Dynamic Modelling of CDOs Thorsten Schmidt Technische Universit at - - PowerPoint PPT Presentation

Dynamic Modelling of CDOs

Thorsten Schmidt

Technische Universit¨ at Chemnitz www.tu-chemnitz.de/mathematik/fima thorsten.schmidt@mathematik.tu-chemnitz.de

Vienna, July 2010 joint work with J. Zabczyk

Thorsten Schmidt, TU Chemnitz 1

The top-down approach

Introduction Essentials of securitization

Consider a CDO as a pool of m defaultable entities. Default i occurs at τi with associated loss qi Cumulative loss At =

m

• i=1

qi1{τi ≤t}. Normalize the total nominal to 1, set I := [0, 1]. Loss is split into tranches: a tranche refers to an interval (xi, xi−1] ⊂ I, 0 = x0 < x1 < · · · < xk = 1

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The top-down approach

Partition of losses into tranches

Examples: Traded indicies (iTraxx, CDX)

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The top-down approach

Single tranche CDOs

A STCDO is specified by a number of future dates T0 < T1 < · · · < Tm, a tranche with lower and upper detachment points x1 < x2 , a fixed spread S. We write H(x) := (x2 − x)+ − (x1 − x)+ =

• (x1,x2]

1{x≤y}dy. An investor in this STCDO receives SH(ATk ) at Tk, k = 1, . . . , m − 1 (payment leg), pays H(At) − H(At−) at any time when ∆At = 0. (default leg)

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The top-down approach

Filipovi´ c, Overbeck and Schmidt (2009)

A security which pays 1{AT ≤x} at T is called (T, x)-bond. Its price at time t ≤ T is denoted by P(t, T, x).

Proposition

The value of the STCDO at time t ≤ T1 is Γ(t, S) =

• (x1,x2]
• S

n

• i=1

P(t, Ti, y) + P(t, Tn, y) − P(t, T0, y) + γ(t, y)

• dy

with γ(t, y) = Tn

T0

I E

• rue−

u

t rs ds1{Au≤y} | Ft

• du.

Solving Γ = 0 for S gives the fair spread.

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The top-down approach

Drift condition

(A1) At =

s≤t ∆As is an increasing marked point process with compensator

νA(t, dx) dt and values in [0, 1]. Consider λ(t, x), such that Mx

t = 1{At≤x} +

t 1{As≤x}λ(s, x) ds is a martingale. Consider a d-dimensional L´ evy process Z such that I E( e−u,Zt) = etJ(u) u ∈ Rd with J(u) = m, u + 1 2Σu, u +

• Rd
• e−u,z − 1 + 1{|z|≤1}(z) u, z
• ˜

ν(dz). (1)

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The top-down approach

Forward-rate approach:

We consider P(t, T, x) = 1{At≤x} exp

• −

T

t

f (t, u, x) du

• where

f (t, T, x) = f (0, T, x) + t a(s, T, x)ds + t b(s, T, x), dZs + t

• I

c(s, T, x; y) µA(ds, dy) (2)

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The top-down approach

No-arbitrage condition

e−

t

0 rsdsP(t, T, x) are local martingales for all (T, x).

(3) Under some technical assumptions, we have that

Theorem

(3) is equivalent to s

t

a(t, u, x)du = J s

t

b(t, u, x)du

• +
• I
• e−

s

t c(t,u,x;y) du − 1

• 1{Lt+y≤x}νA(t, dy)

(4) f (t, t, x) = rt + λ(t, x), (5) where (4) and (5) hold on {At ≤ x}, Q ⊗ dt-a.s.

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The top-down approach

1

In Filipovi´ c, Overbeck, Schmidt (2009) also tractable affine models are developed.

2

Variance-Minimizing Hedging Strategies are developed in Filipovi´ c, Schmidt (2010) which lead to explicit strategies in a affine one-factor model

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Market models

Market models

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Market models

Forward rate modelling

Denote T := {T0, . . . , Tn}, δk := Tk+1 − Tk and let P(t, T, x) = p(t, T, x)1{At≤x}, (6) (p(t, T, x))0≤t≤T a strictly positive special semimartingale with p(T, T, x) = 1.

Definition

The (Tk, Tk+1, x)-spread is given by F(t, Tk, Tk+1, x) := P(t, Tk, x) P(t, Tk+1, x). (7)

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Market models

Proposition.

Forward spreads given on {At ≤ xi} by dF(t, Tk, Tk+1, xi) F(t−, Tk, Tk+1, xi) = αki(t)dt + βki(t), dW (t) +

• Rd
• eβki (t),z − 1
• µ(dt, dz) +
• I
• eγki (t,At−;y) − 1
• 1{At−+y≤xi }µA(dt, dy),

(Tk, xi) ∈ S, k < n and zero on {At > xi} constitute an arbitrage-free market if αki(t) = −λ(t, xi) +

k

• j=η(t)

βji(t), Σβki(t) +

• Rd
• eβki (t),z +
• e−βki (t),z − 1

k−1

• j=η(t)

e−βji (t),z − 1

• ν(dz)

+

• I
• eγki (t,At−;y) +
• e−γki (t,At−;y) − 1

k−1

• j=η(t)

e−γji (t,At−;y)

• 1{At+y≤xi }νA(t, dy)

for all (Tk, xi), (Tk+1, xi) ∈ S.

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Market models

Similar techniques as in interest rate markets can be applied to value CDOs and options on CDOs. Grbac, Eberlein, Schmidt (2010) study directly discrete rates. Statistical results show that in shorter time periods affine models are appropriate.

Further issues

Model risk Measuring the risk of credit and market risk simultaneously.

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Market models [1] A Brace, D. Gatarek, M. Musiela: The market model of interest rate dynamics (1995). [2] E Eberlein, Z Grbac, T Schmidt: Market Models for CDOs driven by L´ evy processes (2010). arXiv:1006.2012 [3] D Filipovi´ c, L Overbeck, T Schmidt: Dynamic CDO term structure modelling (2009). [4] D Filipovi´ c, T Schmidt: ”Pricing and Hedging of CDOs: A Top-Down Approach”, (2010). Contemporary Quantitative Finance,Chiarella, C. and Novikov, A. (Eds.) [5] T Schmidt, J Zabczyk: CDO term structure modelling with L´ evy processes and the relation to market models (2010). arXiv:1007.1706

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