JOHANNES KEPLERUN IVERS IT �AT L INZNe t zw e r k f �u r F o r s h u n g , L e h r e u n d P r a x i sIdenti� ation of Volatility Smiles in the Bla kS holes Equation via Tikhonov RegularizationDiplomarbeit zur Erlangung des akademis hen GradesDiplomingenieurin der Studienri htung Te hnis he MathematikAngefertigt am Institut f�ur IndustriemathematikBetreuung:Univ.-Prof. Dipl.-Ing. Dr. H. W. EnglEingerei ht von:Herbert EggerLinz, November 2001

Johannes Kepler Universit�atA-4040 Linz � Altenbergerstrae 69 � Internet: http://www.uni-linz.a .at � DVR 0093696

A knowledgementsI most gratefully thank Prof. Engl for giving me the oportunity to write mythesis about this interesting and quite hallenging problem and for his helpand patien e in answering my questions.I also would like to thank Dr. A. Binder for his introdu tion in mathemati al�nan e and some very inspiring dis ussions on numeri al methods for solvingthe Bla k S holes problem.Sin e most of the work on this thesis was done during my stay in Oxford inspring 2001, I would like to express my gratitude to my hosts at OCIAM,Dr. Hillary and Dr. John O kendon, again to Prof. Engl for giving me theopportunity for this stay and to the Austrian Sien e Foundation FWF, whosupported the stay in Oxford.Spe ial thanks to Dr. Dewynne and Dr. Howison for a dis ussion of thepossible problems in the identi� ation and the latter for a very inspiring dis- ussion of the inverse Bla k-S holes problem in terms of perturbation theoryduring the ISAM Summer hool on Industrial Mathemati s in Siena 2001.Thanks also to my olleagues Benjamin, Mark and Philipp for their e�ortsin �nding answers to my questions.Finally I would like to thank my girl friend, Elisabeth, for her patien e withmy going abroad for almost six months.

Contents1 Introdu tion 11.1 Finan ial Preliminaries . . . . . . . . . . . . . . . . . . . . . . 21.2 The Bla k-S holes Equation . . . . . . . . . . . . . . . . . . . 41.3 Inverse and Ill-posed Problems . . . . . . . . . . . . . . . . . . 121.4 Tikhonov Regularization for Nonlinear Problems . . . . . . . . 162 Paraboli Equations 202.1 Classi al Theory . . . . . . . . . . . . . . . . . . . . . . . . . 212.1.1 Cau hy Problem . . . . . . . . . . . . . . . . . . . . . 212.1.2 Fundamental Solutions . . . . . . . . . . . . . . . . . . 222.1.3 Existen e and Uniqueness for the Cau hy Problem . . 252.1.4 The Adjoint Equation and a Uniqueness Result . . . . 252.1.5 Maximum Prin iple and Appli ation to the Cau hyProblem . . . . . . . . . . . . . . . . . . . . . . . . . . 262.2 Variational Formulation . . . . . . . . . . . . . . . . . . . . . 282.2.1 Fun tions and Distributions Valued in Bana h Spa es . 282.2.2 Integration . . . . . . . . . . . . . . . . . . . . . . . . 282.3 The Abstra t Cau hy Problem . . . . . . . . . . . . . . . . . . 302.3.1 Existen e, Uniqueness and Estimates . . . . . . . . . . 323 Adjoimt Equation; Parameter-to-Solution Map 353.1 An Adjoint Problem . . . . . . . . . . . . . . . . . . . . . . . 363.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . 423.3 Existen e, Uniqueness and Estimates . . . . . . . . . . . . . . 433.4 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44i

3.5 The Parameter-To-Solution Map . . . . . . . . . . . . . . . . . 463.5.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 463.5.2 Basi Properties of the Parameter-To-Solution Map . . 474 Inverse Problem; Tikhonov Regularization 514.1 Regularized Output Least Squares Problem . . . . . . . . . . 544.2 Existen e of Minimizer . . . . . . . . . . . . . . . . . . . . . . 554.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.4 Convergen e . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.5 Convergen e Rates . . . . . . . . . . . . . . . . . . . . . . . . 614.6 Remarks on the Convergen e Rate Results . . . . . . . . . . . 674.7 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . 685 Numeri al Realization 705.1 Restri tion to Bounded Domains . . . . . . . . . . . . . . . . 725.2 Crank-Ni holson FEM S heme . . . . . . . . . . . . . . . . . . 735.3 Quasi-Newton Method and BFGS-Algorithm . . . . . . . . . . 755.4 Gradients and the Adjoint Approa h . . . . . . . . . . . . . . 765.5 Spline-Representation of the Parameter . . . . . . . . . . . . . 795.6 Numeri al Tests . . . . . . . . . . . . . . . . . . . . . . . . . . 806 Con lusions 98

