Credit Risk Models with Filtered Market Information

Transcription

1 Credit Risk Models with Filtered Market Information Rüdiger Frey Universität Leipzig Bressanone, July joint with Abdel Gabih and Thorsten Schmidt

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4 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

5 Existing model classes Models with conditionally independent defaults and observable factors such as [Duffie and Garleanu, 2001], or [Graziano and Rogers, 20 + 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

6 (Factor) copula models such as [Laurent and Gregory, 2005] [Schönbucher, 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 creditspreads - Tractability, at least for non-homogeneous portfolios. 5

7 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 {τi 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

8 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

9 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

10 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

11 2. Model and Notation We work on probability space (Ω, F, Q) with filtration F. processes will be F adapted. All Consider portfolio of m firms with default state Y t = (Y t,1,..., Y t,m ) for Y t,i = 1 {τi t}. Yt i 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. r(t) 0. Here we let 10

12 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 M t,j := Y t,j t τj 0 λ j (X s )ds (1) 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( e T 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

13 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 Û the optional projection of U w.r.t. the market filtration F M ; recall that Û is a right continuous process and Ût = E(U t Ft M ) for all t 0. 12

14 3. Dynamics of Traded security Traded securities. We consider N liquidly traded credit derivatives (eg. corporate bonds) with maturity T and FT I -measurable payoff P T,1,..., P T,N. A3 (Martingale modelling) The observed prices of traded securities are given by E ( ) P T,i Ft M =: pt,i (expectation wrt. Q). 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 p t,i = E ( E(P T,i F t ) F M t ) = E ( pi (t, X t, Y t ) F M t ). (3) Hence we need to obtain the conditional distribution of X given F M t (a nonlinear filtering problem). 13

15 Security-price dynamics We introduce the innovations processes as follows: M t,j := Y t,j t τj µ t,i := B t,i = Z t,i 0 λ j (X s )ds t 0 for j = 1,, m â i (X s ) ds for i = 1,, l. Note that M j is an F M -martingale and µ is F M -Brownian motion. Lemma 1. Every square integrable F M -martingale (Ût) t [0,T ] has the representation Û T = Û0 + T 0 γ s d M s + T 0 α s d µ s, (4) for R m respectively R l -valued F M -predictable processes γ and α such that E T 0 γ s 2 ds + E T 0 α s 2 ds <. 14

16 General filtering equations Proposition 2 (General filtering equations). Consider a realvalued F-semimartingale ξ t = ξ 0 + t 0 A sds + M t, where M t is an F-martingale with [ M, B] = 0. Then the optional projection ξ t has the following representation ξ t = ξ 0 + t 0 Â s ds + t 0 γ s d M s + t 0 α s dµ s. (5) The square-integrable predictable processes γ and α are given by α t = ξ t a(x t ) ξ t â(x t ), (6) γ t,j = (1 Y t,j ) ( E(ξ t F M t {τ j = t}) E(ξ t F M t ) ). (7) The proof uses the standard arguments from the innovations approach to nonlinear filtering. 15

17 Security-price dynamics Theorem 3. Under A1 - A3 the (discounted) price process of the traded securities has the martingale representation p t,i = p 0,i + α p i t = t 0 p t,i a t p t,i â t γ p i t,j = (1 Y t,j ) γ p i, s d M s + t 0 α p i, s dµ s, with ( E ( p i (t, X t, Y j t ) F M t ) E ( pi (t, X t, Y t ) F M t ) ). The predictable quadratic variations of the asset prices with respect to the market information F M satisfy d p i, p j M t = v ij t dt with v ij t = m n=1 γ p i t,n γ p j t,n λ n + l n=1 α p i t,nα p j t,n. (8) 16

18 Filtering Define the conditional probability vector π t = (πt 1,..., πt K ) with πt k := Q(X t = k Ft M ). π t is the natural state variable; under market information F M all quantities of interest are functions of π t. Dynamics of π t. Updating at a default time τ i. One has Q(X t = k F M t {τ i = t}) = λ i (k)π k t K n=1 λ i(n)π n t. Kushner-Stratonovich equation (K-dim SDE-system for π t ) K dπt k = q(ι, k)πt dt ι + γ π, t d M t + α π, t dµ t, with (9) γ π t,i = πk t ι=1 ( ) λ i (k) Kn=1 1, α π λ i (n)πt n t = πt k ( a(k) ) K i=1 πi t a(i). 17

