Conditional survival probabilities under partial information: a recursive quantization approach with applications
CConditional survival probabilities under partial information:a recursive quantization approach with applications
Cheikh MBAYE ∗ Abass SAGNA † Frédéric VRINS ‡ Abstract
We consider a structural model where the survival/default state is observed together with anoisy version of the firm value process. This assumption makes the model more realistic thanmost of the existing alternatives, but triggers important challenges related to the computation ofconditional default probabilities. In order to deal with general diffusions as firm value process,we derive a numerical procedure based on the recursive quantization method to approximate it.Then, we investigate the error approximation induced by our procedure. Eventually, numericaltests are performed to evaluate the performance of the method, and an application is proposedto the pricing of CDS options.
Keywords: default model, structural model, noisy information, non-linear filtering, credit risk.
In the recent decades, credit risk received an increasing attention from academics and practitioners.In particular, the 2008 financial crisis shed the light on the importance of having sound credit riskmodels to better asses the default likelihood of firms and counterparties. The structural approach isone of the two most popular frameworks. It is originated to the seminal work of Merton [14] anduses the dynamics of structural variables of a firm, such as asset and debt, to determine whether thefirm defaulted before a given maturity. To better deal with the actual timing of the default event,first passage time models were then introduced. Among them is the celebrated Black and Coxmodel [1] which adds a time-dependent barrier, among others. Yet, the Black and Cox model hasfew parameters and is not easily calibrated to structural data such as CDS quotes along differentmaturities. To that end, extensions of the same models called AT1P and SBTV were introducedin [2] and [3] allowing exact calibration to credit spreads using efficient closed-form formulas fordefault probabilities.In practice however, it is difficult for investors to perfectly assess the value of the firm’s assets.In this case, modeling the firm value in a Black-Cox framework is problematic, since the modelassumes that the firm’s underlying assets are observable. Moreover, in such a framework, the defaulttime is predictable, leading to vanishing credit spreads for short maturities. In order to address ∗ Louvain Finance Center, UCLouvain, Voie du Roman Pays 34, 1348 Louvain-la-Neuve, Belgium, e-mail: [email protected] . The research of C. Mbaye is funded by the National Bank of Belgium and anFSR grant. † Fédération de Mathématiques d’Evry, Laboratoire Analyse et Probabilités, 23 Boulevard de France, 91037 Evry, &ENSIIE, e-mail: [email protected] . This research benefited from the support of the “ Chaire Marchés enMutation”, Fédération Bancaire Française. ‡ Louvain Finance Center, UCLouvain, Voie du Roman Pays 34, 1348 Louvain-la-Neuve, Belgium, e-mail: [email protected] . a r X i v : . [ q -f i n . M F ] S e p hese drawbacks, Duffie and Lando [7] proposed a model where the investors have only partialinformation on the firm value and observe at discrete time intervals a noisy accounting report. Thedefault time becomes totally inaccessible in the market filtration. As a result, the correspondingshort term spreads are always higher compared to the complete information short term spreads.Alternatively, some extensions of this model based on noisy information in continuous time can befound among others in [6] or recently in [9].In this paper, we both generalize and improve results derived in earlier studies. The modelspresented in [5] and [20] can deal with arbitrary firm-value diffusions, but are heavy. Moreover, theconsidered information flow is only made of a noisy version of the firm-value. This is not realisticas in practice, investors can obviously observe the default state of the firm. This larger filtration isconsidered in [6], but under the restrictive assumption that the firm-value process is a continuousand invertible function of a Gaussian martingale. In this work, we consider the same information setas [6] but relax the restriction regarding the firm-value dynamics. To deal with this general case, wepropose a numerical scheme based on fast quantization recently introduced in [17]. This techniqueis faster compared to [5] and [20] as there is no need to rely on Monte-Carlo simulations to computethe conditional survival probabilities. A detailed analysis of the error induced by the approximationis provided. Eventually, we illustrated our method on the pricing of CDS option credit derivatives.The reminder of the paper is organized as follows. In Section 2, we introduce the model.Different information flows and corresponding survival probabilities will be discussed. Section 3presents the estimations of the survival probabilities using the recursive quantization method andthe stochastic filtering theory. We then give a brief introduction to the quantization method beforederiving the error analysis pf these estimations. Section 4 is devoted to the results of the numericalexperiments. Assume a probability space (Ω , F , P ) , modeling the uncertainty of our economy. We consider astructural default model, and represent the default time τ X of a reference entity as the first passagetime of firm value process,below a default threshold. More precisely, we consider a Black and Coxsetup [1], where the stochastic process X represents the actual value of the firm and a ∈ R standsfor the default barrier. Assuming τ X > , we have: τ X := inf { u ≥ X u ≤ a } , < a < X (1)where inf ∅ := + ∞ , as usual. We restrict ourselves to consider ≤ t ≤ T where T is a finite timehorizon.We consider a partial information model where the true firm value X (called signal process hereafter) is not observable and we only observe Y ( observation process ), which is correlated with X . We suppose that the dynamics of X and Y are governed by the following stochastic differentialequations (SDEs) : (cid:40) dX t = b ( t, X t ) dt + σ ( t, X t ) dW t , X = x ,dY t = h ( t, Y t , X t ) dt + ν ( t, Y t ) dW t + δ ( t, Y t ) d (cid:102) W t , Y = y , (2)where ( W, (cid:102) W ) is a standard two-dimensional Brownian motion. We suppose that the functions b, σ, ν, δ : [0 , + ∞ ) × R → R are Lipschitz in x uniformly in t and that σ ( t, x ) > for every ( t, x ) ∈ [0 , + ∞ ) × R . These conditions ensure that the above SDEs admit a unique strong solution.Moreover we assume that h is locally bounded and Lipschitz in ( y, x ) , uniformly in t and that ν ( t, y ) > and σ ( t, y ) > for every ( t, y ) ∈ [0 , + ∞ ) × R .2 .1 Information flows One of the major critiques of such models is that in practice, the firm value is not observable. It istherefore not realistic to consider that the information available to the investor is F X := ( F Xt ) t ≥ , F Xt := σ ( X u , ≤ u ≤ t ) . A more realistic framework has been proposed in [5] where the investorinformation is given by the natural filtration F Y of a noisy version Y of the process X . However,one might argue that this way of modeling the information is not realistic either since given F Yt , theinvestor is unable to know whether the reference entity defaulted or not by time t . In other words,the default indicator process H = ( H t ) t ≥ , H t := { t ≥ τ X } , t ≥ , is not adapted to F Y .In this paper, we address this point by considering a more realistic information flow, defined asthe progressive enlargement of F Y with F H , the natural filtration of the default indicator process, F Ht := σ ( H u , ≤ u ≤ t ) , t ≥ . In other words, we have two investors’ information flows. On the one hand we have the informationavailable to the common investor, defined as the progressive enlargement of the natural filtration ofthe default indicator process with that of the noisy firm-value, noted F = ( F t ) t ≥ where F t := F Yt ∨ F Ht , t ≥ . In this setup, the following relationships hold: F H (cid:40) F X = F W (cid:40) G and F Y (cid:40) G . where G := ( G t ) t ≥ is the full information, i.e., the information available for example to a smallnumber of stock holders of the company, who have access to Y and X .On the other hand, the natural filtration of the actual (i.e. noise-free) firm-value process, F X ,could be seen as the information available to insiders. Economically, X would represent the valueof the firm, which is unobservable to the common investors, while Y might be the market price ofan asset issued by the firm, accessible to all market participants, and a would stand for the solvencycapital requirement imposed by regulators. Remark 2.1.
