A Quantization Approach to the Credit Exposure Estimation Michele Bonollo, Credito Trevigiano and IMT Lucca - [email protected] Luca Di Persio, Dept. Informatics - University of Verona - [email protected] Immacolata Oliva, Dept. Informatics - University of Verona - [email protected] Andrea Semmoloni - [email protected] Abstract The present paper aims at giving a rigorous approach to Credit Counterparty Risk Estimation exploiting quantization techniques. 1 Introduction and Scope of the Study The financial crisis in 2007-2008, along with an increasing awareness about the different sources of risk, allowed to focus at a deeper level the counterparty credit risk (CCR). CCR refers to the situation when the counterparty A has a deal, mainly a derivative type suc as an option or a swap, subscribed with the counterparty B. We suppose that according to a valuation criteria based on market prices in the A perspective we observe positive Mark to Market (MtM). It follows that A has a credit exposure with B, hence if B defaults and no future recovery rates and no collaterals were posted, then A loses exactly MtM, the cost for the replacement of the defaulted position. Such type of risk is of particular interest within the so called Over The Counter (OTC) derivatives markets. We recall that OTC markets are characterized by having all the transactions which are not listed in the stock exchange, but, instead, they are settled directly between the two counterpaties. A slightly different perspective of CCR has to taken into account when in the risk management field A evaluates ex-ante the risk belonging to the financial position underwritten with B. In such a case, since the possible default for B is a random event, both in time and also concerning its magnitude, it turns out that the current MtM is a too rough measure of the credit exposure for A. On the basis of the latter consideration, in what follows we will focus our attention on what is called Exposure At Deafult (EAD). EAD parameter can be seen as a sort of conservative expected value of the future MtM at the default time. An official way to estimate the EAD in various contest, has been given in the Basel framework, see [1], namely within the set of recommendations on banking laws and regulations issued by the Basel Committee on Banking Supervision. Such an approach is based on the Expected Positive Exposure (EPE) evaluation, namely on a prudent probabilistic time average of the future MtM. EAD follows just as a multple, i.e. EAD = α · EP E. Moreover, we recall that the international accounting standards require that in the derivatives evaluation a full fair value principle, see, e.g., [15], has to be satisfied. If the the counterparty solvency level falls, we observe a downgrade in its rating and/or an increase in its spread, therefore the related OTC balance sheet evaluation has to embody this effect. The latter implies that we have to adjust the MtM since it may decrease not only due to the usual market parameters, e.g. underlying price, underlying volatility, free risk rate, etc., but also for the credit spread volatility. We refer to such an MtM adjustment, as Credit value Adjustment (CVA), and the related adjusted fair value is sometimes called the full fair value. The adjusted MtM will be denoted by M tM A and we have M tM A = M tM − CV A. Even if the derivative has not been closed, the CVA effect can cause a loss in the balance sheet, namely an unrealized loss. Some studies of the Basel Committe estimate that 2/3 of the losses in the financial crisis years in the OTC sector were unrealized losses in the evaluation process. The CVA loss is (or should be) absorbed by the balance sheet, while the CVA volatility must be faced by the regulatory capital. To this end, a new capital charge, the CVA charge, was introduced with the Basel III framework. For the accounting principles, see [14], while a detailed discussion about the capital charge can be found in [2]. The EAD and the CVA computations pose a lot of methodological, financial and numerical issues, as witnessed by huge amount of literature developed so far, see, e.g., [9], for a detailed review of these subjects. The present paper aims not at discussing the usefulness of EAD/CVA measures, nor the related underlyings or volatility models, moreover we will not concern with the analysis of data quality and data availability. Indeed we aim at studying the feasibility and the trade off accuracy vs. computational effort of the quantization approach for the EAD-estimation (EPE). Therefore our main goal is the numerical CCR analysis, while we addressed the CVA issue in a future work. In particular we will consider a simple Black and Scholes model, without taking into consideration collateral parameters in order to focus the attention of the implemented numerical techniques. The paper is organized as follows: Section 2 is a review of the EPE definition given by the Basel Committee, in Section 3 we will give a description of the quantization approach to the EPE with some theoretical results, in Section 4 we describe some practical cases and we give the set up of the associated numerical experiments, while in Section 5 we report the obtained numerical results along with their interpretation. 2 The Basel EPE definition In what follows we shall give a review about the Basel Committee guide lines concerning the estimation of the Exposure at Default, namely the EPE parameter. Let us set the following notations that will be used throughout the paper • taking into account a derivative maturity time 0 < T < +∞, we consider K ∈ N+ time steps 0 < t1 < t2 < · · · < tK which constitute the so called buckets array, denoted by BT,K , where usually, but not mandatory, tK = T . • for every tk ∈ BT,K we denote by M tM (tk , Sk ) := M tM (tk , Stk ) the fair value (mark to market) of a derivative at time bucket tk , with respect to the underlying value Sk at time tk . For the sake of simplicity, we denote by t0 = 0 the starting time of our evaluation problem, hence considering the european case. • for every tk ∈ BT,K we denote by M tM tk , S k := M tM (tk , S tk ) the the fair value (mark to market) of a derivative at time bucket tk , with respect to the whole sample path S k := {St : 0 ≤ t ≤ tk }, again considering