A Quantization Approach to the Counterparty Credit Exposure Estimation
AA Quantization Approach to the Counterparty CreditExposure Estimation
M. Bonollo ∗ , L. Di Persio † , I. Oliva ‡ , and A. Semmoloni § Iason Ltd and IMT Lucca
Dept. of Computer Sciences, University of Verona, Italy Banca Profilo
Abstract
During recent years the counterparty risk subject has received a growing attentionbecause of the so called
Basel Accord . In particular the
Basel III Accord asks the banksto fulfill finer conditions concerning counterparty credit exposures arising from banks’derivatives, securities financing transactions, default and downgrade risks characterizingthe Over The Counter (OTC) derivatives market, etc. Consequently the developmentof effective and more accurate measures of risk have been pushed, particularly focusingon the estimate of the future fair value of derivatives with respect to prescribed timehorizon and fixed grid of time buckets . Standard methods used to treat the latterscenario are mainly based on ad hoc implementations of the classic Monte Carlo (MC)approach, which is characterized by a high computational time, strongly dependent onthe number of considered assets. This is why many financial players moved to moreenhanced Technologies, e.g., grid computing and
Graphics Processing Units (GPUs)capabilities. In this paper we show how to implement the quantization technique, inorder to accurately estimate both pricing and volatility values. Our approach is testedto produce effective results for the counterparty risk evaluation, with a big improvementconcerning required time to run when compared to MC approach.
The financial crisis in 2007-2008, along with a consequent increasing awareness about thedifferent sources of risk, has suggested to the various financial players to give a greaterattention to the counterparty credit risk (CCR). CCR refers to the situation when the coun-terparty A has a deal, mainly of derivative type, such as an option or a swap, subscribedwith the counterparty B . We suppose that, according to a valuation criteria based on mar-ket prices, A observes a positive fair value , namely the so called Mark-to-Market ( MtM ).It follows that A has a credit exposure with B , hence, if B defaults and no future recovery ∗ [email protected] † [email protected] ‡ [email protected] § andrea.semmoloni@bancaprofilo.it a r X i v : . [ q -f i n . P R ] M a r ates or collateral was posted, then A loses exactly MtM, which is the cost for the replace-ment of the defaulted position. Such type of risk is of particular interest within the OverThe Counter (OTC) derivatives markets, namely those markets which are characterized byhaving transactions settled directly between the two counterparties and outside the stockexchange.A slightly different perspective of CCR needs to be taken into account when, in the riskmanagement field, A wants to assess ex-ante the risk belonging to the financial positionunderwritten with B . In such a case, considering the possible default for B as a randomevent both in time and in its magnitude, it turns out that the current MtM is a ratherrough measure of the credit exposure of A . A better approach is given by considering the Exposure At Default (EAD) parameter which can be seen as conservative expected value ofthe future MtM at the (random) default time. EAD parameter can be seen as conservativeexpected value of the future MtM at the default time. An official way to estimate theEAD in various contexts is given in the Basel framework, see [1], namely within the setof recommendations on banking laws and regulations issued by the Basel Committee onBanking 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 followsjust as a multiple, i.e.
EAD = α · EP E .Moreover, we recall that the international accounting standards require that in thederivatives evaluation a full fair value principle has to be satisfied, see, e.g., [15].If the the counterparty solvency level falls, we observe a downgrade in its rating and/oran increase in its spread , therefore the related OTC balance sheet evaluation has to embodythis effect. This implies that we have to adjust the MtM since it may decrease not only dueto the usual market parameters, e.g. underlying price, underlying volatility, free risk rate,etc., but also because of the credit spread volatility.We refer to such an MtM adjustment as
Credit value Adjustment (CVA), and the relatedadjusted fair value is sometimes called the full fair value . The adjusted MtM will be denotedby
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 balancesheet, namely an unrealized loss. The Basel Committee estimates that / of the losses inthe financial crisis years in the OTC sector were unrealized losses in the evaluation process.The CVA (expected) loss is (or should be) absorbed by the balance sheet, while the CVAvolatility must be faced by the regulatory capital. To this end, a new capital charge, the CVA charge , was introduced within the Basel III framework. We refer the reader to [14] fora skillful analysis of the accounting principles and to [2] for a detailed discussion about thecapital charge.The EAD and the CVA computations pose a lot of methodological, financial and numer-ical issues, as witnessed by a huge amount of literature developed so far, see, e.g., [9], for adetailed review.The present paper aims at studying the feasibility and the trade off accuracy vs. compu-tational effort of the quantization approach for the EAD-estimation (EPE), not at discussingthe usefulness of EAD/CVA measures, nor the related underlying or volatility models, noreven at analyzing data quality and data availability. Therefore, our main goal is the numer-ical CCR analysis, while we will address the CVA issue in a future work.In particular, we will consider a simple Black and Scholes model, without taking into2onsideration collateral parameters in order to focus the attention on the implemented nu-merical techniques.The paper is organized as follows: Section 2 is a review of the EPE definition given bythe Basel Committee, while in Section 3 we give a description of the quantization approachto the EPE with some theoretical results. Section 4 describes some practical cases andcontains the set up of the associated numerical experiments, finally in Section 5 we reportthe obtained numerical results along with their interpretation.
In what follows we shall give a review about the Basel Committee guidelines concerning theestimation of the Exposure at Default, i.e. the EPE parameter. Let us set the followingnotations that will be used throughout the paper. • Given a derivative maturity time < T < + ∞ , we consider K ∈ N + time steps < t < t < · · · < t K which constitute the so called buckets array , denoted by B T,K ,where usually, but not mandatory, t K = T . • For every t k ∈ B T,K we denote by
M tM ( t k , S k ) := M tM ( t k , S t k ) the fair value( Mark-to-Market ) of a derivative at time bucket t k , with respect to the underlyingvalue S k , at time t k .For the sake of simplicity, we denote by t = 0 the starting time of the evaluationproblem, by considering the European case. • For every t k ∈ B T,K we denote by
M tM (cid:0) t k , S k (cid:1) := M tM (cid:0) t k , S t k (cid:1) the fair value( Mark-to-Market ) of a derivative at time bucket t k , with respect to the whole samplepath S k := { S t : 0 ≤ t ≤ t k } , with initial time t = 0 . • Taking into account previous definitions, we indicate by ϕ = ϕ ( T − t k , S k , Θ) thepricing function for the given derivative, where Θ represents the set of parametersfrom which such a pricing function may depends, e.g., the free risk rate r or thevolatility σ. • We will use the notation φ MtM to denote the Mark-to-Market value pricing function.
Remark 1.
We would like to underline that, in the Black-Scholes framework, the volatilitysurface has to be flat, which does not occur when real financial time series are considered. Itfollows that the above-mentioned pricing function φ most likely depends on more than the twoconsidered parameters r and σ . In particular, usually Θ ∈ R n , with n > , where the extraparameters characterize the specific geometric structure of the volatility surface associated tothe considered contingent claim. As usual in the counterparty credit risk EAD estimation, we stress the role of the un-derlying, understood as the only stochastic market parameter, while the others are deemedto be given, specifically, we assume that they are deterministic and constant or substitutedby their deterministic forward values. 3enceforward, we shall often use the notation k to indicate quantities of interest evalu-ated at the k − th time bucket t k . Besides, we give an account of the main amounts that willbe used in the following for EAD estimation, as they are defined in Basel III, [2]. • We denote the
Expected Exposure (EE) of the derivative by EE k := 1 N N (cid:88) n =1 M tM ( t k , S k,n ) + , N ∈ N + , which is nothing but the arithmetic mean of N Monte Carlo simulated MtM values,computed at the k − th time bucket t k , with respect to the underlying S . The positive part operator ( · ) + is effective if we are managing a symmetric derivative,such as an interest rate swap or a portfolio of derivatives. For a single option, it isredundant, as the fair value of the option is always positive from the buy side situation.We want to stress that the sell side does not imply counterparty risk, hence it is outof context. • We evaluate the
Expected Positive Exposure (EPE) by
EP E := (cid:80) Kk =1 EE k · ∆ k T , where ∆ k = t k − t k − indicates the time space between two consecutive time bucketsat k -th level. If the time buckets t k are equally spaced, then the formula reduces to EP E = K (cid:80) Kk =1 EE k . Therefore the EPE value gives the time average of the EE k . • We set
EEE := EE and EEE k := M ax { EE k , EEE k − } , for every k = 1 , . . . , K. Due to its non decreasing property,
EEE k , which is called the Effected Expected Ex-posure, takes into account the fact that, once the time decay effect reduces the MtMas well as the counterparty risk exposure, the bank applies a roll out with some newdeals. • We define the
Effected Expected Positive Exposure (EEPE), by
EEP E := (cid:80) Kk =1 EEE k · ∆ k T .
In order to avoid too many inessential regulatory details, we will work on EE k and EPE,the others being just arithmetic modifications of them. Remark 2.