The results in the paper can be straightforwardly extended in the case when the defaultbarrier a is a piecewise constant function of time a : [0 , ∞ ) → [0 , ∞ ) , with < a (0) < x . Theother extensions are beyong the scope of this paper. Nevertheless, let us mention that in, e.g., [19]crossing probabilities for the Brownian motion are obtained in the case when the (double) barrier isa piecewise linear function on [0 , T ] and approximations for crossing probabilities are obtained forgeneral nonlinear bounds. A fundamental output of a credit model is the survival probability of the firm up to time t conditionalupon F s , s ≤ t ≤ T : P ( τ X > t |F s ) = E (cid:16) { τ X >t } (cid:12)(cid:12)(cid:12) F s (cid:17) (3)Note that this probability collapse to zero whenever { τ X ≤ s } . Recall that the specificity of ourapproach is that, the actual value of the firm X is not revealed in F s ; only a noisy version Y isaccessible.Using the Markov property of X , the fact that the two Brownian motions are independent andthe chain rule of the conditional expectation, we show the following result.3 roposition 2.1. We have, for s ≤ t , P (cid:16) τ X > t (cid:12)(cid:12)(cid:12) F s (cid:17) = { τ X >s } E (cid:2) { τ X >s } F ( s, t, X s ) |F Ys (cid:3) P (cid:16) τ X > s (cid:12)(cid:12)(cid:12) F Ys (cid:17) (4) where, for every x ∈ R , F ( s, t, x ) := P (cid:18) inf s a (cid:12)(cid:12)(cid:12) X s = x (cid:19) (5) is the conditional survival probability under full information. Furthermore, it holds on the set { τ X > s } that P (cid:0) τ X > t |F Ys (cid:1) ≤ P (cid:0) τ X > t |F s (cid:1) . (6) Proof.
Using a key result in the theory of conditional expectations commonly referred to as the
Keylemma (see e.g. Lemma 3.1 in [8]) we have P (cid:16) τ X > t (cid:12)(cid:12)(cid:12) F s (cid:17) = { τ X >s } E (cid:16) { τ X >t } (cid:12)(cid:12)(cid:12) F Ys ∨ F Hs (cid:17) = { τ X >s } E (cid:16) { τ X >t } (cid:12)(cid:12)(cid:12) F Ys (cid:17) P (cid:16) τ X > s (cid:12)(cid:12)(cid:12) F Ys (cid:17) = { τ X >s } P (cid:0) τ X > t |F Ys (cid:1) P (cid:16) τ X > s (cid:12)(cid:12)(cid:12) F Ys (cid:17) . (7)The proof is completed by noting that P (cid:0) τ X > t |F Ys (cid:1) = E (cid:2) { τ X >s } F ( s, t, X s ) |F Ys (cid:3) (see e.g.[20]). The particular case follows from (7) since on the event { τ X > s } , P (cid:0) τ X > t |F Ys (cid:1) = P ( τ X > t |F s ) P (cid:0) τ X > s |F Ys (cid:1) ≤ P ( τ X > t |F s ) or, equivalently, P (cid:0) τ X ≤ t |F s (cid:1) ≤ P (cid:0) τ X ≤ t |F Ys (cid:1) . Remark 2.2.
The inequality (6), confirmed by the numerical experiments in Section 4, means thatthe less we have information on the state of the system ( F Ys ⊂ F Ys ∨ F Hs = F s ), the higher thedefault probability. This also shows the difference with [20] where the quantity of interest is just P (cid:0) τ X > t |F Ys (cid:1) . Note that we have clearly stated the expression of interest, namely the survival probability of thereference entity up to time t conditional upon the investor’s information up to time s , we need toactually compute it. In order to even more comply with real market practice, we further considerthat we can only access to, say, discrete time observations of Y up to time s . To that end, let us startby fixing a time discretization grid over [0 , t ] : t < · · · < t m = s < t m +1 < · · · < t n = t. Our aim is to approximate the right hand side of (4) by recursive quantization. In some specificmodels (those for which (2) admits an explicit solution ( X, Y ) , like in the Black-Scholes frame-work), we will consider the discrete trajectories ( X t k , Y t k ) k =0 ,...,n . In more general models, weneed to make a discrete time approximation of the quantity of interest. To this end, we suppose4hat we have access to a trajectory of Y sampled at m times: ( ¯ Y t , . . . , ¯ Y t m ) , with t = 0 and t m = s (which in practice will be approximated from the paths of the Euler scheme associated tothe stochastic process Y ) and will estimate (3) by P (cid:16) τ X > t (cid:12)(cid:12)(cid:12) F ¯ Ys (cid:17) P (cid:16) τ X > s (cid:12)(cid:12)(cid:12) F ¯ Ys (cid:17) , on the event { τ X > s } , where F ¯ Ys = σ ( ¯ Y t k , t k ≤ s ) = σ ( ¯ Y t , . . . , ¯ Y t m ) . We denote by ¯ X the continuous Euler scheme associated to the process X in Equation (2), namely: ¯ X s = ¯ X s + b ( s, ¯ X s )( s − s ) + σ ( s, ¯ X s )( W s − W s ) , ¯ X = x , with s = t k if s ∈ [ t k , t k +1 ) , for k = 0 , . . . , n . Based on the Euler scheme, we introduce thediscretized version of our state-observation processes ( ¯ X, ¯ Y ) (cid:40) ¯ X t k +1 = ¯ X t k + b ( t k , ¯ X t k )∆ k + σ ( t k , ¯ X t k )( W t k +1 − W t k )¯ Y t k +1 = ¯ Y t k + h ( t k , ¯ Y t k , ¯ X t k )∆ k + ν ( t k , ¯ Y t k )( W t k +1 − W t k ) + δ ( t k , ¯ Y t k )( (cid:102) W t k +1 − (cid:102) W t k ) (8)where k ∈ { , . . . , n − } for the signal process and k ∈ { , . . . , m − } for the observation processand where ∆ k := t k +1 − t k .Supposing that we have access to a discrete trajectory of Y , ( ¯ Y t , . . . , ¯ Y t m ) , our first goal is toapproximate (recall that t m = s ) P (cid:0) τ X > t | F Ys (cid:1) P (cid:0) τ X > s | F Ys (cid:1) by P (cid:0) τ ¯ X > t |F ¯ Ys (cid:1) P (cid:0) τ ¯ X > s |F ¯ Ys (cid:1) , (9)where (recall Equation (1)) τ ¯ X := inf { u ≥ , ¯ X u ≤ a } . Using the Brownian Bridge method and the Markov property of ( ¯ X t k , ¯ Y t k ) k , we show that thequantity (9) can be written in a closed formula. Theorem 2.2.