ii

Abstra tThe Bla k-S holes model is a widely used sto hasti model for valuating op-tions. One very restri tive assumption of this model is that the volatilityof the underlying asset is onstant. This work is on erned with identify-ing volatility smiles (volatility as fun tion of the sto k value) out of optionvalues of European Call options with several strikes and one maturity. Theestimated volatility an then be used for valuating other types of options.The identi� ation is a non-linear ill-posed problem, i.e. small errors in theobserved option values may ause arbitrarily large errors in the identi�edvolatility. We will use an output least squares formulation with Tikhonovregularization to over ome this problem.In Chapter 1 we present some �nan ial preliminaries and derive the Bla k-S holes equation in a formal way. Additionally, we give a short overview overre ent works on erned with this identi� ation problem. We on lude withsome remarks on inverse and ill-posed problems and sket h very brie y themain ideas of the standard theory of Tikhonov regularization for nonlinearill-posed problems.Chapter 2 summarizes some results of the theory of paraboli partial di�er-ential equations. We present the results for lassi al and weak formulations.The lassi al theory is based on the on ept of fundamental solutions. Afundamental solution solves the onsidered paraboli equation and also adual equation. This fa t will be used to derive a dual equation for the Bla kS holes Equation.In Chapter 3 we derive the dual equation to the Bla k-S holes equation andits variational formulation. Option values are given for di�erent strike val-ues, while the Bla k S holes Equation is a relation of option and sto k values.The dual equation is in terms of option and strike values. We onsider thisdual equation in variational formulation and all the problem of determin-ing option values for a given parameter (volatility) the dire t problem. The

dire t problem (option value in dependen e of volatility) is onsidered invariational formulation. The results of Chapter 2 an be applied to thisequation and yield existen e and uniqueness results and estimates for thesolution. Additionally, we show that under smoothness assumptions on theparameter, the solution has higher regularity and the weak and lassi al so-lution oin ide. We on lude this hapter with some onsiderations on theparameter-to-solution map, i.e. we show that this map is Lips hitz ontinu-ous and di�erentiable.Chapter 4 is on erned with the inverse problem of parameter identi� ation.We use an output least squares method with Tikhonov regularization, i.e.we onsider the minimization of an obje tive fun tional of the formf(a) = ku(a)� zk2 + � ka� a�k2�where a is the parameter, a� a given a-priori guess, u(a) is the solution ofthe dire t problem with parameter a and z are the given data. k � k and k � k�are appropriate norms and � is a regularization parameter.For � > 0 this minimization problem is well-posed, i.e we show that thefun tional f has a minimizer a whi h depends in a stable way on the dataz. We also onsider the ase of noisy data for the ase when the data areattainable, that is there exists a parameter ay su h that u(a) = uy and uy arethe data without noise. Following the ideas of [9℄ we show that if the noiselevel goes to zero, the identi�ed parameter onverges (in some sense) to ana�-minimum-norm solution (see Se tion 1.3). Under additional assumptionswe an show even a rate of onvergen e. These results give a theoreti alba kground and a justi� ation for using the regularized output least squaresformulation for � > 0 and the estimated parameter as approximation for thereal volatility.Finally, in Chapter 5 we present a FEM algorithm for solving the dire tproblem and use a BFGS algorithm for minimzing the Tikhonov fun tionalfor � > 0. On some test examples we show that the output least squaresminimization performs very well for identifying the volatiliy. In the aseof data noise, the estimated volatility shows large os illations, although the

number of degrees of freedom for the parameter is hosen very small, whi his regularization by itself. Adding a Tikhonov term (� > 0) results in mu hsmoother volatilities whi h redu e the output error to the same level. We on lude the test examples with estimating the volatility for real option val-ues of European Call options on the S&P500 index.The regularized output least squares formulation we present here gives verysmooth estimates for the volatility and is able to reprodu e market datawith high a ura y. In the ase of data noise the regularisation term playsa ru ial role. The theoreti al onsiderations justify using this method for� > 0.