Let us point out that the definition of EE k is taken from the Basel regulatoryframework. We find it quite strange, since, instead of giving a theoretical principle andsuggesting the Monte Carlo technique just as a possible computational tool, the simulationapproach is officially embedded in the general definition.
4n what follows we shall rewrite previously defined quantities in continuous time, and weadd the index A to indicate the adjusted definitions. Moreover we consider the dynamics ofthe underlying S t := { S t } t ∈ [0 ,T ] , T ∈ R + being some expiration date, as an Itô processes,defined on some filtered probability space (cid:0) Ω , F , F t ∈ [0 ,T ] , P (cid:1) . As an example, S t is thesolution of the stochastic differential equation defining the geometric Brownian motion, F t ∈ [0 ,T ] is the natural filtration generated by a standard Brownian motion W t = ( W t ) t ∈ [0 ,T ] starting from a complete probability space (Ω , F , P ) , where P could be the so called realworld probability measure or an equivalent risk neutral measure in a martingale approachto option pricing, see, e.g., [22, Ch.5].The Adjusted Expected Exposure EE A is given by EE Ak := E P (cid:2) M tM ( t k , S k ) + (cid:3) = (cid:90) ϕ ( t k , S k , Θ) d P ∼ = 1 N N (cid:88) n =1 M tM ( t k , S k,n ) + = (cid:91) EE Ak (1)We define the Adjusted Expected Positive Exposure
EP E A as EP E A := (cid:90) EE At d t = (cid:90) (cid:20)(cid:90) ϕ ( t, S k , Θ) d P (cid:21) d t . (2)In this new formulation, the Basel definition is simply one of the many methods toestimate the expected fair value of the derivative in the future. Remark 3.
We skip any comment about the choice of the most suitable probability measure P to be used in the calculation of EE k , this being beyond the aim of this paper.For a detailed discussion on the role played by the risk neutral probability for the drift S t or the historical real world probability, see, e.g., [8]. Remark 4.
Let us observe that the discount factor, or numeraire , is missing in the EPEdefinition. It was not forgot, but this is one of the several conservative proxies used in therisk regulation.
If we adopt a simulation approach, for the underlying path construction we could gen-erate, for each simulation n , a path with an array of points ( x n,t k ) . This algorithm is called path-dependent simulation (PDS). Alternatively, for each time bucket and for each simula-tion, we could jointly generate our N · K points. This approach is referred as direct-jump tosimulation date (DJS). We will come back on PDS and DJS approaches in Section 5.The figures below, taken from [20], well clarify the difference5igure 1: On the left: an example of PDS approach with six time buckets. On the right: an example ofDJS approach with six time buckets.
Finally, we recall that, in the risk management application, another widely used quantityis the potential future exposure
P F E α , a quantile based figure of the extreme risk. In acontinuous setting, we define the potential future exposure in the following way P F E ( α, t k ) := M tM ∗ such that P (cid:8) M tM ( t k , S k ) + ≥ M tM ∗ (cid:9) = 1 − α . (3) The quantization technique has been known from several decades and it comes from engi-neering, when addressing the issue of converting an analogical signal, e.g. images or sounds,into a discretized digital information. Other important areas of application are data com-pression and statistical multidimensional clustering. For a classical reference concerning thequantization approach, we refer to [11], while [16] gives a survey of the literature concerningfair value pricing problems for plain vanilla options, American and exotic options, basketCDS.In addition, alternative quantization approaches, such as the so-called dual quantization and the treatment of underlying assets driven by more structured stochastic processes, aretaken into consideration in [17] and [18].In this section, we shall give a sketch of the quantization idea, by emphasizing its practicalfeatures, but without giving all the details concerning the mathematical theory behind it.Let X ∈ R d , d ∈ N + , be a d − dimensional continuous random variable, defined overthe probability space (Ω , F , P ) and let P X the measure induced by X . The quantizationapproach is based on the approximation of X by a d -dimensional discrete random variable (cid:98) X , further details of which will be given later, defined by means of a so called quantizationfunction q of X , that is to say, (cid:98) X := q ( X ) , in such a way that (cid:98) X takes N ∈ N + finitelymany values in R d . The finite set of values for (cid:98) X is denoted by q ( R ) := { x , ..., x N } andit is called a quantizer of X, while the application q ( X ) is the related quantization . Todistinguish the one-dimensional case ( d = 1 ) from the d − dimensional one ( d > ), the terms quantization , resp. vector quantization (VQ), are usually used.6uch a set of points in R d can be used as generator points of a Voronoi tessellation.
Letus recall that, if X is a metric space with a distance function d , K is an index and ( P k ) k ∈ K isa tuple of ordered collection of nonempty subsets of X , then the Voronoi cell R k generatedby the site P k is defined as the following the set R k := { x ∈ X | d ( x, P k ) ≤ d ( x, P j ) for all j (cid:54) = k } . Therefore, the Voronoi tessellation is the tuple of cells ( R k ) k ∈ K . In our case such a tessel-lation reduces to substitute the set of cells P k with a finite number x , . . . , x N of distinctpoints in R d , so that the Voronoi cells are convex polytopes.More precisely, we construct the following Voronoi tessels with respect to the euclideannorm (cid:107) · (cid:107) in R d : C ( x i ) = (cid:110) y ∈ R d : | y − x i | < (cid:107) y − x j (cid:107)∀ j (cid:54) = i (cid:111) , with associated quantization function q defined as follows q ( X ) = N (cid:88) i =1 x i · C i ( x ) ( X ) . (4)Such a construction allows us to rigorously define a probabilistic setting for the claimedrandom variable (cid:98) X, by exploiting the probability measure induced by the continuous randomvariable X. In particular, we have a probability space (cid:0) Ω , F , P (cid:98) X (cid:1) , where the set of elementaryevents is given by Ω := { x , . . . , x N } , and the probability measure P (cid:98) X is defined by thefollowing set of relations < P (cid:98) X ( x i ) := P X ( X ∈ C ( x i )) =: p i , for i = 1 , . . . , N . The goal of such an approximation is to deal efficiently with applications that arise whencalculating some functionals of the random vector X , as in the derivative pricing problemcase, in order to evaluate the expectation E [ f ( X )] of a certain payoff function f of X orwhen we have to deal with a quantile base indicator, as it happens in the risk managementfield.We would like to take into account the former case, in particular we will consider thefollowing approximation E [ f ( X )] ∼ = E (cid:104) f (cid:16) (cid:98) X (cid:17)(cid:105) = (cid:88) i f ( x i ) · p i , with control on the accuracy of the chosen quantization.Let us Assume X ∈ L p (Ω , F , P ) , p ∈ (1 , ∞ ) and define the L p error as follows E (cid:104) (cid:107) X − (cid:98) X (cid:107) p (cid:105) := (cid:90) R d min i=1 ,..., N (cid:107) x − x i (cid:107) p d P X ( x ) , (5)where we denote by d P X the probability density function characterizing the random variable X. The integrand in eq. (5) is always well defined, being a minimum with respect to thefinite set of generators x , . . . , x N . the , or, at least, one, optimal quantizer , understoodas the set of Voronoi generators minimizing the value of the integral, once both parameters d, N, p and the probability density of X are given. Even if such a problem could be partic-ularly difficult in the general case, also because it may rise to infinitely many solutions, thisis not the case in our setting. In fact, we aim at considering a standard Black-Sholes frame-work, where the only 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 areinterested in the distribution of the driving random perturbation at maturity time T , whichmeans that we are dealing with the quest of an optimal quantizer for a d -dimensional Gaus-sian random variable, assumed to be standard, up to suitable transformation of coordinates,namely X ∼ N (0 , I d ) .Algorithms to get the optimal quantizer can be found within the aforementioned refer-ences. Moreover, when the dimension d ≤ , there is a well established literature concerninghow to find the optimal quantizers when the Gaussian framework is considered, see, e.g.,the web site and [4, 16, 21].A particularly important quantity related to the choice of the optimal quantizer is rep-resented by the so called distortion parameter D NX ( (cid:98) X ) := E (cid:104) (cid:107) X − (cid:98) X (cid:107) (cid:105) = (cid:90) R d min i=1 ,..., N (cid:107) x − x i (cid:107) · d P X ( x ) , (6)which is defined with respect to the quantizer { x , . . . , x N } . If the quantization function (cid:98) X takes values in the set of optimal generators, then thedistortion parameter admits a minimum, which will be indicated by D NX , with lim N →∞ D NX =0 . The following theorem, originally due to Zador, see [23, 24], generalized by Bucklew andWise in [6] and then revisited in [16] in its non asymptotic version as a reformulation of thePierce lemma, gives a quantitative result about the distortion magnitude.