We have: P (cid:0) τ ¯ X > t |F ¯ Ys (cid:1) P (cid:0) τ ¯ X > s |F ¯ Ys (cid:1) = Ψ( ¯ Y t , . . . , ¯ Y t m ) , (10) where for y = ( y , . . . , y m ) ∈ R m +1 , Ψ( y ) = E (cid:2) ¯ F ( t m , t n , ¯ X t m ) K m a L my (cid:3) E [ K m a L my ] , (11) with K m a = m − (cid:89) k =0 G ¯ X tk , ¯ X tk +1 ∆ k σ k ( a ) , L my = m − (cid:89) k =0 g k ( ¯ X t k , y k ; ¯ X t k +1 , y k +1 ) and where for every x ∈ R , ¯ F ( t m , t n , x ) = E (cid:104) n − (cid:89) k = m G ¯ X tk , ¯ X tk +1 ∆ k σ k ( a ) (cid:12)(cid:12) ¯ X t m = x (cid:105) . (12)5 he function g k is defined by g k ( x k , y k ; x k +1 , y k +1 ) = P (cid:0) ( ¯ X t k +1 , ¯ Y t k +1 ) = ( x k +1 , y k +1 ) | ( ¯ X t k , ¯ Y t k ) = ( x k , y k ) (cid:1) P (cid:0) ¯ X t k +1 = x k +1 | ¯ X t k = x k (cid:1) = 1(2 π ∆ k ) / δ k exp (cid:32) − ν k δ k ∆ k (cid:16) x k +1 − m k σ k − y k +1 − m k ν k (cid:17) (cid:33) (13) with m k := x k + b k ∆ k and m k := y k + h k ∆ k . Finally, G x k ,x k +1 ∆ k σ k ( a ) = P (cid:0) inf u ∈ [ t k ,t k +1 ] ¯ X u ≥ a | ¯ X t k = x k (cid:1) = (cid:18) − exp (cid:18) − x k − a )( x k +1 − a )∆ k σ ( t k , x k ) (cid:19)(cid:19) { x k ≥ a ; x k +1 ≥ a } . (14) Proof.
Following Theorem 2.5. in [20], we have, P (cid:0) τ ¯ X > t |F ¯ Ys (cid:1) = E (cid:2) ¯ F ( t m , t n , ¯ X t m ) K m a L my (cid:3) E [ L my ] and P (cid:0) τ ¯ X > s |F ¯ Ys (cid:1) = E (cid:2) K m a L my (cid:3) E [ L my ] . The question of interest is now to know how to estimate efficiently Ψ( y ) for y = ( ¯ Y t , . . . , ¯ Y t m ) .Owing to the form of the random vector K m a , we may put it together with L my to be reduced tosimilar formula as the filter estimate in a standard nonlinear filtering problem. In other work wemay write for y := ( y , . . . , y m ) , Ψ( y ) = E (cid:2) ¯ F ( t m , t n , ¯ X t m ) L my, a (cid:3) E [ L my, a ] where L my, a = m − (cid:89) k =0 g k ( ¯ X t k , y k ; ¯ X t k +1 , y k +1 ) × G ¯ X tk , ¯ X tk +1 ∆ k σ k ( a ) . Notice that defining the operator π y,m , for every bounded measurable function f , by π y,m f := E (cid:2) f ( ¯ X t m ) L my, a (cid:3) , we have Ψ( y ) = π y,m ¯ F ( t m , t n , · ) π y,m =: Π y,m ¯ F ( t m , t n , · ) , (15)where ( x ) = 1 , for every real x . Then it is enough to tell how to compute the numerator π y,m ¯ F ( t m , t n , · ) = E (cid:104) ¯ F ( t m , t n , ¯ X t m ) m − (cid:89) k =0 g a k ( ¯ X t k , y k ; ¯ X t k +1 , y k +1 ) (cid:105) where g a k ( ¯ X t k , y k ; ¯ X t k +1 , y k +1 ) = g k ( ¯ X t k , y k ; ¯ X t k +1 , y k +1 ) × G ¯ X tk , ¯ X tk +1 ∆ k σ k ( a ) . At this stage, several methods involving Monte Carlo simulations as the particle method can beused to approximate Π y,m . Optimal quantization is an alternative and some times as a substitute6o the Monte Carlo method to approximate such a quantity (we refer to [22] for a comparison ofparticle like methods and optimal quantization methods).To use the optimal quantization methods we have to quantize the marginals of the process ( ¯ X t k ) k , means, to represent every marginal ¯ X t k , k = 0 , . . . , n , by a discrete random variable (cid:98) X Γ k t k (we will simply denote it (cid:98) X t k when there is no ambiguity) taking N k values Γ k = { x k , . . . , x kN k } .As we will see later, we have also need in our context to compute the transition probabilities ˆ p ijk = P ( (cid:98) X t k = x kj | (cid:98) X t k − = x k − i ) , for i = 1 , . . . , N k − ; j = 1 , . . . , N k . To this end, wemay use stochastic algorithms to get the optimal grids and the associated transition probabilities(see e.g. [16, 22]). This method works well but may be very time consuming. The so-calledmarginal functional quantization method (see [5, 20, 21]) is used as an alternative to the previousmethod. It consists to construct the marginal quantizations by considering the ordinary differentialequation (ODE) resulting to the substitution of the Brownian motion appearing in the dynamics of X in (2) by a quadratic quantization of the Brownian motion (see [12]). This procedure performsthe marginal quantizations quite instantaneous and works well enough from the numerical point ofview even if the rate of convergence (which has not been computed yet from the theoretical pointof view) seems to be poor. As an alternative to the two previous methods, we propose the recursivemarginal quantization (also called fast quantization) method introduced in [17]. It consists of quan-tizing the process ( ¯ X t k ) k =0 ,...,n , based on a recursive method involving the conditional distributions ¯ X t k +1 | ¯ X t k , k = 0 , . . . , n − . For the problem of interest, this last method is more performing thanthe previous ones due to its computation speed and to its robustness.On the other hand, the function ¯ F has been estimated by Monte Carlo method in [5, 20]. Forcompetitiveness reasons of the recursive quantization w.r.t. the previously raised methods, we pro-pose here to approximated both quantities Π and ¯ F by the recursive quantization method. Π y,m by recursive quantization Given that the denominator in the right hand side of (15) has a similar form as the numerator, wewill only show how to compute the numerator. We remark that π y,m can be computed from thefollowing recursive formula: π y,k = π y,k − H y,k , k = 1 , . . . , m, (16)where, for every k = 1 , . . . , m , and for every bounded and measurable function f , the transitionkernel H y,k is defined by H y,k f ( z ) = E (cid:2) f ( ¯ X t k ) g a k − ( ¯ X t k − , y k − ; ¯ X t k , y k ) | ¯ X t k − = z (cid:3) with H y, f := E [ f ( ¯ X )] . In fact, for any bounded Borel function f we have π y,k f = E (cid:104) f ( ¯ X t k ) k − (cid:89) (cid:96) =0 g a (cid:96) ( ¯ X t (cid:96) , y (cid:96) ; ¯ X (cid:96) +1 , y (cid:96) +1 ) (cid:105) = E (cid:104) E (cid:16) f ( ¯ X t k ) k − (cid:89) (cid:96) =0 g a (cid:96) ( ¯ X t (cid:96) , y (cid:96) ; ¯ X (cid:96) +1 , y (cid:96) +1 ) (cid:12)(cid:12) F ¯ Xk − (cid:1)(cid:105) . ¯ X still be a Markov