Chapter 1Introdu tionS holes and Merton Win Nobel PrizeProfessors Robert C. Merton and Myron S holes won the Nobelprize for E onomi s on O tober 14, 1997. They were awardedthe prize for their pioneering work, along with the late FisherBla k, in the development of what has be ome known as theBla k-S holes formula. The three Ameri an �nan ial engineerspublished their seminal work in 1973. The Nobel prize winningdis overy has hanged the nature of �nan ial markets worldwide...Finan ial Engineering News, V1,1997Sin e many years the Bla k-S holes model has been used as a tool to value�nan ial derivatives. Although the value of the underlying, for instan e asto k, is assumed to vary sto hasti ally, it turns out that, under some as-sumptions on the market, the value of the derivative is determined by adeterministi di�erential equation.The Bla k-S holes model makes some restri tive assumptions on �nan ialmarkets (e.g. no transa tion osts, trade ontinuous in time, ...), whi h annot be observed in reality. Nevertheless, the model is so widely used thatone might even onsider the in uen e of the model on the markets.In the sequel we introdu e some �nan ial and sto hasti terminologies andthen derive the Bla k-S holes partial di�erential equation (PDE) and the1

Introdu tion 2famous Bla k-S holes formula for pri ing European Call options in a formalway.1.1 Finan ial PreliminariesWe sket h only some terminologies, whi h will be needed in the next se tion.For an overview over mathemati al tools applied to �nan e we refer to [18, 36,23℄. We onsider a �nan ial market, where assets (sto ks, options, bonds,..) an be traded� at any time� in any amount� with no transa tion ostsA bond is a riskless asset. Its value is determined deterministi ally byB(t) = B0 � er t (for the ase of onstant interest rates).We assume the value of a sto k (e.g. share of a ompany) S(t) to follow somesto hasti pro ess, i.e. not to be deterministi .A �nan ial derivative is an instrument whose value depends on the value ofan underlying asset (e.g. sto k).An option is a �nan ial derivative, whi h gives the holder the right, but notthe obligation to buy or sell a ertain amount of a �nan ial asset, by a ertaindate, for a ertain strike pri e. The writer of the option needs to spe ify� the type of option: the option to buy is alled a all while the optionto sell is a put;� the underlying asset: typi ally it an be a sto k, a bond or a urren y;� the amount of the underlying asset to be pur hased or sold;� the expiration date: if the option an be exer ised at any date beforematurity, it is alled an Ameri an option, if it an only be exer ised atmaturity, it is alled a European option;

Introdu tion 3� the exer ise pri e, whi h is the pri e at whi h the transa tion is doneif the option is exer ised.The pri e of an option is alled its premium.A European Call option on a sto k with value St = S(t) with strike K andexpiry T provides its owner the right, to buy the underlying asset at timeT for K (dollars). Obviously, if K > ST , the holder of the option has nointerest in exer ising his right. On the other hand, if ST > K, the holderof the option an make an instantaneous pro�t of ST �K by exer ising hisright, i.e. buying one sto k for K and selling it for ST . Therefore the valueof the option at maturity, its payo�, is given by(ST �K)+ := max(ST �K; 0)If the option is exer ised, the writer must be able to deliver a sto k at pri eK,whi h means that he must have generated an amount (ST �K)+ at maturity.At the time of writing the option (whi h will be onsidered to be the originof time), two questions must be asked:1. How mu h shall the buyer pay for the option? In other words, how anwe pri e at time t = 0 an asset worth (ST �K) at time T ? That is theproblem of pri ing the option.2. How an the writer. who earns the premium initially, generate anamount of (ST �K)+ at time T ? That is the problem of hedging theoption.In order to answer these questions and derive mathemati al formulae forpri ing an option, one has to make additional assumptions on the �nan ialmarket:� no arbitrage: this assumption says, that there is no way to get a higherrate of return (R = V (t)�V (0)V (0) ) than the bond provides, without takingsome risk.� market is omplete: this means that any payo� an be realized by a ombination of a riskless and a risky asset