Theorem 5. [Zador] Let X ∈ L p + ε (Ω , F , P ) , for p ∈ (0 , + ∞ ) , ε > , and let Γ be the N − size tessellation of R d related to the quantizer (cid:98) X. Then, lim N (cid:18) N pd min | Γ |≤ N (cid:107) X − (cid:98) X (cid:107) p P (cid:19) = J p,d (cid:90) R d g dd + p ( x ) d x pd , (7) where we assume d P = g d λ d + d ν, for some suitable function g, and ν ⊥ λ d , d λ d being theLebesgue measure on R d , while the constant J p,d corresponds to the case X ∼ U nif (cid:0) [0 , d (cid:1) . Let us recall that the optimal quantizer is stationary in the sense that E (cid:104) X | (cid:98) X (cid:105) = (cid:98) X ,hence (cid:82) C ( x i ) ( x ∗ i − x ) d P X ( x ) = 0 , i = 1 , . . . , N. In what follows, we focus on the case d = 1 and p = 2 , hence considering a one-dimensional stochastic process and the quadraticdistortion measure, therefore, in terms of Th. (5), we have J p, = p · ( p +1) , hence J , = . Remark 6.
For practical applications, and in order to compare numerical results obtainedby quantization with those produced by standard Monte Carlo techniques, we are mainlyinterested in the order of convergence to zero of the distortion parameter. In particular,thanks to Zador Theorem, we have that the quadratic distortion is of order O (cid:0) N − (cid:1) .
8t is also possible to provide results concerning the accuracy of the approximation E (cid:104) f (cid:16) (cid:98) X (cid:17)(cid:105) , by mean of the distortion’s properties, see [16]. In particular, we can distinguishthe following cases: Lipschitz case if f is assumed to be a Lipschitz function, then (cid:12)(cid:12)(cid:12) E [ f ( X )] − E (cid:104) f (cid:16) (cid:98) X (cid:17)(cid:105)(cid:12)(cid:12)(cid:12) ≤ K f · (cid:13)(cid:13)(cid:13) X − (cid:98) X (cid:13)(cid:13)(cid:13) ≤ KL f · (cid:13)(cid:13)(cid:13) X − (cid:98) X (cid:13)(cid:13)(cid:13) . (8) The smoother Lipschitz derivative case If f is assumed to be continuously differen-tiable with Lipschitz continuous differential Df , then, by performing the quantizationusing an optical quadratic grid Γ and by applying Taylor expansion, we have (cid:12)(cid:12)(cid:12) E [ f ( X )] − E (cid:104) f (cid:16) (cid:98) X (cid:17)(cid:105)(cid:12)(cid:12)(cid:12) ≤ KL Df · (cid:13)(cid:13)(cid:13) X − (cid:98) X (cid:13)(cid:13)(cid:13) . (9) The Convex Case If f is a convex function and (cid:98) X is stationary, then exploiting the Jenseninequality, we have E (cid:104) f (cid:16) (cid:98) X (cid:17)(cid:105) = E (cid:104) f (cid:16) E (cid:104) X | (cid:98) X (cid:105)(cid:17)(cid:105) ≤ E [ f ( X )] , (10)hence, the approximation by the quantization is always a lower bound for the true value of E [ f ( X )] . Remark 7.
We emphasize that the (optimal) quantization grid for given parameter valuesof
N, d and p is calculated off-line once and for all. Then, in the computational effortcomparison vs. a strict sense Monte Carlo approach, we must take into account that withthe quantization one only has to plug-in the points in the numerical model, not to calculateor simulate them. Remark 8.
An increasing literature is devoted to the functional quantization . In this case,the “random variable”, which has to be discretized in an optimal way, belongs to a suitedfunctional space, e.g. one can consider an application X such that X : Ω → (cid:0) L T , (cid:107)·(cid:107) (cid:1) , where L T = L T ([0 , T ] , d t ) . We recall that, even if in mathematical finance applications thestochastic calculus in continuous time is a very useful tool, in practice we have to deal withdiscrete sampling , in fact, Asian options or any other look-back derivatives have to workwith a discrete calendar for the fixing instants. Then, depending on the specific application,one can choose if to approximate the discrete time real-world-problem by optimal quantizationor if it is better to quantize the continuous time setting and then to apply it to the practicalapplication, see, e.g.,[17] for a survey on such a topic. quantization for the EPE calculation
In the following, we focus our attention on the calculation of EPE for option styles derivativesin the Black-Scholes standard setting, see [5]. Strictly speaking, the underlying evolvesaccording to the following stochastic differential equation9 S t = S t · d t + σS t d W t , (11)with related solution S t = S · exp (cid:20)(cid:18) r − σ (cid:19) t + σW t (cid:21) , (12)where r, σ > , ( W t ) t ≥ is a Brownian motion, while S is the initial value of the underlying S t . It is well known that the plain vanilla call and put options have a closed pricing formula.Since we do not want to give here a survey on the several extensions to the model, we contentourselves saying that, in the equity and Forex derivatives markets, the most important modelextensions of eq. (11) are the local volatility models, the
Heston model and the
SABR models,see, e.g., [10], [13], [12], respectively.As usual in a new methodology proposal, as a first step we prefer to check the techniquesin a simple framework, in order to have clear insights about its properties.We guess that the Monte Carlo approach is just one of the many feasible techniques forEE and EPE calculation. After all, it is computationally quite expensive, as shown by thefollowing example. A medium bank easily has D = O (cid:0) (cid:1) derivatives deals. In order tovalidate the internal models for the EPE calculation, usually the central banks require atleast K = 20 time steps and N = 2000 simulations. Finally, let us suppose that the relevantunderlying (often called risk factors ) to be simulated have O (cid:0) (cid:1) order. It is the sum ofequity underlying, FX significant rates and rate curve points. Let U be such a parameter.What about the computational effort for an EPE process task on the whole book? If weadopt a pure Monte Carlo strategy, we must distinguish between the two main steps:1. simulation (and storage) of the underlying paths2. evaluation of the EE quantities. We omit for simplicity the last EPE layer, since it isjust an algebraic recombination of the EEs.The first step implies a grid of G = K · N · U = 4 · points, which work as an input for thestep 2. This one requires N T = K · N · D = 4 · different tasks. By recalling definitionof EE k , each of these tasks is a pricing process, which very often turns to be performed bya Monte Carlo algorithm with several thousands of simulation steps.Generally speaking, the evaluation of EPE by rough Monte Carlo is K · N = O (cid:0) (cid:1) more expensive than the usual daily end-of-day mark to market evaluation of the book.Hence, we argue that the brutal Monte Carlo approach can not be a satisfactory way forthe EPE calculation.For this reason, some banks are exploiting some innovative technological approaches,such as the use of the graphical processing unit (GPU), instead of CPUs, to set up a parallelcalculation system, with some new programming languages such as NVIDIA, while someother banks have been invested a lot buying grid computing platforms.We believe that an algorithmic based improvement could be more efficient than thehardware innovation, or it can be combined with, and much less expensive. In the derivativespricing 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 quan-tization technique. At a first level, we can distinguish between path-dependent derivatives10nd non path-dependent derivatives, in the following pd and npd for brevity. We point outthat this definition is different from the usual plain vs. exotic derivatives. Among the nonpath-dependent derivatives we include not only the plain European options, but also Euro-pean and American style options with exotic payoff, e.g. mixed digital continuous, spreadoptions, etc. In the npd class, we will work with the Asian options, probably the mostpopular one.For the npd derivatives, the quantization for the EPE simply reduces to set up theproblem by selecting the parameters ( N k ) k =1 ,...,K for the quantization size at each timebucket ( t k ) and then to compute the EPE quantized approximation. We use the optimalquantizer case, recalling that S exp (cid:20)(cid:18) r − σ (cid:19) t + σ · W t (cid:21) = S exp (cid:20)(cid:18) r − σ (cid:19) t + σ · √ t · N (0 , (cid:21) . More formally, if we indicate by the exponent Q the quantized EPE, one easily gets EE Qk = N k (cid:88) i =1 M tM (cid:16) t k , S (cid:16) x ki (cid:17)(cid:17) + p ki , (13) EP E Q = (cid:80) EE Qk · ∆ k T = (cid:80) k ∆ k (cid:16)(cid:80) N k i =1 M tM (cid:0) t k , S (cid:0) x ki (cid:1)(cid:1) + · p ki (cid:17) T . (14)Fig. (4), graphically explains the calculation procedure. In particular, the black pointon the left is the underlying level at time t . For each time step t k and for each point of thequantizer x ki , a MtM is calculated and it is weighted by the probability p ki . Hence EE Q and EP E Q straightly follow.Figure 2: evolution of EE Q at every time bucket t k . Concerning the theoretical properties of such an approximation, we provide a usefulresult, which can be easily applied to the case of European style options.
Proposition 9.
Let us consider an option in the Black and Scholes setting, with d = 1 andsuppose that the pricing function ϕ ( · ) is Lipschitz continuous or continuously differentiablewith a Lipschitz continuous differential. oreover, let us define D EP EN := (cid:12)(cid:12) EP E A − EP E Q (cid:12)(cid:12) , where EP E A is the AdjustedExpected Positive Exposure, defined in eq. (2) , and EP E Q is the quantized Expected PositiveExposure, defined in eq. (14) . Then, we have D EP EN ∝ N − · K − , if ϕ Lipschitz continuous ,D EP EN ∝ N − · K − , if ϕ cont. differentiable with Lipschitz cont. differential . Proof.
