DSpace at VNU: Calibration of the purely T-dependent Black-Scholes implied volatility

This article was downloaded by: [Aston University] On: 22 August 2014, At: 00:21 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Applicable Analysis: An International Journal Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/gapa20 Calibration of the purely T-dependent Black–Scholes implied volatility a a b Dang Duc Trong , Dinh Ngoc Thanh , Nguyen Nhu Lan & Pham Hoang Uyen a a Department of Mathematics and Computer Science , University of Science, Vietnam National University , 227 Nguyen Van Cu Q.5, HoChiMinh City , Vietnam b Department of Mathematics , Tay Do University , Can Tho , Vietnam Published online: 14 Jun 2013 To cite this article: Dang Duc Trong , Dinh Ngoc Thanh , Nguyen Nhu Lan & Pham Hoang Uyen (2014) Calibration of the purely T-dependent Black–Scholes implied volatility, Applicable Analysis: An International Journal, 93:4, 859-874, DOI: 10.1080/00036811.2013.800974 To link to this article: http://dx.doi.org/10.1080/00036811.2013.800974 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content This article may be used for research, teaching, and private study purposes Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden Terms & Downloaded by [Aston University] at 00:21 22 August 2014 Conditions of access and use can be found at http://www.tandfonline.com/page/termsand-conditions Applicable Analysis, 2014 Vol 93, No 4, 859–874, http://dx.doi.org/10.1080/00036811.2013.800974 Calibration of the purely T-dependent Black–Scholes implied volatility Dang Duc Tronga∗ , Dinh Ngoc Thanha , Nguyen Nhu Lanb and Pham Hoang Uyena a Department of Mathematics and Computer Science, University of Science, Vietnam National University, 227 Nguyen Van Cu Q.5, HoChiMinh City, Vietnam; b Department of Mathematics, Tay Do University, Can Tho, Vietnam Communicated by R Gilbert Downloaded by [Aston University] at 00:21 22 August 2014 (Received 25 March 2012; final version received 26 April 2013) We consider the problem of recovering an implied volatility σ = σ (t) from the data of option price We improve results of Hein and Hofmann, and Kramer and Richter In fact, we shall construct an elementary regularization Then, we give an explicit formula of the regularization parameter An explicit error estimate is also given Keywords: calibration; volatility; ill-posed; regularization AMS Subject Classifications: 35R30; 65J20; 91B24 Introduction In option pricing, the well-known Black–Scholes (BS for short) formula gives an explicit form of the fair price (arbitrage-free) for European call options The BS formula U B S is the function of variables X, K , r, t, σ where t ∈ [0, T ] is the time variable, T > is the expiration date of the option, K is the strike price of the option, r is the continuously compounded risk-free interest, X = X (t) is the price of the stock at time t, X = X (0) is the current asset price and σ = σ (t, X ) is the volatility of the stock price As known, X, K , r are constants and observable However, the volatility is a market parameter and not directly observable Hence, it is called the implied (or implicit) volatility The volatility plays a crucial role