an initial time t0 = 0. • taking into account previous defintions, we indicate by ϕ = ϕ (T − tk , Sk , Θ) the pricing function for the given derivative, where Θ represents the set of parameters from which such a pricing function depends, e.g. the free risk rate r, the volatility σ, etc. • we will use the notation φM tM to denote the mark to market value pricing function Remark 1. We would like to underline that in the Black-Scholes framework the volatility surface has to be flat, which is not the case when considering real financial time series. It follows that the previously considered pricing function φ most likely depends from more than the two mentioned parameters r and σ. In particular usually Θ ∈ Rn , with n > 2, the extra parameters being those characterizing the specific geometric structure of the volatility surface associated to the considered contingent claim. As usual in the counterparty credit risk EAD estimation, we stress the role of the underlying, while the other market parameters are assumed to be given, namely we assume that they are deterministic and known. As an example of the latter assumption we can consider the case where both the unknown risk free rate and volatility values are substituted by their deterministic forward values. In what follows we shall often use the notation k to indicate quantities of interest evaluated at the k − th time bucket tk , moreover • we denote the Expected Exposure (EE) of the derivative by EEk := 1 X M tM (tk , Sk,n )+ , N n=1..N which is nothing but the arithmetic mean of N ∈ N+ , Monte Carlo simulated MtM values, computed at the k − th time bucket tk , with respect to the underlyng S. The positive part operator (·)+ is effective if we area managing a symmetric derivative, such as an interesest rate swap, or a portfolio of derivatives. For a single option, it is redundant, as the fair value of the option is always postive from the buy side situation, while the sell side does not imply couterparty risk and then it is out of context. PK EE ·∆ • we evaluate the expected positive exposure (EPE) by EP E := k=1 T k k , where ∆k indicates the time space between two consecutive time buckets at k − th level. If P the time buckets tk are equally spaced, then the formula reduces to EP E = K 1 k=1 EEk . Therefore the EP E value gives the time average of the EEk K • we set EEE1 := EE1 and EEEk = M ax {EEk , EEEk−1 }, for every k = 1, . . . , K. In particular EEEk is called the effective expected exposure and its non decreasing rule takes into account the fact that once the time decay effect reduces the MtM, and the counterparty risk exposure accordingly, the bank applies a roll out with some new deals • we define the effected expected positive exposure (EEPE), by PK EEEk · ∆k EEP E := k=1 . T To avoid too many inessential regulatory details, we will work on the first quantities EEk and EP E, the others being just arithmetic modifications of them. Remark 2. Let us point out that the definiton of EEk is taken from the Basel regulatory framework.We find it quite strange, since instead of giving a theoretical principle and suggesting the Monte Carlo technique just as a possible computational tool, the simulation approach is officially embedded in the general definition. In what follows we shall rewrite previously defined quantities in continuous time, and we add the index A to indicate the adjusted definitions. Moreover we consider the dynamic of the underlying St := {St }t∈[0,T ] , T ∈ R+ being some expiration date, to be the one of an It¨ı¿œ processes defined on some filtered probability space Ω, F, Ft∈[0,T ] , P . As an example St being the solution of the stochastic differential equation defining the geometric Brownian motion, Ft∈[0,T ] is the natural filtration generated by a standard Brownian motion Wt = Wt∈[0,T ] starting from a complete probability space (Ω, F, P), where P could be the so called real world probability measure or, in a martingale approach to option pricing, see, e.g., [22, Ch.5], an equivalent risk neutral measure • we set ˆ EEkA := EP M tM (tk , Sk )+ = ϕ (tk , Sk , Θ) dP 1 X [A ∼ M tM (tk , Sk,n )+ = EEk = EE = k N n=1..N • we define EP E A := ´ EEtA · dt = ´ ´ (1) ϕ (t, Sk , Θ) dP · dt In this new formulation the Basel definition is simply (one of the many) method to estimate the expected fair value of the derivative in the future. Remark 3. Which probability P to use in the calculation of EEk , i.e. a risk neutral probability for the drift St vs. an historical real world probability, is beyond the scope of our paper. For a detailed discussion, see, e.g., [8] Remark 4. We can observe in the EPE definition that the discount factor, or numeraire, is missing. It was not forgot, but this is one of the several conservative proxies used in the risk regulation. If we adopt a simulation approach, for the underlying path construction we could, for each simulation n, to generate a path with an array of points (xn,tk ). This algorithm is called path-dependent simulation (PDS). Alternatively we could jointly, for each time bucket and each simulation, generate our N · K points. This approach is referred as direct-jump to simulation date (DJS). The figure below, taken from [20], well clarify the difference Finally, we recall that in the risk managment application another quantity is very popular in the practice, i.e. the P F Eα (potential future esposure) a quantile based figure of the exreme risk. In a continuous setting • P F E (α, tk ) = M tM ∗ : P M tM (tk , Sk )+ ≥ M tM ∗ = 1 − α 3 A short quantization review The quantization technique is kown from several decades, and it comes from engineering, when facing the issue of converting an analogical signal, e.g. images, sounds, etc., in to a discretized digital information. Other relevant fields are those of information compression and statistical multidimensioanl clustering. For a classical reference of quantization we refer to see [11], while we refer to [16] for a survey of the related literature concerning mainly the problems of fair value pricing for plain vanilla options, american and exotic options, and basket CDS. Moreover in [17] and [18] alternative quantization approaches, such as the so called dual quantization, and the treatment of underlyings driven by more structured