Introdu tion 4One immediate onsequen e of the absen e of arbitrage is the so alled Put-Call parity for European type options:Ct � Pt = St �K e�r (T�t)where Ct and Pt denote the values of a Put and a Call option and r isthe ( onstant, risk free) interest rate. To see this, we onsider a portfolio onsisting of one sto k, one Put and minus one Call option. The net valueof this portfolio is given by St + Pt � CtAt time T two out omes are possible:� ST < K: the Call is exer ised, we deliver the sto k, re eive the amountK.� ST � K: the put is exer ised, we sell the sto k for K.In any ase, we generate an amount of K at time T , hen e the value (Vt) ofthe portfolio at time T is equal to K, and by dis ounting the Put-Call parityfollows.1.2 The Bla k-S holes EquationEven though no-arbitrage arguments lead to answers to many interestingquestions, they are not suÆ ient in themselves to develop pri ing formulae.To a hieve this, the sto k pri es must be modeled more pre isely. We intro-du e here very brie y a sto hasti model for the evolution of a sto k valueand then derive formally the Bla k-S holes PDE for pri ing �nan ial deriva-tives. For some remarks on sto hasti pro esses we refer to the appendix, forfurther reading on sto hasti di�erential equations see [17, 16℄.Bla k and S holes were the �rst to suggest a model where one an derive anexpli it formula for the pri e of a European Call on a sto k whi h pays no

Introdu tion 5dividends. A ording to their model, the writer has the possibility to hedgehimself perfe tly, i.e. to build a ompletely risk free portfolio (repli atingportfolio). Following the no-arbitrage prin iple, the value of this portfoliomust evolve like a riskless bond. The derivation of the Bla k-S holes equation an be found in almost any textbook in mathemati al �nan e (for instan e[23, 36, 18℄).By a short look on a hart (Figure 1) on an gather the following information

Figure 1: NEMAX, 6 months; sour e: boerse.de� over long time, the asset value shows some time evolution (drift)� the long-time movement is governed by some sto hasti u tuations.This observation motivates the following sto hasti model for the evolutionof a sto k value. dSS = � dt+ � dW (1.1)where the drift � hara terises the mean evolution, the volatility � spe i�esthe size of u tuations and W (t) is a standard Brownian motion. S(t) againis a sto hasti pro ess. (1.1) has the losed form solutionS(t) = S(0) e(���2=2) t+�W (t)

Introdu tion 6whi h an be veri�ed by using Ito's formula ((1.4)). By de�nition the in- rements W (s) � W (t) are normal distributed with mean 0 and varian es � t. Thus X(t) = (� � �2=2) t + �W (t) is normal distributed with mean(� � �2=2) t and varian e �2 t. Sin e the exponent of a normal distributedrandom variable is log-normal distributed, the relative hangesS(s)S(t) = e(���2=2)(s�t) e� (W (s)�W (t))are log-normal distributed. This is already one of the major drawba ks of thismodel, sin e by analyzing histori al data one an �nd that the probabilityof large u tuations is higher than predi ted, the orresponding probabilitydensities show "fat tails". There are several ways to over ome this problem� variable volatility: the volatility may depend on time (term stru ture)and the sto k value. Sin e the volatility vs. sto k urve often lookslike a smile or a frown, this dependen e is usually denoted as volatilitysmile.� sto hasti volatility: another way to explain the fat tails is to allowthe volatility itself to vary sto hasti ally. A rigorous mathemati altreatment of this ase is mu h more ompli ate.� non-normal distributions: hyperboli distributions mat h histori al datafar better (see [7℄).We restri t our dis ussion here on the ase of variable volatilities, whi h area simple extension to the Bla k-S holes model and an be treated with thesame mathemati al tools.Equation (1.1) still needs some explanation. It is well known (see [16℄) that aBrownian motion is nowhere di�erentiable. A sto hasti di�erential equation(SDE) of the form dX = f(t; X) dt+ g(t; X) dW (1.2)hen e has to be interpreted as the orresponding integral equation