process we deduce that π y,k f = E (cid:104) E (cid:16) f ( ¯ X t k ) g a k − ( ¯ X t k − , y k − ; ¯ X k , y k ) (cid:12)(cid:12) F ¯ Xk − (cid:1) k − (cid:89) (cid:96) =0 g a (cid:96) ( ¯ X t (cid:96) , y (cid:96) ; ¯ X (cid:96) +1 , y (cid:96) +1 ) (cid:105) = E (cid:104) H y,k f ( ¯ X t k − ) k − (cid:89) (cid:96) =0 g a (cid:96) ( ¯ X t (cid:96) , y (cid:96) ; ¯ X (cid:96) +1 , y (cid:96) +1 ) (cid:105) = π y,k − H y,k f. Then, when we have access to the quantization of the marginals of the process ¯ X , the functional π y,k can be approximated recursively by optimal quantization as ˆ π y,k = ˆ π y,k − (cid:98) H y,k where for every k ≥ , (cid:98) H y,k is a matrix N k × N k − which components (cid:98) H i,jy,k read (cid:98) H ijy,k = g a k − ( x ik − , y k − ; x jk , y k ) ˆ p ijk δ x jk where p ijk = P ( (cid:98) X t k = x jk | (cid:98) X t k − = x ik − ) and ( (cid:98) X t k ) k is the quantization of the process ( ¯ X t ) t ≥ over the time steps t k , k = 1 , . . . , m : on thegrids Γ k = { x k , . . . , x N k k } , of sizes N k .As a consequence, the quantity of interest Π y,m ¯ F ( t m , t n , · ) is estimated by (cid:98) Π y,m ¯ F ( t m , t n , · ) = N m (cid:88) i =1 (cid:98) Π iy,m ¯ F ( t m , t n , x im ) . (17)where (cid:98) Π iy,m := (cid:98) π iy,m (cid:80) N m j =1 (cid:98) π iy,m , i = 1 , . . . , N m and where (cid:98) π y,m is the estimation (by optimal quantization) of π y,m defined recursively by (cid:98) π y, = (cid:98) H y, (cid:98) π y,k = (cid:98) π y,k − (cid:98) H y,k := (cid:104) (cid:80) N k − i =1 (cid:98) H i,jy,k (cid:98) π iy,k − (cid:105) j =1 ,...,N k , k = 1 , . . . , m (18)with (cid:98) H ijy,k = g a k − ( x ik − , y k − ; x jk , y k ) ˆ p ijk δ x jk . (19)Our aim is now to use the (marginal) recursive quantization method to estimate the F ( t m , t n , x im ) (cid:48) s. ¯ F ( t m , t n , · ) by recursive quantization Recall that for every x , ¯ F ( t m , t n , x ) = E (cid:16) n − (cid:89) k = m G ¯ X tk , ¯ X tk +1 ∆ k σ k ( a ) (cid:12)(cid:12) ¯ X t m = x (cid:17) . As previously, we remark that if we define the functional π n,m by (cid:0) π n,m f (cid:1) ( x ) = E (cid:16) f ( ¯ X t n ) n − (cid:89) k = m G ¯ X tk , ¯ X tk +1 ∆ k σ k ( a ) (cid:12)(cid:12) ¯ X t m = x (cid:17) , f , then F ( t m , t n , x ) reads ¯ F ( t m , t n , x ) = (cid:0) π n,m (cid:1) ( x ) . Now, for every bounded and measurable function f , defining as previously the transition kernel H k as, (cid:0) H k f (cid:1) ( z ) = E (cid:18) f ( ¯ X t k ) G ¯ X tk − , ¯ X tk ∆ k − σ k − ( a ) | ¯ X t k − = z (cid:19) , for every k = m + 1 , . . . , n and setting H m f = E (cid:2) f ( ¯ X t m ) (cid:3) (20)yields for every k = m + 1 , . . . , n , ( π k,m f )( x ) = E (cid:16) E (cid:16) f ( ¯ X t k ) k − (cid:89) i = m G ¯ X tk , ¯ X tk +1 ∆ k σ k ( a ) (cid:12)(cid:12) ( ¯ X t (cid:96) ) (cid:96) = m,...,k − (cid:17)(cid:12)(cid:12) ¯ X t m = x (cid:17) = E (cid:16) E (cid:16) f ( ¯ X t k ) G ¯ X tk − , ¯ X tk ∆ k − σ k − ( a ) | ( ¯ X t (cid:96) ) (cid:96) = m,...,k − (cid:17) k − (cid:89) (cid:96) = m G ¯ X t(cid:96) , ¯ X t(cid:96) +1 ∆ (cid:96) σ (cid:96) ( a ) | ¯ X t m = x (cid:17) = E (cid:16) E (cid:16) f ( ¯ X t k ) G ¯ X tk − , ¯ X tk ∆ k − σ k − ( a ) | ¯ X t k − (cid:17) k − (cid:89) (cid:96) = m G ¯ X t(cid:96) , ¯ X t(cid:96) +1 ∆ (cid:96) σ (cid:96) ( a ) | ¯ X t m = x (cid:17) = ( π k − ,m H k f )( x ) . Consequently, if one has access to the recursive quantizations ( (cid:98) X t k ) k = m,...,n and the transitionprobabilities { ˆ p ijk , k = m + 1 , . . . , n } of the process ( ¯ X t k ) k = m,...,n , the quantity F ( t m , t n , x ) willbe estimated by (cid:98) F ( t m , t n , x ) = N n (cid:88) j =1 (cid:98) π n,m δ { x jm = x } , (21)where the (cid:98) π n,m ’s are defined from the following recursive formula (cid:98) π m,m = (cid:98) H m (cid:98) π k,m = (cid:98) π k − ,m (cid:98) H k := (cid:104) (cid:80) N k − i =1 (cid:98) H i,jk (cid:98) π k − ,m (cid:105) j =1 ,...,N k , k = m + 1 , . . . , n (22)with (cid:98) H ijk = G x ik − ,x jk ∆ k − σ k − ( a ) ˆ p ijk δ x jk , i = 1 , . . . , N k − ; j = 1 , . . . , N k . Π y,m ¯ F ( t m , t n , · ) by recursive quantization Combining equations (17) and (21), the conditional survival probability Π y,m F ( t m , t n , · ) will beestimated (for a fixed trajectory ( y , . . . , y m ) of the observation process ( Y t , . . . , Y t m ) ) by (cid:98) Π y,m (cid:98) F ( t m , t n , · ) = N m (cid:88) i =1 N n (cid:88) j =1 (cid:98) Π iy,m (cid:98) π n,m δ x jm . (23)9 emark 3.1. In Section 4, the formula (23) will be compared to the one of interest in [20]: P (cid:0) τ ¯ X >t |F ¯ Ys (cid:1) , which reads (following the previous notations) P (cid:0) τ ¯ X > t | ( ¯ Y t , . . . , ¯ Y t m ) = y (cid:1) = E (cid:2) ¯ F ( t m , t n , ¯ X t m ) L my, a (cid:3) E [ L my ] . (24)The conditional probability has been approximated in [20] via an hybrid Monte Carlo - optimalquantization method. It may be approximated following the procedure we propose using onlyoptimal quantization method as (cid:98) (cid:36) y,m (cid:98) F ( t m , t n , · ) = N m (cid:88) i =1 N n (cid:88) j =1 (cid:98) (cid:36) iy,m (cid:98) π n,m δ x jm (25)where the (cid:98) (cid:36) iy,m ’s are obtained from (18) by replacing the function g a k by g k of equation (13).Let us say now how to quantize the signal process X from the recursive quantization method. Recall first that for a given R d -valued random vector X defined on (Ω , F , P ) with distribution P X , the L r ( P X ) -optimal quantization problem of size N for X (or for the distribution P X ) aimsto approximate X by a Borel function of X taking at most N values. If X ∈ L r ( P ) and defining (cid:107) X (cid:107) r := ( E | X | r ) /r where |·| denotes an arbitrary norm on R d , this turns out to solve the followingoptimization problem (see e.g. [11]): e N,r ( X ) = inf {(cid:107) X − (cid:98) X Γ (cid:107) r , Γ ⊂ R d , card (Γ) ≤ N } = inf Γ ⊂ R d card (Γ) ≤ N (cid:18)(cid:90) R