stochastic processes are taken into consideration. In this section we shall give a sketch of the quantization idea, by emphasizing its practical features without giving all the details concerning the mathematical theory behind it. Let X ∈ Rd , d ∈ N+ , be a d-dimensional, continuous random variable defined over the probability space (Ω, F, P) and let PX the measure induced by X. The quantization approach is based on the approximation of X by a d-dimensional discrete random variable b see later, defined by mean of a so called quantization function q of X, namely X b := X, + d b q (X), in such a way that X takes N ∈ N , finitely many values in R . The finite set of b will be denoted by q (R) := {x1 , ..., xN }, and it is called a quantizer of X, values for X while the application q (X) is the related quantization. Latter defined set of points in Rd can be used as generators points of a Voronoi tesselletion. Let us recall that if X is a metric space with a distance function d, K is an indices and (Pk )k∈K is a tuple of ordered collection of nonempty subsets of X, then the Voronoi cell Rk generated by the site Pk is the defined as the set Rk := {x ∈ X | d(x, Pk ) ≤ d(x, Pj ) for all j 6= k} . It follows that the Voronoi tesselletion is the tuple of cells (Rk )k∈K . In our case such a tesselletion reduces to substitute the set of cells Pk with a finite number x1 , . . . , xN of distinct points in Rd , so that the Voronoi cells are convex polytopes. Taking into accout previous definitions, we construct the following Voronoi tessels with respect to the euclidean norm k·k in Rd , C (xi ) = y ∈ Rd : |y − xi | < ky − xj k∀j 6= i , with associated quantization function q defined as follows q (X) = N X xi · 1Ci (x) (X) (2) i=1 On the basis of previous definition, we can therefore rigorously define a probabilistic b exploiting the probability measure induced by setting for the claimed random variable X the continuous random variable X, In particular we have a probability space Ω, F, PXb , where the set of elementary events is as follows Ω := {x1 , . . . , xN }, and the probability measure PXb is defined by the following set of relations 0 < PXb (xi ) := PX (X ∈ C (xi )) =: pi , for i = 1, . . . , N , The goal of such an approximation is to deal efficiently with applications that arise when calculating some functionals of the random vector X, as in the case we would take into consideration the derivative pricing problem evaluating the expectation of a certain payoff function f of X, i.e. E [f (X)], or we have to deal with a quantile base indicator, as in the risk management field, etc. Applying previously defined Voronoi approach to the pricing problem, we shall consider approximation of the following type h i X b = E [f (X)] ∼ f (xi ) · pi , =E f X i with control on the accuracy of the chosen quantization. Assuming that X ∈ Lp (Ω, F, P) , p ∈ (1, ∞), we define the Lp error as follows i ˆ h p b min kx − xi kp · dPX (x) , E kX − Xk = Rd i=1,...,N (3) where we have indicated with dPX the probability density function characterizing the random variable X, and we note that the integrand is always well defined as a minimum taken with respect to the finite set of generators x1 , . . . , xN . Concerning eq. (3), a general question is to find the, or at least one, optimal quantizer, namely the set of Voronoi generators minimizing the value of the integral, once both the parameters d, N, p and the probability density of X are given. Such a problem turns out to be particularly difficult in general case, and it may rise to infinitely many solutions. Nevertheless, since we aim at considering a standard Black-Sholes framework, hence the source of randomness is given by a Brownian motion. In particular we shall deal with the pricing problem related to a European style option, therefore we are interested in the distribution of the driving random perturbation at maturity time T , which means that we are dealing with the quest of an optimal quantizer for a d-dimensional Gaussian random variable which we assume to be standard, up to suitable transformation of coordinates, namely X ∼ N (0, Id ). For the algorithms to get the optimal quantizer, see the above references. Anyway, for a large set of , d ≤ 10, There is a well established literature concerning the quest for optimal quantizers in case of Gaussain distribution, see, e.g., [4, 16, 21], and the web site http://www.quantize.maths-fi.com/. A particularly important quantity related to the choice of the optimal quantizer is represented by the so called distortion parameter which is defined as follows h i ˆ N b 2 b DX (X) := E kX − Xk = min kx − xi k2 · dPX (x) , (4) Rd i=1,...,N with respect to the quantizer {x1 , . . . , xN }. We have a minimum in eq. (4), let us indicate it N b by DN X , if the X takes value in the optimal generators set, moreover limN →∞ D X = 0. The following theorem, originally due to Zador, see [23, 24], and then generalized by Bucklew and Wise, see [6], and revisited in its non asymptotic version, as a reformulation of the Pierce lemma, by , see [16], gives a quantitative result about the distortion magnitude Theorem 5. [Zador] Let X ∈ Lp+ε (Ω, F, P), for p ∈ (0, +∞), ε > 0, and be Γ the N b then size tessellation of Rd related to the quantizer X, 1+ pd ˆ d p b p = Jp,d g d+p lim N d min kX − Xk (x) dx (5) P N |Γ|≤N Rd where we have assumed that dP = gdλd + dν, for some suitable function g, and with ν ⊥ λd , dλd beign the Lebesgue measure on Rd , while the constant Jp,d corresponds to d the case X ∼ U nif [0, 1] . h i b b Let us recall that the optimal quantizer is stationary in the sense that E X|X = X, ´ hence C(xi ) (x∗i − x) · dPX (x) = 0, i = 1..N In what follows we focus our attention on the case d = 1 and p = 2, namely we consider a oe dimensional stochastic process, and the 1 1 , hence J2,1 = 12 . quadratic distortion measure. In the latter setting we have Jp,1 = 2p ·(p+1) Remark 6. For practical applications, and in order to compare numerical results obtained by quantization with those produced by standard Monte Carlo techniques, we are mainly interested on the order of convergence to zero for the distortion parameter. In particular, tahnks to the Zador theorem, we have that he quadratic distortion is of order O (N −2 ). also possible to