in option pricing We can use the volatility to measure the uncertainty of the future return of assets [1, p 377] If the volatility σ (t, X ) is given, the problem of calculating options prices is called the direct (or forward) one of option pricing In the present paper, we consider the inverse problem of finding the (implied) volatility from the data of option price The problem is called by a lot of names as the inverse problem of option pricing, the model calibrating problem (see, e.g [2]) The literature of the inverse problem is very rich In almost papers, the authors considered the problem of recovering the implied volatility which are depended on (t, X ), i.e σ = σ (t, X ) (see [2–14]) In [4,8], the authors considered the implied volatility having the separable form σ = σ0 (X )ρ(t) The purely t-dependent volatility σ = σ (t) is a more restricted model However, it was considered in [9–12] since it can be seen as a benchmark model for studying the nature of ill-posedness of the inverse problem In the present paper, we also investigate the latter model in case of continuous volatilities ∗ Corresponding author Email: ddtrong@hcmus.edu.vn © 2013 Taylor & Francis 860 D.D Trong et al In fact, let U B S (X, K , r, t, s) be the Black–Scholes function, one has X (d1 ) − K e−r t (d2 ), (X − K e−r t )+ , U B S (X, K , r, t, s) = s > 0, s = 0, where √ ν + r t + 2s , d2 = d1 − s, √ s X ν = ln , K z (z) = √ e−x /2 d x 2π −∞ Downloaded by [Aston University] at 00:21 22 August 2014 d1 = Let u(t) be the fair price of the option at time t We consider the problem of finding an unknown continuous volatility σ (t) from u(t) satisfying UB S t X, K , r, t, σ (τ )dτ = u(t), ≤ t ≤ T (1) We recall that, as in [9], the problem can be divided into two problems The first one, called the outer problem, is of finding a function S(u) from the option price u such that U B S (X, K , r, t, S(u)(t)) = u(t) From the solution S(u) of the outer problem, we shall find the implied volatility σ (t) such that t σ (τ )dτ = S(u)(t) (2) The problem is called the inner problem As shown in the paper by Hein and Hofmann [9], the problem is ill-posed Hence, a regularization is in order For discussing deeply, we state precisely our problem Calibration Problem (CP) Let u ex , σex ∈ C[0, T ] satisfy (1), let C[0, T ] be inexact data satisfying > and let u ∈ u − u ex ≤ From u , we have to construct a function σ ∈ C[0, T ] such that lim →0+ σ − σex = where f = sup0≤t≤T | f (t)| We have three questions (see, e.g [15] or [16]) (1) How to construct the regularization scheme ? (2) How to choose the regularization parameter? (3) What is the size of the error? We first consider the question (1) The inner problem is ill-posed and there are many ways to regularize it Now, we discuss the outer problem In [9], the authors did not give a regularization scheme for the C-space setting case In [12], the authors proved that, in Applicable Analysis 861 the case of continuous solutions, the outer problem is well-posed But, from (2), we have S(u)(t) → as t → Hence, as discussed in [9,12], the outer problem is still instable near t = by the ill-conditioning effect |S(u)(t) − S(v)(t)| = φ(t)|u(t) − v(t)| with lim φ(t) = ∞ Downloaded by [Aston University] at 00:21 22 August 2014 t→0 In [12], using the monotonization algorithm, the authors gave a method to eliminate the latter effect In the present paper, we shall eliminate the ill-conditioning effect by ‘truncating’ small values of the function S(u) Using this idea, we shall give a regularization for the outer problem in the C-space setting case Now, we consider the second and the third questions In [9,10] and in other papers in the references (see [2–14,18]), the dominant method of regularization is the Tikhonov one (or Tikhonov-type one as in [10]) The method was often used for the case of an L -space setting (or Hilbert space setting) Hence, it is possible that we cannot use the method to regularize in C-spaces On the other hand, in these results, an explicitly form of the regularization parameter was not given In the present paper, explicit formulae of the regularization parameter and explicit error estimates are given Moreover, we shall prove that the order of our error estimate for the outer problem is optimal in an appropriate sense The remain of the paper divided into four sections In Section 2, we shall state the precise form of the problem and give existence results for the outer and for the (CP) problems We also discuss the ill-conditioning of the outer problem in case of at-the-money options and in other cases In Section 3, we investigate the construction of regularization scheme for the outer problem, give the stability and the error estimate for the outer problem We also give a lower bound of the estimate which shows the optimality of the order of our estimate In Section 4, we present the inner problem of finding the implied volatility σ (t) from S(t) Finally, in Section 5, we give a total result combining the outer and the inner problems The existence and the ill-conditioning of the solution of the outer problem 2.1 Statement of the problem The function U B S is defined in the domain X > 0, K > 0, r ≥ 0, t ≥ 0, s ≥ In the formula of U B S , the quantities X, K , r can be seen as constants Hence, for short, we shall denote U (t, s) = U B S (X, K , r, t, s) From u ∈ C[0, T ], we consider the problem of finding the function S(u) such that U (t, S(u)(t)) = u(t) (3) 2.2 Existence results Before considering the outer problem, we state some properties of the Black–Scholes function 862 D.D Trong et al Lem m a For t ≥ 0, s > we have ∂U X ν + rt s (ν + r t)2 = √ − − exp − ∂s 2s 2π s > From the above equality, one has U (t, u) ≤ U (t, v) if and only if ≤ u ≤ v Downloaded by [Aston University] at 00:21 22 August 2014 The proofs of these properties are elementary and it can be given directly by computation Hence, we omit it (see also [9,12]) We shall consider the existence of the solution S(u) of the outer problem (3) In fact, we have Proposition The outer problem has a unique solution S(u) ∈ C[0, T ], S(u)(t) ≥ for ≤ t ≤ T if and only if u ∈ F, where F = {u ∈ C[0, T ] : (X − K e−r t )+ ≤ u(t) < X } (4) Moreover, for fixed t ∈ [0, T ], S(u)(t) = if and only if u(t) = (X − K e−r t )+ Proof We have lim U (t, s) = X, s→+∞ lim U (t, s) = (X − K e−r t )+ s→0+ On the other hand, from Lemma 1, the function U (t, s) is strictly monotonically