d d ( x, Γ) r d P X ( x ) (cid:19) /r (26)where (cid:98) X Γ , the quantization of X on the subset Γ = { x , . . . , x N } ⊂ R d (called a codebook, an N -quantizer or a grid) is defined by (cid:98) X Γ = Proj Γ ( X ) := N (cid:88) i =1 x i { X ∈ C i (Γ) } and where ( C i (Γ)) i =1 ,...,N is a Borel partition (Voronoi partition) of R d satisfying for every i ∈{ , . . . , N } , C i (Γ) ⊂ { x ∈ R d : | x − x i | = min j =1 ,...,N | x − x j |} . Keep in mind that for every N ≥ , the infimum in (26) is reached at one grid at least. Any N -quantizer realizing this infimum is called an L r -optimal N -quantizer. Moreover, if card(supp ( P X )) ≥ N then the optimal N -quantizer is of size N (see [11] or [15]). On the other hand, the quantiza-tion error, e N,r ( X ) , decreases to zero at an N − /d -rate as the grid size N goes to infinity. Thisconvergence rate (known as Zador Theorem) has been investigated in [4] and [23] for absolutelycontinuous probability measures under the quadratic norm on R d . A detailed study of the conver-gence rate under an arbitrary norm on R d and for both absolutely continuous and singular measuresmay be found in [11].The recursive quantization of the Euler scheme of an R d -valued diffusion process has been in-troduced in [17]. The method allows to speak of fast online quantization and consists on a sequenceof quantizations ( (cid:98) X Γ k t k ) k =0 ,...,n of the Euler scheme ( ¯ X t k ) k =0 ,...,n defined recursively as (cid:101) X = ¯ X , (cid:98) X Γ k t k = Proj Γ k ( (cid:101) X t k ) and (cid:101) X t k +1 = E k ( (cid:98) X Γ k t k , Z k +1 ) , k = 0 , . . . , n − , (27)10here ( Z k ) k =1 ,...,n is an i.i.d. sequence of N (0; I q ) -distributed random vectors, independent from ¯ X and E k ( y, z ) = y + ∆ b ( t k , y ) + √ ∆ σ ( t k , y ) z, y ∈ R d , z ∈ R q , k = 0 , . . . , n − . The sequence of quantizers satisfies for every k ∈ { , . . . , n } , Γ k ∈ argmin { (cid:101) D k (Γ) , Γ ⊂ R d , card(Γ) ≤ N k } , where for every grid Γ ⊂ R d , (cid:101) D k +1 (Γ) := E (cid:2) dist( (cid:101) X t k +1 , Γ) (cid:3) .This recursive quantization method raises some problems among which the computation of thequadratic error bound (cid:107) ¯ X t k − (cid:98) X Γ k t k (cid:107) := (cid:0) E | ¯ X t k − (cid:98) X Γ k t k | (cid:1) / , for every k = 0 , . . . , n . It has beenshown in [17, 18] that for any sequences of (quadratic) optimal quantizers Γ k for (cid:101) X Γ k t k , for every k = 0 , . . . , n − , the quantization error (cid:107) ¯ X t k − (cid:98) X Γ k t k (cid:107) is bounded by the cumulative quantizationerrors (cid:107) (cid:101) X t i − (cid:98) X Γ i t i (cid:107) , for i = 0 , . . . , k . This result is obtained under the following assumptions andis stated in Proposition 3.1 below:1. L -Lipschitz assumption . The mappings x (cid:55)→ E k ( x, Z k +1 ) from R d to L (Ω , A , P ) , k = 1 : n are Lipschitz continuous i.e. (Lip) ≡ ∀ x, x (cid:48) ∈ R d , (cid:13)(cid:13) E k ( x, Z k +1 ) − E k ( x (cid:48) , Z k +1 ) (cid:13)(cid:13) ≤ [ E k ] Lip | x − x (cid:48) | , k = 1 : n. L p -linear growth assumption . Let p ∈ (2 , . (SL) p ≡ ∀ k ∈ { , . . . , n } , ∀ x ∈ R d , E |E k ( x, Z k +1 ) | p ≤ α p,k + β p,k | x | p . Proposition 3.1.
Let (cid:98) X = ( (cid:98) X t k ) k =0: n be defined by (27) and suppose that all the grids Γ k arequadratic optimal. Assume that both assumptions (Lip) and (SL) p (for some p ∈ (2 , ) hold andthat X ∈ L p ( P ) . Then, (cid:13)(cid:13) ¯ X t k − (cid:98) X t k (cid:13)(cid:13) ≤ C d,p k (cid:88) i =0 [ E i +1: k ] Lip (cid:34) i (cid:88) (cid:96) =0 α p,(cid:96) β p,(cid:96) +1: i (cid:35) p N − d i (28) where C d,p > and α p, = E | X | p = (cid:107) X (cid:107) pp , β p,(cid:96) : i = (cid:81) im = (cid:96) β p,m (with (cid:81) ∅ = 1 ) and [ E i : k ] Lip := k (cid:89) (cid:96) = i [ E (cid:96) ] Lip , ≤ (cid:96) ≤ k ≤ n and [ E k +1: k ] Lip = 1 . The associated probability weights and transition probabilities are computed from explicit for-mulas we recall in the following result.
Proposition 3.2.
Let Γ k +1 be a quadratic optimal quantizer for the marginal random variable (cid:101) X t k +1 . Suppose that the quadratic optimal quantizer Γ k for (cid:101) X t k is already computed and that wehave access to its associated weights P ( (cid:101) X t k ∈ C i (Γ k )) , i = 1 , . . . , N k . The transition probability ˆ p ijk = P ( (cid:101) X t k +1 ∈ C j (Γ k +1 ) | (cid:101) X t k ∈ C i (Γ k )) = P ( (cid:98) X t k +1 = x jk +1 | (cid:98) X t k = x ik ) is given by ˆ p ijk = Φ (cid:0) x k +1 ,j + ( x ki ) (cid:1) − Φ (cid:0) x k +1 ,j − ( x ki ) (cid:1) , (29) where Φ( · ) is the cumulative distribution function of the standard Gaussian distribution, x k +1 ,j − ( x ) := x j − / k +1 − m k ( x ) v k ( x ) and x k +1 ,j + ( x ) := x j +1 / k +1 − m k ( x ) v k ( x ) , ith m k ( x ) = x + ∆ b ( t k , x ) , v k ( x ) = √ ∆ σ ( t k , x ) and, for k = 0 , . . . , n − and for j =1 , . . . , N k +1 , x j − / k +1 = x jk +1 + x j − k +1 , x j +1 / k +1 = x jk +1 + x j +1 k +1 , with x / k +1 = −∞ , x N k +1 +1 / k +1 = + ∞ . Once we have access to the marginal quantizations and to its associated transition probabilities,the right hand side of (23) can be computed explicitly.
Our aim in this section is to investigate the error resulting from the approximation of Π y,m ¯ F ( t m , t n , · ) = (cid:2) Π y,m (cid:0) π n,m (cid:1)(cid:3) ( · ) by (cid:2) (cid:98) Π y,m (cid:0) ˆ π n,m (cid:1)(cid:3) ( · ) . This error is an aggregation of three terms (see the proofof Theorem 3.6) involving the approximation errors | Π y,m ¯ F ( t m , t n , · ) − (cid:98) Π y,m ¯ F ( t m , t n , · ) | and | (cid:0) π n,m (cid:1) ( x ) − (cid:0) ˆ π n,m (cid:1) ( x ) | , x ∈ R . The two following results give the bounds associated twothe former approximation errors. Both are (carefully) adjustments of Theorem 4.1. and Lemma 4.1.in [16] to our context so that we refer to the former paper for their detailed proofs. Theorem 3.3.
