Credit Risk Models with Filtered Market Information R¨ udiger Frey Universit¨ at Leipzig Bressanone, July 2007 ruediger.frey@math.uni-leipzig.de www.math.uni-leipzig.de/~frey joint with Abdel Gabih and Thorsten Schmidt
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1. Introduction Market for portfolio credit derivatives increases and becomes more and more liquid ⇒ suitable models for pricing and hedging these products are needed. Criteria for a good model (idealistic wish-list) • Realistic dynamics of credit spread allowing for spread risk and contagion. • Realistic dependence structure of default times in order to capture observed properties of credit derivative prices (in particular the correlation-skew on CDO markets) • Tractability. computation of prices/hedge ratios and calibration with reasonable computational effort. 3
Existing model classes • Models with conditionally independent defaults and observable factors such as [Duffie and Garleanu, 2001], or [Graziano and Rogers, 2006 + Tractability, both theoretically and - for simple parameterizations - also numerically; + Inherently dynamic models; – no default contagion; – Realistic levels of default correlation and credit spread dynamics require complicated models; 4
• (Factor) copula models such as [Laurent and Gregory, 2005] [Sch¨ onbucher, 2003] or [Hull and White, 2004] (market standard.) + Easy to calibrate to defaultable term structure (CDS-spreads). + Sophisticated parameterizations can explain CDO prices. – Inherently static models; in particular hedging is based on ad-hoc methods. • Models with interacting intensities. Default contagion and is explicitly modelled, often using Markov chains. Examples include [Jarrow and Yu, 2001], [Frey and Backhaus, 2006]. + Dynamic framework allowing for with rich dynamics of credit- spreads - Tractability, at least for non-homogeneous portfolios. 5
Our information-based approach We model the default evolution of a portfolio of m firms; τ i denotes the default time of firm i , Y t,i = 1 { τ i ≤ t } is the corresponding default indicator, and Y t = ( Y t, 1 , . . . , Y t,m ) gives current default state; default history is F Y . We work directly under risk-neutral measure Q . Three layers of information: 1. Fundamental Model. Here default times τ i are conditionally independent doubly stochastic random times driven by a finite-state Markov chain X with state space S X = { 1 , . . . , K } . Fundamental model is a theoretical device for model-construction. 6
2. Market information. Prices of traded credit derivatives are determined by informed market-participants. These investors observe the default history and some process Z giving X in additive Gaussian noise. Discounted prices of traded securities will be martingales wrt so-called market information F M := F Y ∨ F Z ; filtering results wrt F M will be used to obtain asset price dynamics. 3. Investor information The process Z represents an abstract form of ‘insider information’ and is not directly observable. We therefore study pricing and hedging of (non-traded) credit derivatives from the viewpoint of secondary-market investors with information set F I ⊂ F M (investor information). We assume that F I contains the default history F Y and (noisily observed) prices of traded credit derivatives. 7
Advantages • Prices are weighted averages of full-information value (the theoretical price wrt F X ∨ F Y ) so that most computations are done in the full-information model. Since the latter has a simple structure, computations become straightforward. • Rich credit-spread dynamics with spread risk (as credit spreads fluctuate in response to fluctuations in Z ) and default contagion (as defaults of firms in the portfolio lead to an update of the conditional distribution of X given F M t ). • Model has has a natural factor structure with factors given by the conditional probabilities π k t = Q ( X t = k | F M ) , 1 ≤ k ≤ K . • Great flexibility for calibration. In particular, we may view observed prices as noisy observation of the state X t and apply calibration via filtering. 8
Some related work • [Frey and Runggaldier, 2006]. Relation between credit risk and nonlinear filtering and analysis of corresponding filtering problems; dynamics of credit risky securities is not studied • [Gombani et al., 2005] Calibration via filtering for default-free Gaussian term-structure models. • [Frey and Schmidt, 2006] Firm-value models with unobservable asset-value, extending [Duffie and Lando, 2001] • and many more... 9