By definition, we have (cid:12)(cid:12)
EP E A − EP E Q (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:80) EE Ak ∆ k T − (cid:80) EE Qk ∆ k T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 1 T (cid:12)(cid:12)(cid:12)(cid:88) ∆ k · (cid:16) EE Ak − EE Qk (cid:17)(cid:12)(cid:12)(cid:12) , hence rearranging the terms and recalling that the CCR is effective just for the buy sideposition, we can skip the positive part obtaining (cid:12)(cid:12)(cid:12)(cid:88) ∆ k (cid:16) EE Ak − EE Qk (cid:17)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:88) ∆ k · (cid:0) E P [ M tM ( t k , S k )] − E (cid:98) P [ M tM ( t k , S k )] (cid:1)(cid:12)(cid:12)(cid:12) ≤ (cid:88) ∆ k (cid:0) E P [ M tM ( t k , S k )] − E (cid:98) P [ M tM ( t k , S k )] (cid:1) . (15)Moreover, we have (cid:12)(cid:12)(cid:12)(cid:88) ∆ k (cid:16) EE Ak − EE Qk (cid:17)(cid:12)(cid:12)(cid:12) ≤ (cid:16)(cid:88) ∆ k (cid:12)(cid:12) E P [ M tM ( t k , S k )] − E (cid:98) P [ M tM ( t k , S k )] (cid:12)(cid:12)(cid:17) . In a more explicit fashion, let us work on the single element k of the summation, i.e. (cid:0) E P [ M tM ( t k , S k )] − E (cid:98) P [ M tM ( t k , S k )] (cid:1) . If we consider
M tM ( t k , S k ) = ϕ ( t k , S k ( X )) asthe function appearing in eq. (8) and eq. (9), we get, respectively, (cid:0) E P [ M tM ( t k , S k )] − E (cid:98) P [ M tM ( t k , S k )] (cid:1) ≤ KL f,k · (cid:13)(cid:13)(cid:13) X k − (cid:99) X k (cid:13)(cid:13)(cid:13) (16) (cid:12)(cid:12) E P [ M tM ( t k , S k )] − E (cid:98) P [ M tM ( t k , S k )] (cid:12)(cid:12) ≤ KL Df,k · (cid:13)(cid:13)(cid:13) X k − (cid:99) X k (cid:13)(cid:13)(cid:13) . (17)By replacing eq. (16) and eq. (17) in eq. (15), we obtain (cid:12)(cid:12)(cid:12)(cid:88) ∆ k (cid:16) EE Ak − EE Qk (cid:17)(cid:12)(cid:12)(cid:12) ≤ (cid:88) ∆ k KL f,k · (cid:13)(cid:13)(cid:13) X k − (cid:99) X k (cid:13)(cid:13)(cid:13) , (18) (cid:12)(cid:12)(cid:12)(cid:88) ∆ k (cid:16) EE Ak − EE Qk (cid:17)(cid:12)(cid:12)(cid:12) ≤ (cid:88) ∆ k KL Df,k · (cid:13)(cid:13)(cid:13) X k − (cid:99) X k (cid:13)(cid:13)(cid:13) . (19)Let us consider eq. (18), the calculation for eq. (19) being the same.Set KL equal to the mean of ( KL · ,k ) , for all k, and suppose that the mesh t k is regularenough, i.e. we require that lim K ∆ k = 0 , O (∆ k ) = K − ∀ k. Thanks to the Zador Theoremand taking N → + ∞ , we have T · (cid:12)(cid:12)(cid:12)(cid:88) ∆ k (cid:16) EE Ak − EE Qk (cid:17)(cid:12)(cid:12)(cid:12) ≤ T (cid:88) ∆ k KL f,k (cid:13)(cid:13)(cid:13) X k − (cid:99) X k (cid:13)(cid:13)(cid:13) ∝ T · K · KL · T K · N − .
12y simplifying, we get the result. Similar calculations provide the result when the pricingfunction is assumed to be continuously differentiable with Lipschitz continuous differential.For a more abstract setting, see [16, Sec.2.4].
Remark 10.
Let us discuss the hypothesis under which the result holds. The central roleis played by the function ϕ ( x, · ) as a function of the Brownian motion W t , that is, of thequantity x · √ t , x sampled from the N (0 , . We recall in fact that the usefulness of thequantization is to infer the properties of the approximation in the specific problem, startingfrom the Gaussian optimized discretization. As an example, for a put option the pricingfunction is bounded, Lipschitz continuous and twice differentiable, since the Black-Scholesformula is C ∞ . Then, the above proposition holds.For a call style option, the convexity properties easily holds, hence the quantization givesus a lower bound to the EPE estimation.
Remark 11.
If we consider the pricing function ϕ ( · ) as the elementary unit of our EPEcomputational work-flow, the computational complexity of the quantized approach is (cid:80) N kk , to be compared with the global number of Monte Carlo scenarios simulations. For path-dependent derivatives, for each time t k , at least in a discrete sampling sense,one needs the whole path of the underlying. Hence, the above approach is not satisfactory,as the pricing function depends not only on the current level S k , but also on some functions,e.g. the average, min, max, etc., of the underlying level until t k . Latter task can be efficientlystudied by an approach like the PDS one, as in figure 2.Let N k be a positive integer for the quantization , and q k ( R ) = ( x , x , ..., x N k ) is thequantizer of size N k , namely the random variable (cid:99) X k = q ( X k ) that maps the randomvariable to an optimal discrete version. If we refer to the Black-Scholes model, we want toquantize at each step the Brownian motion W t that generates the log-normal underlyingprocess. We recall that W t ∼ N (0 , t ) is a normal centered random variable and that W t − W s ∼ N (0 , t − s ) . Again, we define C ( x i ) as the i − th Voronoi tessel such that C ( x i ) := (cid:110) y ∈ R d : | y − x i | < | y − x j | ∀ j (cid:54) = i (cid:111) , with the so-called nearest neighborhood principle. A set of probability masses vectors isassigned to the N k − tuple, let be p k = (cid:16) p k , p k , . . . , p kN k (cid:17) , where p i = P ( C ( x i )) under theoriginal probability P X , for all i = 1 , . . . , N k . The following questions naturally arise: howand where to use the quantizers for the EE k calculation ? The quantization tree is a discretespace, discrete time process, defined by p ki = P (cid:16) (cid:98) X k = x ki (cid:17) = P (cid:16) X k ∈ C i (cid:16) x k (cid:17)(cid:17) , (20) π kij = P (cid:16) (cid:98) X k +1 = x k +1 j | (cid:98) X k = x ki (cid:17) = P (cid:16) X k +1 ∈ C j (cid:16) x k +1 (cid:17) | X k ∈ C i (cid:16) x k (cid:17)(cid:17) = P (cid:0) X k +1 ∈ C j (cid:0) x k +1 (cid:1) , X k ∈ C i (cid:0) x k (cid:1)(cid:1) p ki (21)13he following theoretical result allows us to explicitly solve the transition probabilityformula (4), by recalling that X k is the original Brownian motion sampled at a given time t k . Proposition 12.
Let us use denote by ∆ k := t k +1 − t k the time space between two consecutivetime buckets and by φ the density of the N (0 , random variable. Furthermore, let usindicate by U k , U k +1 , resp. by L k , L k +1 , the upper, resp. lower, bounds of the − dimensionaltessels C i (cid:0) x k (cid:1) and C j (cid:0) x k +1 (cid:1) . Then, the transition probability π kij assumes the following expression π kij = (cid:90) C j,k +1 f ( x k +1 | C i,k ) d x k = (cid:82) U k L k (cid:16)(cid:82) U k +1 L k +1 φ (cid:16) ( x − y ) √ ∆ k (cid:17) d x (cid:17) φ (cid:16) y √ t k (cid:17) · dyp ki . (22) Proof.
The result is a straightforward calculation, indeed let us start considering a morespecific problem, namely we aim at calculate P (cid:0) X k +1 ∈ C j (cid:0) x k +1 (cid:1) | X k = y, y ∈ C (cid:0) x k (cid:1)(cid:1) . For given x ∈ C j (cid:0) x k +1 (cid:1) , y ∈ C j (cid:0) x k (cid:1) , by recalling the scaling property and the indepen-dence of the Brownian motion increments, we easily get P (d x ) = φ (cid:16) ( x − y ) √ ∆ k (cid:17) · d x , and theresult follows by extending such a fact to tessels C i , C j .The picture below shows the practical features of the formula eq. (22).Figure 3: a graphical representation of the transition probability function Remark 13.
Even if the proposition comes from elementary probability, this result is auseful improvement to the procedure in Pagès et al (2009), where a Monte Carlo approachfor the π kij was suggested. Remark 14.