Suppose that Assumption (
Lip ) holds true. Then, for any bounded Lipschitz contin-uous function f on R d we have, | Π y,m f − (cid:98) Π y,m f | ≤ K mg φ m ∨ ˆ φ m m (cid:88) k =0 A mk ( f, y ) (cid:107) ¯ X t k − (cid:98) X Γ k t k (cid:107) , where φ m := π y,m , (cid:98) φ m := (cid:98) π y,m ,A mk ( f, y ) := 2 (cid:107) f (cid:107) ∞ K mg [ g k ] Lip ( y k − , y k ) + 2 (cid:107) f (cid:107) ∞ K mg m (cid:88) j = k +1 [ E ] j − k − Lip (cid:16) [ g j ] Lip ( y j − , y j )+ [ E ] Lip [ g j ] Lip ( y j − , y j ) (cid:17) , and, for every k ∈ { , . . . , m } , [ g k ] Lip ( y, y (cid:48) ) and [ g k ] Lip ( y, y (cid:48) ) are such that for every x, x (cid:48) , ˆ x, ˆ x (cid:48) ∈ R d , | g ak ( x, y ; x (cid:48) , y (cid:48) ) − g ak ( (cid:98) x, y ; (cid:98) x (cid:48) , y (cid:48) ) | ≤ [ g k ] Lip ( y, y (cid:48) ) | x − (cid:98) x | + [ g k ] Lip ( y, y (cid:48) ) | x (cid:48) − (cid:98) x (cid:48) | . The quantities K g and [ E ] Lip are defined as K g = max k =1 , ··· ,m (cid:107) g ak (cid:107) ∞ and [ E ] Lip = max k =1 ,...,m [ E k ] Lip . Remark that the existence of [ g k ] Lip ( y k − , y k ) and [ g k ] Lip ( y k − , y k ) is guaranteed by the factthat the function g ak ( x, y ; x (cid:48) , y (cid:48) ) is Lipschitz with respect to ( x, x (cid:48) ) .Let us give now the error bound associated to the approximation of π n,m . Proposition 3.4.
Let y = ( y , . . . , y m ) ∈ ( R q ) m +1 . Then, we have for any x ∈ R . (cid:12)(cid:12)(cid:0) π n,m (cid:1) ( x ) − (cid:0) ˆ π n,m (cid:1) ( x ) (cid:12)(cid:12) ≤ n (cid:88) k = m +1 B k ( G ) (cid:107) ¯ X t k − (cid:98) X Γ k t k (cid:107) (30)12 here B k ( G ) = Λ n − m − (cid:16) [ G m ] Lip δ { k = m } + (cid:0) [ G k ] Lip ∨ [ G k − ] Lip (cid:1) δ { k ∈{ m +1 ,...,n − }} + [ G n ] Lip δ { k = n } (cid:17) with Λ = max k = m +1 ,...,n (cid:13)(cid:13) G ( • , • )∆ k σ k ( a ) (cid:13)(cid:13) ∞ Proof.
Recall that (cid:0) π n,m (cid:1) ( x ) = E (cid:16) Λ m ( ¯ X t m : n ) (cid:12)(cid:12) ¯ X t m = x (cid:17) where for every k ≥ m , Λ m ( ¯ X t m : k ) := k − (cid:89) (cid:96) = m G ¯ X tk , ¯ X tk +1 ∆ k σ k ( a ) with the convention that Λ m ( ¯ X t m : m ) = 1 . Now, we have for any k ≥ m , Λ m ( ¯ X t m : k ) − Λ m ( ˆ X t m : k ) = (cid:16) G ¯ X tk − , ¯ X tk ∆ k − σ k − ( a ) − G ˆ X tk − , ˆ X tk ∆ k − σ k − ( a ) (cid:17) Λ m ( ¯ X t m : k − )+ G ˆ X tk − , ˆ X tk ∆ k − σ k − ( a ) (cid:0) Λ m ( ¯ X t m : k − ) − Λ m ( ˆ X t m : k − ) (cid:1) . Since the function G ( • , • )∆ k σ k ( a ) is Lipschitz and bounded and that for any k ≥ m + 1 , Λ m ( ¯ X t m : k − ) ≤ Λ k − m − , we have | Λ m ( ¯ X t m : k ) − Λ m ( ˆ X t m : k ) | ≤ (cid:16) [ G k ] Lip | ¯ X t k − − ˆ X t k − | + [ G k ] Lip | ¯ X t k − ˆ X t k | (cid:17) Λ k − m − + Λ | Λ m ( ¯ X t m : k − ) − Λ m ( ˆ X t m : k − ) | . Keeping in mind that Λ m ( ¯ X t m : m ) = Λ m ( ˆ X t m : m ) = 1 , we deduce from an induction on k that | Λ m ( ¯ X t m : n ) − Λ m ( ˆ X t m : n ) | ≤ Λ n − m − n (cid:88) k = m +1 [ G k ] Lip | ¯ X t k − − ˆ X t k − | + [ G k ] Lip | ¯ X t k − ˆ X t k | . The result follows by noting that (cid:12)(cid:12)(cid:0) π n,m (cid:1) ( x ) − (cid:0) ˆ π n,m (cid:1) ( x ) (cid:12)(cid:12) ≤ E (cid:0) | Λ m ( ¯ X t m : n ) − Λ m ( ˆ X t m : n ) | (cid:12)(cid:12) ¯ X t m = x (cid:1) ≤ Λ n − m − n (cid:88) k = m +1 [ G k ] Lip | ¯ X t k − − ˆ X t k − | + [ G k ] Lip | ¯ X t k − ˆ X t k | and by using the non-decreasing property of the L p -norm.We may deduce now the global error induced by our procedure, means, the error deriving fromthe estimation of Π y,m F ( t m , t n , · ) = P (cid:0) τ X > t n | ( Y t , . . . , Y t m ) = y (cid:1) by Equation (23). To this end, we need the following additional assumptions which will be usedto compute (see [10]) the convergence rate of the quantity E (cid:12)(cid:12) { τ ¯ X >t } − { τ X >t } (cid:12)(cid:12) towards . Wesuppose that the diffusion is homogeneous and( H1 ) b is a C ∞ b ( R ) function and σ is in C ∞ b ( R ) .( H2 ) There exists σ > such that ∀ x ∈ R , σ ( x ) ≥ σ ( uniform ellipticity ).13e have the following result. Proposition 3.5 (See [10]) . Let t > . Suppose that Assumptions ( H1 ) and ( H2 ) are fulfilled. Then,for every η ∈ (0 , [ there exists an increasing function K ( T ) such that for every t ∈ [0 , T ] and forevery x ∈ R , E x (cid:2)(cid:12)(cid:12) { τ X >t } − { τ ¯ X >t } (cid:12)(cid:12)(cid:3) ≤ n − η K ( T ) √ t , where n is the number of discretization time steps over [0 , t ] . Theorem 3.6.