increasing in the variable s So we have for every t, s ≥ (X − K e−r t )+ ≤ U (t, s) < X, i.e the condition u ∈ F is necessary Inversely, let u ∈ F and let t be fixed in [0, T ], if α = u(t) satisfies (X − K e−r t )+ ≤ α < X then there exists uniquely a ξ ≥ such that U (t, ξ ) = α We shall put ξ = S(u)(t) This completes the proof of our proposition From Proposition 1, we therefore have the operator S : F → C[0, T ] in which S(u) satisfy the outer problem for each u ∈ F and we can get Proposition Problem (1) has a unique nonnegative solution σ ∈ C[0, T ] if and only if the fair price u satisfies u ∈ F ∩ C [0, T ], u(0) = (X − K )+ and u (t) − K r e−r t ln X K + r t − S(u)(t) √ S(u)(t) ≥ where S(u) is the unique solution of the outer problem (3) We not prove Proposition 2, the proof can be seen in [9,12] (5) Applicable Analysis 863 2.3 The ill-conditioning of the solution We put some notations Letting s¯ > 0, φ ∈ C[0, T ] satisfy ≤ φ(t) ≤ s¯ for t ∈ (0, T ], we denote F(φ, s¯ ) = {u ∈ F : U (t, φ(t)) ≤ u(t) ≤ U (t, s¯ ), t ∈ (0, T ]}, F(φ, ∞) = {u ∈ F : U (t, φ(t)) ≤ u(t) < X, t ∈ (0, T ]} We note that F(0, ∞) = F For the operator S defined in Subsection 2.2, we shall investigate its modulus of continuity in some cases We first put some notations For B0 ⊂ B ⊂ C[0, T ], and N : B → C[0, T ] being an operator, we put Downloaded by [Aston University] at 00:21 22 August 2014 ω(N , B0 , ) = sup{ N (u ) − N (u ) : u , u ∈ B0 , u − u ≤ } Now, we discuss about the ill-conditioning in several cases We consider the case ν = ln KX = (X = K ) which is called the one of at-the-money options (AMO for short) The case X = K can be called the non-AMO one As mentioned in [9], the solution of the outer problem is ill-conditioned in the non-AMO case We investigate the AMO case If r = then √ s/2 X U (t, s) = √ e−x /2 d x √ 2π − s/2 From the formula, we can get directly that ω(S, F(0, s¯ ), ) ≤ C and S is Lipschitz on F(0, s¯ ) (see also Theorem 4.6 in [11]) So, the problem is well-conditioning in this case Hence, in the rest of the paper, we always assume r > in the AMO case To get some more ideas for discussing the ill-conditioning of the outer problem, we consider Lem m a Let s¯ > 0, ≤ μ < s¯ For every s1 , s2 ∈ [μ, s¯ ], one has |s1 − s2 | ≤ M(μ, s¯ , t)|U (t, s1 ) − U (t, s2 )|, where ν = ln X K and √ (ν + r t)2 (ν + r t) s 2π s exp + + , M(μ, s¯ , t) = sup X 2s μ (6) then we can find a c p > such that ψ p (t) := c p t p ≤ S(u ex )(t) for t ∈ (0, T ] (7) In the following proposition, we give a result for Lipschitz stability of the operator S in the latter case Proposition Let X = K , r > 0, > If ≤ p ≤ 2, c p > then ω(S, F(ψ p , s¯ ), ) ≤ C and S is Lipschitz on F(ψ p , s¯ ) Proof Since ν = 0, r > we get M(μ, s¯ , t) = sup μ of σex shall assume that Volatility assumption (VA) The exact solution σex ∈ C[0, T ] satisfies ≤ σex (t) ≤ K0, for ≤ t ≤ T From (VA), we can get a bound for S(u ex )(t) In fact, one has by (2) t S(u ex )(t) = σex (τ )dτ Hence, (VA) implies ≤ S(u ex )(t) ≤ K T, for ≤ t ≤ T (8) From now on, we put M0 (μ, t) = M(μ, K T, t) Lemma suggests several ideas for constructing a regularization scheme If the assumptions of Proposition not hold then the function