From a computational complexity point of view, we observe that the abovecoefficients π kij can be calculated off line, once and for all, given the time discretizationparameter K and the chosen granularities { N , N , ..., N K } . N P, could be too many from a theo-retical perspective. In fact, they amount to
N P = (cid:81) k N k . If we set, as usual, K = O (cid:0) (cid:1) and N k = O (cid:0) (cid:1) , then N P = O (cid:0) (cid:1) , which seems to be intractable for practical pur-poses. Fortunately, this does not occur, from a practical point of view. Many paths have anegligible probability, as very often we have π kij (cid:39) , so we can skip a large fraction of thecombinatorial cases by some heuristic a priori rule that avoids calculation depending fromthe distance (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) x k +1 j − x ki √ ∆ k (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . In this section we will give an application of quantization method in CCR with respect toa portfolio consisting of a bank account and one risky asset which is the underlying of aEuropean type option. We suppose that the dynamics of the underlying S t := { S t } t ∈ [0 ,T ] , forsome maturity time T ∈ R + , is given accordingly to a geometric Brownian motion, namely d S t = rS t d t + σS t d W t , (23)where r is the risk free interest rate, e.g. associated to a bank account, σ is the volatilityparameter characterizing the underlying behaviour, while W t := { W t } t ∈ [0 ,T ] is a R − val-ued Brownian motion on the filtered probability space (Ω , F , F t , P ) , {F t } t ∈ [0 ,T ] being thefiltration generated by W t . We recall that the stochastic differential equation (23) admitsan explicit solution, see, e.g.,[22, Ch.3], given by S t = S exp (cid:26) σW t + (cid:18) r − σ (cid:19) t (cid:27) , (24) S being the initial value of the underlying S t .If we consider a European call option with strike price K ∈ R + snd maturity time T ,written on S t described as above, then its fair value C eu = C eu ( S , r, K, σ, T ) , with respectto the unique martingale equivalent measure, is given by C eu ( S , r, K, σ, T ) := e − rT E (cid:2) ( S T − K ) + (cid:3) = e − rT E (cid:34)(cid:18) S exp (cid:26)(cid:18) r − σ (cid:19) T + σW T (cid:27) − K (cid:19) + (cid:35) , (25)with explicit solution given by the Black-Scholes formula.Before enter into details about a numerical application of our proposal, see Sec. (4), toa concrete case, let us underline some points. In practical applications, the computationalchallenges are very often much harder than one believes from a theoretical perspective.Referring this general principle to the CCR, the portfolio of derivatives of a counterparty A with B may consist of several hundreds of derivatives j = 1 ...J , then the Mark-to-Market isgiven by M tM A ( t ) = (cid:80) j M tM Aj ( t ) . These derivatives could be bought options, s old options,15 waps and so on. A collateral of value V t is usually posted. Hence, at the current time, theexposure of the counterparty A to B is given by (cid:88) j M tM Aj ( t ) − V t + , (26)an expression which is similar to the one describing a call option written on a derivativesportfolio. In the CCR applications, one wants to check several quantities related to thecurrent exposure, such as EE, EPE, PFE, and so on. In calculating the expected exposureof quantities as in (26), i.e. EE j ( · ) , because of non linearity, one can not calculate separatelythe single deal quantities, i.e. the EE j ( t k ) , to aggregate them by summation in a secondmoment. A fortiori, in the PFE quantile framework, one can not infer easily the quantileby the specific quantiles.It follows that all banks, as a general strategy, first calculate a large set of scenarios forthe underlyings, coherently with respect to the considered probabilistic structure for it, andthen they evaluate and store a large set of
MtM, from which to pick any kind of statisticsand risk figures. In this field quantization can play a role as a competitive methodology,particularly to what concerns saving computational costs. Nevertheless, since the CCR fora whole book comes from the single deals computations, we aim to test at a low level thequantization, in a plain vanilla context. In further research we will move to Exotic deals aswell as effective management of large portfolios, will be the subject of our next studies.
For the market valuation of financial statements, the generation of potential market scenariosis required. In Sec. (2), we definded two achievable approaches, namely the path-dependentsimulations method (PDS) and the direct jump to simulation data (DJS) technique: in firstcase one simulates a whole path-wise possible trajectory, while in the second each time pointis directly computed.More practically, we could apply the DJS approach by selecting a grid size N k for eachtime t k and then using K different 1-dimension quantizers q k ( R ) .Alternatively, by using the PDS approach, we can work just with one d -dimension q (cid:0) R d (cid:1) quantizer of cardinality N , hence each dimension works for one time step.We choose the latter approach for our application. The steps are as follows.1. Selection of the grid size N and the dimension d, according respectively to the com-putational effort constraints and to the time buckets cardinality K, i.e. d = K.
2. By obvious dilatation, each point of the ( N × K ) grid of the vector quantizer is mappedin order to get a proper to a Brownian motion increment realization, i.e. x i,k → x i,k (cid:112) ∆ k = ∆ (cid:102) W , ∀ i ∈ { , . . . , N } , k ∈ { , . . . , d } .
3. The above increments are inserted in the Black-Scholes diffusion to get N “possible”underlying paths S t k .
16. Payoff, MtMs and all the related quantities are calculated, by using the probabilitymasses p i . Although we know that it is not possible to get an exhaustive comparison, anyhow we tryto make the exercise quite general, by choosing some different parameter combinations, e.g. • spot price ( S ) : 90 , , • interest rate r : 3% • volatility σ : 15% , , • strike price R : 100 (we do not use the usual K notation to avoid confusion with thetime buckets set { t k } ) • time to maturity T : one yearAccording to the choice of several banks to consider an increasing sampling frequency overtime, because of accuracy decreasing over large horizons, we decide to set time buckets inthe following range (cid:26) , , , , , , , , , (cid:27) , namely, we are considering the first, the second, the third and the fourth week and then thesecond, the third, the sixth and the twelfth month of the year. In order to evaluate the Expected Positive Exposure (EPE), we compare the quantizationapproach with standard Monte Carlo method. We distinguish several cases, depending onthe moneyness , i.e. the relative position of S t versus the strike price K of the consideredcall option, and volatility parameters.Each case will be analyzed with respect to the Monte Carlo-Sobol approach, see, e.g., [7],with N = 10 , as well as considering the Monte Carlo simulations (MC) and the quantizationgrids (Q), with N = 10 . Concerning 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 driftof the geometric Brownian motion has to be equal to the risk free rate. Under such anassumption, the Expected Exposure (EE) admits a closed form, i.e. EE Ak = E P (cid:2) M tM ( t k , S k ) + (cid:3) = M tM ( t , S ) · exp ( t k − t ) , (27)which is implied by the martingale property of the discounted fair value, see [22, Ch.5]for further details.In a more general setting, the simple approach characterized by eq.