provide results concerning the accuracy of the approximation h It is i b E f X , by mean of the distortion’s properties, see [16]. In particular can distinguish the following cases • Lipschitz case: if f is assumed to be a Lipschitz function, then h i b ≤ Kf · b b E [f (X)] − E f X X − X ≤ KLf · X − X 1 (6) 2 • The smoother Lipschitz derivative case: if f is assumed to be is continuously differentiable with Lipschitz continuous differential Df , then, performing the quantization by mean of an optical quadratic grid Γ, then, by Taylor expansion, we have h i b ≤ KLDf · b (7) E [f (X)] − E f X X − X 1 b is stationary, then exploiting the • The Convex Case: if f is a convex function and X Jensen inequality, we have h i h h ii b = E f E X|X b E f X ≤ E [f (X)] (8) Therefore the approximation by the quantization is alqays a lower bound for the true value of E [f (X)]. Remark 7. We emphasize that the (optimal) quantization grid for given values of N, d and p, is calculated off-line once and for all. Then in the computational effort comparison vs. a strict sense Monte Carlo approach, we must take into account that with the quantization one only has to plug-in the points in the numerical model, not to calculate or simulate them. Remark 8. An increasing literature has been devoted to the functional quantization. In this case, the “random variabile” to be dicretized in an optimal way belongs to a suited functional space, e.g. an application X : Ω → (L2T , kk2 ), where L2T = L2T ([0, T ] , dt). We recall that in mathematical finance applications the stochastic calculus in the continuous time is a very useful tool, but on the other hand in every real world payoffs we have discrete sampling. Asian options or any other lookback derivatives has to work with a discrete calendar for the fixing istants. Then, depending from the specific application, one can choose if to approximate by optimal quantization the discrete time, real world, problem, or if it is better to quantize the continuous time setting and then to apply it to the practical application, see, e.g.,[17] for a survey on such a topic. 4 The Proposal: quantization for the EPE calculation In the following, we focus our attention about the calcualtion of EPE for option styles derivatives in the Black-Scholes standard setting, see [5] . Namely we have for the underlying the follwoing stochastic differential equation with the related closed for solution dSt = St · dt + σ · St · dBt (9) σ2 St = S0 · exp r − · t + σ · Bt 2 (10) As well known, the plain vanilla call and put options have the closed procing formula, see ( ). We can not here give an survy of the severla extensions to the model. Briefly, in the equity anf forex derivatives markets, the most important modesl extensions are the local volatility models, the Heston models and the SABR models, see, e.g., [10], resp. [13], resp. [12]. As usual in a new methodology proposal, we prefer as a first research to check it in a simpe framework to have clear insights about its properties. We guess that the Monte Carlo approach is just one of the many feasible techinques for the EE and EP E calculation. Moreover, it is quite expensive. Infact, let us give the magnitude of the computational complexity. A medium bank easily has D = O (104 )derivatives deals. In order to validate the internal models for the EP E calculation, usually the central banks require at least K = 20 time steps and N = 2000 simulations. Finally let us suppose that the relevant underlying (oftne called risk factors) to be simulated have an O (103 )order. It is the sum of equity underlying, FX significant rates and rate curve points. Let be U this parameter. What about the computational effort for an EP E process task on the whole book? If we adopt a pure Monte Carlo strategy, we must distinguish between the two main steps: 1. to simulate (and to store) the underlying paths; 2. to calculate the EE quantities. We omit for simplicity the last EP E layer, since it is just an algebraic recombination of the EEs. The first step implies a grid of G = K · N · U = 4 · 106 points which work as an input for the step 2. This one requires N T = K · N · D = 4 · 108 different tasks. We recall (see the definition of EEk ) that each one of these tasks is a pricing process, then very often it could be it self a Monte Carlo algorithm with several thousands of simulation steps. Generally speaking, to calculate EP E by rough Monte Carlo is K · N = O (104 )more expensive thtan the usal daily end-of-day mark to market evaluation of the book. Then we argue that the brutal Monte Carlo approach can not be a satisfactory way for the EP E calculation. For this reason, some banks are eploiting some innovative technological approaches, e.g. using the graphical processing unit (GPU) instead of CPUs to set up a parallel calculation system, with some new programming languages such as NVIDIA. Some other banks are investing a lot for the grid computing platforms. We believe that an algorithmic based improvement can be more efficient than the hardware innovation (or it can be combined with) and much less expensive. In the derivatives pricing field, such a mixed approach is well explained in [19]. Coming back to our credit exposure estimation, we try to figure how to use the quantization tecnique. At a first level, we can distinguish between path dependent derivatives and non path dependent derivatives, in the following pd and npd for brevity. . We point out that this definition is different from the usal plain vs. exotic derivatives. In the non path dependent derivatives we include not only the plain europaen options, but also european and american style options with exotic payoff (mixed digital continuous, spread options, ..). In the npd class, we will work with the asian options, probably the most popular one. For the npd derivatives the quantization for the EPE simply reduces to set up the problem by selecting the parameters (Nk )k=1..K for the quantization size at each time bucket (tk )and then to compute the approximation. We use the quantizer i i h optimal√ hEP E 2quantized σ2 σ case, recalling that S0 ·exp r − 2 · t + σ · Bt = S0 ·exp r − 2 · t + σ · t · N (0, 1) . More formally, if we notate with the exponent Q the quantized EPE one easily gets: EEkQ Nk X + k = M