M0 (μ, t) tends to +∞ as μ → We therefore shall “truncate” all of S(u ex )(t) ≈ In fact, let δ > 0, to get a strictly positive approximation of S(u ex )(t) we can simply use the function max{S(u ex )(t), δ} Combining the latter idea with (8), we construct an approximation of the solution of (3) as follows S δ (u)(t) = min{K T, max{S(u)(t), δ}} where < δ < min{K T, 1} In the next two subsections, we shall verify that it is a regularization scheme 866 D.D Trong et al 3.3 Stability estimate Proposition Let < δ < min{1, K T } and let φ ∈ C[0, T ] satisfy ≤ φ(t) < K T for t ∈ (0, T ] For two functions u, v ∈ F(φ, ∞) one has |S δ (u)(t) − S δ (v)(t)| ≤ M0 (max{δ, φ(t)}, t)|u(t) − v(t)| Proof that For u, v ∈ F(φ, ∞), we give the estimate of |S δ (u)(t) − S δ (v)(t)| We first note u(t) = U (t, S(u)(t)), v(t) = U (t, S(v)(t)) ≥ U (t, φ(t)) From Lemma 1, we get S(u)(t), S(v)(t) ≥ φ(t) Downloaded by [Aston University] at 00:21 22 August 2014 Now, we consider several cases Case δ ≤ S(u), S(v) ≤ K T We have S δ (u)(t) = S(u)(t), S δ (v)(t) = S(v)(t) On the other hand, by the definition of S(u), S(v), one has U (t, S(u)(t)) − U (t, S(v)(t)) = u(t) − v(t) From (9), we have max{δ, φ(t)} ≤ u(t), v(t) ≤ K T Hence, using Lemma 2, one has |S δ (u)(t) − S δ (v)(t)| ≤ M0 (max{δ, φ(t)}, t)|u(t) − v(t)| Case ≤ S(u)(t), S(v)(t) ≤ δ or S(u)(t), S(v)(t) ≥ K T We have S δ (u)(t) = S δ (v)(t) Hence, the statement holds in this case Case ≤ S(u)(t) ≤ δ ≤ S(v)(t) ≤ K T We have S δ (u)(t) = δ ≤ S δ (v)(t) = S(v)(t) So u(t) = U (t, S(u)(t)) ≤ U (t, δ) ≤ U (t, S(v)(t)) = v(t) Hence, we have |U (t, δ) − U (t, S(v)(t))| ≤ |u(t) − v(t)| On the other hand, from Lemma we get |δ − S(v)(t)| ≤ M0 (max{δ, φ(t)}, t)|U (t, δ) − U (t, S(v)(t))| It follows that |S δ (u)(t) − S δ (v)(t)| = |δ − S(v)(t)| ≤ M0 (max{δ, φ(t)}, t)|U (t, δ) − U (t, S(v)(t))| ≤ M0 (max{δ, φ(t)}, t)|u(t) − v(t)| (9) Applicable Analysis 867 Case δ ≤ S(u)(t) ≤ K T ≤ S(v)(t) We can prove similarly as Case to get the desired inequality Case ≤ S(u)(t) ≤ δ ≤ K T ≤ S(v)(t) We have S δ (u)(t) = δ, S δ (v)(t) = K T, and u(t) = U (t, S(u)(t)) ≤ U (t, δ) ≤ U (t, K T ) ≤ U (t, S(v)(t)) = v(t) Hence, |U (t, δ) − U (t, K T )| ≤ |v(t) − u(t)| On the other hand, one has Downloaded by [Aston University] at 00:21 22 August 2014 |S δ (u)(t) − S δ (v)(t)| = |δ − K T | ≤ M0 (max{δ, φ(t)}, t)|U (t, δ) − U (t, K T )| This completes the proof of Proposition 3.4 Error estimates Now, we consider the error estimate between the exact function S(u ex )(t) and the regularization function S δ (u )(t) Proposition Let δ, φ be as in Proposition and let u ex ∈ F(φ, K T ) be the exact fair price of the option Let u ∈ F(φ, ∞) be noise data of u ex such that u − u ex ≤ Then S(u ex ) − S δ (u ) ≤ δ + sup M0 (max{δ, φ(t)}, t) 0≤t≤T Proof One has |S(u ex )(t) − S δ (u )(t)| ≤ |S(u ex )(t) − S δ (u ex )(t)| + |S δ (u ex )(t) − S δ (u )(t)| We first have in view of Proposition |S δ (u ex )(t) − S δ (u )(t)| ≤ M0 (max{δ, φ(t)}, t)u (t) − u ex (t)| ≤ M0 (max{δ, φ(t)}, t) Now, we have to estimate the term |S(u ex )(t) − S δ (u ex )(t)| The volatility assumption (VA) give ≤ S(u ex )(t) ≤ K T Hence, one has two cases Case δ ≤ S(u ex )(t) ≤ K T In this case S(u ex )(t) = S δ (u