(27) can not beapplied, because of more involved payoffs. Moreover, the Mark-to-Market function does notexist in an analytical form and the drift can assume values different from the risk free rate r. Besides, we are often interested in calculating a measure of the possible worst exposurewith a certain level of confidence. Such a measure is expressed in terms of p − percentile of17he exposure’s distribution, the so-called Potential Future Exposure (PFE), defined in eq.(3).As already stressed, since we aim at testing the efficiency of quantization techniques,we refer here to a simple problem, while the case of more complex financial models is thesubject of our next research.In what follows we always consider a ( D + 1) × matrix, D being the number of timebuckets taken into account, hence D = 9 , since we start from t = 0 . Each matrix entryrepresents the value of the Expected Exposure (EE), EE k := N (cid:80) Nn =1 M tM ( t k , S k,n ) + ,obtained by applying eq. (13). Last row gives the value of the Expected Positive Exposure(EPE), EP E := (cid:80) Kk =1 EE k · ∆ k T , calculated according to eq. (14).The efficiency evaluation of exploited procedures requires a comparison between thevalue obtained by simulations and a benchmark. In the case of quantization approach, sucha comparison is given by the evaluation of the (percent) relative error ε with respect tothe Black-Scholes price obtained using formula (27). As regards the Monte Carlo approach,the analyzed quantity is the (percent) relative standard error (RSD). By the Law of largenumbers , it is well known that the Monte Carlo approach always converges to the truevalue, hence its standard deviation is more informative than the single execution error.The numerical calculation in the tables stands for the numerical integration of formula (1),i.e. the expected value of the possibles MtMs over the different underlying prices. Theintegration has been done considering a simple rectangle scheme, with points. Finally,we also tested the Monte Carlo-Sobol technique, based on binary digits that well fill the [0 , interval. To summarize, all the different techniques were compared with the same numberof points and avoiding too sophisticated versions, to keep the comparison as fair as possible.Tables 1, 2 and 3, contain numerical results in the ITM case with S = 110 , whiletables 4,5 and 6, refer to the ATM case with S = 100 , finally tables 7,8, and 9, report theperformances in the OTM case, with S = 90 . Analytic Numerical Quantization Monte Carlo MC-Sobolt EE EE ε EE ε EE RSD EE ε
1w 14,711 14,710 -0,007% 14,711 0,000% 14,649 0,004% 14,710 -0,010%2w 14,719 14,717 -0,007% 14,719 0,000% 14,725 0,006% 14,717 -0,014%3w 14,726 14,726 -0,008% 14,727 0,000% 14,815 0,007% 14,725 -0,012%1m 14,776 14,734 -0,008% 14,736 0,000% 14,869 0,008% 14,735 -0,002%2m 14,813 14,774 -0,010% 14,775 0,000% 15,003 0,012% 14,775 -0,005%3m 14,924 14,811 -0,011% 14,812 0,000% 14,801 0,014% 14,808 -0,030%6m 15,036 14,921 -0,016% 14,924 0,000% 14,916 0,020% 14,924 -0,001%9m 15,149 15,033 -0,019% 15,036 0,000% 15,157 0,026% 14,994 -0,282%1y 15,149 15,145 -0,023% 15,149 -0,001% 14,870 0,031% 15,049 -0,659%EPE 14,970 14,967 -0,017% 14,970 0,000% 14,951 -0,125% 14,934 -0,241%
Table 1:
EPE: − ITM European call. σ = 15% . nalytic Numerical Quantization Monte Carlo MC-Sobolt EE EE ε EE ε EE RSD EE ε
1w 18,0448 18,0435 -0,007% 18,0447 0,000% 17,9530 0,495% 18,0427 -0,012%2w 18,0551 18,0538 -0,008% 18,0552 0,000% 18,0637 0,693% 18,0524 -0,015%3w 18,0656 18,0640 -0,009% 18,0656 0,000% 18,2050 0,880% 18,0628 -0,015%1m 18,0760 18,0743 -0,009% 18,0760 0,000% 18,2792 0,996% 18,0754 -0,004%2m 18,1248 18,1225 -0,013% 18,1247 0,000% 18,4457 1,420% 18,1234 -0,007%3m 18,1701 18,1673 -0,015% 18,1701 0,000% 18,1189 1,764% 18,1627 -0,041%6m 18,3069 18,3027 -0,023% 18,3069 0,000% 18,2130 2,568% 18,3091 0,012%9m 18,4447 18,4392 -0,030% 18,4447 0,000% 18,6685 3,359% 18,3983 -0,252%1y 18,5836 18,5763 -0,036% 18,5836 0,000% 18,2320 3,985% 18,5664 -0,090%EPE 18,3638 18,3590 -0,025% 18,3638 0,000% 18,33791 -0,140% 18,34755 -0,088%
Table 2:
EPE: − ITM European call. σ = 25% .Analytic Numerical Quantization Monte Carlo MC-Sobolt EE EE ε EE ε EE RSD EE ε
1w 19,884 19,883 -0,007% 19,884 0,000% 19,777 0,525% 19,882 -0,012%2w 19,896 19,894 -0,008% 19,895 0,000% 19,905 0,735% 19,892 -0,016%3w 19,907 19,905 -0,009% 19,907 0,000% 20,073 0,935% 19,904 -0,016%1m 19,918 19,917 -0,010% 19,918 0,000% 20,157 1,058% 19,918 -0,004%2m 19,972 19,969 -0,014% 19,972 0,000% 20,338 1,510% 19,971 -0,008%3m 20,022 20,019 -0,017% 20,022 0,000% 19,947 1,881% 20,013 -0,045%6m 20,173 20,168 -0,026% 20,173 0,000% 20,028 2,758% 20,177 0,018%9m 20,325 20,318 -0,035% 20,325 0,000% 20,608 3,622% 20,278 -0,230%1y 20,478 20,468 -0,044% 20,478 0,000% 20,083 4,309% 20,509 0,156%EPE 20,236 20,229 -0,030% 20,236 0,000% 20,204 -0,155% 20,232 -0,019%
Table 3:
EPE: − ITM European call. σ = 30% .Analytic Numerical Quantization Monte Carlo MC-Sobolt EE EE ε EE ε EE RSD EE ε
1w 7,48940 7,4889 -0,007% 7,4894 0,000% 7,4479 0,539% 7,4885 -0,012%2w 7,49372 7,4931 -0,008% 7,4937 0,000% 7,4974 0,755% 7,4925 -0,016%3w 7,49804 7,4974 -0,009% 7,4981 0,000% 7,5623 0,960% 7,4968 -0,016%1m 7,50237 7,5016 -0,010% 7,5024 0,000% 7,5947 1,086% 7,5021 -0,004%2m 7,52260 7,5216 -0,013% 7,5226 0,000% 7,6646 1,552% 7,5219 -0,009%3m 7,54143 7,5402 -0,017% 7,5414 0,000% 7,5128 1,935% 7,5379 -0,046%6m 7,59820 7,5964 -0,024% 7,5982 0,000% 7,5412 2,842% 7,6011 0,039%9m 7,65540 7,6530 -0,031% 7,6554 0,000% 7,7662 3,735% 7,6354 -0,262%1y 7,7130 7,7099 -0,037% 7,7130 0,000% 7,5699 4,473% 7,7345 0,282%EPE 7,6218 7,61975 -0,026% 7,6218 0,000% 7,61211 -0,127% 7,62250 0,010%
Table 4:
EPE: ATM European call. σ = 15% . nalytic Numerical Quantization Monte Carlo MC-Sobolt EE EE ε EE ε EE RSD EE ε
1w 11,3550 11,3542 -0,007% 11,3550 0,000% 11,2873 0,583% 11,3536 -0,013%2w 11,3616 11,3606 -0,009% 11,3616 0,000% 11,3670 0,815% 11,3597 -0,016%3w 11,3681 11,3670 -0,010% 11,3681 0,000% 11,4762 1,039% 11,3661 -0,018%1m 11,3747 11,3735 -0,011% 11,3747 0,000% 11,5270 1,176% 11,3742 -0,004%2m 11,4054 11,4036 -0,016% 11,4054 0,000% 11,6289 1,684% 11,4042 -0,010%3m 11,4339 11,4317 -0,020% 11,4339 0,000% 11,3723 2,107% 11,4279 -0,053%6m 11,5200 11,5165 -0,030% 11,5200 0,000% 11,3929 3,132% 11,5262 0,054%9m 11,6067 11,6020 -0,040% 11,6067 0,000% 11,8133 4,142% 11,5793 -0,236%1y 11,6941 11,6879 -0,050% 11,6941 0,000% 11,4752 4,978% 11,7172 0,201%EPE 11,5558 11,5518 -0,034% 11,5558 0,000% 11,53970 -0,139% 11,55554 -0,001%
Table 5:
EPE: ATM European call. σ = 25% .Analytic Numerical Quantization Monte Carlo MC-Sobolt EE EE ε EE ε EE RSD EE ε
1w 13,291 13,290 -0,008% 13,291 0,000% 13,209 0,600% 13,289 -0,013%2w 13,298 13,297 -0,009% 13,298 0,000% 13,305 0,839% 13,296 -0,016%3w 13,306 13,305 -0,010% 13,306 0,000% 13,438 1,070% 13,303 -0,019%1m 13,314 13,312 -0,011% 13,314 0,000% 13,498 1,211% 13,313 -0,005%2m 13,349 13,347 -0,016% 13,349 0,000% 13,614 1,736% 13,348 -0,010%3m 13,383 13,380 -0,021% 13,383 0,000% 13,301 2,176% 13,375 -0,056%6m 13,484 13,479 -0,034% 13,484 0,000% 13,314 3,251% 13,491 0,057%9m 13,585 13,579 -0,045% 13,585 0,000% 13,846 4,312% 13,555 -0,220%1y 13,687 13,679 -0,057% 13,687 0,000% 13,422 5,193% 13,711 0,172%EPE 13,526 13,520 -0,038% 13,526 0,000% 13,504 -0,161% 13,525 -0,004%