Introdu tion 7X(t) = X(0) + Z t0 f(�;X(�)) d� + Z t0 g(�;X(�)) dW (�) (1.3)Sin eW (t) is not of bounded variation, this integral does not exist as Riemann-Stieltjes integral. A de�nition of the se ond term in (1.3) asI(g) := Z t0 g(�;X(�)) dW (�) := limn!1 tXk=1 g(tk�1)(W (tk)�W (tk�1))leads to the on ept of Ito integrals. In the time interval [tk�1; tk℄, the in-tegrated fun tion g is weighted with its values at time tk�1. The followingresult, the Ito formula, is fundamental for the understanding of sto hasti di�erential equations. If X(t) is an Ito pro esses of the form (1.3) with thedi�erential (1.2) and F (t; x) is a deterministi fun tion with ontinuous par-tial derivatives, then the sto hasti pro ess F (t; X(t)) has the di�erential(see [16℄)Ito's formuladF (t; X(t)) = [�F�t + �F�x f + 12 �2F�x2 g℄ dt+ �F�x g dW (t) (1.4)The appli ation of Ito's formula on the value of an option V (S; t) makes itpossible to derive di�erential equations for the premium.We turn now to a formal derivation of the Bla k-S holes PDE. Let V (t; S)be the value of an option and let the value of a sto k S(t) satisfy the SDEdS(t) = �S dt+ � dW (t) (1.5)In addition letB(t) be an asset with deterministi value (bond)B(t) = B0 ert.By Ito's lemma, we get immediatelydV = (�V�t + �S�V�S + 12� S2�2V�x2 dt) + �2 S�V�S dWWe now build a portfolio onsisting of one option and an amount �� ofsto ks. The value of the portfolio is then given by�(t; S(t)) = 1 � V (t; S(t))��(t) � S(t)

Introdu tion 8We assume � to be �xed for a very short time interval dt and use Ito'sformula to getd� = (�V�t + �S �V�S + 12�2 S2�2V�S2 ���S) dt+ � S (�V�S ��) dWTo eliminate the sto hasti part we hoose � = �v�S . To adapt the amountof sto ks �� ontinuously is alled \Delta-Hedging". The in rement of thevalue of our portfolio be omes deterministi , i.e. free of any risk.d� = (�V�t + 12� S2 �2V�S2 ) dtNow we an use the assumption of "no arbitrage". Hen e the expe ted returnof this risk less portfolio must be the same as that of the bond, or equivalently(�V�t + 12� S2 �2V�S2 ) dt = r� dt = r (V � �V�S S) dtSummarizing this yieldsProposition 1.1. The value of an European Call option with strike K andexpiry T is given by the solution to the Bla k-S holes Di�erential Equation�V�t + 12�2 S2�2V�S2 + r S �V�S � r V = 0 (1.6)where the value V at expiry time T is given by the payo�V (T ) = (S �K)+and (�)+ = max(�; 0).For onstant �, r one an derive an analyti formula for the value of a Euro-pean Call option C with strike K, the famous Bla k-S holes formulaC(S; t) = S �( ln(S=K)+(r+�2=2)(T�t)�pT�t )� K e�r(T�t) �( ln(S=K)+(r+�2=2)(T�t)�pT�t ) (1.7)where � denotes the umulative Normal distribution.

Introdu tion 9Although the Bla k-S holes model has been very widely used for severalyears, its assumptions are quite restri tive and do not really mat h reality(e.g. no transa tion osts, no dividends, onstant volatility and drift,...).As already mentioned, one very intuitive generalization of the Bla k-S holesmodel is to drop the assumptions of onstant volatility and drift rate. Hen e,the sto k value is assumed to follow the SDEdSS = �(t) dt+ �(t; S(t)) dWIn addition ontinuous dividend payments an be invoked, whi h leads to amore general form of (1.6)�V�t + 12�2(S; t)S2�2V�S2 + (r(S; t)� q(S; t))S�V�S � r(S; t)V = 0 (1.8)with the dividend rate q.European options are usually traded very frequently and their values an beassumed to be given orre tly by the market values. But there are manyderivatives (e.g. exoti options), whi h need to be valued orre tly. In prin- iple, one an derive partial di�erential equations in quite the same wayfor many options. In order to determine the right values for their premia,the volatility of the underlying must be known. Here an interesting inverseproblem arises:Determine the value of the volatility out of market pri es of Eu-ropean optionsIn [4℄ the authors refer to this problem as the Inverse Problem of OptionPri ing (IPOP).In prin iple there are two di�erent ways of determining values for �� histori al volatility : The volatility an be estimated by analyzing his-tori al data. It is questionable if the volatility of the past des ribes thebehaviour of the future in a good way.� implied volatility : It is assumed that the market "knows" the volatilityand uses this information to value options. Therefore urrent option

