Pricing and Hedging of Credit Derivatives via Nonlinear Filtering Rüdiger Frey Universität Leipzig May 2008 ruediger.frey@math.uni-leipzig.de www.math.uni-leipzig.de/~frey based on work with T. Schmidt, W. Runggaldier, H. Mühlichen and A. Gabih
Overview 1. Introduction: credit risk under incomplete information 2. Pricing and hedging credit derivatives via nonlinear filtering: the [Frey et al., 2007] model. Main ideas: We model evolution of investors believes about credit quality, as those are driving credit spreads. We use innovations approach to nonlinear filtering for deriving dynamics of traded credit derivatives. 1
Attainable Correlations correlation -1.0-0.5 0.0 0.5 1.0 min. correlation max. correlation 0 1 2 3 4 5 sigma Attainable correlations for two lognormal variables, X 1 Ln(0, 1), X 2 Ln(0, σ 2 ); (from McNeil, Frey, Embrechts, Quantitative Risk Management, Princeton University Press 2005) 2
1. Credit Risk and Incomplete Information Basically we have two classes of dynamic credit risk models. Structural models: Default occurs if the asset value V i of firm i falls below some threshold K i, interpreted as liability, so that default time is τ i := inf{t 0: V t,i K i }. τ i is (typically) predictable; dependence between defaults via dependence of the V i. Reduced form models: Default occurs at the first jump of some point process, typically with stochastic intensity λ t,i. (τ i is totally inaccessible.) Usually λ t,i = λ i (t, X t ), where X is a common state variable process introducing dependence between default times. 3
Incomplete information In both model-classes it makes sense to assume that investors have only limited information about state variables of the model Asset value V i is hard to observe precisely consider firm-value models with noisy information about V (see for instance [Duffie and Lando, 2001], [Jarrow and Protter, 2004], [Coculescu et al., 2006] or [Frey and Schmidt, 2006]). In reduced-form models state variable process X is usually not associated with observable economic quantities and needs to be backed out from observables such as prices. 4
Implications of incomplete information Under incomplete informations τ i typically admits an intensity. Natural two-step-procedure for pricing: prices are first computed under full information (using Markov property) and then projected on the investor filtration Pricing and model calibration naturally lead to nonlinear filtering problems. Information-driven default contagion. In real markets one frequently observes contagion effects, i.e. spreads of non-defaulted firms jump(upward) in reaction to default events. Models with incomplete information mimic this effect: given that firm i defaults, conditional distribution of the state-variable is updated, default intensity of surviving firms increases ([Schönbucher, 2004], [Collin-Dufresne et al., 2003],....) 5
Some literature (mainly reduced-form models) Simple doubly-stochastic models with incomplete information such as [Schönbucher, 2004], [Duffie et al., 2006], extensions in recent work by Giesecke. [Frey and Runggaldier, 2007]. Relation between credit risk and nonlinear filtering and analysis of filtering problems in very general reduced-form model; dynamics of credit risky securities not studied. Default-free term-structure models: [Landen, 2001]: construction of short-rate model via nonlinear filtering; [Gombani et al., 2005]: calibration of bond prices via filtering. [Frey and Runggaldier, 2008] A general overview over nonlinear filtering in term-structure and credit risk models. 6
2. Our information-based model Overview. Three layers of information: 1. Underlying default model (full information) Default times τ i are conditionally independent doubly-stochastic random times; intensities are driven by a finite-state Markov chain X. 2. Market information. Prices of traded credit derivatives are determined by informed market-participants who observe default history and some (abstract) process Z giving X in additive Gaussian noise (market information F M := F Y F Z ); Filtering results wrt F M are used to obtain asset price dynamics. 3. Investor information. Z represents abstract form of insider information and is not directly observable. study pricing and hedging of credit derivatives for secondary-market investors with investor information F I F M. 7
Advantages Prices are weighted averages of full-information values (the theoretical price wrt F X F Y ), so that most computations are done in the underlying Markov model. Since the latter has a simple structure, computations become relatively easy. Rich credit-spread dynamics with spread risk (spreads fluctuate in response to fluctuations in Z) and default contagion (as defaults lead to an update of the conditional distribution of X t 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
Notation We work on probability space (Ω, F, Q), Q the risk-neutral measure, with filtration F. All processes will be F adapted. We 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. Here r(t) 0. 9
The underlying full-information model Consider a finite-state Markov chain X with S X := {1,..., K} and generator Q X. A1 The default times are conditionally independent, doubly stochastic random times with (Q, F)-default intensity (λ i (X t )). Implications. The processes Y t,j t τ j 0 λ j (X s )ds, 1 j m, are F- martingales. τ 1,..., τ m are conditionally independent given F X ; in particular no joint defaults. The pair process (X, Y) is Markov wrt F. 10