tM tk , S xki · pi (11) i=1 P ∆k · N Pk xki + pki M tM tk , S · EEkQ · ∆k i=1 k = (12) EP E = T T The figure below explains the calculation procedure. The black point on the left is the underlying level at time t0 . For each time step tk and for each point of the quantizer xki , a M tM is calculated and it is weighted by the probability pki . Hence EE Q and EP E Q straightly follow. About the theoretical properies of such an approximation, we have an useful result. Q P Proposition 9. For an option in the Black-Scholes setting let us consider derivatives payoffs such that the pricing function ϕ () is (a) Lipschitz continuous or (b) continuously differentiable with a Lipschitz continuous differential. This is the case for many european styles options. Let be EP E Q defined as in (10), and let K, Nk , tk the related parameters, d = 1. EP E A Q 2 EP E EP E ∝ N −4 ·K −1 ∝ N −2 ·K −1 and DN Let be DN = E EP E − EP E . Then DN respectively. Proof. From the definitions we have P A P 2 2 EEk ·∆k P EEkQ ·∆k 2 Q 1 A E EP E A − EP E Q = E − = ·E ∆ · EE − EE k k k T T T2 By rearranging and recalling that the CCR is effectve only in the buy case, we can skip the positive part and we get P 2 P 2 Q ∆k · EEkA − EEk = ∆k · EP [M tM (tk , Sk )] − EPb [M tM (tk , Sk )] ≤ 2 P 2 ≤ ∆k · EP [M tM (tk , Sk )] − EPb [M tM (tk , Sk )] Moreover P we have 2 2 P Q ∆k · EP [M tM (tk , Sk )] − EPb [M tM (tk , Sk )] ∆k · EEkA − EEk ≤ In a more explicit fashion, let us work on the single element k of the summation, i.e. EP [M tM (tk , Sk )] − EPb [M tM (tk , Sk )] . If we look at M tM (tk , Sk ) = ϕ (tk , Sk (X)) as the function f () of the results in (6), (7) one gets respectively 2 2 2 c EP [M tM (tk , Sk )] − E b [M tM (tk , Sk )] ≤ KL · Xk − Xk P f,k 1 and 2 c EP [M tM (tk , Sk )] − E b [M tM (tk , Sk )] ≤ KL2 · X − X k Df,k k P 2 Finally we easily obtain 2 P 2 P Q 2 2 A c ∆k · EEk − EEk ≤ ∆k · KLf,k · Xk − Xk 1 P 2 P 4 Q 2 2 A c ∆k · EEk − EEk ≤ ∆k · KLDf,k · Xk − Xk 2 As N goes to infinity and if the mesh tk is regular enough, e.g. limK ∆k = 0, O (∆k ) = K −1 ∀k, Letting KL = mean (KL·,k ) and by recalling the Zador theorem we finally have 2 P 2 P 2 Q 1 1 A 2 c ∆k · EE − EE ≤ 2 · ∆ · KL · Xk − Xk ∝ 12 · K · KL · 2 ·E k T T2 k T k f,k 2 T −2 ·N . By simplifying we get the result. In the (b) case, the calculations are the same. For a more abtract setting, see [16, Sec.2.4]. K2 . Remark 10. Let us discuss the hypothesis under which the result holds. The key point√is the function ϕ (x, ·) as a function of the brownian motion Bt , namely of the quantity x · t, x sampled from the N (0, 1). We recall infact that usefulness of the quantization is to infer the properties of the approximation in the specific problem from the starting guassian optimized discretization. As an example, for the put option the pricing function is Lipschitz continuous and twice differentiable, since the Black-Scholes formula is C ∞ . Then the above proposition holds. Otherwise, for the call style options just the convexity properties easily holds, hence the quantization furnishes a lower bound to the EPE estimation. Remark 11. If we consider the pricing function ϕ ()the elementary piece of our EPE P computational workflow, the computational complexity of the quantized approach is Nk , to k be compared with the global number of Monte Carlo scenarios simulations. For the path-dependent derivatives one needs for each time tk the whole (at least in a discrete sampling sense) path of the underlying. Hence the above approach is not satisfactory, as the pricing function depends not only from the current level Sk , but from some function (average, min, max, ..) of the underlying level until tk . For this goal, an approach like the PDS in the first figure well fits the problem. Now, let Nk a positive integer for the quantization and let qk (R) = (x1 , x2, ..., xNk )the ck = q (Xk )that maps the random variable in quantizer of size, i.e. the random variable X an optimal discrete version. If we refer to the Black-Scholes model, we want to quantize at each step the brownian motion Bt that generates the lognormal underlying process. We recal that Bt ∼ N (0, t) is a normal centerd random variable and that Bt − Bs ∼ N (0, t − s). Again, we define C (xi )the ithVoronoi tessel for the quantization,e.g. the in a general dimension d one has C (xi ) = y ∈ Rd : |y − xi | < |y − xj | ∀j 6= i with the so called nearest neighborhood principle. A matrix of probability masses vectors is assigned to the Nk − tuple , let be pk = p1k , p2,k ..., p k Nk . Of course, the mass probability is set by pi = P (C (xi )) How to use the quantizers for the EEk calculation? and where? The quantization tree is a discrete space discrete time process defined as follows pki πijk k b = P Xk = xi = P Xk ∈ Ci xk k+1 b k b = P Xk+1 = xj |Xk = xi = P Xk+1 ∈ Cj xk+1 |Xk ∈ Ci xk πijk P Xk+1 ∈ Cj xk+1 , Xk ∈ Ci xk = pki (13) (14) (15) By recalling that Xk is the original brownian motion sampled at a given time tk we can solve “explicitly” the transition probability formula (2) by the following Proposition 12. Let us use the below notations to alleviate the expressions • ∆k = tk+1 − tk • Uk , Uk+1 are the upper bounds of the 1 dimensional tessels Ci xk and Cj xk+1 respectively • Lk , Lk+1 are the lower bounds of the tessels Ci xk and Cj xk+1 respectively • φ (·) is the density of the N (0, 1) standard random variable Then we have for the transition probability πijk the result Cˆ j,k+1 πijk = f (xk+1 |Ci,k ) · dxk = ´ Uk ´ Uk+1 (x−y) · dx · φ tyk · dy φ ∆k Lk+1 Lk pki (16) Proof. The result is obtained by straight calculations. We start form a more specific prob k+1 k lem, e.g. to calculate P Xk+1 ∈ Cj x |Xk = y, y ∈ C x . For a given x ∈ Cj xk+1 , by recalling the scaling property and the independence of the brownian motion increments (x−y) we easily get P (dx) = φ ∆k · dx. Then the proposition follows by extension this elementary result to the tessels Ci , Cj . In the below figure we show the practical explanation of the formula Remark 13. Even if the proposition comes for elementary probability, this result is a useful improvement to the procedure in Pag¨ı¿œs et al (2009), where a Monte Carlo approach for the πijk was suggested. Remark 14. From a computationl complexity, we observe that the above coefficients πijk can be calculated off line, once and for all, given the time discretization parameter K and the target granularities {N1 , N2 , ..., NK }. Nevertheless, the possible paths of the quantization tree, letQ be N P, could be too many from a theoretical perspective. Infact they amount to N P = k Nk . If we set as usual K = O (101 )and Nk = O (102 ) the value of N P seems to be untractable for practical purposes, infact N P = O (1020 ). This is not true practically. Many of the paths have a negligible probability, as very often we have πijk ' 0 and we van skip a large fraction of the combinatorial cases. Further details in the application sections. 