Introdu tion 10values an be used to re onstru t �. If the volatility is assumed to be onstant, the implied Bla k-S holes volatility an be derived dire tlyfrom (1.7). But even a dependen e of the volatility on time (termstru ture) and on the sto k values (smile) an be estimated out ofmarket data.The identi� ation of the volatility stru ture has been addressed re ently sev-eral times. To understand the arguments of some of the authors, we justnote here that the value of a European Call option an also be derived asC(S; t;K; T ) = Z 10 (S �K)+ �T (s) ds(see [6, 5, 23℄), where �T = �2C�K2 an be understood as the probability densityfun tion of the underlying random pro ess and orresponds to a fundamentalsolution of (1.6). In order to re onstru t �t (whi h orresponds dire tly toidentifying �) option pri es for all possible strikes K are needed.D. Shimko suggests in [29℄ to evaluate Bla k-S holes implied volatilities forseveral strikes and then �nd a least squares approximation for � in the formof a quadrati parabola. We will see in Chapter 5 that this pro edure isnot re ommendable, sin e it tends to weigh strikes far o� the spot pri e toomu h.In [6, 5℄ the authors show, that the option value C also satis�es the equation�C�T = 12�2K2 �2C�K2 � r K �C�K (1.9)We present the derivation of this result in the ontext of dual equations in thefollowing se tion, whi h an be found in [4℄. As an immediate onsequen e,one an solve expli itly for � and gets� =r2 CT + rK CKK2CKKIn prin ipal, the volatility is uniquely determined by this formula as longas the denominator is bounded away from zero and the option values for all

Introdu tion 11possible strikes are available. In [27℄ the authors show that CK and CKK tendto zero exponentially forK !1 orK ! 0. Even small errors in market datawill be �rst ampli�ed by di�erentiation and probably ause CKK to be ome0. the problems arising in the analyti al formula indi ate the ill-posednessof the problem.In [4℄ Isakov and Bou houev re onstru t �T dire tly using properties of fun-damental solutions (i.e. the parametri expansion, duality relation). We willpresent their derivation of the dual equation (1.9) in the setting of funda-mental solutions in Chapter 3.In [2℄ Avellaneda et al. determine a volatility surfa e by means of a minimumentropy formulation. Their goal is to minimize the relative entropy of thedistribution density implied by � with respe t to some given a-priori guess �0subje t to the onstraints given by the a tual option values for some strikes.Their method treats the sto hasti ontrol prolblemminimize EQ(�(�2))subje t to EQ(e�r TiGi(STi)) = Ci i = 1; ::;Mwhere EQ denotes the expe tation with respe t to the probability distributionimplied by � and �(�2) is in the simplest ase given by �(�2) = �2��20 . Thevolatility stru ture an �nally be determined as�2 = �0(e�rt2 S2WSS)where W (S; t) is de�ned as the solution of the equationWt+ert �(e�rt2 S2WSS)+�SWS�rW = � Xt<Ti�T �i Æ(t�Ti)Gi(S) S > 0; t � Tand � is the Legendre dual of �. At the minimum option pri es are mat hedperfe tly, a ording to the ne essary �rst order onditions for minimizing the orresponding Lagrange fun tional.