Examples 1. Homogeneous model (default intensities of all firms are identical). Default intensities are modelled by some increasing function λ : {1,..., K} (0, ) of the states of the economy. Elements of S X thus represent different states of the economy (1 is the best state and K the worst state). Various possibilities for generator Q X ; a very simple model takes X to be constant (Bayesian analysis instead of filtering). 2. Global- and industry factors. Assume that we have r different industry groups. Let S X = {1,..., κ} {0, 1} r ; write X 0,..., X r for the components of X, modelled as independent Markov chains. X r is the state of industry r which is good (X r = 0) or bad (X r = 1); X 0 represents the global factor. Default intensity of firm i from industry group r takes the form λ i (x) = g i (x 0 ) + f i (x r ) for increasing functions f i and g i. 11
Full-information-values Define the full-information value of a FT Y typical credit derivative) by -measurable claim H (a E Q( H F t ) =: h(t, X t, Y t ) ; (1) the last definition makes sense since (X, Y ) is Markov w.r.t. F. Computation of full-information values. Many possibilities: Bond prices or legs of a CDS can be computed via Feynman-Kac For portfolio products such as CDOs we can use conditional independence and compute Laplace transform of portfolio loss, (as in [Graziano and Rogers, 2006]) or use Poisson- and normal approximations, combined with Monte Carlo. 12
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 RCLL 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 with Ût = E(U t F M t ) for all t 0. 13
3. Dynamics of Security Prices Traded securities. We consider N liquidly traded credit derivatives (eg. corporate bonds) with maturity T and FT Y -measurable payoff P T,1,..., P T,N. We use martingale modelling: A3 Prices of traded securities are given by p t,i := E Q( P T,i F M t ). Market-pricing. Denote by p i (t, X t, Y t ) the full-information value of security i. We get from iterated conditional expectations p t,i = E ( E(P T,i F t ) F M t ) = E ( pi (t, X t, Y t ) F M t ). (2) Note that this is solved if we know the conditional distribution of X t given Ft M (a nonlinear filtering problem). Goal. Study the dynamics of traded security prices p t,i ; this is a prerequisite for hedging and risk management. 14
Innovations processes As a first towards determining the dynamics of the traded security prices step we introduce the innovations processes: M t,j := Y t,j µ t,i := Z t,i t τj 0 t 0 λ j (X s )ds, j = 1,, m â i (X s ) ds, i = 1,, l. Properties. M j is an F M -martingale and µ is F M -Brownian motion. Every F M -martingale can be represented as stochastic integral wrt M and µ. 15
General filtering equations Proposition 1 (General filtering equations). Consider a F- semimartingale of the form J t = J 0 + t 0 A sds + Mt J, M J an F- martingale with [M J, B] = 0. Suppose that [J, Y i ] t = t 0 RJ,i s dy s,i. Then Ĵ has the representation Ĵ t = Ĵ0 + t 0 Â s ds + t 0 γ s dm s + t 0 α s dµ s ; (3) γ and α are given by α t = J t a(x t ) Ĵtâ(X t), (4) γ t,i = 1 ( ( λ i ) (Ĵλ i) t + Ĵt ( λ i ) t + ( R ) J,i λ i ) t. (5) t Proof based on innovations approach to nonlinear filtering. 16
Security-price dynamics Theorem 2. Under A1 - A3 the (discounted) price process of the traded securities has the martingale representation p t,i = p 0,i + t 0 γ p i, s dm s + α p i t = p t,i a t p t,i â t γ p i t,j = as in (5) with R p i,j t t 0 α p i, s dµ s, with = p i (t, X t, Y i t ) p(t, X t, Y 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 t,n + l n=1 α p i t,nα p j t,n. (6) 17
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. Kushner-Stratonovich equation. (K-dim SDE-system for π) Let q(ι, k), 1 ι, k K denote generator matrix of X. Then K dπt k = q(ι, k)πtdt ι + (γ k (π t )) dm t + (α k (π t )) dµ t, with ι=1 (7) ( γj k λ j (k) ) (π) = π k K n=1 λ 1, j(n)π n 1 j m, (8) α k (π) = π k (a(k) K n=1 ) π n a(n). (9) 18
Default contagion Updating at the default time τ j. ( πτ k j = πτ k λ j (k) ) j K n=1 λ 1. j(n)πτ n j Default contagion. At τ j default intensity of firm i jumps: λ τj,i λ τj,i = K ( ) λ i (k) πτ k λ j (k) j K l=1 λ 1 = covπ ( ) τ j λ i, λ j j(l)πτ l j E π τ j (λ j ) k=1. 19
The filter in action 20
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. 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 ) F I t ) K = E(πt k Ft I ) h t (k), k=1 i.e. pricing wrt F I reduces to finding E(π k t F I t ). 21
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 ν t (dπ), the conditional distribution of π given F I t. 22
Modelling F I and Calibration Strategies Pragmatic calibration. Here 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 )) of fundamental values has full rank, the vector π t could be implied by standard 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. 23
Preliminary numerical results Left: itraxx spreads from last winter for different maturities; Right: homogeneous model with 3 states and state probabilities calibrated to itraxx; note that probability of worst state increases over time. 24
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 (7). Hence computation of the conditional distribution of π t given Ft I is a nonlinear filtering problem with signal process π and observation process U and Y. 25
Filtering problem for secondary-market investors Challenging problem: Observations of mixed type; Joint jumps of state process π and observation Y at defaults (see for instance [Frey and Runggaldier, 2007]) Typically high-dimensional problem use particle filtering as in [Crisan and Lyons, 1999] Numerical analysis work in progress. 26
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