5 The numerical Application In this section we will give an application of quantization method in CCR with respect to a portfolio consisting of a bank account and one risky asset which is the underlying of a European type option. We suppose that the dynamics of the underlying St := {St }t∈[0,T ] , for some maturity time T ∈ R+ , is given accordingly to a geometric Brownian motion, namely dSt = rSt dt + σSt dWt , (17) where r, resp. σ, indicates the risk free, resp. the volatility, of the model, while Bt := {Bt }t ∈ [0,T ] is a R− valued Brownian motion on the filtered probability space (Ω, F, Ft , P), {Ft }t∈[0,T ] being the filtration generated by Bt . We recall that the stochastic differential equation (17) admits an explicit solution, see, e.g.,[22, Ch.3], is given by σ2 t . (18) St = S0 exp σBt + r − 2 If we consider a European call option with strike price K ∈ R+ , maturity time T , risk free interest rate r, written on the underlying St with initial value S0 , then its fair value Ceu = Ceu (S0 , r, K, σ, T ), with respect to the unique martingale equivalent measure, is given by " + # 2 σ + T + σBT − K , Ceu (S0 , r, K, σ, T ) := e−rT E (ST − K) = e−rT E S0 exp r− 2 (19) The above expectation is solved by the celebrated Black&Scholes formula. It could seem too unrealistic show as an “application” the standard call option, hence we want to stress some points. In the practical applcations, the computational challenges are very often much harder than one believes in a theoretical perspective. Referring this general principle to the CCR, the portfolio of derivatives of a counterpaty A with B may conisist of several of Phundreds derivatives j = 1...J, then the mark to market is given by M tM A (t) = M tMjA (t). j These derivatives could be bought options, sold options, swaps and so on. A collateral of value Vt is usually posted. Hence at the current time the exposure of the counterparty A to !+ P B is given by M tMjA (t) − Vt . In the CCR applications one wants to check several j quantities related to the current exposure, such as EE, EPE, PFE, and so on. Because of non linearity in no case one can calculate separately the single deal quantities, i.e. the EEj (tk ), and then to aggregate them by summation. By fortiori in the PFE quantile situation, one can not infer the quantile in a trivial way by the specific quantiles. For this reason in all banks the general strategy is to calculate a large set of coherent (with respect to the probabilistic sructure) scenarios for the underlyings and then to calculate and to store a large set of MtM from which to pick any kind of statistics and risk figures. In this field quantization can play a role as a competitive methodology. Nevertheless, since the CCR for a whole book comes from the single deals computations, we aim to test at a “low” level the quantization, in a plain vanilla context. In further research we will move to exotoc deals and to an effective whole portfolio management. 5.1 Set-up and the practical cases For the market valuation of financial statements, the generation of potential market scenarios is required. We distinguish between two achievable approaches, the path-dependent simulations (PDS) and the direct jump to simulation data (DJS), according as one would to simulate a whole pathwise possible trajectory or directly each time point, respectively. Referring to the case study, since we are dealing with european option pricing, we choose to use the latter approach, namely DJS. In order to obtain numerical results for counterparty credit risk, we set parameters involved in simulations. In particular, we consider the following values: • spot price (S0 ) : 90, 100, 110 • interest rate r : 3% • volatility σ : 15%, 25%, 30% • strike price K : 100 • time to maturity T : one year According to the choice of several banks to consider weekly time intervals up to a month and monthly time intervals up to a year, we decide to set time buckets ranging in 1 2 3 4 2 3 6 9 0, , , , , , , , , 1 , 52 52 52 52 12 12 12 12 namely, we are considering the first, the second, the third and the fourth week and then the second, the third, the sixth and the twelfth month in one year. 5.2 Results and Discussion In order to evaluate the Expected Positive Exposure (EPE), we compare the quantization approach with standard Monte Carlo method. We distinguish three cases, depending on the moneyness parameter, namely with respect to the relative position of St versus the strike price K of the considered call option. In each case, we analyze the performance, by considering the Monte Carlo-Sobol approach, see, e.g., [7], with N = 103 while we run Monte Carlo simulations (MC) and quantization grids (Q) with N = 103 . As regards the choice of the benchmark, let us note that, in the Black-Scholes setting, in order to price a European call option, we work in a risk neutral context where the drift of the geometric Brownian motion has to be equal to the risk free rate. Under such an assumption, the Expected Exposure EE admits a closed form, that is, EEkA = EP M tM (tk , Sk )+ = M tM (t0 , S0 ) · exp (tk − t0 ) . . (20) This is due to the martingale property of the discounted fair value, see [22]for further details. In a more general setting, the mark-to-market function does not exist in an analytical form and the drift can assume values different to the risk free rate r. What is more, we are interested in calculating a measure of the possible worst exposure with a certain level of confidence. Such a measure is expressed in terms of p−percentile of the exposure’s distribution, the so-called Potential Future Exposure