -norm. We deal with initial conditions of the form v(0, x) = 1 N N δ(x x i ), i=1 Crank-Nicolson timestepping gives spurious oscillations for Dirac initial conditions, and reduces the convergence order for discontinuous intitial conditions. C. Reisinger p.31

39 Rannacher timestepping Rannacher proposed the following modification: Replace first Crank-Nicolson steps with backward Euler steps. Need to balance between accuracy and stability. Analysis by Giles and Carter of the heat equation suggests to replace the first two Crank-Nicolson steps by four backward Euler steps of half the stepsize. We do this at t = 0, and also at t = T k where the interface conditions introduce discontinuities at x = 0. This restores second order convergence in time. C. Reisinger p.32

40 Averaging The sum of δ-distributions needs approximation on the grid. Could collect the firms into symmetric intervals of width x around grid points, v 0 j = 1 N x xj + x/2 x j x/2 δ(x i x) dx, however, this reduces the overall order of the finite difference scheme to 1: The approximation cannot distinguish between initial positions x i in an interval of length x. C. Reisinger p.33

41 Projection To achieve higher (i. e. second) order, the δ s are split between adjacent grid points. The correct weighting for a single firm with distance-to-default x i in the interval [x j, x j+1 ), is v 0 k = x 2 (x j+1 x i ) k = j x 2 (x i x j ) k = j else This can be written more elegantly as L 2 -projection onto the basis of hat functions Φ k 0 k N where Φ k (x) = 1 x min (max(x x k + x), max(x k x,0)), then v 0 k = Φ k, v 0 = xmax x min Φ k (x)v 0 (x) dx. Note that x N k=0 v0 k = 1. C. Reisinger p.34

42 Interface condition At t = T k, we have to evaluate the grid function at shifted arguments that do not normally coincide with the grid. First, define a piecewise linear reconstruction from the approximation vk I obtained in the last step over the previous interval [T k 1, T k ], as N v x (T k, x) = Φ j (x M)vj. I j=0 Then, approximate the shift by setting v 0 j = xj + x/2 max(x j x/2,0) v x (T k, x) dx, with M = M Tk M Tk 1. C. Reisinger p.35

43 Interface condition Use this as initial condition for the next interval (the integral is understood to be 0 if the lower limit is larger than the upper limit). This ensures that x J vj 0 = j=0 xmax 0 v h (T k, x) dx, so the cumulative density of firms with firm values greater than 0 is preserved. Also, the solution is smoothened at x = 0 C. Reisinger p.36

44 Simulations For a given realisation of the market factor, we can approximate the loss functional L Tk at time T k by L x T k = 1 xmax 0 v x (T k, x) dx Explicitly include the dependency L x T k (Φ), where Φ i are drawn independently from a standard normal distribution, Then for N sims simulations with samples Φ l = (Φ l k ) 1 k n, 1 l N sims, E Q [X Tk ] E Q [max(d L x T k (Φ), 0) max(a L x T k (Φ), 0)] 1 N sims N sims l=1 ( max(d L x T k (Φ l ), 0) max(a L x T k (Φ l ), 0) ) C. Reisinger p.37

45 Data The simulation error has two components, the discretisation error and the variance of the Monte Carlo estimate. For the following simulations we have used these data: T = 5, r = 0.027, σ = 0.24, R = 0.7, ρ = Initial positions for individual firms, calibrated to their individual CDS spreads, were well within the range [x min, x max ] = [ 10, 20]. C. Reisinger p.38

46 Grid convergence First consider the discretisation error in t and x ǫj J ǫi I Extrapolation-based estimator for the discretisation error of L Tn for increasing J (left) and I (right) for a single realisation of the path of the market factor. We clearly see second order convergence in t and x. C. Reisinger p.39

47 MC convergence s [0, 3%] [3%, 6%] [6%, 9%] [9%, 12%] [12%, 22%] [22%, e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e Monte Carlo estimates for expected outstanding tranche notionals for N sims = 64 4 s 1 Monte Carlo runs, s = 1,..., 10. The finite difference parameters were fixed at J = 256, I = 4. C. Reisinger p.40

48 MC convergence ], a = 0, b = 0.03 E Q [XTn log 4 N sims ], a = 0.06, b = x E Q [XTn log 4 N sims ], a = 0.12, b = 0.22 E Q [XTn x 10 7 log 4 N sims Monte Carlo estimates with standard error bars for expected losses in tranches [0, 3%], [6%, 9%], [12%, 22%] for N sims = 16 4 k 1, k = 1,...,10, J = 256, I = 4. C. Reisinger p.41