ex )(t) So |S(u ex )(t) − S δ (u ex )(t)| = Case ≤ S(u ex )(t) ≤ δ In this case S δ (u ex )(t) = δ So, we have |S(u ex )(t) − S δ (u ex )(t)| ≤ δ − S(u ex )(t) ≤ δ 868 D.D Trong et al In any case, we have |S(u ex )(t) − S δ (u ex )(t)| ≤ δ From the above estimate, we get |S(u)(t) − S u δ (t)| ≤ δ + M0 (max{δ, φ(t)}, t) This completes the proof of Proposition Using the proposition, we can choose an explicit form of regularization parameter In fact, we shall give an estimate for M0 (max{δ, φ(t)}, t) We have Lem m a Let p > and let c p , ψ p be as in (7) Put Downloaded by [Aston University] at 00:21 22 August 2014 −1/ p ν0 = max {|ν|, |ν + r T |} , ν p = r c p √ ν0 K0T 2π K T exp + K1 = X Then ν02 2δ ≤ M0 (δ, t) ≤ K exp If X = K then M0 (max{δ, ψ p (t)}, t) ≤ K exp Proof We have M0 (δ, t) = sup δ≤s≤K T , ν 2p 2δ ( p−2)/ p √ (ν + r t)2 (ν + r t) s 2π s exp + + X 2s But, for ≤ t ≤ T , one has |ν + r t| ≤ max{|ν|, |ν + r T |} = ν0 Hence, √ ν2 ν0 K0T 2π K T exp + + ≤ M0 (δ, t) ≤ X 2δ Now, when X = K , we have M0 (max{δ, ψ p (t)}, t) = sup max{δ,ψ p (t)} max{δ, ψ p (t)}, Applicable Analysis 869 −2/ p r 2c p r 2t r2 ≤ ≤ s c p t p−2 δ ( p−2)/ p −1/ p If ψ p (t) < δ then max{δ, ψ p (t)} = δ and t ≤ c p δ 1/ p In this case, for s > max{δ, ψ p (t)} we have −2/ p −2/ p r c p δ 2/ p r 2c p r 2t ≤ = ( p−2)/ p s δ δ Combining two cases completes the proof of Lemma From the above lemmas and propositions, we get the main theorem of our paper Downloaded by [Aston University] at 00:21 22 August 2014 Theorem Let < < 12 In the non-AMO and the SIAMO cases, let u ∈ F be noise K1 data of u ex ∈ F(0, K T ) such that u − u ex ≤ If we choose ν02 δ = δ0 ( ) = ln then S(u ex ) − S δ0 ( ) (u ) ≤ √ K 12 + δ0 ( ) := β0 ( ) In the OIAMO case, let u ex ∈ F(ψ p , K T ), u ∈ F(ψ p , ∞) ( p > 2) If we choose ⎞ ⎛ ⎜ δp( ) = ⎜ ⎝ ν 2p ln then S(u ex ) − S δ p ( ) (u ) ≤ K 12 √ p p−2 ⎟ ⎟ ⎠ + δ p ( ) := β p ( ) Proof We give the proof for the non-AMO and the SIAMO cases From the definition of δ( ), we have ν02 K exp =√ 2δ0 ( ) From Lemma and Proposition 5, we get in view of the latter equality S(u ex )(t) − S u δ0 ( ) (t) ≤ δ0 ( ) + K exp = δ0 ( ) + √ ν02 2δ( ) The proof for the OIAMO case is similar This completes the proof of Theorem 870 D.D Trong et al 3.5 The optimality of the order of error estimate in case X = K (non-AMO) In Subsection 3.4, for the non-AMO and SIAMO cases we get the error estimate β0 ( ) = p O ln −1 The error estimate in case of OIAMO is β p ( ) = O ln 2− p which is better than the one of the previous cases In the SIAMO case, it is possible that we can establish some error estimates better than β0 ( ) But, we not go into details of this case In the non-AMO case (X = K ), we shall prove that the order of the error estimate is optimal in an appropriate sense To this end, we first give a result giving lower bounds for the modulus of continuity of S Proposition Let s¯ , , r > 0, X = K and let S : F → C[0, T ] be the operator as in Proposition with U (t, S(u)(t) = u(t) for u ∈ F Put Downloaded by [Aston University] at 00:21 22 August 2014 F+ (0, s¯ ) = {u ∈ F : u(0) = (X − K )+ , (X − K e−r t )+ < u(t) ≤ U (t, s¯ ), t ∈ (0, T ]} Then there are c0 , > such that ω(S, F+ (0, s¯ ), ) ≥ c0 , ln(1/ ) 0< < Proof As in [11], we put the Nemyskii operator N : C[0, 1] → C[0, 1] by N (h)(t) = U (t, h(t)) for every h ∈ C[0, T ], h(t) ≥ for t ∈ [0, T ] From the definition of S(u), one has N −1 |F = S Put D+ = {g ∈ C[0, T ] : g(0) = 0, g(t) > for t > 0}, D+,¯s = {g ∈ D+ : D+,¯s (N −1 g ≤ s¯ }, ) = N (D+,¯s ) We claim that D+,¯s (N −1 ) = F+ (0, s¯ ) In fact, we first verify that D+,¯s (N −1 ) ⊂ F+ (0, s¯ ) Let u ∈ D+,¯s (N −1 ) We have N −1 (u ) = S(u ) ∈ D+,¯s It follows that S(u )(0) = and < S(u )(t) ≤ s¯ t ∈ (0, T ] Since S(u )(0) = 0, we have u (0) = (X − K )+ On the other hand, from the latter inequalities, we have (X − K e−r t )+ < U (t, S(u )(t)) = u (t) ≤ U (t, s¯ ) Combining the above facts, we get u ∈ F+ (0, s¯ ) It follows that D+,¯s (N −1 ) ⊂ F+ (0, s¯ ) as desired Inversely, let u ∈ F+ (0, s¯ ) Using Proposition 1, there exists S(u ) ∈ D+ such that U (t, S(u )(t)) = u (t) But u (t) ≤ U (t, s¯ ) Hence, Lemma gives S(u )(t) ≤ s¯ It follows that F+ (0, s¯ ) ⊂ D+,¯s (N −1 ) Hence, D+,¯s (N −1 ) = F+ (0, s¯ ) So we have ω(S, F+ (0, s¯ ), ) = ω(N −1 , D+,¯s (N −1 ), ) From Proposition 4.7 in [11], there are c0 , ω N −1 , D+,¯s N −1 , > such that c0 , 0< ≥ ln(1/ ) < Combining two latter inequalities, we get the desired inequality This completes the proof of the proposition Applicable Analysis 871 Now we prove the optimality of the order of error estimate in the non-AMO case (X = K ) Let δ0 > We consider the set G of regularization schemes R : F × (0, δ0 ) → C[0, T ] such that there are δ R ( ) > and β R ( ) > satisfying lim δ R ( ) = lim β R ( ) = →0 →0 and R(v, δ R ( )) − S(u) ≤ β R ( ) (10) for all u, v ∈ F+ (0, s¯ ) with v − u ≤ We have Proposition There are c¯0 , > such that inf β R ( ) ≥ Downloaded by [Aston University] at 00:21 22 August 2014 R∈G Proof c¯0 , ln(1/ ) 0< < We assume that u, v ∈ F+ (0, s¯ ) and v − u ≤ From (10), we have R(v, δ R ( )) − S(u) ≤ β R ( ) and R(v, δ R ( )) − S(v) ≤ β R ( ) Using the triangle inequality, we get S(u) − S(v) ≤ 2β R ( ) The inequality gives ω(S, F+ (0, s¯ ), ) ≤ 2β R ( ) Combining with Proposition 6, we have βR ( ) ≥ c0 , ln(1/ ) 0< < This completes the proof of the proposition The inner problem We consider the problem of recovery a derivative from its integral Precisely, let μ > and let g − S(u ex ) ≤ μ, we consider the problem of recovering the function σex (t) from the approximate value g(t) of S(u ex )(t) The problem is ill-posed and there are many well-known regularization schemes We give here only a simple way to recovery the function σex (t) Here, we recall t that S(u ex )(t) = aex (τ )dτ where aex = σex Proposition Let α ∈ (0, 1] and let σex satisfy (VA) For h ∈ (0, T ), put aμh (t) = g(t+h)−g(t) , h g(T )−g(T −h) , h ≤ t ≤ T − h, T −h as in Theorem For the OIAMO case, letting p > and u ex ∈ F(ψ p , K T ), u ∈ F(ψ p , ∞), we choose the regularization parameter δ p ( ) > as in the same theorem After that, we put for j = ( non-AMO case and SIAMO case) or j = p (OIAMO case) g j (t) := S δ j ( ) (u )(t) = min{K T, max{δ j ( ), S(u )(t)}} Now, for β j ( ) defined in Theorem 1, put h j ( ) = (β j ( ))α , ⎧ ⎨ g j (t+h j ( ))−g j (t) , ≤ t ≤ T − h, hj( ) a j (t) = g j (T )−g (T −h ( )) j j ⎩ , T −h