PFE, such that P (max(M tM (t), 0) > P F E(t, p)) = 1 − p . As already stressed before, since we just aim to test the efficiency of quantization techniques, we refer here to a very streamlined problem, intending to leave for further research the generalization of the implemented procedures to more complex financial models. In what follows we always consider a (D + 1) × 1 matrix, D being the number of time buckets taken into account, hence D = 9, since we start from t0 =P 0. Each matrix entry 1 M tM (tk , Sk,n )+ , represents the value of the Expected Exposure (EE) EEk := N n=1..N PK EE ·∆ obtained by applying eq. (11). Last row gives the value of EPE EP E := k=1 T k k , calculated according to eq. (12). In order to evaluate the efficiency of procedures involved, it is required to compare the value obtained by simulations and a benchmark. In the case of quantization approach, such a comparison is given by the evaluation of the (percent) relative error ε with respect to the Black-Scholes price obtained by calculating formula (20). As regards the brutal Monte Carlo approach, the analyzed quantity is the (percent) relative standard error (RSD). Tables 12,3 contain numerical results in the ITM case with S0 = 110, tables 4,5,6 refer to the ATM case with S0 = 100, finally tables 7,8,9 report the performances in the OTM case, with S0 = 90. t 1w 2w 3w 1m 2m 3m 6m 9m 1y EPE Analytic EE 14,7105 14,7190 14,7263 14,7757 14,8127 14,9242 15,0366 15,1493 14,9705 14,97049 numerical EE ε 14,7095 -0,007% 14,7179 -0,007% 14,7263 -0,008% 14,7348 -0,008% 14,7743 -0,010% 14,8110 -0,011% 14,9219 -0,016% 15,0336 -0,019% 15,1458 -0,023% 14,96795 -0,017% Quantization EE ε 14,7105 0,000% 14,7190 0,000% 14,7275 0,000% 14,7360 0,000% 14,7757 0,000% 14,8127 0,000% 14,9242 0,000% 15,0366 0,000% 15,1498 0,003% 14,97061 0,001% MonteCarlo EE RSD 14,6491 0,004% 14,7255 0,006% 14,8159 0,007% 14,8699 0,008% 15,0038 0,012% 14,8014 0,014% 14,9164 0,020% 15,1573 0,026% 14,8702 0,031% 14,95171 -0,125% Table 1: EPE: 10%−ITM European call. σ = 15%. MC sobol EE ε 14,7090 -0,010% 14,7170 -0,014% 14,7257 -0,012% 14,7357 -0,002% 14,7750 -0,005% 14,8083 -0,030% 14,9241 -0,001% 14,9941 -0,282% 15,0495 -0,659% 14,93437 -0,241% t 1w 2w 3w 1m 2m 3m 6m 9m 1y EPE Analytic EE 18,0448 18,0551 18,0656 18,0760 18,1248 18,1701 18,3069 18,4447 18,5830 18,36368 numerical EE ε 18,0435 -0,007% 18,0538 -0,008% 18,0640 -0,009% 18,0743 -0,009% 18,1225 -0,013% 18,1673 -0,015% 18,3027 -0,023% 18,4392 -0,030% 18,5763 -0,036% 18,3590 -0,025% Quantization EE ε 18,0447 0,000% 18,0552 0,000% 18,0656 0,000% 18,0760 0,000% 18,1247 0,000% 18,1701 0,000% 18,3069 0,000% 18,4447 0,000% 18,5836 0,003% 18,36380 0,001% MonteCarlo EE RSD 17,9530 0,495% 18,0637 0,693% 18,2050 0,880% 18,2792 0,996% 18,4457 1,420% 18,1189 1,764% 18,2130 2,568% 18,6685 3,359% 18,2320 3,985% 18,33791 -0,140% MC sobol EE ε 18,0427 -0,012% 18,0524 -0,015% 18,0628 -0,015% 18,0754 -0,004% 18,1234 -0,007% 18,1627 -0,041% 18,3091 0,012% 18,3983 -0,252% 18,5664 -0,090% 18,34755 -0,088% Table 2: EPE: 10%−ITM European call. σ = 25%. t 1w 2w 3w 1m 2m 3m 6m 9m 1y EPE Analytic EE 19,8845 19,8960 19,9074 19,9184 19,9726 20,0226 20,1734 20,3252 20,4776 20,23592 numerical EE ε 19,8831 -0,007% 19,8943 -0,008% 19,9057 -0,009% 19,9170 -0,010% 19,9699 -0,014% 20,0192 -0,017% 20,1681 -0,026% 20,3181 -0,035% 20,4687 -0,044% 20,22994 -0,030% Quantization EE ε 19,8845 0,000% 19,8956 0,000% 19,9074 0,000% 19,9189 0,000% 19,9726 0,000% 20,0226 0,000% 20,1733 0,000% 20,3252 0,000% 20,4782 0,003% 20,23605 0,001% MonteCarlo EE RSD 19,7774 0,525% 19,9054 0,735% 20,0735 0,935% 20,1576 1,058% 20,3389 1,510% 19,9478 1,881% 20,0284 2,758% 20,6088 3,622% 20,0838 4,309% 20,20466 -0,155% MC sobol EE ε 19,8821 -0,012% 19,8929 -0,016% 19,9042 -0,016% 19,9182 -0,004% 19,9710 -0,008% 20,0136 -0,045% 20,1770 0,018% 20,2785 -0,230% 20,5096 0,156% 20,23208 -0,019% Table 3: EPE: 10%−ITM European call. σ = 30%. t 1w 2w 3w 1m 2m 3m 6m 9m 1y EPE Analytic EE 7,48940 7,49372 7,49804 7,50237 7,52260 7,54143 7,59820 7,65540 7,71280 7,62176 numerical EE ε 7,4889 -0,007% 7,4931 -0,008% 7,4974 -0,009% 7,5016 -0,010% 7,5216 -0,013% 7,5402 -0,017% 7,5964 -0,024% 7,6530 -0,031% 7,7099 -0,037% 7,61975 -0,026% Quantization EE ε 7,4894 0,000% 7,4937 0,000% 7,4981 0,000% 7,5024 0,000% 7,5226 0,000% 7,5414 0,000% 7,5982 0,000% 7,6554 0,000% 7,7130 0,003% 7,62183 0,001% MonteCarlo EE RSD 7,4479 0,539% 7,4974 0,755% 7,5623 0,960% 7,5947 1,086% 7,6646 1,552% 7,5128 1,935% 7,5412 2,842% 7,7662 3,735% 7,5699 4,473% 7,61211 -0,127% Table 4: EPE: ATM European call. σ = 15%. MC sobol EE ε 7,4885 -0,012% 7,4925 -0,016% 7,4968 -0,016% 7,5021 -0,004% 7,5219 -0,009% 7,5379 -0,046% 7,6011 0,039% 7,6354 -0,262% 7,7345 0,282% 7,62250 0,010% t 1w 2w 3w 1m 2m 3m 6m 9m 1y EPE Analytic EE 11,3550 11,3616 11,3681 11,3747 11,4054 11,4339 11,5200 11,6067 11,6937 11,55571 numerical EE ε 11,3542 -0,007% 11,3606 -0,009% 11,3670 -0,010% 11,3735 -0,011% 11,4036 -0,016% 11,4317 -0,020% 11,5165 -0,030% 11,6020 -0,040% 11,6879 -0,050% 11,5518 -0,034% Quantization EE ε 11,3550 0,000% 11,3616 0,000% 11,3681 0,000% 11,3747 0,000% 11,4054 0,000% 11,4339 0,000% 11,5200 0,000% 11,6067 0,000% 11,6941 0,003% 11,55579 0,001% MonteCarlo EE RSD 11,2873 0,583% 11,3670 0,815% 11,4762 1,039% 11,5270 1,176% 11,6289 1,684% 11,3723 2,107% 11,3929 3,132% 11,8133 4,142% 11,4752 4,978% 11,53970 -0,139% MC sobol EE ε 11,3536 -0,013% 11,3597 -0,016% 11,3661 -0,018% 11,3742 -0,004% 11,4042 -0,010% 11,4279 -0,053% 11,5262 0,054% 11,5793 -0,236% 11,7172 0,201% 11,55554 -0,001% Table 5: EPE: ATM European call. σ = 25%. t 1w 2w 3w 1m 2m 3m 6m 9m 1y EPE Analytic EE 13,2910 13,2986 13,3063 13,3140 13,3499 13,3833 13,4841 13,5856 13,6874 13,52588 numerical EE ε 13,2900 -0,008% 13,2975 -0,009% 13,3050 -0,010% 13,3125 -0,011% 13,3477 -0,016% 13,3805 -0,021% 13,4795 -0,034% 13,5794 -0,045% 13,6796 -0,057% 13,52072 -0,038% Quantization EE ε 13,2910 0,000% 13,2986 0,000% 13,3063 0,000% 13,3140 0,000% 13,3499 0,000% 13,3833 0,000% 13,4841 0,000% 13,5856 0,000% 13,6878 0,003% 13,52596 0,001% MonteCarlo EE RSD 13,2095 0,600% 13,3050 0,839% 13,4380 1,070% 13,4980 1,211% 13,6147 1,736% 13,3018 2,176% 13,3145 3,251% 13,8465 4,312% 13,4224 5,193% 13,50407 -0,161% MC sobol EE ε 13,2892 -0,013% 13,2965 -0,016% 13,3038 -0,019% 