49 Index spreads The fixed coupons, traded spreads and model spreads for the itraxx Main Series 10 index. Maturity Date Fixed Coupon (bp) Traded Spread (bp) Model Spread (bp) 20/12/ /12/ /12/ February 22, Parameters used for the model spreads are r = 0.042, σ = 0.22, R = 0.4. Maturity Date Fixed Coupon (bp) Traded Spread (bp) Model Spread (bp) 20/12/ /12/ /12/ December 5, Parameters used for the model spreads are r = 0.033, σ = 0.136, R = 0.4. C. Reisinger p.42

50 Implied correlation Implied Correlation Skew for itraxx Main Series 6 Tranches, Feb 22, The implied correlation for each tranche is the value of correlation that gives a model tranche spread equal to the market tranche spread. Implied Corrleation (%) Year 7 Year 10 Year % 6% 9% 12% 22% 100% Tranche Detachment Point Model parameters are r = 0.042, σ = 0.22, R = 0.4. C. Reisinger p.43

51 5 Year Tranche Market ρ = 0.3 ρ = 0.4 ρ = 0.5 ρ = 0.6 ρ = 0.7 ρ = 0.8 ρ 0%-3% 71.5 % % 75.9 % % % % % 4 3%-6% %-9% %-12% %-22% %-100% Model tranche spreads (bp) for varying values of the correlation parameter. The equity tranches are quoted as an upfront assuming a 500bp running spread. The model is calibrated to the itraxx Main Series 10 index for Dec 5, Market levels shown are for this date; model parameters are r = 0.033, σ = 0.136, R = 0.4. C. Reisinger p.44

52 7 Year Tranche Market ρ = 0.3 ρ = 0.4 ρ = 0.5 ρ = 0.6 ρ = 0.7 ρ = 0.8 ρ 0%-3% 72.9 % % % % % 60 % % 4 3%-6% %-9% %-12% %-22% %-100% Model tranche spreads (bp) for varying values of the correlation parameter. The equity tranches are quoted as an upfront assuming a 500bp running spread. The model is calibrated to the itraxx Main Series 10 index for Dec 5, Market levels shown are for this date; model parameters are r = 0.033, σ = 0.136, R = 0.4. C. Reisinger p.45

53 10 Year Tranche Market ρ = 0.3 ρ = 0.4 ρ = 0.5 ρ = 0.6 ρ = 0.7 ρ = 0.8 0%-3% 73.8 % % % % % % % 3%-6% %-9% %-12% %-22% %-100% Model tranche spreads (bp) for varying values of the correlation parameter. The equity tranches are quoted as an upfront assuming a 500bp running spread. The model is calibrated to the itraxx Main Series 10 index for Dec 5, Market levels shown are for this date; model parameters are r = 0.033, σ = 0.136, R = 0.4. C. Reisinger p.46

54 Dynamic properties, forward starting CDOs, options on tranches,... Simulation of SPDEs driven by jumps, with Karolina Bujok. Calibration, including jumps. Analytic work by Hambly/Jin on jumps, contagion, granularity. Continuous defaults, with Mike Giles, via Multi-Level Monte Carlo. Convergence analysis through mean-square stability. Importance sampling for senior tranches, with Tom Dean. C. Reisinger p.47

55 Importance sampling Using large deviations theory, see also [Glasserman et al, 1999, 2007], preliminary observations: without importance sampling with importance sampling Probability of more than 3% defaults occuring for n Monte Carlo paths C. Reisinger p.48

56 Importance sampling Using large deviations theory, see also [Glasserman et al, 1999, 2007], preliminary observations: without importance sampling 1.8 x with importance sampling 1.8 x Probability of more than 10% defaults occuring for n Monte Carlo paths C. Reisinger p.48

57 Literature References [Black & Cox(1976)] Black, F. & Cox, J. (1976) Valuing Corporate Securities: Some Effects of Bond Indenture Provisions,J. Finance, 31, [Cameron & Martin(1947)] Cameron, R. H., & Martin, W. T. (1947) The Orthogonal Development of Non-Linear Functionals in Series of Fourier-Hermite Functionals, Ann. Math, 48, [Giles & Carter(2006)] Giles, M. & Carter, R. (2006) Convergence analysis of Crank-Nicolson and Rannacher time-marching, J. Comp. Fin., 9(4), [Hambly & Jin (2008)] Hambly, B.M. & Jin, L. (2008) SPDE approximations for large basket portfolio credit modelling, in preparation. [Haworth & Reisinger(2007)] Haworth, H., & Reisinger, C. (2007) Modelling Basket Credit Default Swaps with Default Contagion, J. Credit Risk, 3(4), C. Reisinger p.49