2. Model and Notation • We work on probability space (Ω , F , Q ) with filtration F . All processes will be F adapted. • Consider portfolio of firms with default state = m Y t Y i ( Y t, 1 , . . . , Y t,m ) for Y t,i = 1 { τ i ≤ t } . t is obtained from Y t by flipping ith coordinate. Ordered default times denoted by T 0 < T 1 < . . . < T m ; ξ n ∈ { 1 , . . . , m } gives identity of the firm defaulting at T n . • Default-free interest rate r(t), t ≥ 0 deterministic. Here we let r ( t ) ≡ 0 . 10
The fundamental model Consider a finite-state Markov chain X with generator Q X and S X := { 1 , . . . , K } We assume that A1 The default times have ( Q, F ) -default intensity ( λ i ( X t )) , i.e. there are functions λ j : S X �→ (0 , ∞ ) , such that the processes � t ∧ τ j M t,j := Y t,j − λ j ( X s − ) ds (1) 0 are F -martingales, 1 ≤ j ≤ m . Moreover, τ 1 , . . . , τ m are conditionally independent given F X ∞ = σ ( X s : s ≥ 0) . Define the full-information value of a F Y T -measurable claim H by E Q � � � T e − t r s ds H | F t =: h ( t, X t , Y t ) ; (2) the last definition makes sense since ( X, Y ) is Markov w.r.t. F . 11
Market information Recall that the informational advantage of informed market participants is modelled via observations of a process Z . Formally, A2 F M = F Y ∨ F Z , where the l -dim. process Z solves the SDE dZ t = a ( X t ) dt + dB t . Here, B is an l -dim standard F -Brownian motion independent of X and Y , and a ( · ) is a function from S X to R l . Notation. Given a generic process U , we denote by � U the optional projection of U w.r.t. the market filtration F M ; recall that � U is a right continuous process and � U t = E ( U t |F M t ) for all t ≥ 0 . 12
3. Dynamics of Traded security Traded securities. We consider N liquidly traded credit derivatives (eg. corporate bonds) with maturity T and F I T -measurable payoff P T, 1 , . . . , P T,N . A3 (Martingale modelling) The observed prices of traded securities � � P T,i |F M are given by E =: � p t,i (expectation wrt. Q ). t Market-pricing as a nonlinear filtering problem. Denote by p i ( t, X t , Y t ) the full-information value of security i . We get from iterated conditional expectations we therefore obtain � � � � E ( P T,i |F t ) | F M p i ( t, X t , Y t ) |F M p t,i = E = E (3) � . t t Hence we need to obtain the conditional distribution of X given F M t (a nonlinear filtering problem). 13
Security-price dynamics We introduce the innovations processes as follows: � t ∧ τ j � � M t,j := Y t,j − λ j ( X s − ) ds for j = 1 , · · · , m 0 � t � µ t,i := � B t,i = Z t,i − a i ( X s ) ds for i = 1 , · · · , l. 0 Note that � M j is an F M -martingale and µ is F M -Brownian motion. Lemma 1. Every square integrable F M -martingale ( � U t ) t ∈ [0 ,T ] has the representation � T � T U T = � � s d � γ ⊤ α ⊤ U 0 + M s + (4) s d � µ s , 0 0 for R m respectively R l -valued F M -predictable processes γ and α such � T � T 0 | γ s | 2 ds + E 0 | α s | 2 ds < ∞ . that E 14
General filtering equations Proposition 2 (General filtering equations). Consider a real- � t 0 A s ds + ˜ M t , where ˜ valued F -semimartingale ξ t = ξ 0 + M t is an F -martingale with [ ˜ M, B ] = 0 . Then the optional projection � ξ t has the following representation � t � t � t ξ t = � � � s d � γ ⊤ α ⊤ ξ 0 + A s ds + M s + (5) s dµ s . 0 0 0 The square-integrable predictable processes γ and α are given by α t = � ξ t � ξ t a ( X t ) − � a ( X t ) , (6) � � E ( ξ t |F M t − ∨ { τ j = t } ) − E ( ξ t |F M γ t,j = (1 − Y t − ,j ) t − ) . (7) The proof uses the standard arguments from the innovations approach to nonlinear filtering. 15
Security-price dynamics Theorem 3. Under A1 - A3 the (discounted) price process of the traded securities has the martingale representation � t � t d � p i , ⊤ p i , ⊤ γ � α � p t,i = p 0 ,i + M s + dµ s , with � � s s 0 0 α � p i � = p t,i · a t − � p t,i � a t t � �� � � � γ � p i p i ( t, X t , Y j t − ) |F M p i ( t, X t , Y t − ) |F M E − E = (1 − Y t − ,j ) . t − t − t,j The predictable quadratic variations of the asset prices with respect to the market information F M satisfy d � � t = v ij p j � M t dt with p i , � m l � � p j � p j � p i p i v ij γ � t,n � α � t = λ n + (8) t,n γ t,n α t,n . n =1 n =1 16
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