19 4. Secondary market investors Recall that secondary market investors do not observe Z. Their information set is given by F I F M ; typically F I contains default history and noisy price information. Put ν k t := Q(X t = k F M t ), 1 k K. Pricing. Consider non-traded FT Y -measurable claim H. Define its secondary-market value as E(H Ft I ). Let h t (X t ) = E(H F t ) (full-information value of H). We get from iterated conditional expectations E(H F I t ) = E ( E(H F M t ) ) ( K F I t = E πt k h t (k) ) K F I t = νt k h t (k), k=1 k=1 i.e. pricing for secondary-market investors reduces to finding ν t. 18

20 Hedging. We look for risk-minimizing strategies under restricted information in the sense of [Schweizer, 1994]. Quadratic criterion combines well with incomplete information On credit markets it is natural to minimize risk wrt martingale measure Q as historical default intensities are hard to determine. The risk-minimizing strategy θ H can be computed by suitably projecting the F m -risk-minimizing hedging strategy ξt H on the set of F I -predictable strategies. For instance we get with only one traded asset that θ t is left-continuous version of E(v t ξ H t F I t ) / E(v t F I t ). Recall that v t and ξ t are nonlinear functions of π t. We need to determine conditional distribution of π t given F I t. 19

21 Modelling F I and Calibration Strategies Simple calibration. In this scenario F I = F p F Y (prices of traded securities are observable). Recall that p t,i = K k=1 πk t p i (t, k, Y t ). If N K (more securities than states) and if the matrix p(t, Y t ) := (p i (t, k, Y t )) 1 i N,1 k K of fundamental values has full rank, the vector π t could be computed by simple calibration: π t = argmin {π 0, K k=1 π k =1} N w n ( p t,n K p n (t, k, Y t )π k ) 2, n=1 k=1 for suitable weights w 1,..., w N. In that case pricing and hedging for secondary market investors and informed market participants coincides. 20

22 Calibration via filtering Alternatively, assume that F I = F Y F U where the N-dim process U solves the SDE du t = p t dt + dw t = p(t, Y t )π t dt + dw t for a Brownian motion W independent of X, Y, Z. U can be viewed as cumulative noisy price information of the traded assets p 1,..., p N ; noise reflects observation errors and model errors. Recall that π solves the KS-equation (9). Hence computation of the conditional distribution of π t given Ft I is a standard nonlinear filtering problem with signal process π and observation U. Note that π is typically high-dimensional; particle filtering might be used (numerical analysis work in progress). 21

23 References [Duffie and Garleanu, 2001] Duffie, D. and Garleanu, N. (2001). Risk and valuation of collateralized debt obligations. Financial Analyst s Journal, 57(1): [Duffie and Lando, 2001] Duffie, D. and Lando, D. (2001). Term structure of credit risk with incomplete accounting observations. Econometrica, 69: [Frey and Backhaus, 2006] Frey, R. and Backhaus, J. (2006). Credit derivatives in models with interacting default intensities: a Markovian approach. Preprint, Universität Leipzig. available from [Frey and Runggaldier, 2006] Frey, R. and Runggaldier, W. (2006). 22

24 Credit risk and incomplete information: approach. preprint, Universität Leipzig. a nonlinear filtering [Frey and Schmidt, 2006] Frey, R. and Schmidt, T. (2006). Pricing corporate securities under noisy asset information. preprint, Universität Leipzig, submitted. [Gombani et al., 2005] Gombani, A., Jaschke, S., and Runggaldier, W. (2005). A filtered no arbitrage model for term structures with noisy data. Stochastic Processes and Applications, 115: [Graziano and Rogers, 2006] Graziano, G. and Rogers, C. (2006). A dynamic approach to the modelling of correlation credit derivatives using Markov chains. working paper, Statistical Laboratory, University of Cambridge. [Hull and White, 2004] Hull, J. and White, A. (2004). Valuation of 23

25 a CDO and a nth to default CDS without monte carlo simulation. Journal of Derivatives, 12:8 23. [Jarrow and Yu, 2001] Jarrow, R. and Yu, F. (2001). Counterparty risk and the pricing of defaultable securities. J. Finance, 53: [Laurent and Gregory, 2005] Laurent, J. and Gregory, J. (2005). Basket default swaps, CDOs and factor copulas. Journal of Risk, 7: [Schönbucher, 2003] Schönbucher, P. (2003). Credit Derivatives Pricing Models. Wiley. [Schweizer, 1994] Schweizer, M. (1994). Risk minimizing hedging strategies under restricted information. Math. Finance, 4: