RE S E A R C H PA P E R
Q U A N T I T A T I V E F I N A N C E V O L U M E 3 (2003) 451–457 INSTITUTE O F PHYSICS PUBLISHING
quant.iop.org
A new well-posed algorithm to recover implied local volatility Lishang Jiang1 , Qihong Chen2 , Lijun Wang3 , Jin E Zhang4,5 1
Institute of Mathematics, Tongji University, Shanghai 200092, People’s Republic of China 2 Department of Applied Mathematics, Shanghai University of Finance and Economics, Shanghai 200433, People’s Republic of China 3 Department of Applied Mathematics, Tongji University, Shanghai 200092, People’s Republic of China 4 Department of Finance, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, People’s Republic of China E-mail:
[email protected],
[email protected],
[email protected] and
[email protected] Received 11 July 2002, in final form 16 July 2003 Published 5 September 2003 Online at stacks.iop.org/Quant/3/451
Abstract This paper presents a new algorithm to calibrate the option pricing model, i.e. the algorithm that recovers the implied local volatility function from market option prices in the optimal control framework. A unique optimal control is shown to exist. Our algorithm is well-posed. Our numerical experiments show that, with the help of the techniques developed in the field of optimal control, the local volatility function is recovered very well.
1. Introduction Based on the assumption of constant volatility, the famous Black–Scholes formula can be used to evaluate European options simply and quickly by using the estimated or forecast volatility constant as an input [2]. The value of the option is monotonic in the volatility parameter. Then, most option traders invert the Black–Scholes formula to determine the volatility (called the implied volatility) from the market option price. Actually, the Black–Scholes formula has been used not so much as a pricing tool but as a means to switch back and forth between market option prices and their associated implied volatilities. If the model were perfectly realistic, the implied value would be the same for all options on the same underlying with different strikes and maturities. Unfortunately, this is not the case. Implied Black–Scholes volatilities vary with strikes and time to maturity, which are respectively known as the smile effect and the term structure [21]. 5
Author to whom any correspondence should be addressed.
1469-7688/03/060451+07$30.00
© 2003 IOP Publishing Ltd
There have been various attempts to extend the Black– Scholes theory to account for the volatility smile effect and the term structure. One class of models introduces a non-traded source of risk such as jumps [19] or stochastic volatility [13]. A newer class of models introduces a deterministic volatility function that varies with the spot price and time. Rubinstein [22], Dupire [9] and Derman and Kani [8] have independently constructed a discrete approximation to the riskneutral process for the underlying in the form of a bi/trinomial tree, which are extensions of the original Cox et al [7] binomial trees. These implied trees are compatible with the observed smiles at all maturities and also keep the model complete. Bouchouev and Isakov [4, 5] reduce the identification of volatility to an inverse parabolic problem with the final observation and establish uniqueness and stability results under certain assumptions. Then, they obtain a non-linear Fredholm integral equation for unknown volatility after dropping terms of the higher order of time to maturity and solve the equation iteratively. This approach is applicable to finding short-term volatility. If long-term volatility is the main interest, then one
PII: S1469-7688(03)39369-8
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may be better off using an optimization-based algorithm. Lagnado and Osher [16, 17] suggest minimizing the gradient of the volatility function subject to a finite number of constraints. The gradient descent procedure is used to carry out minimization. The variational derivative has to be computed on each iteration step for all strikes and times to maturity and for each point on the finite difference grid. In other words, the fundamental solution for the Black–Scholes equation must be computed in order to obtain the next volatility approximation. Consequently the computation could be highly demanding. Avellaneda et al [1] use the dynamic programming approach and minimize the entropy functional. A review of a few other optimization approaches can be found in [5]. This paper is a sequel to the paper by Jiang and Tao [14]. They use an optimal control framework to determine the implied local volatility and rigorously analyse the inverse problem. With a solid mathematical foundation well-established, we present a new algorithm to implement the procedure of calibrating the option pricing model, i.e. recovering the local volatility function from market option prices in the optimal control framework [10,11]. With the help of the techniques developed in the field of optimal control, we have successfully recovered the volatility function. It turns out that, as an optimal control, the unknown volatility solves an elliptic bilateral variational inequality coupled with a forward and backward parabolic system. Our numerical experiments show that the volatility function is recovered very well.
Problem I. Determine the coefficient, σ , such that the solution of (2.2) fits the current market prices of options at (s ∗ , t ∗ ) for different strikes, K, and/or maturities T .
2. Related optimal control problem
In principle, we can infer the volatility function from the complete knowledge of the option price. That is to say, if the current market prices of the options are known for all conceivable strikes, K, and maturities, T , then the volatility function can be theoretically found directly from (2.4). But this method is not reliable for two reasons. The first is that we need to evaluate numerical derivatives of u(K, T ) with respect to T and K, especially the second derivative with respect to K. A small change in u may result in a big change in its derivatives. Therefore, the computation might not be stable. The second reason is that the market option data only provide a set of discrete data points at (Km , Tn ), m = 1, 2, . . . , M, n = 1, 2, . . . , N. The value of u at some other point is obtained with an inter/extrapolation technique. It is therefore subject to some kind of error, which affects the final value of the local volatility. Dupire’s formula (2.4) is found to be difficult to apply in practice. It therefore needs to be improved. We study the problem in two steps. First, we study the case when the local volatility is independent of time and use only option prices with different strikes and a fixed maturity date in this paper. The case when the local volatility is also a function of time is left for future research. Thus, from above, this problem can be formulated as a typical inverse problem of parabolic equation with a terminal observation.
As is well known [2, 12, 18, 23], when the stock price, s, follows a general diffusion of the form ds = µ dt + σ dWt , s
(2.1)
where µ is the drift, σ is the stock volatility, and Wt is a standard Wiener process, the European call option premium, u = u(s, t; K, T ), satisfies the following Black–Scholes equation: ut + 21 s 2 σ 2 uss + (r − q)sus − ru = 0 (s, t) ∈ R + × [0, T ), u(0, t) = 0, u(s, T ) = (s − K)+ ,
(2.2)
where T is the maturity, K is the strike, r is the risk-free interest rate, q is the dividend yield (on the stock) and σ is the stock volatility as in (2.1), which is the only parameter (in the pricing model) that is not directly observable. In recent years, many options became liquid (actually, we can now find as many as a hundred simultaneously traded options on the S&P 500 index, all differing in strike price and/or time to maturity). With such plentiful data, the option market has become a rich source of information. It is thus natural to desire to recover the unknown volatility from the observed market price of options for different K and/or T at current time t ∗ with current stock price s ∗ . In the continuous-time setting, this amounts to the following problem.
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From the mathematical point of view, this is an inverse problem of partial differential equation (PDE). But it is not a standard one, since it requires us to determine the coefficient, σ , of the pricing equation by means of a series of observed values of the solution corresponding to certain parameters (K and/or T , which are usually discrete) at a fixed point (s ∗ , t ∗ ). Such a problem is ill-posed in general. The fundamental issue of ill-posedness is the focus of Bodurtha and Jermakyan [3], who treat the problem from the classic Tikhonov perspective. With the derivation of a dual problem, the problem can be reduced to a standard parabolic equation with new variables: uτ = 21 K 2 σ 2 uKK − (r − q)KuK − qu (K, τ ) ∈ R + × R + , u|K=0 = e−qτ s, u|τ =0 = (s − K)+ ,
(2.3)
where τ = T − t is the time remaining to maturity. By rearranging the terms, we have Dupire’s [9] formula for the implied local volatility function: uτ + (r − q)KuK + qu . (2.4) σ (K, T ) = 1 2 K uKK 2
Problem II. Find σ (K) such that the solution of (2.3) satisfies u(s ∗ , t ∗ ; K, τ )|τ =τ ∗ = u∗ (K)
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A new well-posed algorithm to recover implied local volatility
where τ ∗ = T − t ∗ and u∗ (K) is the current market price of options for different K > 0 at current time t ∗ with stock price s∗.
Together with (2.7), (2.8) ensures that |v(y, τ ) − v ∗ (y)|2 dy < ∞, ∀τ ∈ [0, τ ∗ ]. R
In this paper, for the sake of compatibility, u∗ (K) is assumed to be a continuous function satisfying ∗
lim u∗ (K) = e−qτ s ∗ ,
lim u∗ (K) = 0.
K→∞
K→0
(2.5)
The above problem is not well-posed either. However, with the standardized form, we are able to solve it in an optimal control framework. First, to remove the singularity at K = 0, we make a change of variables in (2.4): y = ln
K , s
v=
(2.6)
v|τ =0 = (1 − e ) , y +
where a(y) =
1 2 σ (K). 2
Under the above transformation, problem II becomes Problem II . Find a(y) such that the solution of (2.6) satisfies
lim v ∗ (y) = 1,
y→−∞
where v is the solution of (2.6) corresponding to a. We now introduce the following optimal control problem. Problem III. Find an a¯ ∈ A, such that
Such an a, ¯ if it exists, is called an optimal control. The unusual feature of our problem is that the control variable lies in the coefficient of the second-order partial derivative in the pricing PDE and, moreover, the equation is in a non-divergent form. It is easily seen that J (a) is a non-negative lower semicontinuous functional and A is bounded in H¨older space, 1 C 2 (R). Based on the state analysis, we can establish the existence and uniqueness for the above optimal control problem. Theorem 2.1. Problem III admits a unique optimal control, a¯ ∈ A. A rigorous proof of the theorem is available in [14]. Recovering the unknown volatility is reduced to finding the optimal control of problem III. Now, we derive the necessary condition for the optimal control. Let a¯ ∈ A be an optimal control of problem III and v¯ be the solution of (2.6) corresponding to a. ¯ Note that A is a convex set, for any h ∈ A,
v(y, τ )|τ =τ ∗ = v ∗ (y), 1 qτ ∗ ∗ e u (K) s∗
satisfying (by (2.5)) lim v ∗ (y) = 0.
y→+∞
The solution to the above inverse problem, a(y), is dependent on s ∗ and τ ∗ just as v ∗ (y) is. Moreover, we further impose the following condition on the given data: |v ∗ (y) − H (−y)|2 dy < +∞, (2.7) R
(1 − λ)a¯ + λh ∈ A
j (λ) = J ((1 − λ)a¯ + λh)
be the control set, where a0 and a1 are the lower and upper bounds of half the volatility squared respectively. The known theory for parabolic equations [15] guarantees that, for any given a ∈ A, there is a unique solution, v(y, τ ), to the Cauchy problem (2.6) with the property (y → ∞).
(2.8)
λ ∈ [0, 1]
is well defined and reaches its minimum at λ = 0. Then, we must have j (0) =
A = {a ∈ C(R)| 0 < a0 a(y) a1 , ∇a ∈ L2 (R)}
∀λ ∈ [0, 1].
For any h ∈ A, the function
where H (·) is the well-known Heaviside function. Let
|v(y, τ ) − H (−y)| = O(e−|y| )
a ∈ A, (2.9)
a∈A
(y, τ ) ∈ Q = R × (0, τ ∗ ],
where v ∗ (y) =
1 N v(·, τ ∗ )−v ∗ (·)2L2 (R) + ∇a2L2 (R) , 2 2
J (a) ¯ = inf J (a).
vτ = a(y)(vyy − vy ) − (r − q)vy
1 qτ e u(K, τ ), s
J (a) =
1 qτ e u, s
which leads to a Cauchy problem,
v(y, τ ) =
This makes it possible for us to define a meaningful cost functional:
d J ((1 − λ)a¯ + λh)|λ=0 0 dλ
i.e. d [|v λ (y, τ ∗ ) − v ∗ (y)|2 dλ R + N |∇((1 − λ)a¯ + λh)|2 ] dy |λ=0 0
∀h ∈ A, (2.10)
where v (y, τ ) is the solution of (2.6) corresponding to a = (1 − λ)a¯ + λh (λ ∈ [0, 1]). λ
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By direct differentiation with respect to λ on both sides of (2.6) for v λ , it can be verified that dv λ (y, τ ) = ξ(y, τ ), dλ λ=0 where ξ satisfies the following variational equation: ¯ yy − ξy ) + (r − q)ξy = (h − a)( ¯ v¯yy − v¯y ), Lξ ≡ ξτ − a(ξ
3. Numerical simulations 3.1. Algorithm After summarizing the results of the last section, we have the following three sets of equations, state equation (3.1), adjoint equation (3.2) and variational inequality (3.3): vτ = a(vyy −vy )−(r −q)vy ,
ξ |τ =0 = 0.
−ϕτ = (aϕ)yy + (aϕ)y + (r − q)ϕy ,
(2.11) Thus, (2.10) gives [v(y, ¯ τ ∗ ) − v ∗ (y)]ξ(y, τ ∗ ) dy R +N ∇ a¯ · ∇(h − a) ¯ dy 0
ϕ|τ =τ ∗ = v(y, τ ∗ ) − v ∗ (y);
∀h ∈ A.
(−ayy + f (y; v, ϕ))(a − a0 ) 0,
(2.12) where
f (y; v, ϕ) =
∗
L ϕ ≡ −ϕτ − (aϕ) ¯ yy − (aϕ) ¯ y − (r − q)ϕy = 0, ϕ|τ =τ ∗ = v(y, ¯ τ ∗ ) − v ∗ (y).
(2.13)
Then, from (2.11), (2.13) and Green’s formula, we may deduce that τ∗ (ϕLξ ¯ − ξ L∗ ϕ) ¯ dy dτ 0 R = [v(y, ¯ τ ∗ ) − v ∗ (y)]ξ(y, τ ∗ ) dy, R
ϕ(h ¯ − a)( ¯ v¯yy − v¯y ) dy dτ
R
=
[v(y, ¯ τ ∗ ) − v ∗ (y)]ξ(y, τ ∗ ) dy.
(2.14)
R
Combining (2.12) and (2.14), we get
∇ a¯ · ∇(h − a) ¯ dy
N R
τ∗
ϕ( ¯ v¯yy − v¯y )(h − a) ¯ dy dτ 0
+ 0
∀h ∈ A.
R
(2.15) Equation (2.15) shows that a¯ is a weak solution to the following elliptic bilateral variational inequality [6]: a0 a a1 , (−ayy + f (y; v, ¯ ϕ))(a ¯ − a0 ) 0,
(2.16)
(−ayy + f (y; v, ¯ ϕ))(a ¯ − a1 ) 0, where
∗ 1 τ ϕ( ¯ v¯yy − v¯y ) dτ. (2.17) N 0 Consequently, we have obtained the optimality condition for our optimal control problem, as follows. f (y; v, ¯ ϕ) ¯ =
Theorem 2.2. Let a¯ ∈ A be an optimal control of problem III and v¯ be the corresponding solution of (2.6). Then, there exists a function ϕ¯ solving the adjoint equation (2.13) such that a¯ is a solution of the elliptic bilateral variational inequality (2.16).
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(3.3)
(−ayy + f (y; v, ϕ))(a − a1 ) 0,
R
0
(3.2)
a0 a a1 ,
Let ϕ¯ be the generalized solution of the following adjoint equation:
i.e., τ∗
v|τ =0 = (1−ey )+ ; (3.1)
1 N
τ∗
ϕ(vyy − vy ) dτ.
(3.4)
0
Given the market price of option u∗ (K) and the underlying s at time t ∗ , we would then determine v ∗ (y) by its definition ∗ v ∗ (y) = eq(T −t ) u∗ (K)/s ∗ . Our task becomes solving for a(y) from the three sets of equations. Once we have a(y), √ we would be able to get the volatility function by σ (s) = 2 a(ln s/s ∗ ). Given τ ∗ and v ∗ (y), the task of solving a(y) can be achieved by the following iteration procedure: ∗
(1) We make an initial value for a(y), say, a(y) = A0 (y). Solving equation (3.1) gives v(y, τ ). The algorithm for solving PDEs such as (3.1) and (3.2) is given in the appendix. (2) Assigning τ ∗ in v(y, τ ) obtained in step 1 gives v(y, τ ∗ ). (3) The difference between v(y, τ ∗ ) and v ∗ (y) gives the final condition of ϕ|τ =τ ∗ . With this ϕ|τ =τ ∗ and the a(y) given in step 1, we can solve equation (3.2) to obtain ϕ(y, τ ). (4) With v(y, τ ) and ϕ(y, τ ), we can solve the variational inequality (3.3) to obtain a new function of a(y), denoted by A1 (y). The algorithm is also given in the appendix. (5) Justify the convergence of the iteration. Suppose ε is a prespecified error bound. If |A1 (y) − A0 (y)| < ε, the iteration is finished. Otherwise we use A1 (y) as a new value for a(y) and return to step 1 and repeat the procedure until |An+1 (y) − An (y)| < ε.
3.2. Numerical experiments We have performed three numerical experiments in which the ‘true’ volatility functions (denoted by σp (s)) are prespecified as follows. (1) ‘flat’ volatility: σp (s) = 0.25; (2) ‘smile’ volatility: σp (s) = (ln s − ln 10)2 /40 + 0.2; (3) ‘skew’ volatility: σp (s) = −(ln s − ln 10)3 /80 + 0.2. In all three experiments, we use a flat line, σ0 (s) = 0.20, as an initial value. A few basic parameters are: q = 0, r = 0.12, a0 = 0.005, a1 = 0.045, ε 10−4 , s ∗ = 10 and τ ∗ = 1. In our numerical experiments, we first solve equation (3.1) with a(y) = 21 σ 2 (s ∗ ey ) to get V (y, τ ). We treat u∗ (K) =
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A new well-posed algorithm to recover implied local volatility
Table 1. Numerical results for the ‘flat’ volatility function. s
4
6
8
10
12
14
16
18
20
22
24
100 × σ (s) 100 × σp (s)
24.99 25.00
24.99 25.00
24.99 25.00
24.99 25.00
24.99 25.00
24.99 25.00
24.99 25.00
24.99 25.00
24.99 25.00
25.00 25.00
25.00 25.00
Table 2. Numerical results for the ‘smile’ volatility function. s
4
6
8
10
12
14
16
18
20
22
24
100 × σ (s) 100 × σp (s)
22.30 22.10
20.99 20.61
20.52 20.13
20.38 20.00
20.43 20.08
20.60 20.28
20.83 20.56
21.10 20.86
21.39 21.20
21.72 21.55
22.06 21.91
s
4
6
8
10
12
14
16
18
20
22
24
100 × σ (s) 100 × σp (s)
21.28 20.96
20.34 20.17
20.09 20.14
19.99 20.00
19.92 19.99
19.86 19.95
19.78 19.87
19.69 19.74
19.57 19.59
19.44 19.39
19.27 19.16
0.275
0.275
0.250
0.250
0.225
0.225
Volatility
Volatility
Table 3. Numerical results for the ‘skew’ volatility function.
0.200 0.175
0.200 0.175
0.150
0.150
0.125
0.125 5
10
15 Stock price
20
5
25
Figure 1. The recovery of ‘flat’ volatility. The dashed line is the
initial condition σ0 = 0.2, the solid line is the true volatility σp = 0.25, the dots are the volatility that is recovered numerically by using our algorithm from the option prices generated with the true volatility.
10
15 Stock price
20
25
Figure 2. The recovery of ‘smile’ volatility. The dashed line is the
initial condition σ0 = 0.2, the solid curve is the true volatility σp = (ln s − ln 10)2 /40 + 0.2, the dots are the volatility that is recovered numerically by using our algorithm from the option prices generated with the true volatility.
∗
0.275 0.250 Volatility
s ∗ e−qτ V (ln sK∗ , τ ∗ ) as the given market value. Then, following the iteration procedure presented in the previous section, we solve the system (3.1)–(3.3) with v ∗ (y) = V (y, τ ∗ ) to obtain A(y). By comparing the computed volatility function σ (s) = √ 2 A(ln s/s ∗ ) with the true function σp (s), we are able to justify the accuracy of our method. Table 1 shows that, in the first experiment, the relative difference between σ (s) and σp (s) is uniformly smaller than 1×10−4 . In the second experiment, the numerical computation converges quickly and the accuracy is on the order of 4 × 10−3 (see table 2). The third experiment works as well as the last experiment (see table 3). Figures 1–3 show the results of the three experiments.
0.225 0.200 0.175 0.150 0.125 5
10
15 Stock price
20
25
Figure 3. The recovery of ‘skew’ volatility. The dashed line is the
4. Concluding remarks Calibrating the option pricing model is still an outstanding issue in finance, especially in the continuous-time setting. The difficulty is due to the limitation of mathematical tools in dealing with the related inverse problem.
initial condition σ0 = 0.2, the solid curve is the true volatility σp = −(ln s − ln 10)3 /80 + 0.2, the dots are the volatility that is recovered numerically by using our algorithm from the option prices generated with the true volatility.
In this paper, we solve the inverse problem of recovering the volatility function in an optimal control framework. With
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the existence and uniqueness of the optimal control established in [14], we present a necessary condition in a set of variational inequalities. We then propose an iteration procedure to solve the volatility function. Numerical experiments show that the numerical algorithm is stable and that it converges quickly. With the theoretical foundation well established, our method is now ready for both practitioners and researchers to test with market data and to price options.
A.2. Numerical scheme for variational inequality (3.3) Given A0 (y), τ ∗ , vn (y) and ϕn (y), we are now solving equation (3.3) for a(y), in which f (y; v, ϕ) is defined by (3.4). Rearranging equation (3.1) gives vyy − vy = [vτ + (r − q)vy ]/a. Therefore, we have
Acknowledgments We wish to thank two anonymous referees for helpful comments and suggestions. JEZ is supported by the Research Grants Council of Hong Kong under grant CERG HKUST1068/01H.
f (y; v, ϕ) =
1 N
For equation (3.1), for any a(y) = A0 (y), there always exist large numbers, L 1 and M 1, such that v(−L) = 1 and v(M) = 0. By assuming vτ = (vn − vn−1 )/τ , vn = v(y, nτ ), x = y + L, equation (3.1) is changed to vn (x) − vn−1 (x) − A0 (y)[vn (x) − vn (x)] τ + (r − q)vn (x) = 0, x−L +
) ,
(A.1) vn (0) = 1,
vn (L + M) = 0. Now, we use Meyer’s methods of invariant embedding [20] to solve this equation numerically. Let zn (x) = vn (x) and vn (x) = Un (x)zn (x) + un (x).
(A.2)
We have zn (x) = vn (x) = Un (x)zn (x) + Un (x)zn (x) + un (x). Substituting this into equation (A.1) and comparing the zeroorder and first-order coefficients of zn (x) gives the following three ordinary differential equations: 1 r − q + A0 (y) Un2 (x) − Un (x), =1− A0 (y)τ A0 (y) Un (0) = 0; (A.3) −1 1 un (x) = Un (x)un (x) + v1 (x)Un (x), A0 (y)τ A0 (y)τ un (0) = 1;
(A.4)
1 r − q + A0 (y) Un (x) + zn (x) A0 (y)τ A0 (y) 1 + [un (x) − vn−1 (x)], (A.5) A0 (y)τ un (L + M) zn (L + M) = − . Un (L + M) Solving the above four sets of equations (A.2)–(A.5) gives us vn (x) or v(y, τ ). The numerical procedure for equation (3.2) is similar to that for equation (3.1) and is thus omitted. zn (x) =
456
ϕ[vτ + (r − q)vy ]/a dτ.
0
a0 a a1 , (at − ayy + f (y; v, ϕ))(a − a0 ) 0,
A.1. Numerical schemes for PDEs (3.1) and (3.2)
Un (x)
τ∗
For variational inequality (3.3), we introduce a ‘false’ time variable t and obtain a parabolic bilateral variational inequality
Appendix
v0 (x) = (1 − e
(A.6)
(at − ayy + f (y; v, ϕ))(a − a1 ) 0 with initial value a|t=0 = A0 (y). The asymptotic steady-state solution of parabolic variational inequality (A.6) will be the solution of elliptic variational inequality (3.3). Now, we implement the explicit finite difference method on (A.6): a0 ain+1 a1 , n n ([(ain+1 − ain )/t] − [(ai+1 − 2ain + ai−1 )/(y)2 ] + fi )
× (ain − a0 ) 0, ([(ain+1 ×
n − ain )/t] − [(ai+1 (ain − a1 ) 0.
(A.7) −
2ain
+
n ai−1 )/(y)2 ]
+ fi )
Let n n +(1−2α)ain + αai−1 −fi t, Bin = αai+1
α = t/(y)2 .
Then, (A.7) becomes a0 ain+1 a1 ,
(ain+1 − Bin )(ain − a0 ) 0,
(ain+1 − Bin )(ain − a1 ) 0. Therefore, we have n Bi n+1 ai = a0 a1
if a0 < ain < a1 ; if ain = a0 and Bin < a0 ; if ain = a1 and Bin > a1 ,
where ai0 = A0 (yi ), α = t/(y)2 1 and n is taken to be larger than 103 . By the computation above, we obtain the first iteration, A1 (y), of a(y). If the difference between A1 (y) and A0 (y) is larger than the error bound, ε, we continue the iteration according to the same procedure.
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