13,3134 -0,005% 13,3486 -0,010% 13,3759 -0,056% 13,4917 0,057% 13,5557 -0,220% 13,7110 0,172% 13,52533 -0,004% Table 6: EPE: ATM European call. σ = 30%. t 1w 2w 3w 1m 2m 3m 6m 9m 1y EPE Analytic EE 2,7600 2,7616 2,7632 2,7649 2,7723 2,7792 2,8001 2,8212 2,8424 2,80882 numerical EE ε 2,7598 -0,008% 2,76135 -0,010% 2,7629 -0,012% 2,7644 -0,014% 2,7716 -0,023% 2,7784 -0,030% 2,7988 -0,049% 2,8194 -0,065% 2,8401 -0,080% 2,80730 -0,054% Quantization EE ε 2,7600 0,000% 2,7616 0,000% 2,7632 0,000% 2,7649 0,000% 2,7723 0,000% 2,7792 0,000% 2,8001 0,000% 2,8212 0,000% 2,8424 0,003% 2,80883 0,000% MonteCarlo EE RSD 2,7396 0,731% 2,7628 1,023% 2,7987 1,310% 2,8121 1,483% 2,8325 2,145% 2,7476 2,725% 2,7319 4,239% 2,9162 5,719% 2,8243 7,036% 2,81496 0,218% Table 7: EPE: 10%−OTM European call. σ = 15%. MC sobol EE ε 2,7597 -0,012% 2,7612 -0,016% 2,7626 -0,016% 2,7647 -0,004% 2,7720 -0,009% 2,7772 -0,046% 2,8055 0,039% 2,8119 -0,262% 2,8258 0,282% 2,80347 -0,191% t 1w 2w 3w 1m 2m 3m 6m 9m 1y EPE Analytic EE 6,2016 6,2052 6,2088 6,2124 6,2291 6,2447 6,2917 6,3391 6,3866 6,31125 numerical EE ε 6,2011 -0,008% 6,2046 -0,010% 6,2081 -0,012% 6,2116 -0,013% 6,2278 -0,021% 6,2430 -0,027% 6,2889 -0,045% 6,3352 -0,062% 6,3817 -0,077% 6,3080 -0,051% Quantization EE ε 6,2016 0,000% 6,2052 0,000% 6,2088 0,000% 6,2124 0,000% 6,2291 0,000% 6,2447 0,000% 6,2917 0,000% 6,3391 0,000% 6,3868 0,003% 6,31129 0,001% MonteCarlo EE RSD 6,1579 0,695% 6,2080 0,973% 6,2835 1,245% 6,3131 1,409% 6,3616 2,033% 6,1831 2,573% 6,1594 3,956% 6,5221 5,314% 6,3051 6,501% 6,31287 0,026% MC sobol EE ε 6,2008 -0,013% 6,2042 -0,016% 6,2074 -0,018% 6,2121 -0,004% 6,2285 -0,010% 6,2405 -0,053% 6,3005 0,054% 6,3217 -0,236% 6,3418 0,201% 6,29739 -0,220% Table 8: EPE: 10%−OTM European call. σ = 25%. t 1w 2w 3w 1m 2m 3m 6m 9m 1y EPE Analytic EE 7,9807 7,9853 7,9899 7,9945 8,0160 8,0361 8,0966 8,1575 8,2187 8,12170 numerical EE ε 7,9800 -0,008% 7,9845 -0,010% 7,9889 -0,011% 7,9934 -0,013% 8,0144 -0,021% 8,0339 -0,028% 8,0928 -0,046% 8,1523 -0,064% 8,2120 -0,082% 8,11738 -0,053% Quantization EE ε 7,9807 0,000% 7,9853 0,000% 7,9899 0,000% 7,9945 0,000% 8,0160 0,000% 8,0361 0,000% 8,0966 0,000% 8,1575 0,000% 8,2189 0,000% 8,12176 0,001% MonteCarlo EE RSD 7,9245 0,693% 7,9889 0,970% 8,0857 1,241% 8,1237 1,405% 8,1858 2,027% 7,9568 2,564% 7,9268 3,942% 8,3893 5,295% 8,0964 6,475% 8,11858 -0,038% MC sobol EE ε 7,9795 -0,013% 7,9839 -0,016% 7,9881 -0,019% 7,9941 -0,005% 8,0153 -0,010% 8,0306 -0,056% 8,1067 0,057% 8,1376 -0,220% 8,1760 0,172% 8,10792 -0,170% Table 9: EPE: 10%−OTM European call. σ = 30%. By comparing EPE values reported in tables, we gather that the quantization approach performs well for ATM, ITM, OTM European call options. For the sake of completeness and in order to stress how the quantization technique perform better than Monte Carlo method, we introduce a couple of figures showing the error ε for Monte Carlo and quantization performances. Figure 1: MC: EPE error ε for European Call. Figure 2: VQ: error ε for European Call. References [1] BCBS(2006), “International Convergence of Capital Measurement and Capital Standards”, Basel Committe Paper 128. 1 [2] BCBS (2011), “Basel III: A global regulatory framework for more resilient banks and banking systems”, Basel Committe Paper, 189 1 [3] BCBS (2014), “The standardised approach for measuring counterparty credit risk exposures”, Basel Commitee Paper, 279 [4] Bally V., Pag¨ı¿œs G., Printemps J. (2010), “A quantization tree method for pricing and hedging multi-dimensional American options”, Working Paper 3 [5] Black F., Scholes M. (1973), The Pricing of Options and Corporate Liabilities Journal of Political Economy 81 (3), 637–654. 4 [6] Bucklew, J.A. and Wise, G.L. (1982) Multidimensional asymptotic quanti- zation theory with rth power distortion, IEEE Trans. Inform. Theory, Vol.28, NO.2 3 [7] Caflisch, R.E. and Morokoff, W.J., (1995) Quasi-Monte Carlo integration, J. Comput. Phys. Vol.122 , no. 2 5.2 [8] Castagna A. (2012), “Fast computing in the CCR and CVA measurement”, IASON ALGO, Working Paper. 3 [9] Cesari G. et al. (2009), Modeling, Pricing and Hedging Counterparty Credit Exposure, Springer Finance Editor. 1 [10] Dupire B. (1994), “Pricing with a smile”, Risk, , January, 18-20. 4 [11] Gersho A., Gray R. (1982), “IEEE on Information Theory, Special Issue on Quantization”, No.28. 3 [12] Hagan P. et al (2002), “Managing Smile Risk”, Wilmott Magazine 4 [13] Heston S. (1993), A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options The Review of Financial Studies 6 (2), 327–,343. 4 [14] IFRS (2013), “IFRS 13 Fair Value Measurement”, IFRS Technical Paper 1 [15] A.Mikes, (2013), The Appeal of the Appropriate: Accounting, Risk Management, and the Competition for the Supply of Control Systems, Harward Business School working papers, No.12, Vol.115 1 [16] Pag¨ı¿œs G., Printemps J., Pham H. (2004), “Optimal quantization methods and applications to numerical problems in finance”, Handbook on Numerical Methods in Finance, Birkh¨ı¿œuser, 253-298. 3, 3, 3, 3, 4 [17] Pag¨ı¿œs G., Luschgy H. (2006), “Functional quantization of a class of Brownian diffusions: A constructive approach”, Stochastic processes and their Applications, Elsevier, No.116, 310-336. 3, 8 [18] Pag¨ı¿œs G., Wilbertz B. (2012), “Intrinsic stationarity for vector quantization: Foundation of dual quantization”, SIAM Journal on Numerical Analysis, 747-780. 3 [19] Pag¨ı¿œs G., Wilbertz B. (2011), “GPGPUs in computational finance: Massive parallel computing for American style options”, Working paper. 4 [20] Pykhtin M., Zhu S. (2007), “A Guide to Modelling Counterparty Credit Risk”, GARP publication. 2 [21] Sellami, A. (2005) M¨ı¿œthodes de quantification optimale pour le filtrage et applications ¨ı¿œ la finance, Applied mathematics: Universit¨ı¿œ Paris Dauphine. 3 [22] Shreve, S., (2004) Stochastic Calculus for Finance II. Continuous time models, Springer Finance series 2, 5, 5.2 [23] Zador, P.L. (1963), Development and evaluation of procedures for quantizing multivariate distributions, Ph.D. dissertation, Stanford Univ. (USA). 3 [24] Zador, P.L. (1982), Asymptotic quantization error of continuous signals and the quantization dimension. IEEE Trans. Inform. Theory, Vol.28, No.2. 3

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