Ludovic Goudenège
email: ludovic.goudenege@math.cnrs.fr

Andrea Molent
email: andrea.molent@uniud.it

Antonino Zanette
email: antonino.zanette@uniud.it

Computing Credit Valuation Adjustment solving coupled PIDEs in the Bates model

Keywords: Credit Value Adjustment, Hybrid methods, PIDE, Monte Carlo, Bates model

. Moreover, the direct application of the hybrid techniques in the original FDMC approach is also considered for comparison purposes. Several numerical tests prove the effectiveness and the reliability of the proposed approach when both European and American options are considered.

Introduction

Financial institutions suffer several risks, one of them is the counterparty credit risk (CCR). This risk arises from the possibility that the counterparty of a financial contract may default. This risk was often overlooked, but in the last decades, after the financial crisis of 2007 and the Lehman Brothers failure in 2008, it gained more and more interest by practitioners and academics. In particular, according to the Basel III framework of 2010, financial institutions must charge a premium to their counterparty according to its credit reliability in order to compensate for a possible counterparty default. Also IFRS 13 in 2013 requires the fair value of financial products to be measured based on counterparty credit risk. For these reasons, financial institutions charge to the counterparty a premium called Credit Valuation Adjustment (CVA), which is the difference between the risky value and the current risk-free value of derivatives contract. Estimating the good value of the CVA can be a demanding effort due to its complicated definition and to its dependence by the underlying stochastic model. As no specific method is prescribed in the accounting literature, various approaches are used in practice by derivatives dealers and by end users to estimate the effect of credit risk on the fair value of financial derivatives. The common approach is to price the CVA through the so called expected exposure, which is the mean exposure distribution at a future date. Usually, this exposure is calculated by means of Monte Carlo approaches which are computationally expensive. In particular, nested Monte Carlo simulations or least squares techniques are employed by Joshi and Kwon [START_REF] Joshi | Least squares Monte Carlo credit value adjustment with small and unidirectional bias[END_REF]. These techniques have been improved through the use of stochastic grid bundling method by Jain and Oosterlee [START_REF] Jain | The stochastic grid bundling method: Efficient pricing of Bermudan options and their Greeks[END_REF] and by Karlson et al. [START_REF] Karlsson | Counterparty credit exposures for interest rate derivatives using the stochastic grid bundling method[END_REF]. An interesting approach to compute the CVA -when the Heston model is assumed -is the so called finite-difference Monte Carlo (FDMC) method, proposed by de Graaf et al. [START_REF] De Graaf | Efficient estimation of sensitivities for counterparty credit risk with the finite difference Monte Carlo method[END_REF]. Such an approach combines the finite-difference method and the Monte Carlo method to solve a partial differential equation (PDE) and to estimate the mean exposure respectively. Feng [START_REF] Feng | CVA under Bates model with stochastic default intensity[END_REF] adapted the FDMC approach to deal with the case of an underlying evolving according to the Bates model: in this particular case, the PDE to be solved is replaced by a partial integral differential equation (PIDE), which implies an additional computational effort. Other recently introduced alternative approaches consist in employing the fast Fourier transform (Borovykh et al. [START_REF] Borovykh | Efficient computation of various valuation adjustments under local Lévy models[END_REF]) or marked branching diffusions (Henry-Labordère [START_REF] Henry-Labordère | Cutting CVA's complexity[END_REF]).

In this paper we focus on the computation of the CVA when the Bates model is considered and we propose an efficient method which improves the results of the FDMC method. Specifically, the Bates model considers an underlying affected by both stochastic volatility and random jumps: the dynamics of the underlying asset price is driven by both a Heston stochastic volatility [START_REF] Heston | A closed-form solution for options with stochastic volatility with applications to bond and currency options[END_REF] and a compound Poisson jump process of the type originally introduced by Merton [START_REF] Merton | Option pricing when underlying stock returns are discontinuous[END_REF]. Such a model was introduced by Bates [START_REF] Bates | Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche mark options[END_REF] in the foreign exchange option market in order to tackle the well-known phenomenon of the volatility smile behavior. In the case of plain vanilla European options, Fourier inversion methods, employed by Carr and Madan [START_REF] Carr | Option valuation using the fast Fourier transform[END_REF], lead to closed-form formulas to compute the price under the Bates model. Two innovative and efficient approaches to price derivatives when the Bates model is considered are proposed by Briani et al. [START_REF] Briani | On a hybrid method using trees and finitedifferences for pricing options in complex models[END_REF], the so called Hybrid Tree-Finite Difference method and Hybrid MC method.

Our main result consists in developing an efficient numerical method to estimate the CVA. In particular, the method we propose consists in computing the CVA value as the solution of two coupled PIDE -one for the risk free price and one for the risk adjusted price -which are solved by means of the Hybrid Tree-Finite Difference method. That is, we replace the Monte Carlo step in the FDMC method with the computation of the solution of a PIDE and this improves clearly the computational efficiency.

Moreover, the direct application of the hybrid techniques in the original FDMC approach is considered for comparison purposes. Specifically, we apply the two hybrid methods developed by Briani et al. to perform both the finite difference step and the Monte Carlo step of the FDMC approach.

Several numerical experiments show that the proposed method is efficient and reliable as it provides accurate approximations for the CVA value with a low computational cost.

The reminder of the paper is organized as follows. Section 2 introduces the Bates stochastic model and the partial integro-differential equation that allows one to compute option prices. Section 3 presents the CVA definition and some useful properties used in the following. Section 4 outlines the main features of the hybrid methods and how to employ them in option pricing. Section 5 describes the numerical methods for computing the CVA. Section 6 presents and discusses the results of the numerical simulations. Section 7 concludes.

The Model

In the Bates model the volatility is assumed to follow the CIR process (Cox et al. [START_REF] Cox | A theory of the term structure of interest rates[END_REF]) while the underlying asset price process contains a further noise from a jump process. In particular, the model for the stock price and its volatility is given by the following relations:

dS t S t - = (r -η)dt + V t dZ S t + dH t , dV t = κ(θ -V t )dt + σ √ V t dZ V t , (2.1)
where η denotes the continuous dividend rate, S 0 , V 0 are positive values, Z S , Z V are correlated Brownian motions such that dZ s t , dZ V t = ρdt and H is a compound Poisson process with intensity λ and i.i.d. jumps

{J k } k , that is H t = Kt k=1 J k , (2.2)
K denoting a Poisson process with intensity λ. As in the model of Merton [START_REF] Merton | Option pricing when underlying stock returns are discontinuous[END_REF], we assume that the jumps are i.i.d. log-normal random variables, described by

log (1 + J) ∼ N α - β 2 2 , β 2 , (2.3)
with α and β real parameters.

We consider an option with maturity T > 0, payoff function ψ (S) and we denote with V (t, S, V ) its fair price at time t, if S t = S and V t = V . Then, the price of the derivative V is then given by

V (t, S, V ) = E e -r(T -t) ψ (S T ) |S 0 = S, V 0 = V . (2.4)
Using a martingale approach for an European option, it is possible to show that V (t, S, V ) satisfies the following partial integro-differential Equation (PIDE) (Salmi et al [START_REF] Salmi | An IMEX-scheme for pricing options under stochastic volatility models with jumps[END_REF]):

∂V (t, S, V ) ∂t + 1 2 V S 2 ∂ 2 V (t, S, V ) ∂S 2 +ρσV S ∂ 2 V (t, S, V ) ∂S∂V + 1 2 σ 2 V ∂ 2 V (t, S, V ) ∂V 2 +(r -q -λ (e γ -1)) ∂V (t, S, V ) ∂S + κ (θ -V ) ∂V (t, S, V ) ∂V -(r + λ) V (t, S, V ) + λ ∞ 0 V (t, xS, V ) p J (x) dx = 0, (2.5)
with the terminal condition is

V (T, S, V ) = ψ (S) , (2.6)
where p J (x) is the log-normal probability density function of the jump variable J.

We define the following operators:

L C V = 1 2 V S 2 ∂ 2 V ∂S 2 + ρσV S ∂ 2 V ∂S∂V + 1 2 σ 2 V ∂ 2 V ∂V 2 + (r -q -λ (e γ -1)) ∂V ∂S + κ (θ -V ) ∂V ∂V -(r + λ) V, (2.7)
and

L J V = λ ∞ 0 V (t, xS, V ) p J (x) dx. (2.8)
Therefore, by using (2.7) and (2.8), the PIDE (2.5) can be rewritten in the more compact form as follows:

∂V (t, S, V ) ∂t + L C V (t, S, V ) + L J V (t, S, V ) = 0.
(2.9)

3 The CVA Let us define the exposure (or financial exposure) E (t) towards a counterparty at time t as the positive side of a contract (or portfolio) value V (t, S t , V t ), that is

E (t) = max [V (t, S t , V t ) , 0] . (3.1)
This amount represents the maximum loss if the counterparty defaults at t: an economic loss would occur if the transactions with the counterparty has a positive economic value at the time of default (see the Basel Committee [START_REF]International convergence of capital measurement and capital standards: a revised framework[END_REF]).

Let us define the present Expected Exposure (EE) at a future time t < T as

EE (t) = E [E (t) |F 0 ] (3.2)
where F 0 is the filtration at time t = 0. In particular, in the case of a long position in a Call or Put option, the price is always positive and thus the EE is equal to the future option price. Let τ C denote the counterparty's default time: such a time is a random variable which is supposed to be independent from the stochastic processes Z S , Z V and H. Moreover, we describe its cumulative distribution function as follows:

P D (t) = 1 -exp - t 0 δ (s) ds . (3.3)
Here, δ (t), the so called hazard rate, is a non-negative function such that +∞ 0 δ (s) ds = +∞ (Promislow [START_REF] Promislow | Fundamentals of Actuarial Mathematics[END_REF]). If the counterparty has not defaulted yet, then δ (t) dt is the probability that it would default between t and t + dt. The possibility of counterparty default reduces the value of the option as in case of default the holder does not receive the whole value of the option. In particular, the holder can recover only a percentage of the contract value, which is called the recovery rate R. The CVA is defined as the difference between the risk free price and the risk adjusted price. More precisely, as stated by Gregory [START_REF] Gregory | Counterparty credit risk: the new challenge for global financial markets[END_REF], the CVA is given by:

CVA = (1 -R) T 0 D (0, s) EE (s) dP D (s) , (3.4)
where D (0, t) is the risk-free discount factor. This mean that the CVA is the expected value of the possible losses due to counterparty default. We stress out that definition requires the exposure and the counterparties default probability to be independent and the discount factor to be also independent of the exposure. Using equations (3.2) and (3.4), we obtain the following relation:

CVA = E T 0 D (0, s) (1 -R) E (s) dP D (s) |F 0 . (3.5)
According to (3.5), the CVA is the price (mean value of the future cash flows) of a financial derivative which pays (1 -R) max [V (τ C , S τC , V τC ) , 0] at time τ C . Therefore, we can consider the CVA as a derivative itself and we denote its financial value at time t with C (t, S t , V t ), that is

C (t, S t , V t ) = E T t D (0, s) (1 -R) E (s) dP D s |F t ,
having CV A = C (0, S 0 , V 0 ). It is possible to show that C (t, S t , V t ) solves the following PIDE (see the Appendix for more details):

∂C (t, S, V ) ∂t + L C C (t, S, V ) + L J C (t, S, V ) + (1 -R) max [V (t, S t , V t ) , 0] ∂P D ∂t (t) = 0, (3.6)
with the terminal condition C (T, S, V ) = 0.

(3.7)

We stress out that equation (3.6) depends on the value V (t, S t , V t ) which has to be computed previously by solving equation (2.9).

Hybrid methods in the Bates model

The Hybrid Tree Finite Difference (HTFD) and the Hybrid Tree Monte Carlo (HTMC) methods are two innovative and efficient approaches to price derivatives when the Bates model is considered. In this Section we present the main ideas that underlie these two methods: the interested reader can find more details about the numerical procedures in Briani et al. [START_REF] Briani | On a hybrid method using trees and finitedifferences for pricing options in complex models[END_REF].

The HTFD method

The HTFD method is a backward induction algorithm that works following a finite difference PIDE method in the direction of the share process and following a tree method in the direction of the other random sources, that is volatility in the case of Bates model. Specifically, the method is based on the following steps. First of all, a binomial tree for the CIR volatility process V is considered according to Apolloni et al. [START_REF] Appolloni | A robust tree method for pricing American options with the Cox-Ingersoll-Ross interest rate model[END_REF]. Then, a transformation which keeps the diffusion processes S and V uncorrelated is applied. Finally, a finite difference approach in the S-direction is developed.

In particular, consider a large integer value N , a time horizon [0, T ] and define h = T /N . For n = 0, 1, . . . , N , define

V h n = {v n,k } k=0,1,...,n (4.1)
with

v n,k = V 0 + σ 2 (2k -n) √ h 2 1 √ V0+ σ 2 (2k-n) √ h>0 . (4.2)
We define the multiple jumps

k h d (n, k) = max {k * : 0 ≤ k * ≤ j and v n,k + µ V (v n,k ) h ≥ v n+1,k * } , k h u (n, k) = min {k * : k + 1 ≤ k * ≤ n + 1 and v n,k + µ V (v n,k ) h ≤ v n+1,k * } in which µ V denotes the drift coefficient of V , that is µ V (v) = κ (θ -v).
Starting from the node (n, k), the discrete process can reach the up-node n + 1, k h u (n, k) or the down-jump node n + 1, k h d (n, k) with transition probability given by up-jump:

p h k h u (n,k) = 0 ∨ µ V (v n,k ) h + v n,k -v n+1,k h d (n,k) v n+1,k h u (n,k) -v n+1,k h d (n,k) ∧ 1, (4.3)
down-jump :

p h k h d (n,k) = 1 -p h k h u (n,k) . (4.4)
Multiple jumps and jump probabilities are set in order to match the first local moment of the tree and of the process V up to order one with respect to h. As a consequence, as h approaches to 0, the weak convergence on the path space is guaranteed. Moreover, in order to obtain the convergence, the Feller condition (Albrecher et al. [START_REF] Albrecher | The little Heston trap[END_REF]) is not required. Let us consider the diffusion pair (Y, V ), where Y is a stochastic process defined by

Y t = log (S t ) - ρ σ V t . (4.5)
Clearly the couple (S, V ) can be retraced by (Y, V ) by inverting relation (4.5). We set ρ = 1 -ρ 2 and we consider (W, Z) as a standard Brownian motion in R 2 . Then, the dynamics of the couple (Y, V ) is given by

dY t = r -η - 1 2 V t - ρ σ κ (θ -V t ) dt + ρ V t dZ t + dN t , (4.6
)

dV t = κ (θ -V t ) dt + σ V t dW t , (4.7)
with Y 0 = log S 0 -ρ σ V 0 . Here, N t is the compound Poisson process written through the Poisson process K, with intensity λ, and the i.i.d. jumps {log (1 + J k )}, that is

N t = Kt k=1 log (1 + J k ) . (4.8)
We set

µ Y (v) = r -η - 1 2 v - ρ σ κ (θ -v) (4.9) and µ V (v) = κ (θ -v) . (4.10)
Let V h = V h n n=0,...,N denote the tree process approximating V and set V h t = V h ⌊t/h⌋ , t ∈ [0, T ], the associated piecewise constant and cà dlà g approximation path. In order to approximate Y , Briani et al. construct a Markov chain from the finite difference method. Starting from the Euler scheme Y h 0 = Y 0 and for t ∈ (nh, (n + 1) h], n = 0, . . . , N , it is possible to set

Y h t = Y h nh + µ Y V h nh (t -nh) + ρ V h nh (Z t -Z nh ) + (N t -N nh ) , (4.11)
Z being independent of the noise driving V h . Now, let W (t, Y, V ) = V t, exp Y + ρ σ V , V the function which gives the price of the financial derivative at time t in terms of the couple (Y, V ) . Then,

E W (n + 1) h, Y (n+1)h , V (n+1)h |Y nh = y, V nh = v ≈ E W (n + 1) h, Y h (n+1)h , V h (n+1)h |Y nh = y, V nh = v (4.12) = E u h nh, y; v, V h (n+1)h |V h nh = v (4.13)
where The key point of the HTFD method is that the function (s, y) → u h (s, y; v, z) solves the following PIDE in the time interval nh < s < (n + 1) h:

u h (nh, y; v, z) = E W (n + 1) h, Y h (n+1)h , z |Y h nh = y, V h nh = v (4.
∂u h ∂s + µ Y (v) ∂u h ∂y + 1 2 ρ2 v ∂ 2 u h ∂y 2 + +∞ -∞ u h (s, y + x; v, z) -u h (s, y; v, z) p J (x) dx = 0, (4.16)
for y ∈ R, and with terminal condition given by u h ((n + 1) h, y; v, z) = W ((n + 1) h, y, v) . (4.17)

In order to numerically solve the PIDE (4.16) by using a finite difference scheme, we first localize the variables and the integral term to bound the computational domains. For this purpose, we use the estimates for the localization domain and the truncation of large jumps given by Voltchkova and Tankov [START_REF] Voltchkova | Deterministic methods for option pricing in exponential Lévy models[END_REF]. Then, the derivatives of the solution are replaced by finite differences and the integral terms are approximated using the trapezoidal rule. Finally, the problem is solved by using an explicit-implicit scheme.

We observe that the above algorithm is referred to a European option, can easily be adapted to consider an American option. Specifically, we approximate the American option with a Bermudan option with exercise dates given by the set {nh} n=0,...,N and replacing equation (4.17) with the following one:

u h ((n + 1) h, y; v, z) = max W ((n + 1) h, y, v) , ψ exp y + ρ σ v (4.18)
We stress out that the computation of u h as the solution of (4.16) is a one dimensional problem with constant coefficients, thus it can be solved in a very efficient way, with a low computational cost.

The HTMC method

The HTMC method is a Monte Carlo algorithm which is based on the approximations (4.3) and (4.11). In particular, the HTMC method consists in simulating a continuous process in space (the component Y ) starting from a discrete process in space (the 1-dimensional tree for V ). In particular, we set Ŷ h 0 = Y 0 for t ∈ [nh, (n + 1) h] with n = 0, 1, . . . , N -1. Then, we compute Ŷ h n+1 recursively by the following relation:

Ŷ h n+1 = Ŷ h n + µ Y V h n h + ρ h V h n ∆ n+1 + N (n+1)h -N nh (4.19)
where ∆ 1 , . . . , ∆ N are i.i.d. standard normal random variables which are independent of the noise driving the Markov chain V and N (n+1)h -N nh is the compound Poisson increment. Roughly speaking, one let the pair (Y, V ) evolve on the tree and simulate the process Y at time nh by using relation (4.19).

5 Numerical method for computing the CVA

In this section we present the proposed approach to compute the CVA, namely the Coupled-Hybrid Tree Finite Differences (C-HTFD), which is based on the resolution of two coupled PIDE. Moreover, we start presenting the Hybrid Tree Finite Difference-Hybrid Monte Carlo (HTFD-HTMC) method which is based on the direct application of the hybrid techniques in the original FDMC (we consider this method for comparison purposes mainly). The two approaches are both based on the application of the hybrid algorithms of Briani et al. [START_REF] Briani | On a hybrid method using trees and finitedifferences for pricing options in complex models[END_REF], presented in Section 4.

The HTFD-HTMC approach

This method is based on the direct application of the hybrid techniques to the Finite Difference Monte Carlo (FDMC) method, which was first developed by de Graaf et al. [START_REF] De Graaf | Efficient estimation of sensitivities for counterparty credit risk with the finite difference Monte Carlo method[END_REF] for the Heston model and then adapted for the Bates model by Feng [START_REF] Feng | CVA under Bates model with stochastic default intensity[END_REF]. First of all, an estimation of the value function V (t, S, V ) is computed through a grid of values by solving equation (2.5) by employing the Alternating Direction Implicit (ADI) method and specifically the scheme proposed by Haentjens and In't Hout [START_REF] Haentjens | Alternating direction implicit finite difference schemes for the Heston-Hull-White partial differential equation[END_REF]. Then a Monte Carlo simulation is employed to estimate the expectation in (3.4) and thus estimate the CVA.

In the HTFD-HTMC approach we employ the HTFD method to solve (2.5) and the HTMC method to estimate (3.4). Specifically, let h = T /N as in Subsection 4.1. First of all, the HTFD method is used to compute the risk free price {V (nh, S nh , V nh )} n=0,...,N at discrete times {0, h, . . . , N h}. Then, we estimate the expected exposure (3.2) via a set of N MC Monte Carlo simulations S j nh , V j nh , n = 0, . . . , N and j = 1, . . . , N MC generated via the HTMC method, that is

EE (nh) ≈ 1 N MC NMC j=1 max V nh, S j nh , V j nh . (5.1)
Finally, we compute the CVA by approximating the integral in (3.4) using the uniform partition {0, h, . . . , N h} of [0, T ], the estimated value {EE (nh) , n = 0, . . . , N } and the trapezoidal rule.

The C-HTFD approach

First of all, an estimation of the value function V (t, S, V ) is computed through a grid of values by solving equation (2.5) by means of the HTFD method. Then, the CVA is computed as the solution of the PIDE (3.6), which is done again by employing the HTFD method. We stress that C-HTFD method and the HTFD-HTMC method share the first step, but in the C-HTFD method the Monte Carlo step is replaced with a second finite difference like step.

Numerical Results

In this section we propose the results of some numerical experiments, which aim to compare the goodness of the proposed methods. In particular, we compare the standard FDMC method employed by Feng [START_REF] Feng | CVA under Bates model with stochastic default intensity[END_REF], the HTFD-HTMC approach and the C-HTFD approach. We employ the numerical methods according to 4 configurations (A, B, C, D), each of them with an increasing number of steps, determined in order to achieve approximately these run times1 : (A) 0.25 s, (B) 1 s, (C) 4 s, (D) 16 s. The employed mesh and configuration parameters are reported in Table 1. Specifically, for the FDMC method with the sequence (time steps -points for S -points for V -MC simulations) and for the hybrid methods with the sequence (time steps -points for Y -MC simulations). Moreover, we reported the 95% confidence intervals for the FDMC method and for the HTFD-HTMC. Furthermore, we employ the FDMC method with a huge number of steps and paths 250; 500; 200; 10 6 as a benchmark (BM ). We consider the following parameters for the Bates model: S 0 = 80, 100, 120, K = 100 , T = 1, r = 0.03, η = 0.00, V 0 = 0.01 κ = 2, θ = 0.01, σ = 0.2, λ = 0.1, α = 0.1, β 2 = 0.1 and ρ = 0.5. According to the default parameters, we consider δ (t) = δ = 0.03 and R = 0.4. We compute the CVA for a European and an American a Put option with strike equal to K = 100.

Results are available in Tables 2 and3, which show that all the implemented methods give similar results. Values calculated via HTFD-HTMC and values calculated via FDMC have similar accuracy. We emphasize that, despite the similarity in numerical results of these two methods, we prefer the HTFD-HTMC method because of its simplicity of implementation.

The values returned by C-HTFD are the most accurate since they are very close to the benchmark for the whole parameter configurations.

Conclusions

In this paper we have proposed a numerical method to compute the CVA of European and American options when the underlying is assumed to evolve according to the Bates model. This method is based on the the resolution of two coupled PIDE which is done by employing the Hybrid Tree-Finite Difference algorithm developed by Briani et al.. Specifically, the C-HTFD approach consist in replacing the Monte Carlo step of the FDMC method with the resolution of the PIDE followed by the CVA cost, which is done by employing the Hybrid technique for the Bates model.

Numerical results show that our method is very stable and robust. In particular, numerical tests reveals that the values returned by C-HTFD method are very accurate, much more than the results provided by other methods which involve a Monte Carlo step, since the use of a PIDE approach in place of a Monte Carlo one dramatically improves the computational efficiency. Thus, the C-HTFD is efficient and reliable and the use of the two coupled PIDE represents a relevant improvement with respect to the standard pricing techniques of the CVA when the Bates model is considered. A 0.058209 ± 0.003071 0.063808 ± 0.003809 0.060724 0.060359 ± 0.000125 B 0.059096 ± 0.002635 0.062193 ± 0.003005 0.060613 C 0.059838 ± 0.002178 0.061673 ± 0.002385 0.060507 D 0.060978 ± 0.001664 0.060749 ± 0.001678 0.060467 S 0 = 120 A 0.005658 ± 0.001697 0.007251 ± 0.001976 0.005633 0.005589 ± 0.000059 B 0.005406 ± 0.001289 0.006296 ± 0.001461 0.005610 C 0.005334 ± 0.000954 0.006299 ± 0.001328 0.005592 D 0.005660 ± 0.000758 0.005616 ± 0.000852 0.005584 A 0.336835 ± 0.005078 0.342798 ± 0.006316 0.338987 0.339054 ± 0.000208 B 0.337859 ± 0.004630 0.342703 ± 0.005295 0.339084 C 0.338970 ± 0.003560 0.341533 ± 0.004139 0.339134 D 0.340182 ± 0.002755 0.340498 ± 0.003019 0.339165 S 0 = 100 A 0.059803 ± 0.003168 0.065669 ± 0.003937 0.062466 0.062145 ± 0.000130 B 0.060742 ± 0.002723 0.063979 ± 0.003103 0.062364 C 0.061522 ± 0.002248 0.063446 ± 0.002458 0.062260 D 0.062717 ± 0.001718 0.062501 ± 0.001732 0.062221 S 0 = 120 A 0.005798 ± 0.001745 0.005449 ± 0.002043 0.005782 0.005740 ± 0.000061 B 0.005537 ± 0.001327 0.006466 ± 0.001509 0.005760 C 0.005473 ± 0.000983 0.006467 ± 0.001368 0.005742 D 0.005812 ± 0.000784 0.005763 ± 0.000877 0.005735

  14) and u h (nh, y; v, z) = u h (s, y; v, z) | s=nh . (4.15)

FDMC

Table 1 :

 1 Configuration parameters for the numerical methods.

			FDMC		HTFD-HTMC C-HTFD
		A	50; 80; 15; 1500	50; 100; 1500	50; 100
		B	75; 110; 30; 2000	75; 150; 2000	75; 150
		C 100; 200; 50; 3300	100; 250; 3300	100; 250
		D 125; 300; 100; 6000 125; 350; 6000	125; 350
		FDMC	HTFD-HTMC	C-HTFD	BM
	S 0 = 80	A 0.320071 ± 0.005172 0.327462 ± 0.006098 0.323732 B 0.322874 ± 0.004469 0.327165 ± 0.005107 0.323713 C 0.323805 ± 0.003435 0.326023 ± 0.003993 0.323707 D 0.324926 ± 0.002660 0.324871 ± 0.002911 0.323703	0.323724 ± 0.000200
	S 0 = 100

Table 2 :

 2 CVA for European Put options.

Table 3 :

 3 CVA for American Put options. -R) E (t) dt (A.[START_REF] Borovykh | Efficient computation of various valuation adjustments under local Lévy models[END_REF] where the last term reflects the cash-flows from the fraction of the original counterparties who default between t and t + dt. Setting x = ∂C ∂t in equation (A.5) gives Since the portfolio is now (locally) riskless, it must earn the risk-free interest rate r. Setting dΠ = rΠdt results in

	Then, by Itô's lemma,
	dΠ = -	∂C ∂t	+	σ 2 S 2 2	∂ 2 C ∂S 2 + µS (1 dΠ = -∂C ∂S dt -σS ∂C ∂S dW + xµSdt + xσSdW -dP D (t) dt ∂C ∂t + σ 2 S 2 2 ∂ 2 C ∂S 2 dt -dP D (t) dt (1 -R) E (t) dt.	(A.6)
				-	∂C ∂t	+	σ 2 S 2 2	∂ 2 C ∂S 2 dt -	dP D (t) dt	(1 -R) E (t) dt = r -C +	∂C ∂t	S dt.	(A.7)
	Therefore, we get			∂C ∂t	+	σ 2 S 2 2	∂ 2 C ∂S 2 + rS	∂C ∂S	-rC + (1 -R) E (t)	dP D (t) dt	= 0.	(A.8)

We performed the numerical tests using a personal computer with the following features. CPU: Intel(R) Core(TM) i5-7200

2.50 GHz; RAM: 8GB, DDR4

Appendix A Proof of PIDE (3.6) This appendix provides a derivation of the PIDE (3.6) followed by the the CVA price. First of all, we present the proof for the simple case of an underlying evolving according to the Black-Scholes model. Then we consider the Bates model.

Black-Scholes model

Let S denote the underlying, which is assumed to evolve according to

where η denotes the continuous dividend rate, S 0 is a positive value and W t is a Brownian motion. We assume that counterparty credit risk is diversifiable across a large number of counterparties. In the case that this assumption is not justified, then the risk-neutral value of the contract can be adjusted using an actuarial premium principle (Gaillardetz and Lakhmiri [START_REF] Gaillardetz | A new premium principle for equity-indexed annuities[END_REF]). Moreover, the fraction of the original counterparties of the contract who have defaulted before time t is given by

Let us consider a financial product which pays (1 -R) E (t) if the counterparty defaults at time t and 0 otherwise. Then the value of such a product at time 0 -the discounted value of future cash-flows -is equal to the CVA. Therefore, we can consider the CVA as a derivative itself and we denote its financial value at time t with C (t, S t , V t ), that is

Suppose that the writer of a CVA derivative forms a self-financing portfolio portfolio Π which, in addition to being short to the CVA, is long x units of the index S, i.e. Π = -C (t, S, V ) + xS.

(A.4)

Bates model

The proof of PIDE (3.6) in the case of the Bates model may be done by employing the same approach followed in the previous Sub Section for the Black-Scholes model. Alternatively, we can obtain the PIDE (3.6) by a straightforward application of the Feynman-Kac formula for Lévy processes (Rong [START_REF] Rong | On solutions of backward stochastic differential equations with jumps and applications[END_REF], Glau [START_REF] Glau | A feynman-kac-type formula for Lévy processes with discontinuous killing rates[END_REF]), which can be applied since the CVA is defined as the expected value of a time integral of function depending on a Lévy process.


domain: math shs

text:
Anna Löwenhielm
email: anna.lowenhielm@mnd.su.se

Gosia Marschall
email: gosia.marschall@mnd.su.se

Judy Sayers
email: judy.sayers@mnd.su.se

Paul Andrews
email: paul.andrews@mnd.su.se

Opportunities to acquire foundational number sense: A quantitative comparison of popular English and Swedish textbooks

Keywords: Foundational number sense, mathematics textbooks, England, Sweden, grade one. Thematic Working Group 02

In this paper, we present analyses of popular grade one textbooks, one from each of England and Sweden. Focused on Foundational Number Sense, we examine how each book's tasks facilitate children's learning of those number-related competences that require instruction and which underpin later mathematical learning. Analyses identified both similarities and differences. Similarities lay in books' extensive opportunities for children to recognise and write numbers and undertake simple arithmetical operations. However, neither offered more than a few tasks related to estimation or simple number patterns. Differences lay in the Swedish book's greater emphases on different representations of number, quantity discrimination and relating numbers to quantity, highlighting conceptual emphases on number. The English book offers substantially more opportunity for students to count systematically, highlighting procedural emphases.

Introduction

In this paper we offer a comparative analysis of how commonly used textbooks, one from each of England and Sweden, enable year one pupils' acquisition of foundational number sense (FoNS). FoNS, which has been discussed in earlier CERME papers [START_REF] Back | The development of foundational number sense in England and Hungary: A case study comparison[END_REF][START_REF] Andrews | Developing foundational number sense: Number line examples from Poland and Russia[END_REF]Sayers & Andrews, 2015), comprises those number-related competences that underpin later mathematical learning, both in the short and the long term, and require instruction. Derived from a systematic review of the literature (Andrews & Sayers, 2015), FoNS comprises the eight broad categories shown in Table 1. Focused on the FoNS-related opportunities initiated during whole class teaching, the framework has structured analyses of grade one lessons in various European countries [START_REF] Back | The development of foundational number sense in England and Hungary: A case study comparison[END_REF]Andrews et al., 2015;[START_REF] Sayers | The role of conceptual subitising in the development of foundational number sense[END_REF] and identified didactical emphases commensurate with earlier research undertaken in the same countries.

Until now, we have not examined the framework's effectiveness with respect to identifying FoNSrelated opportunities in textbooks. This is a significant omission, particularly as both textbook production and deployment are unregulated in England and Sweden. This significance is heightened by uncertainty with respect to pre-school students' likely FoNS-related experiences. On the one hand, the English pre-school curriculum specifies that children should "count reliably with numbers from 1 to 20, place them in order and say which number is one more or one less than a given number. Using quantities and objects, they add and subtract two single-digit numbers and count on or back to find the answer" (Department for Education, 2014, p.11). On the other hand, the Swedish pre-school curriculum, which specifies no such detail, expects children to develop an understanding of the basic properties of quantity, number and number concepts [START_REF] Skolverket | Läroplan för förskolan Lpfö 98: Reviderad 2016[END_REF]. Thus, while there are no explicit FoNS-related expectations in the Swedish pre-school curriculum, a number, but not all, are addressed in the English. Of particular interest to this paper is [START_REF] Bierhoff | Laying the foundations of numeracy: A comparison of primary school textbooks in Britain, Germany and Switzerland[END_REF] comparison of the number-related opportunities offered in commonly used English, German and Swiss textbooks. Focused on the transition from "working with numbers up to 20… to working with two-digit numbers" (p. 143), she found that English textbooks were the least coherently structured. Also, students were expected to calculate with large numbers before consolidating their understanding of the integers up to 20, a situation made problematic by the English overemphasis on place value. Turning more explicitly to studies focused solely on English textbooks, [START_REF] Newton | Could elementary mathematics textbooks help give attention to reasons in the classroom[END_REF], in an evaluation of the professional support school textbooks might afford primary teachers, examined eighteen textbooks written for use with English 7-11 students. They found few tasks that would facilitate mathematical reasoning, being primarily focused on skills acquisition.

With respect to Sweden, as in England, the production of textbooks has been unregulated since 1991 [START_REF] Ahl | Research findings' impact on the representation of proportional reasoning in Swedish mathematics textbooks[END_REF] and several recent studies have examined Swedish mathematics textbooks against various criteria. For example, at the university level, [START_REF] Lithner | Mathematical reasoning in calculus textbook exercises[END_REF] found tasks typically promoting low levels of imitative reasoning. At the upper secondary, or post-compulsory, level [START_REF] Nordström | Proof in Swedish upper secondary school mathematics textbooks: The issue of transparency[END_REF] analysed the extent to which students were offered opportunities to engage with proof in two commonly used sets of textbooks. They found little evidence of proof in any of their examined topics, although there were many implicit opportunities in many of the tasks analysed. In similar vein, Lundberg (2011) compared three of the most commonly used textbooks from the perspective of proportional reasoning and found not only that direct proportion dominated but also that while both dynamic and static notions of proportion were present in all three textbooks, justifications were rare.

With respect to the final years of compulsory school, [START_REF] Ahl | Research findings' impact on the representation of proportional reasoning in Swedish mathematics textbooks[END_REF] examined the proportional reasoning in two popular textbooks. She found that "the impact of research findings on the representation of proportional reasoning is scant" in both (Ahl, 2016, p. 198) and that the books failed to encourage learners to understand the distinction between additive and multiplicative situations. In short, the limited available evidence indicates that textbooks written for older Swedish students present few opportunities for them to make mathematical connections or engage in mathematical reasoning.

However, little is yet known about the ways in which textbooks written for young children present mathematical ideas.

This study is a first attempt to evaluate the FoNS framework as a tool for analysing grade one textbooks. Thus, while it is not an explicit attempt to evaluate the content of the books themselves, it is an important first comparison of textbooks from the two countries. In making this comparison, we acknowledge Rezat's (2006, p. 482) position that the mathematics textbook "can be regarded as an artefact in the broad sense of the term. It is historically developed, culturally formed, produced for certain ends and used with particular intentions". In other words, comparative analyses of this nature highlight well cultural differences in expected learning outcomes.

Methods

Two popular textbooks, one from England and one from Sweden, were identified for analytical purposes. In focusing on popular textbooks, we believed we would gain insight into not only how a reasonably high proportion of children in both countries experiences FoNS but also what teachers and schools value in their choice of textbooks. Before formal analyses were undertaken, all four authors met for two days to discuss and evaluate a range of textbook tasks in order to operationalise the FoNS categories. Drawing on the studies of [START_REF] Li | A comparison of problems that follow selected content presentations in American and Chinese mathematics textbooks[END_REF] and others, only those tasks explicitly addressed to the student were analysed. For example, both of the examined textbooks included instructions or suggested activities that teachers might use. However, these were not analysed as they did not explicitly address the learner and typically included too little detail to show how they might have been used with children. For similar reasons, since tasks included in teacher guides were not focused directly on students, teacher guides were not included in the analyses. After this first pass, each of the first two authors took responsibility for analyses of the Swedish and English textbooks respectively. In these roles, each was supported by the third and fourth authors with respect to ambiguous or difficult to interpret tasks. In addition, random exercises from each textbook were also coded by both the third and fourth authors as part of a moderation process. In one of several similar tasks in one exercise, Swedish students were asked to "compare the number of dots" and then "write either = or ≠" in the box. This particular task, which occurred before the introduction of addition, was thought to encourage completion by counting and coded for systematic counting. The expectation that students would address issues of equality or inequality led to its also being coded for quantity discrimination. In addition, the dot patterns not only offered different representations of number but allowed for subitising and an awareness of the relationship between number and quantity. The goal of the English task, based on a coat hanger with ten pegs of which some of which had been covered with a cloth, was to identify the number of hidden pegs. The way in which the task was presented explicitly involved number recognition, while its focus was on simple arithmetic. In addition, its allusion to cardinality led to its being coded for awareness of the relationship between number and quantity. In short, many tasks attracted multiple codes. Some FoNS categories, as shown in Table 3, were rare in both textbooks. In this respect, Figure 2 shows tasks, one from each textbook, with explicit foci on number patterns. The Swedish task on the left was based on a section of a hundred square, with students being expected to complete the missing values. In addition to being coded for number patterns, the explicit focus of the task, it was also coded for systematic counting, number recognition and, implicitly, simple arithmetical operations. These decisions drew on the facts that the task required students to count on, recognise numbers and, in moving from one row to another, add or subtract ten. The English task on the right was one of several based around a section of a multiplication table torn from a longer strip of paper that invited students to count on in fives and enter the missing numbers. In addition to being coded for number patterns, these tasks were also coded for number recognition, systematic counting and simple arithmetical operations.

Operationalising the codes

Results

Below we present two analyses offering similar but importantly different perspectives on the data. The first is based on frequencies and the second on proportions.

A frequency analysis

The figures of Table 2 show the distribution of the eight FoNS categories across the two textbooks, one from England and one from Sweden. The first thing to notice, acknowledging that both books are intended to provide the complete learning experience for year one students, is that the Swedish book offered 444 tasks appropriate for FoNS coding, while the English only 257. That is, while both figures represented similar proportions of the totality of tasks within their respective books, the Swedish textbook comprised 187 (73%) more FoNS-related tasks than the English. When data are compared, some interesting results emerge. On the one hand the English books comprised significantly higher proportions of tasks involving number recognition (χ 2 =31.8, p<0.0005) and systematic counting (χ 2 =50.9, p<0.0005) than the Swedish. On the other hand, the Swedish books offered significantly higher proportions of tasks involving opportunities for students to relate numbers to quantity (χ 2 =17.9, p<0.0005), engage in quantity discrimination (χ 2 =7.33, p=0.007) and experience different representations of number (χ 2 =88.9, p<0.0005). Proportionally, the figures of Table 2 show no significant differences with respect to estimation, simple arithmetical operations or number patterns. These results take us to the second step of the analysis.

A proportional analysis

A second perspective on the data can be seen in Table 3. Firstly, several FoNS categories were found in similar proportions in both textbooks. These included relatively high occurrences of simple arithmetical operations, implicated in just under half of all tasks in both textbooks. In smaller proportions, around a quarter of all tasks in both books, were opportunities for students to relate number to quantities. In very small proportions in both books, were found number patterns and estimation. Secondly, several categories distinguished the expectations found in one book from the other. On the one hand, the English textbook comprised a significantly higher percentage of number recognition tasks (89%) than the Swedish (70%) (t=6.31, p<0.0005). Also, almost half of all English tasks involved systematic counting in comparison with less than a fifth in the Swedish (t=6.95, p<0.0005). Alternatively, the Swedish textbook comprised nearly three times as many tasks involving different representations of number as the English (t=10.57, p<0.0005), twice as many tasks focused on quantity discrimination (t=2.92, p=0.004) and almost twice as many tasks relating numbers to quantity (t=4.42, p<0.0005). Finally, Table 3 shows that the percentage of tasks coded for estimation, simple arithmetical operations and number patterns were comparable in both books, confirming that the two analyses, one essentially parametric and the other non-parametric, yielded equivalent results.

Discussion

In this paper our objective was to examine the efficacy of the FoNS framework as tool for evaluating the learning opportunities embedded in commonly used textbooks and to undertake a comparative analysis to determine the framework's sensitivity to different cultural expectations. In both cases, we believe the study to have been successful. For example, with respect to the identification of the different FoNS categories, very few tasks were identified with an emphasis on estimation, a finding resonating closely with earlier classroom observations showing no evidence of teachers in England, Hungary, Poland, Russia or Sweden emphasising it in their teaching [START_REF] Back | The development of foundational number sense in England and Hungary: A case study comparison[END_REF]Andrews et al., 2015;[START_REF] Sayers | The role of conceptual subitising in the development of foundational number sense[END_REF]. This, it seems to us, is an issue of some concern and the basis of further systematic inquiry. Indeed, acknowledging that estimation skills are important indicators of later mathematical competence (Booth & Sigler, 2006), that both older students [START_REF] Sowder | The development of concepts and strategies used in computational estimation[END_REF]) and many otherwise competent adults [START_REF] Hanson | Computational estimation skill of college students[END_REF] are uncomfortable with estimation tasks, it seems sensible to ask; why does estimation play such a lowly role in the classroom practice and textbooks of these two countries? This, we argue, is particularly pertinent in light of evidence from other countries that teachers see little relevance in teaching estimation [START_REF] Alajmi | Addressing computational estimation in the Kuwaiti curriculum: teachers' views[END_REF]. Furthermore, the similar frequencies of other FoNS categories are unsurprising. For example, it is reasonable to assume that the relative lack, in both textbooks, of tasks focused on number patterns may be explained by the fact that most year one curricular goals emphasise learners' number recognition, relating number to quantity and the beginnings of arithmetic. In other words, while number patterns are important in preparing students for later mathematical learning [START_REF] Lembke | Identifying early numeracy indicators for kindergarten and firstgrade students[END_REF], they may be subordinated in children's early number experiences to more pressing developmental needs.

With respect to cultural sensitivity the data yielded several hitherto uncovered insights. For example, on the one hand, the higher proportions of Swedish tasks coded for different representations of number, relating number to quantity and quantity discrimination allude to a book focused on conceptual understanding. On the other hand, the apparent lack of a conceptual emphasis in the English book finds further support in the high proportions of tasks coded for systematic counting and extremely high proportions of tasks addressing number recognition, which tend to suggest a book focused on the development of procedural knowledge commensurate with the low levels of mathematical challenge found in earlier studies of English textbooks (Bierhoff, 1999;[START_REF] Haggarty | An investigation of mathematics textbooks and their use in English, French and German classrooms: who gets an opportunity to learn what[END_REF][START_REF] Newton | Could elementary mathematics textbooks help give attention to reasons in the classroom[END_REF]. However, the conceptual emphasis found in the Swedish textbooks seemed not to match the generally negative findings of earlier Swedish studies [START_REF] Ahl | Research findings' impact on the representation of proportional reasoning in Swedish mathematics textbooks[END_REF][START_REF] Lundberg | Proportion in mathematics textbooks in upper secondary school[END_REF][START_REF] Nordström | Proof in Swedish upper secondary school mathematics textbooks: The issue of transparency[END_REF]. In this respect, it is not improbable that these differences may be because these earlier studies addressed textbooks for students in grades 7 and upward rather than on those for young children. Finally, drawing on [START_REF] Bernstein | Class, codes and control[END_REF] notion of curricular framing, it is interesting to note that the weakly framed Swedish pre-school curriculum seems to have prompted a conceptually focused textbook, while the strongly framed English preschool curriculum seems to have precipitated a procedurally focused textbook. Such matters allude to research beyond the scope of this paper but which will form a key aspect of any further analyses we make.

Figure 1 :

 1 Figure 1: Additive tasks from the Swedish and English textbooks respectively Figure 1 shows one example from each of the textbooks, Swedish on the left and English on the right.In one of several similar tasks in one exercise, Swedish students were asked to "compare the number of dots" and then "write either = or ≠" in the box. This particular task, which occurred before the introduction of addition, was thought to encourage completion by counting and coded for systematic counting. The expectation that students would address issues of equality or inequality led to its also being coded for quantity discrimination. In addition, the dot patterns not only offered different

Figure 2 :

 2 Figure 2: Number patterns tasks from the Swedish and English textbooks respectively

Table 1 : Summaries of the eight FoNS categories

 1

	FoNS Characteristic	Learners are encouraged to
	Number recognition	Identify, name and write particular number symbol
	Systematic counting	Count systematically, forwards and backwards, from arbitrary starting points
	Number and quantity	Understand the one-to-one correspondence between number and quantity
	Quantity discrimination Compare magnitudes and deploy language like 'bigger than' or 'smaller than'
	Different representations	Recognise and make connections between different representations of number
	Estimation	Estimate, whether it be the size of a set or an object
	Simple arithmetic	Perform simple addition and subtraction operations
	Number patterns	Recognise and extend number patterns, identify a missing number

Table 2 : Frequencies and chi square tests for each category for each country.

 2 Table2also shows that of the eight FoNS categories, number recognition was the most frequently observed, with 532 out of 691 tasks providing opportunities for learners to recognise, write and say numbers. In similar vein, simple arithmetical operations were common occurrences throughout both books. Neither of these results, we suggest, is surprising as arithmetical competence is an unequivocal curricular goal, which relies extensively on number recognition. The least commonly observed FoNS category was estimation, with just 18 occurrences.

		Category present in task (444 Swedish tasks and 257 English tasks)
		No	Yes	No	Yes	No	Yes
		Number recognition	Systematic counting	Number and quantity
	England	29	228	145	112	194	63
	Sweden	130	304	354	80	259	175
		159	532	499	192	453	238
		χ 2 = 31.8 (A)	χ 2 = 50.9 (A)	χ 2 = 17.9 (A)
		Quantity discrimination Different representations	Estimation
	England	237	20	202	55	250	7
	Sweden	370	64	181	253	423	11
		607	84	383	308	672	18
		χ 2 = 7.33 (B)	χ 2 = 88.9 (A)	χ 2 = 0.03 (C)
		Simple arithmetic	Number patterns
	England	154	103	232	25
	Sweden	232	202	406	28
		386	305	638	53
		χ 2 = 2.74 (C)	χ 2 = 2.45 (C)
	Chi squares marked A yielded p<0.0005, B yielded p<0.01 and C were not significant

Table 3 : Percentage of all tasks coded for each FoNS category along with t-tests

 3

		E%	S%	t	p
	Number recognition	89	70	6.31	0.000
	Systematic counting	44	18	6.95	0.000
	Relating number to quantity	25	40	-4.42 0.000
	Quantity discrimination	8	15	-2.92 0.004
	Different representations	21	58	-10.57 0.000
	Estimation	3	3	0.16	0.874
	Simple arithmetical operations	40	47	-1.66 0.097
	Number patterns	10	6	1.49	0.136

Proceedings of CERME10


text:
Beatriz Vargas Dorneles
email: beatriz.dorneles@ufrgs.br

Mariana Lima Duro

Nohemy Marcela

Bedoya Rios

Camila Peres Nogues

Clarissa Dos

Santos Pereira

Marcela Nohemy

Bedoya

; Nohemy

Marcela Bedoya Rios

Camila Peres

Number estimation in children: An assessment study with number line estimation and numerosity tasks

Keywords: Number line estimation, numerical development, spatial representation of number magnitude

come   L'archive ouverte pluridisciplinaire

Introduction

The decimal number system is used to establish exact quantifications, in contrast to quantity estimation processes. In everyday situations, we often use either exact quantification or number estimation (Feigenson, Libertus, & Halberd, 2013), and sometimes estimation can be easier than exact quantification [START_REF] Siegler | Development of numerical estimation in young children[END_REF]. Number estimation is a cognitive process used for quick or approximate answers or, for example, to calculate the duration of a movie or the distance between two places. It can be used as a feedback tool to check performance in different areas of mathematics including those requiring exact quantification. From our point of view, two complementary ideas define number estimation: a non-counting based quantitative answer to represent a set of objects; and a translation [START_REF] Siegler | Development of numerical estimation in young children[END_REF] between two different ways of representing a number. We know that mathematical competence involves a group of abilities and cognitive processes. Number estimation has been considered one such process [START_REF] Levine | Strategy use and estimation ability of college students[END_REF] despite the fact it has been less studied than exact quantification [START_REF] Piazza | Exact and approximate judgements of visual and auditory numerosity: An fMRI study[END_REF] although its importance has been highlighted [START_REF] Rousselle | The development of automatic numerosity processing in preschoolers: Evidence for numerosity-perceptual interference[END_REF]. This can be explained by the variability of the tasks used to assess estimation skills in children, adolescents and adults as well as the different situations in which we use estimation [START_REF] Siegler | Development of numerical estimation in young children[END_REF]. The recognition of small amounts may be related to the ability to represent quantities in a mental number line and this ability would assist in comparing the magnitude between two numbers [START_REF] Schneider | Mental number line, number line estimation, and mathematical achievement: Their interrelations in grades 5 and 6[END_REF].The estimation performance can be necessary in solving some mathematical tasks and the development of estimation is also considered a good predictor of later symbolic math skills [START_REF] Park | Training the approximate number system improves math proficiency[END_REF]. Despite the importance of its use, number estimation is not part of the school curriculum in many countries, including Brazil. Changes designed to improve mathematical achievement, including the introduction of number estimation in the curriculum, are currently being introduced in Brazil. For teachers, assessing the ability to estimate using different tasks might be a good starting point to analyze the importance of the ability as well as to highlight the topic´s importance in mathematics education. One of the most consistent conclusions reached by studies about the development of estimation is that children are not very able estimators, even when estimation is used in various daily applications. However, some researchers have hypothesized that children's estimations reflect their internal representation of numbers [START_REF] Siegler | The development of numerical estimation: Evidence for multiple representations of numerical quantity[END_REF].

Moreover, there is some evidence to suggest estimation is related to mathematical competence in general and arithmetical performance in particular [START_REF] Siegler | Development of numerical estimation in young children[END_REF][START_REF] Booth | Developmental and individual differences in pure numerical estimation[END_REF][START_REF] Schneider | Mental number line, number line estimation, and mathematical achievement: Their interrelations in grades 5 and 6[END_REF][START_REF] Mazzocco | Preschoolers' precision of the approximate number system predicts later school mathematics performance[END_REF][START_REF] Laski | Is 27 a big number? Correlational and causal connections among numerical categorization, number line estimation, and numerical magnitude comparison[END_REF] and can be improved [START_REF] Park | Training the approximate number system improves math proficiency[END_REF]. Recent research has highlighted the importance of estimation for mathematical development [START_REF] Link | On the relation between the mental number line and arithmetic competencies[END_REF][START_REF] Laski | Number line estimation and mental addition: Examining the potential roles of language and education[END_REF]. This research indicated that the better the students' accuracy in mental number line is, the better their performance in other numerical and arithmetic tasks [START_REF] Link | On the relation between the mental number line and arithmetic competencies[END_REF]. Hence, it is important to understand the estimation process, the abilities involved, how to assess them and its role in mathematical performance, especially regarding the proposed changes to the curriculum in Brazil. It can be said that even though the estimation processes have been studied for the past twenty years, there is no consensus regarding the assumption that the estimations assessed by different tasks reflect a pure mental representation of numbers as proposed by [START_REF] Siegler | Development of numerical estimation in young children[END_REF] and some new data indicate that it is affected by the limited knowledge of numbers [START_REF] Ebersbach | The relationship between children's familiarity with numbers and their performance in bounded and unbounded number line estimations[END_REF], as well as by visuospatial abilities [START_REF] Crollen | Spatial and numerical processing in children with high and low visuospatial abilities[END_REF].

In the current scientific debate, among the explanations for the development of number estimation in children, two stand out. The first, the multiple representations of numbers model [START_REF] Siegler | Development of numerical estimation in young children[END_REF], assumes that children initially represent numbers in a less accurate algorithmic way and develop a more accurate linear representation with age and experience. That is, in children the mental number line is compressed and they tend to maximize the distance between the magnitudes of numbers at the low end of the range and minimize the distance between the magnitudes of numbers in the middle and upper ends of the range. This tendency was named logarithmic representation. Gradually, children develop a linear representation, which maintains the same distance between the numerical magnitudes. Empirical evidence to support this logarithmic-to-linear shift model has largely come from the Number Line Estimation Task (NLET) proposed by [START_REF] Siegler | Development of numerical estimation in young children[END_REF] in which children or adults must estimate the magnitude of a number by marking its proper position on a number line. However, the logarithmic-to-linear-shift hypothesis has been questioned by researchers studying number-line estimation [START_REF] Barth | The development of numerical estimation: evidence against a representational shift[END_REF][START_REF] Ebersbach | The relationship between children's familiarity with numbers and their performance in bounded and unbounded number line estimations[END_REF]. Some of the issues raised led to the development of a second explanation, the proportion-judgement strategies model [START_REF] Barth | The development of numerical estimation: evidence against a representational shift[END_REF], which suggests that in the NLET children estimate the size of a part, the numerical magnitude of a specific number, relative to the size of the whole, thus making a judgement about the proportion of the size of the former. Hence, according to the numerical range, the more reference points (landmarks) made available, the more accurate the estimation will be, especially close to the landmarks. In other words, estimation performance reflects the strategies chosen to solve the tasks. Recently, a third model, the Two-Linear Account has been proposed as a plausible alternative [START_REF] Ebersbach | The relationship between the shape of the mental number line and familiarity with numbers in 5-to 9-year-old children: Evidence for a segmented linear model[END_REF] and explains the developmental changes in number estimation as a result of children's familiarity with numbers. In this model, the mental numerical representation can be alternatively described as a combination of two linear patterns with different slopes, depending on number familiarity. In other words, this linear representation of numbers changes according to the age and numerical range known by children; the unknown numbers have a slower linear representation than the known numbers. A recent paper [START_REF] Dackermann | An integration of competing accounts on children's number line estimation[END_REF] proposes the integration of these accounts, which is a line of reasoning that we support as they introduce the idea that aspects of all three accounts may complement each other and facilitate a more comprehensive understanding of children´s development of number line estimation.

Among the tasks most widely used to assess the ability to estimate are the NLET, described above, and the Numerosity Estimation Task (NET), which requires the subject to estimate the quantity of objects in a set (Luwel, Verschaffel, Onghena & De Corte, 2003;Barth, Starr & Sulivan, 2009). To the best of our knowledge, there is no research that indicates which of these tasks best assesses students' accuracy in estimation and therefore which would be best for application in research and in schools. Thus, the purpose of the study was to compare the accuracy in the numerical estimation of 60 children from the 2 nd and 3 rd grades of a public school in the city of Porto Alegre (Brazil) in order to determine which task (NLET or NET) best assesses the students' accuracy in number estimation. As the children in both grades were used to manipulating objects and completing tasks similar to the NET and admittedly had no contact with the number line before, we assumed the students would perform better in the NET than in the NLET. We chose the two tasks because the NLET is the most frequently used in estimation research and is widely used to examine how the human mind represents numbers [START_REF] Barth | The development of numerical estimation: evidence against a representational shift[END_REF][START_REF] Ebersbach | The relationship between children's familiarity with numbers and their performance in bounded and unbounded number line estimations[END_REF], while the NET is similar to another task often used by some research groups that have a slightly different theoretical viewpoint regarding estimation, for example, Barth, Starr & Sulivan (2009). Both tasks are assumed to assess the same numerical estimation ability.

Method

Using two different tasks, we compared performance in number estimation within a group of sixty children (mean age = 8.4, SD = .69, age range from 7.4 to 11.2 years) who were recruited from one public school, 37 boys and 23 girls: 28 from 2 nd grade, (M = 7.8, SD = .29) and 32 from 3 rd grade (M = 8.9, SD =.50). Two tasks were used: the NLET [START_REF] Siegler | Development of numerical estimation in young children[END_REF]) and the NET, adapted from Luwel, Verschaffel, Onghena & De Corte (2003). We used only one criterion to determine the students' accuracy in the presented tasks: the measure proposed by [START_REF] Siegler | Development of numerical estimation in young children[END_REF], described below. The task was applied collectively in the classroom, the workplace and the school activities were affected as little as possible. The time to perform the activities in each class was about 40 minutes per task. The solution to both tasks involved the use of pen and paper.

The NLET requires the subjects to mark points corresponding to specific numbers along a number line bounded by 0 and 100. Children were asked to mark the place they considered most suitable for the position of the number to be estimated. Before each item, the experimenter said, "This number line goes from 0 at this end to 100 at this end. If this is 0 and this is 100, where would you put n?" (n being the number specified in the trial). 29 number estimations were required, one at a time. Each number was presented twice. Each child received a booklet with a number line drawn on each sheet to mark their answers. The difference between the two estimations of the same number provided a measure of the variability of the estimations. The 29 numbers comprised the 24 proposed by [START_REF] Siegler | Development of numerical estimation in young children[END_REF], plus another 5 numbers that were also used in the second task. The 29 numbers presented were 3,4,6,7,8,9,12,17,21,23,25,29,33,39,43,48,49,52,57,61,64,72,78,79,81,84,90,95,96. They were presented in random order and then repeated in the same random order. Children had no predetermined time to finish the task, each one could complete the task in the time they wanted.

The NET requires the children to estimate the amount of dots distributed in a checkered 10x10 grid. Before starting the task, the students were told the empty grid contained 0 dots and the full grid contained 100 dots. To reduce the possibility of verbal counting, the stimuli were presented quickly (1 second for each group of ten dots presented) and immediately followed by a white screen. Students were asked to perform a numerical estimation of the amount observed, writing them down in a notebook. They could not use any additional tool to solve the task. In the task, eight numbers from 0 to100 were randomly matched (4,7,9,17,25,49,78,95) in two different ways. In the first, the dots were presented in clusters and, in the second, they were presented dispersed. Both tasks were carried out collectively on different days. There was no feedback for correct or wrong answers.

Results

To calculate the accuracy of the estimations given, the calculation of absolute percentage error of each child was used, adapted from [START_REF] Siegler | Development of numerical estimation in young children[END_REF], and represented by the formula: Mean Estimation -Estimated Quantity _______________________________ Scale of Estimations (100)

To illustrate how this measure works, if a child was asked to estimate the location corresponding to the number 60 (or quantities of dots) in a number line from 0 to 100 (or 10X10 grid) and his/her answers were 65 in the first estimation and 75 in the second, we calculated the mean between the estimated values in the two attempts (in this example, (65+75)/2=70). The absolute percentage error would be 10%, corresponding to the result of (70 -60)/100, according to the above formula. We used this calculation because previous analysis showed that the difference between two answers for each number in both tasks was not significant (p=.24) for NLET or (p=.06) for NET.

After that, a descriptive analysis of the accuracy of each child was conducted to identify the general standard of performance for each task. These analyzes show the students' accuracy is higher in the NLET.

The reported performance tends to be more cohesive in the NLET (Table 1). To determine the correlation between the two tasks, Pearson´s correlation coefficient was carried out (r =.67, p<.01), and indicated a positive correlation between both tasks, suggesting that either demand similar cognitive functions or both are related to other skills that were not measured.

Mean

To test for differences in the children's estimation accuracy when required to estimate smaller and larger amounts, we considered the same numbers estimated in both tasks. A paired-samples t-test only indicated difference when estimating the same number in each task in larger quantities ( This analysis showed that the students' estimations were all more accurate in the NLET, however, these differences were only statistically significant with the numbers 49, 78 and 95 (Table 2), while there is greater variability in the children's estimation in the NET. This may suggest important differences between the tasks performed. Maybe the three numbers (49, 78 and 95) are closer to "quarters" (50, 75 and 100), which would help children to identify the position of the numbers in the number line. Hypotheses to explain these variations will be discussed later. In the NLET, one of the most common strategies used by students was to fill the number line with marks that represent the numbers before or after the number proposed, as illustrated in Figure 1. In the NET, some students tried counting groups of dots and imagining how many similar groups could be in the whole. Progressions related to the speed with which the children performed the estimations were not tested in this analysis, considering that the time for execution was the same for all participants.

Discussion and conclusions

As we described above, number line estimation is related to both basic and complex arithmetical abilities. Moreover, there is evidence to suggest number estimation is related to mathematical achievement. Despite this, number estimation is rarely taught in Brazil and other countries. In our research, we observed children more accurately estimate numbers on a number line than dots on a grid. Although the two tasks measure the ability to estimate, as indicated by the general correlation between the students' accuracy in the two tasks, they may be linked to different cognitive functions, as suggested by the differential performance when the tasks involve numbers over 25. Although speculative in nature, some ideas help us to understand the results. The NLET requires transposition from numerical knowledge to a position on a line, whereas the NET requires transposition from a perceptual estimation (linked to quantities) to numerical knowledge. Both demand translation between different representations. However, unlike the NET, the NLET allows children to try a discrete quantity representation, sometimes marking the number line with lines or dots from the beginning to the point that could represent the required number. This marking strategy was the most widely used by the children and helped them identify an almost correct answer. Children used different strategies or representations when estimating. The linear distance of numbers along the number line seems to be an important support for the estimations, as [START_REF] Siegler | Development of numerical estimation in young children[END_REF] have described. Additionally, children used decision-based strategies considering the proximity of the extremities, for example, to represent number 78, they made marks from 100 to 78 in descending order. Alternatively, some decided to begin from 50 and made marks from 50 to 78 in an ascending sequence. Another strategy was to mark the quarters (e.g. 25, 50, 75) as landmarks. All these strategies have been identified as means to improve the way children estimate [START_REF] Siegler | The development of numerical estimation: Evidence for multiple representations of numerical quantity[END_REF][START_REF] Siegler | Development of numerical estimation in young children[END_REF]. The use of these strategies suggests the children did not estimate the numbers by chance, but instead coordinated mathematical knowledge and spatial skills to assess the place to mark. This tactic fits very well with the proportionjudgement strategies model [START_REF] Barth | The development of numerical estimation: evidence against a representational shift[END_REF]. The central number (50) was understood by some students as a reference mark used to estimate the other numbers.

The difference found in the tasks with the numbers 49, 78 and 95 can be explained by the fact the three numbers are close to "quarters", which would help children to identify the position of the numbers on the number line. Also, we must remember that, generally, younger children tend to overestimate small numbers and compress large numbers toward the end of the scale, whereas older children, using the same number range, tend to estimate more accurately [START_REF] Siegler | Development of numerical estimation in young children[END_REF]. Moreover, in our research, it may be the case that due to lack of familiarity with the number line task, the older children continued to overestimate small numbers, as described in the logarithmic model. Another possible factor influencing the results was the time. In the NLET, children have time to think about the relation between the number and the place on the line, while in the NET they have only a few seconds to observe the quantities and more time to estimate the quantity represented. The NLET requires the ability to coordinate the knowledge of number systems with a kind of spatial graphic representation on a line. Maybe the effort made by children to coordinate these two cognitive demands helps them to estimate more accurately, despite having little experience with number line tasks. Moreover, the NET may not help students to access their knowledge of the number system, since it provides no clues that would allow students to adopt some proportion-judgement strategies coordinated with their number system knowledge. We suggest the NET requires more "guessing" than the NLET because there is no opportunity to mark "quarters" or anything like that to help them to estimate. One limitation of the research is that only eight numbers were repeated in both tasks. For future research, we suggest amplifying the analysis of the cognitive processes involved in both tasks. We did not control the number system knowledge of the subjects. We do not discuss whether or not the number line task reflects an internal mental number line, since this was not the subject of the paper. However, considering the results of our research, we can say that the NLET is a relevant measure even if it does not reflect an internal mental number line. We support this idea because, although the children were not familiar with the number line estimation task and were more used to NET-like activities, they performed better in the NLET. This surprising result provides the opportunity to introduce the discussion about estimation in the Brazilian curriculum. Furthermore, considering that the ability to estimate is correlated to many aspects of mathematics, for example, number comparison, addition, and subtraction, it is important to take in consideration the assessment of this ability in mathematics education as well as the ways to improve it.

Figure 1 .

 1 Figure 1. Example of marking strategy in two numbers of NLET

Table 1 : Comparison of the Mean of Percentage Error in each task

 1

	SD	Significance

Table 2

 2

	).

Table 2 : Comparison of estimations between the same numbers

 2

Proceedings of CERME10