Introdu tion 12In this work we will try to identify volatility smiles for one period ( onstantin time) out of observed market data, i.e. we assume � to be onstant in timeand depend only on S. A generalization for a volatility, pie ewise onstantin time, is straight forward and will be suggested in a remark in Chapter 4.The Bla k-S holes PDE (1.6) is a paraboli equation, and � a kind of di�usion oeÆ ient. Thus, we onsider here a parameter estimation problem for aparaboli equation in non-divergen e form and use an output least squaresformulation with a Tikhonov term for regularization, whi h orresponds tominimizing a fun tional of the formf(�) = kV (�)� Ck2� + � k� � �0k2 (1.10)with appropriate (semi-) norms k � k� and k � k, where V (�) and C denote theestimated and observed option values.In [27℄ the authors addressed this problem (also using a Tikhonov regulariza-tion term) in the lassi al formulation and derived optimality onditions anda uniqueness result. In [3℄ the same problem of identifying term and smilestru ture with a H1-regularization term is studied. The authors remark thatunder their assumptions, the existen e of a minimizer and the validity oftheir approa h annot be guaranteed. Their method of iteratively solvingthe nonlinear equation given by the ne essary �rst order ondition for mini-mizing the Tikhonov fun tional orresponds to a method of steepest des entfor minimizing the Tikhonov fun tional (1.10).We will study here the problem in weak formulation and the ase of noisydata, whi h has not been addressed so far. In addition to existen e of a min-imizer for the Tikhonov fun tional and ontinuous dependen e on the datawe an prove onvergen e of the identi�ed volatility and even onvergen erates for the ase of attainable data. The main results will be derived in thetheoreti al framework presented in [9℄.1.3 Inverse and Ill-posed ProblemsUsing the term "inverse problem" immediatly leads to the question "inverseto what?" Following J.B. Keller [21℄ one alls two problems inverse to ea h

Introdu tion 13other, when the formulation of one problem involves the other one. For somereasons one of the problems may be alled dire t problem, the other inverseproblem. In many ases there is a quite natural distin tion between the dire tand the inverse problem. Usually the determination of the future behaviourof a physi al system is alled the dire t problem, whereas the re onstru tionof the present state by means of future observations or the identi� ation ofphysi al parameters from observations of the evolution or the steady stateof a system (parameter identi� ation) is usually alled an inverse problem.Thus, one might say that inverse problems are on erned with determining auses for observed or desired e�e ts. Many inverse problems turn out to beill-posed.A ording to Hadamard, a mathemati al problem is alled well posed, if thefollowing onditions are satis�ed� for any admissible data, a solution exists� the solution is unique� the solution depends ontinuously on the dataIf now any of these onditions is violated, a problem is alled ill-posed. Theproblem of existen e of a solution an be addressed in the framework ofgeneralised solutions. If for example a solution to a linear equationT x = ydoes not exist, one might ask for x su h that T x approximates y as wellas possible in some given norm. For details see [9℄. The problem of non-uniqueness an be solved by hoosing some sele tion riterion, for examplethe approximate solution with minimal norm. For linear bounded operatorson Hilbert spa es this orresponds to �nding an approximate solution via aMoore-Penrose inverse.Maybe the most diÆ ult problem to handle is the problem of ontinuousdependen e on the data. If for example T is a ompa t operator on in�nitedimensional Hilbert spa es, its range annot be losed, unless it has �nitedimension (see [10℄). Consequently its inverse annot be bounded, and thus

Introdu tion 14the generalized solution x of T x = y will in general not depend ontinuouslyon y, i.e. small perturbations in y may have arbitrarily large in uen e on x.We give here two simple examples for ill-posed inverse problems:� Di�erentiation: Consider T : C(R) ! C(R) given by T (x)(t) = R t0 x(s) ds.The a ording inverse problem of solving T (x) = y for some y 2 C(R)is then di�erentiation. If now for example yn = sin(n2 x)n , then the a - ording solution of the inverse problem of di�erentiation xn is given byxn = n os(n2 x). Small errors in y may be ampli�ed arbitrarily.� Consider the ordinary di�erential equationu0(t) = �a(t) u(t); u(0) = 1; 0 � twith solution u(t) = e� R t0 a(t) dt. Let a(t) > 0 be given. The inverseproblem of parameter identi� ation would now be, to determine a(t)out of given measurements for u. If the real parameter is for examplea0 = 1 then u0 = e�t. We havea(t) = � u(t)u0(t)If now the measurements u(t) are disturbed by some noise, for exampleun(t) = u0(t) + sin(nt)n , we geta(t) = �e�t + sin(nt)=n�e�t + sin(nt) (1.11)Now for any n the denominator will vanish for some t large enough.Hen e the formula for re onstru ting the parameter a(t) is not wellde�ned, and even where it is de�ned, small data errors may be ampli�edarbitrarily large.In the �rst example, the dire t problem of integration is a smoothing pro ess.High frequen ies are smoothed out. The inverse problem has the opposite