Suppose that the coefficients b and σ of the continuous signal process X are suchthat Assumptions (H1) and (H2) are satisfied and let η ∈ (0 , ] . We also suppose that Assumption( Lip ) holds. Then, for any bounded Lipschitz continuous function f : R d (cid:55)→ R and for any fixedobservation y = ( y , . . . , y m ) we have (cid:12)(cid:12)(cid:12) P (cid:0) τ X > t n | ( ¯ Y t , . . . , ¯ Y t m ) = y (cid:1) − N n (cid:88) i =1 N m (cid:88) j =1 (cid:98) Π iy,m (cid:98) π n,m δ x jm (cid:12)(cid:12)(cid:12) ≤ O (cid:16) n − + η (cid:17) + n (cid:88) k =0 C nk ( ¯ F ( s, t, · ) , y ) (cid:107) ¯ X t k − (cid:98) X Γ k t k (cid:107) , where C nk ( ¯ F ( s, t, · ) , y ) = K mg φ m ∨ ˆ φ m A mk ( f, y ) δ { k ≤ m } + B k ( G ) δ { k ≥ m +1 } and where K mg , φ m , ˆ φ m , A mk ( f, y ) and B k ( G ) are defined in Theorem 3.3 and in Proposition 3.4. Proof . We have (cid:12)(cid:12)(cid:12) P ( τ X > t n | ( ¯ Y t , . . . , ¯ Y t m ) = y ) − N n (cid:88) i =1 N m (cid:88) j =1 (cid:98) Π iy,m (cid:98) π n,m δ x jm (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12) P ( τ X > t n | ( ¯ Y t , . . . , ¯ Y t m ) = y ) − P ( τ ¯ X > t n | ( ¯ Y t , . . . , ¯ Y t m ) = y ) (cid:12)(cid:12) + (cid:12)(cid:12) Π y,m ¯ F ( t m , t n , · ) − (cid:98) Π y,m ¯ F ( t m , t n , · ) (cid:12)(cid:12) + (cid:12)(cid:12) (cid:98) Π y,m ¯ F ( t m , t n , · ) − (cid:98) Π y,m ˆ F ( t m , t n , · ) (cid:12)(cid:12) . (31)Now, we have (cid:12)(cid:12) (cid:98) Π y,m ¯ F ( t m , t n , · ) − (cid:98) Π y,m ˆ F ( t m , t n , · ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) N m (cid:88) i =1 (cid:16) (cid:98) Π iy,m ¯ F ( t m , t n , x im ) − (cid:98) Π iy,m ˆ F ( t m , t n , x im ) (cid:17)(cid:12)(cid:12)(cid:12) ≤ sup x ∈ R (cid:12)(cid:12) ¯ F ( t m , t n , x ) − ˆ F ( t m , t n , x ) (cid:12)(cid:12) N m (cid:88) i =1 (cid:98) Π iy,m = sup x ∈ R (cid:12)(cid:12) ¯ F ( t m , t n , x ) − ˆ F ( t m , t n , x ) (cid:12)(cid:12) = sup x ∈ R (cid:12)(cid:12) ( π n,m )( x ) − (ˆ π n,m )( x ) (cid:12)(cid:12) . The error bound for | ( π n,m )( x ) − (ˆ π n,m )( x ) | is independent from x and is given by (30). On theother hand we have (cid:12)(cid:12) P ( τ X > t n | ( ¯ Y t , . . . , ¯ Y t m ) = y ) − P ( τ ¯ X > t n | ( ¯ Y t , . . . , ¯ Y t m ) = y (cid:12)(cid:12) ≤ E (cid:0) | { τ X >t n } − { τ ¯ X >t n } | (cid:12)(cid:12) ( ¯ Y t , . . . , ¯ Y t m = y ) (cid:1) ≤ P (( ¯ Y t , . . . , ¯ Y t m ) = y ) E (cid:12)(cid:12) { τ X >t n } − { τ ¯ X >t n } (cid:12)(cid:12) . We conclude by Proposition 3.5. 14
Numerical results
We illustrate the numerical part by considering a Black-Scholes model. More specifically, thedynamics of the signal process X and observation process Y are in the form (cid:40) dX t = X t ( µdt + σdW t ) , X = x ,dY t = Y t ( µdt + σdW t + δd (cid:102) W t ) , Y = y (32)meaning that dY t Y t = dX t X t + δd (cid:102) W t which can be interpreted as the yields of the observation process Y are noised yields of the signalprocess X with magnitude δ . Intuitively, in order to deal with CDS option implied volatilitiesbelow, we will play with the parameter δ . Before tackling the CDS examples, we first test the performance of our method in two setups:- By comparing the function F ( s, t, x ) = P (inf s ≤ u ≤ t X u > a | X s = x ) and its quantizedversion (cid:98) F ( s, t, x ) defined in (21), keeping in mind that in the model (32), F ( s, t, x ) = Φ( h ( x, t − s )) − (cid:16) a x (cid:17) σ − ( µ − σ / Φ( h ( x, t − s )) , (33)where h ( x, u ) = 1 σ √ u (cid:18) log (cid:16) x a (cid:17) + (cid:18) µ − σ (cid:19) u (cid:19) ,h ( x, u ) = 1 σ √ u (cid:18) log (cid:16) a x (cid:17) + (cid:18) µ − σ (cid:19) u (cid:19) . This comparison allows us to test our method given benchmark values.- By comparing the conditional default probabilities P ( τ ¯ X ≤ t |F ¯ Ys ∨ F ¯ Hs ) and P ( τ ¯ X ≤ t |F ¯ Ys ) respectively estimated by (23) and (25), where F ( s, t, · ) is computed using the exact for-mula (33). Our aim here is to confirm the impact of the additional information F ¯ Hs on theconditional probability P ( τ ¯ X ≤ t |F ¯ Ys ) .Notice that we deal in this paper with a general framework where the signal and the observationprocesses have no closed formula. Even if in our model both processes have explicit solutions andwe may deduce a similar formula to (11) using these closed formulae (the only change will comefrom the function g k which involves the conditional density of ( X t k +1 , Y t k +1 ) given ( X t k , Y t k ) ), westill consider their associated Euler schemes processes ¯ X and ¯ Y in order to stay in the scope of theproposed numerical method.To compare the functions F ( s, t, · ) and (cid:98) F ( s, t, · ) , we choose the following set of parameters(like those of [20]): µ = 0 . , σ = 0 . , δ = 0 . , x = y = 86 . and a = 76 . Figure 1 shows theconvergence of the quantized function (cid:98) F ( t m , t n , x ) toward the exact one F ( t m , t n , x ) with t m = 1 , t n ∈ [1 . , and where x is one point, say x (cid:63)m , on the grid { x im , i = 1 , . . . , N m } (see equation(21)). Once we fix t m , F ( t m , t n , x ) depends on both t n and x . Therefore, to show the convergence,we fix x and plot both F ( t m , t n , x ) and (cid:98) F ( t m , t n , x ) with respect to t n . The number of discretiza-tion points m is set to and the convergence is achieved by increasing the number of quantization15oints N n . Since the fixed quantization point x (cid:63)m can differ when moving N n , the correspondingfigures can take different shapes but, we have only to make sure that the convergence is achievedwhen increasing N n . 16 .0 1.5 2.0 2.5 3.0 . . . . . t n F ( t m ,t n ,. ) (a) N n = 50 . . . . . . t n F ( t m ,t n ,. ) (b) N n = 100 . . . . t n F ( t m ,t n ,. ) (c) N n = 50 . . . . . t n F ( t m ,t n ,. ) (d) N n = 400 Figure 1: Convergence of ¯ F ( t m , t n , · ) estimated by (cid:98) F ( t m , t n , · ) (dotted blue) toward F ( t m , t n , · ) (solid magenta).We now proceed to the numerical comparison between the conditional default probabilities P ( τ ¯ X ≤ t |F ¯ Ys ∨ F ¯ Hs ) and P ( τ ¯ X ≤ t |F ¯ Ys ) , respectively estimated by (23) and (25), in order to check17he statements of Remark 3.1. Setting s = t m = 1 , t = t n and considering the same parameter setas in the previous figure but with t n ∈ [1 . , , Figure 2 depicts the trajectories of the observationprocess ¯ Y from to t m and the associated conditional default probabilities as a function of t n . First,we notice that equation (6) is fulfilled as given a trajectory of the observation process ¯ Y representedin red, P ( τ ¯ X ≤ t |F ¯ Ys ) lined up in dots is always above P ( τ ¯ X ≤ t |F ¯ Ys ∨ F ¯ Hs ) in magenta. Second,the gab between the two quantities is larger for an downward movement of Y compared to anupward movement for which the firm is less exposed to default. This can be understood by the factthat the more the firm is creditworthy, the less the default information is important and the less thedefault probability is. Then, the model preserves the memory of all the observed path of the process Y when computing default probabilities. This path-dependent future of the default probabilitieshas already been shown in [6] and is known to be very important as it is implicit in reduced-formmodels for calibration purpose to historical data.18 .0 0.2 0.4 0.6 0.8 1.0 (a) ¯ Y Up . . . . . . (b) Default probability (c) ¯ Y Down . . . . . . . . (d) Default probability Figure 2: Trajectories of the observation process ¯ Y (solid red), Y up: panel (a), ¯ Y Down: panel (c)and the associated conditional default probabilities functions P ( τ ¯ X ≤ t |F ¯ Ys ∨ F ¯ Hs ) (solid magenta)and P ( τ ¯ X ≤ t |F ¯ Ys ) (dotted blue): panels (b) and (d), with t m = 1 and t n ∈ [1 . , and number ofquantization points N n = N m = 30 . 