Table 6:
EPE: ATM European call. σ = 30% .Analytic Numerical Quantization Monte Carlo MC-Sobolt EE EE ε EE ε EE RSD EE ε
1w 2,7600 2,7598 -0,008% 2,7600 0,000% 2,7396 0,731% 2,7597 -0,012%2w 2,7616 2,76135 -0,010% 2,7616 0,000% 2,7628 1,023% 2,7612 -0,016%3w 2,7632 2,7629 -0,012% 2,7632 0,000% 2,7987 1,310% 2,7626 -0,016%1m 2,7649 2,7644 -0,014% 2,7649 0,000% 2,8121 1,483% 2,7647 -0,004%2m 2,7723 2,7716 -0,023% 2,7723 0,000% 2,8325 2,145% 2,7720 -0,009%3m 2,7792 2,7784 -0,030% 2,7792 0,000% 2,7476 2,725% 2,7772 -0,046%6m 2,8001 2,7988 -0,049% 2,8001 0,000% 2,7319 4,239% 2,8055 0,039%9m 2,8212 2,8194 -0,065% 2,8212 0,000% 2,9162 5,719% 2,8119 -0,262%1y 2,8424 2,8401 -0,080% 2,8424 0,000% 2,8243 7,036% 2,8258 0,282%EPE 2,8088 2,80730 -0,054% 2,8088 0,000% 2,81496 0,218% 2,80347 -0,191%
Table 7:
EPE: − OTM European call. σ = 15% . nalytic Numerical Quantization Monte Carlo MC-Sobolt EE EE ε EE ε EE RSD EE ε
1w 6,2016 6,2011 -0,008% 6,2016 0,000% 6,1579 0,695% 6,2008 -0,013%2w 6,2052 6,2046 -0,010% 6,2052 0,000% 6,2080 0,973% 6,2042 -0,016%3w 6,2088 6,2081 -0,012% 6,2088 0,000% 6,2835 1,245% 6,2074 -0,018%1m 6,2124 6,2116 -0,013% 6,2124 0,000% 6,3131 1,409% 6,2121 -0,004%2m 6,2291 6,2278 -0,021% 6,2291 0,000% 6,3616 2,033% 6,2285 -0,010%3m 6,2447 6,2430 -0,027% 6,2447 0,000% 6,1831 2,573% 6,2405 -0,053%6m 6,2917 6,2889 -0,045% 6,2917 0,000% 6,1594 3,956% 6,3005 0,054%9m 6,3391 6,3352 -0,062% 6,3391 0,000% 6,5221 5,314% 6,3217 -0,236%1y 6,3868 6,3817 -0,077% 6,3868 0,000% 6,3051 6,501% 6,3418 0,201%EPE 6,3113 6,3080 -0,051% 6,3113 0,000% 6,31287 0,026% 6,29739 -0,220%
Table 8:
EPE: − OTM European call. σ = 25% .Analytic Numerical Quantization Monte Carlo MC-Sobolt EE EE ε EE ε EE RSD EE ε
1w 7,9807 7,9800 -0,008% 7,9807 0,000% 7,9245 0,693% 7,9795 -0,013%2w 7,9853 7,9845 -0,010% 7,9853 0,000% 7,9889 0,970% 7,9839 -0,016%3w 7,9899 7,9889 -0,011% 7,9899 0,000% 8,0857 1,241% 7,9881 -0,019%1m 7,9945 7,9934 -0,013% 7,9945 0,000% 8,1237 1,405% 7,9941 -0,005%2m 8,0160 8,0144 -0,021% 8,0160 0,000% 8,1858 2,027% 8,0153 -0,010%3m 8,0361 8,0339 -0,028% 8,0361 0,000% 7,9568 2,564% 8,0306 -0,056%6m 8,0966 8,0928 -0,046% 8,0966 0,000% 7,9268 3,942% 8,1067 0,057%9m 8,1575 8,1523 -0,064% 8,1575 0,000% 8,3893 5,295% 8,1376 -0,220%1y 8,2189 8,2120 -0,082% 8,2189 0,000% 8,0964 6,475% 8,1760 0,172%EPE 8,1218 8,11738 -0,053% 8,1218 0,000% 8,11858 -0,038% 8,10792 -0,170%
Table 9:
EPE: − OTM European call. σ = 30% . Comparing EPE values reported in tables, we deduce that the quantization approachprovides highly satisfactory results for ATM, ITM, OTM European call options.We also note that the Monte Carlo relative errors increase for the out of the money cases.This is due to the fact that in the considered framework, the true value of EE becomes verysmall. It is worth to note that the VQ also works better than the numerical integration.For the sake of completeness and in order to stress how the quantization techniqueperform better than Monte Carlo method, we report two figures showing the error ε forMonte Carlo and quantization performances. The plots were obtained for a more granularcombination of the couple (Spot,Volatility) values.21igure 4: MC: EPE error ε for European Call. Figure 5:
VQ: error ε for European Call. In order to generalize techniques and results shown in the previous subsection, we are goingto consider now a set of n European options, i.e. a derivative portfolio, for which we willevaluate the Expected Exposure (EE) and the Expected Positive Exposure (EPE).The portfolio may consist of call options and put options, related to a group of trans-actions with a single counterparty, which are subject to a legally enforceable bilateral nettingarrangement. Such a set of derivatives is called netting set.
From a mathematical point of22iew, such a choice means that the expected exposure is given by EE = n (cid:88) j =1 M tM ( t ) − V t + , unlike the case of no-netting portfolio setting, where the sum of the fair prices of Europeanoptions represents the Mark-to-Market of the portfolio. The term V t represent the collateral value posted by the debtor, i.e. the counterparty with the negative Mark-to-Market portfolioat time t . In what follows we set V t = 0 , therefore we deal with the netting agreementsituation, supposing no collateral. Even if the set up can appear simple, this is not true, andthe problem turns out to be rather challenging. That is because, generally speaking, onecan not perform the analytical calculation of EE k . In fact, also if the martingale propertyholds for each derivative in the portfolio, in the present case the positive part operator iseffective, hence the future value is not simply the compounded current MtM.For computational purposes, we consider n = 10 European options, that is call options,the first, the third and the last one are of buy type, while the second and the fourth one areof sell type, and put options, namely, the first, the second and the last one of sell type andthe remaining of buy type. To summarize, we have constructed a table with the features ofthe different options, see Table 5.3.The software code is quite general to deal with any change in quantities, buy/sell, marketand instrument parameters.For the application we consider the following values: • spot price ( S ) : 90 , , • interest rate ( r ) : 3% • volatility ( σ ) : 15% , , • maturity ( T ) : one year • time buckets: (cid:8) , , , , , , , , , (cid:9) . type position strike maturityOption 1 call buy Table 10:
Description of the portfolio. The table summarizes the characteristics of the netting set
To better understand the characteristics of the portfolio, let us consider the followingtwo figures, which show the portfolio profile for the different volatilities. It is a long styleportfolio, with different levels of delta sensitivities.23igure 6:
MtM of the portfolio depending on volatility.
Figure 7:
Delta of the portfolio depending on volatility.
The benchmark value for this application can not be a closed Black-Scholes approach.Hence we use a Monte Carlo-Sobol sequence with a large enough number of points as anacceptable pivot value. We use points. The vector quantization and the Monte Carlotechniques are tested with points.As it was done in Subsection 5.2, in relation to the case of only one option, the comparisonamong the different procedures consists in analyzing the percent relative standard error(RSD) for the Monte Carlo approach and the percent relative error ( ε ) for the quantizationtechnique. 24 C-Sobol Quantization Monte Carlo MC-Sobol t EE EE ε EE RSD EE RSD1w 0,0000 0,0000 NaN 0,0000 NaN 0,0000 NaN2w 0,0000 0,0000 4,184% 0,0000 NaN 0,0002 99,950%3w 0,0006 0,0006 -0,585% 0,0007 99,950% 0,0004 99,950%1m 0,0030 0,0030 -0,021% 0,0022 70,933% 0,0060 58,952%2m 0,0537 0,0537 0,033% 0,0425 22,046% 0,0586 20,217%3m 0,1462 0,1463 0,048% 0,1254 15,505% 0,1552 15,306%6m 0,5045 0,5045 0,010% 0,4539 10,512% 0,5054 10,581%9m 0,8529 0,8529 -0,005% 0,8965 8,428% 0,8926 8,581%1y 1,3863 1,3874 0,078% 1,2717 7,466% 1,4630 7,294%EPE 0,7030 0,7033 0,040% 0,6698 -4,719% 0,7336 4,348%
Table 11:
EPE: S = 90; σ = 15% . MC-Sobol Quantization Monte Carlo MC-Sobol t EE EE ε EE RSD EE RSD1w 0,0001 0,0001 0,701% 0,0001 99,950% 0,0000 NaN2w 0,0075 0,0075 0,032% 0,0074 54,701% 0,0093 64,637%3w 0,0348 0,0348 -0,046% 0,0523 24,373% 0,0427 27,104%1m 0,0817 0,0817 0,014% 0,0939 18,901% 0,0899 23,586%2m 0,4319 0,4319 0,001% 0,4329 11,475% 0,4489 12,248%3m 0,8119 0,8121 0,028% 0,7440 10,323% 0,8226 10,542%6m 1,8836 1,8837 0,005% 1,7520 8,568% 1,9046 8,687%9m 2,7611 2,7612 0,005% 2,9002 7,896% 2,8727 8,103%1y 3,5191 3,5217 0,075% 3,4064 7,868% 3,5674 8,452%EPE 2,1497 2,1505 0,034% 2,1185 -1,454% 2,1977 2,232%
Table 12:
EPE: S = 90 , σ = 25% . MC-Sobol Quantization Monte Carlo MC-Sobol t EE EE ε EE RSD EE RSD1w 0,0013 0,0013 0,145% 0,0040 68,879% 0,0005 99,950%2w 0,0291 0,0291 0,004% 0,0325 30,112% 0,0312 37,205%3w 0,0981 0,0981 -0,013% 0,1321 18,842% 0,1157 19,728%1m 0,1954 0,1954 -0,012% 0,2277 14,700% 0,2049 17,692%2m 0,7796 0,7796 0,004% 0,7821 10,110% 0,7997 10,680%3m 1,3415 1,3418 0,021% 1,2408 9,292% 1,3594 9,441%6m 2,8256 2,8257 0,005% 2,6487 8,089% 2,8590 8,221%9m 4,0074 4,0077 0,008% 4,2019 7,686% 4,1538 7,896%1y 4,9419 4,9447 0,058% 4,8168 7,829% 5,0104 8,410%EPE 3,1317 3,1325 0,027% 3,0981 -1,074% 3,1976 2,106%
Table 13:
EPE: S = 90 , σ = 30% . C-Sobol Quantization Monte Carlo MC sobol t EE EE ε EE RSD EE RSD1w 0,3565 0,3565 0,000% 0,3325 6,223% 0,3562 5,746%2w 0,5758 0,5758 0,000% 0,5681 5,447% 0,5757 5,433%3w 0,7463 0,7463 0,000% 0,7992 5,182% 0,7482 5,275%1m 0,8904 0,8904 -0,003% 0,9439 5,022% 0,8889 5,266%2m 1,3966 1,3966 0,000% 1,4407 4,793% 1,4016 5,007%3m 1,7463 1,7463 0,004% 1,6845 4,950% 1,7351 5,051%6m 2,5014 2,5014 0,003% 2,4279 4,960% 2,5149 5,097%9m 3,0139 3,0138 -0,002% 3,0990 5,225% 3,0437 5,377%1y 3,5845 3,5852 0,018% 3,5369 5,253% 3,6740 5,459%EPE 2,5952 2,5954 0,007% 2,5865 -0,337% 2,6279 1,261%
Table 14:
EPE: S = 100 , σ = 15% . MC-Sobol Quantization Monte Carlo MC-Sobol t EE EE ε EE RSD EE RSD1w 0,5510 0,5510 0,000% 0,5113 7,316% 0,5500 6,632%2w 0,9683 0,9683 0,000% 0,9532 6,113% 0,9677 6,094%3w 1,2999 1,2999 0,000% 1,4062 5,717% 1,2986 5,884%1m 1,5831 1,5831 0,001% 1,6824 5,537% 1,5901 5,777%2m 2,5909 2,5909 0,001% 2,6600 5,273% 2,5990 5,501%3m 3,2975 3,2977 0,004% 3,1569 5,466% 3,2792 5,561%6m 4,8611 4,8614 0,006% 4,6669 5,563% 4,8787 5,676%9m 5,9723 5,9725 0,002% 6,1526 5,816% 6,0292 6,002%1y 6,8363 6,8377 0,020% 6,6731 6,139% 6,9781 6,320%EPE 5,0094 5,0099 0,009% 4,9625 -0,936% 5,0627 1,065%
Table 15:
EPE: S = 100 , σ = 25% . MC-Sobol Quantization Monte Carlo MC-Sobol t EE EE ε EE RSD EE RSD1w 0,7348 0,7348 0,000% 0,6826 7,124% 0,7335 6,474%2w 1,2651 1,2651 0,000% 1,2462 6,034% 1,2643 6,021%3w 1,6844 1,6844 -0,001% 1,8210 5,684% 1,6836 5,844%1m 2,0418 2,0418 0,001% 2,1698 5,522% 2,0507 5,765%2m 3,3126 3,3126 0,001% 3,3997 5,308% 3,3231 5,541%3m 4,2047 4,2049 0,004% 4,0221 5,531% 4,1822 5,630%6m 6,1892 6,1895 0,005% 5,9304 5,685% 6,2124 5,795%9m 7,6209 7,6210 0,002% 7,8537 5,953% 7,7022 6,134%1y 8,6815 8,6828 0,015% 8,4780 6,330% 8,8203 6,549%EPE 6,3807 6,3811 0,007% 6,3196 -0,957% 6,4407 0,941%
Table 16:
EPE: S = 100 , σ = 30% . C-Sobol Quantization Monte Carlo MC-Sobol t EE EE ε EE RSD EE RSD1w 5,9931 5,9931 0,000% 5,9450 0,786% 5,9939 0,777%2w 5,9972 5,9972 0,000% 6,0020 1,095% 5,9974 1,105%3w 6,0053 6,0053 0,000% 6,0699 1,372% 6,0046 1,350%1m 6,0194 6,0194 0,000% 6,1167 1,537% 6,0213 1,546%2m 6,1483 6,1482 -0,001% 6,3028 2,048% 6,1578 2,136%3m 6,3084 6,3083 -0,001% 6,2873 2,407% 6,3192 2,483%6m 6,7919 6,7918 -0,002% 6,7474 3,050% 6,7916 3,188%9m 7,1835 7,1835 0,000% 7,2932 3,677% 7,2228 3,721%1y 7,5898 7,5915 0,022% 7,4585 4,085% 7,5753 4,210%EPE 6,9306 6,9310 0,005% 6,9284 -0,031% 6,9385 0,114%
Table 17:
EPE: S = 110 , σ = 15% . MC-Sobol Quantization Monte Carlo MC-Sobol t EE EE ε EE RSD EE RSD1w 6,5122 6,5122 0,000% 6,4141 1,465% 6,5134 1,439%2w 6,5989 6,5989 0,000% 6,6139 1,926% 6,6015 1,950%3w 6,7315 6,7315 0,000% 6,8607 2,310% 6,7373 2,273%1m 6,8813 6,8813 0,000% 7,0735 2,479% 6,8863 2,514%2m 7,5927 7,5927 0,000% 7,8513 2,975% 7,5995 3,131%3m 8,1894 8,1895 0,001% 8,1085 3,360% 8,2180 3,442%6m 9,6447 9,6446 -0,001% 9,4633 3,984% 9,6789 4,093%9m 10,7365 10,7363 -0,002% 10,9562 4,531% 10,7812 4,626%1y 11,5732 11,5745 0,012% 11,4042 4,941% 11,5706 5,132%EPE 9,8664 9,8666 0,003% 9,8547 -0,118% 9,8887 0,226%
Table 18:
EPE: S = 110 , σ = 25% . MC-Sobol Quantization Monte Carlo MC-Sobol t EE EE ε EE RSD EE RSD1w 6,7056 6,7056 0,000% 6,5844 1,762% 6,7063 1,727%2w 6,8948 6,8948 0,000% 6,9154 2,237% 6,8984 2,270%3w 7,1282 7,1282 0,000% 7,3020 2,627% 7,1311 2,602%1m 7,3679 7,3680 0,000% 7,6149 2,783% 7,3787 2,836%2m 8,3987 8,3987 0,000% 8,6959 3,263% 8,4147 3,434%3m 9,2137 9,2140 0,003% 9,0911 3,657% 9,2276 3,754%6m 11,1493 11,1492 -0,001% 10,8929 4,279% 11,1882 4,389%9m 12,5989 12,5984 -0,003% 12,8948 4,798% 12,6484 4,921%1y 13,6675 13,6689 0,010% 13,4530 5,231% 13,7136 5,432%EPE 11,4158 11,4160 0,002% 11,3947 -0,185% 11,4523 0,320%
Table 19:
EPE: S = 110 , σ = 30% . Below we present a couple of figures which clearly show how the VQ technique performswith an excellent accuracy over the different situations. It is worth to observe that, for somevery deep out of the money situation, the Monte Carlo simulations shows a huge relativestandard error. This occurs when the value of EE is next to zero. We remember that whenthe EE is very small, no effective counterparty risk has to be faced by the bank, hence, themagnitude of the error is not as dramatic as it seems from a numerical perspective.27ventually, we plotted the percent relative error ε and the percent relative standarderror RSD, in order to compare, once again, the quantization technique and the MonteCarlo method, showing the excellent accuracy of the former approach, when compared withthe latter. Figure 8: VQ: EPE error ε for portfolio Figure 9:
MC: EPE relative error
RSD for portfolio Conclusions and Further Research
In the present work we show how the quantization approach outperforms the classical MonteCarlo methods in the CCR field. The counterparty risk field poses a lot of theoretical andpractical challenges. A whole portfolio of derivatives must be evaluated in the future, forseveral time steps and many scenarios, in order to get some useful risk figures. This largeamount of computations can be solved by improving the technologies or the algorithmicstrategies.At the best of our knowledge, this is the first work which exploits the quantizationapproach in this area. The quantization has been intensively tested in the last decade in somepricing problems for different complex situations: American style options, multidimensionalassets, credit derivatives. Despite this first study covers some simple derivatives and smalldimension portfolios, the quantization seems to be very promising. Given an equivalentcomputational effort, it is undoubtedly better than the standard Monte Carlo simulationand some of its refinements such as the Sobol sequences.Nevertheless, further research is needed. As the next step, we aiming at treating moreinvolved portfolios and payoffs, as in the case of path dependent options, where the quan-tization tree could be a competitor of binomial and trinomial trees. Moreover, concerningportfolios depending on many underlyings, there are issues related to the choice of a coherent set of quantized paths that has to be fixed.Finally, numerical extensions as to be taken into account in order to pass from theaccuracy comparison, given the same computational effort, to the search of the quantizationtradeoff, namely an estimate of its relative effort saving, given the same accuracy.
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