19 .2 Application to CDS option pricing In this section, we briefly recall the concept and valuation of credit default swaps and swaptionsbefore analyzing the quantization procedure applied to such models. This will allow to give afull pricing formula of credit swaps derivatives in a firm value approach using partial informationtheory and optimal quantization. In addition, the fact that we add the default filtration in the modelindicating whether default has already taken place or not is very important in this case as it ispointless to price a default swap post-default.A credit default swap (CDS) is an agreement between two counterparties to buy or sell protectionagainst the default risk of a third party called reference entity . We set τ X as the default time ofthe latter. In this case, if the contract is signed at time s , started at time T a with maturity T b ,the protection buyer pays a coupon (or spread) k at payments dates T a +1 , . . . , T b as long as thereference entity does not default or until τ X . If the default occurs at time τ X with T a < τ X ≤ T b ,the protection seller will make a single payment LGD (that we assume to be a known constant) tothe protection buyer. A CDS option (CDSO) or default swaption is an option written on a defaultswap. From this perspective, it requires to recall the no-arbitrage pricing equation of a CDS. Thetime- s price of a general buyer CDS CDS s ( a, b, k ) with unit notional starting at time T a withmaturity T b , s ≤ T a < T b , a spread k and loss given default LGD is given by the difference of theconditional risk-neutral expectations of the protection and the premium discounted cashflows:
CDS s ( a, b, k ) = E (cid:2) LGD { T a <τ X ≤ T b } D s ( τ X ) |F s (cid:3) − k E (cid:34) b (cid:88) i = a +1 (cid:18) { τ X ≥ T i } α i D t ( T i ) + { T i − ≤ τ X
Black ( a, b, k, ¯ σ ) = C ( a, b ) [ k (cid:63) ( a, b )Φ( d ) − k Φ( d )] where d = ln k (cid:63) ( a,b ) k + ¯ σ T a ¯ σ √ T a , d = d − ¯ σ (cid:112) T a . Hence, the CDS option implied volatility ¯ σ can be found by solving the following equation P SO ( a, b, k ) = P SO
Black ( a, b, k, ¯ σ ) . We now assess the numerical results based on the model’s applications to the pricing of CDSoption. The model’s parameter set is the same as before except here we take σ = 5% and δ isvarying. Table 1 shows the estimated values of a European payer CDS option and the correspondingBlack’s volatilities with different strikes and different values of δ . First, we observe that both CDSoption prices and the implied volatilities are increasing with the noise volatility, δ . This can beexplained by the fact that, the higher δ , the noisier the observations are and the higher the defaultprobability. Since δ measures the degree of transparency of the firm, this will have a positiveimpact on the prices of the CDS option, hence on the corresponding implied volatilities. In contrast,while the option prices are always decreasing with the strike, this is not the case with the impliedvolatilities except for δ = 2% and δ = 3% . In the case where δ = 1% , the implied volatilityis increasing with respect to the strike. Hence with the help of the parameter δ , one can observedifferent levels of skewness. k (bps) Payer Implied vol (%) δ = 1 % δ = 2 % δ = 3 % δ = 1 % δ = 2 % δ = 3 %52.9 0.004655 0.007214 0.009276 69.44 133.84 196.8066.2 0.003739 0.006077 0.008032 73.36 124.16 173.7079.4 0.003107 0.005298 0.007081 76.85 121.08 161.57 Table 1: CDS options and corresponding Black volatilities (with spread k (cid:63) ( a, b ) = 66 . pbs, T a = 1 and T b = 3 ) implied by the structural model using Monte Carlo simulation ( . · ) pathsand for various volatility parameter δ and different strikes (80%, 100%, 120%) k (cid:63) ( a, b ) and σ = Recall that this does not mean in any way that the market naively believes that credit spreads exhibit log-normaldynamics. Market participants simply rely on the Black-Scholes machinery to convert a price into a quantity that is moreintuitive to traders, namely implied volatilities. model survival probability curve as a CDS term structure. Calibration issues of the model toreal market data will be investigated in a future work. In this paper, a new structural model for credit risk has been proposed, generalizing earlier works.Our model deals with an incomplete information, where the default state and a noisy observationof the firm valued are accessible to the investor. It is therefore an extension of [6], as the firm-valuetriggering the default is no longer restricted to be a continuous and invertible function of a Gaussianmartingale, but can be any diffusion.This more general framework benefits however from a limited analytical tractability. Therefore,we propose a numerical method that relies on nonlinear filtering theory associated with recursivequantization. Compared to earlier works such as [5] or [20], our numerical procedure is basedon the fast quantization method recently introduced in [17], which avoids the use of Monte Carlosimulations. A rigorous analysis of the global error induced to the estimation of the survival pro-cesses is performed. We analyze the shapes of the default probabilities which are characterized bya path-dependent feature keeping the memory of all the path of the observed process. Eventuallywe quantify the impact of the volatility of the noise impacting the firm-value process on the pric-ing of CDS options and the corresponding implied volatilities using a hybrid Monte Carlo-optimalquantization method.In future research, we will first investigate the calibration issues of the model which can betackled by either using observed prices or CDS quotes. In this case, our model can be easilyextended to other works dealing with exact calibration to survival probabilities such as including aspecific time-dependent barrier [2] or using time change techniques [13]. Another possible researcharea is to deal with the price of general default sensitive securities. While we have derived a fullquantization scheme to estimate the conditional default probabilities, this was not the case in thepricing of CDS option which required additional Monte Carlo simulations in order to be estimated.To derive a full quantization scheme for the pricing of defaultable claims, a possible route is toexploit the functional quantization method. 22 eferences [1] F. Black and J. C. Cox. Valuing corporate securities: Some effects of bonds indenture provi-sions.
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