Introdu tion 15behaviour. High frequen y errors are ampli�ed. For linear ompa t opera-tors (des ribing the dire t problem) this behaviour an be explained by thespe tral properties of the operator (see for example [9, 10℄).The se ond problem is nonlinear, and the problems there are not so mu hfrequen y dependent, but the fa t that the denominator in (1.11) vanishesvery rapidly.The inverse problem of re onstru ting the volatility smile out of market datafor options will have both of the mentioned properties. To see this, just re all� =r2 CT + rK CKK2CKK (1.12)and note that CT vanishes again exponentially with T large or small.In order to solve an ill-posed problem in the presen e of data noise at leastapproximately, one has to apply some sort of regularization. In general terms,regularization is the approximation of an ill-posed problem by a family ofneighbouring well-posed problems, whi h an be done in several ways. Letus, for simpli ity, onsider the problem of �nding an approximate solution ofT x = yfor a spe i� right-hand side y, where only some perturbed measurement yÆof the unknown exa t data y with ky� yÆk < Æ for some known "noise level"Æ is available. Possible ways of regularizations are for instan e� restri tion to ompa t sets: if the x is restri ted to a ompa t set,the dependen e on the data will always be ontinuous. By Tikhonov'slemma the inverse of an inje tive ontinuous operator on a ompa t setis ontinuous.� proje tion: If the sample or parameter spa e is proje ted to a �-nite dimensional spa e, the proje ted inverse will again be ontinu-ous. With in reasing dimension the problem will get more and moreill- onditioned.

Introdu tion 16� Tikhonov regularization: Let T � be the adjoint operator to T . Theequation T � T = T � y is alled normal equation. Note, that T �T willin general (e.g. for in�nite dimensional ompa t operators) not be ontinuously invertible, not even on R(T �). To regularize the problem,we add a term � x and getT �T x+ � x = T � yÆ (1.13)This equation is alled regularised normal equation. For every � > 0,the operator (� I+T �T ) is ontinuously invertible. (1.13) is equivalentto minimizing the Tikhonov fun tionalkTx� yÆk2 + � kxk2 �! min (1.14)The (generalized) solutions x� of the approximated problems an be al ulated in a stable way and it an be shown that if the noise levelÆ ! 0 and �(Æ) is hosen appropriately, the solutions xÆ� onverge (insome sense) to a solution of the original problem, if existent.The minimization problem (1.14) suggests to use Tikhonov regularizationalso for nonlinear problems.1.4 Tikhonov Regularization for Nonlinear Prob-lemsAs already indi ated by the examples (1.11), (1.12) many parameter estima-tion problems are nonlinear inverse problems, even if the dire t problem islinear. While the theory for linear ill-posed problems is rather omplete, theinvestigation of nonlinear problems still leaves some open questions. We willsket h very brie y the main results that an be found in [9℄ whi h we willapply, in prin iple, also to our problem.Consider a nonlinear mapping F : X ! Y with Hilbert spa es X and Y . LetF satisfy

Introdu tion 171. F is ontinuous2. F is weakly (sequentially) losedWe onsider the problem of �nding a x�-minimum-norm solution toF (x) = yfor given y 2 Y , this means of �nding xy 2 X su h thatkxy � x�kX = min fkx� x�k : F (x) = ygSu h a solution may not always exist, and even if it exists, it may (in ontraryto the linear ase) not be unique. In the sequel we will always assume thatsu h a x�-minimum-norm solution exists, i.e. the data y are attainable.Usually, the data y will not be given exa tly, but only by measurements yÆ ontaining some noise (Æ). The minimum knowledge that must be given is abound for the measurement errorsky � yÆk � ÆNow even if there exist a solution to F (x) = y, the problem F (x) = yÆwill not have a solution in general, and even if it has, it may not depend ontinuously on the error level Æ. This is exa tly what we observed in theexample on erning di�erentiation.As already mentioned, the idea of regularization is to repla e the ill-posedproblem by a family of well-posed problems, su h that solutions to the ap-proximate problems xÆ� onverge to a solution of the ill-posed problem xy (ifexistent) for Æ; �! 0. In the ontext of Tikhonov regularization, xÆ� will bedetermined as the minimizer of the Tikhonov fun tionalkF (x)� yÆk2Y + � kx� x�k2X (1.15)The results in [9℄ give answers to questions on erning� existen e of a minimizer of (1.15) for � > 0:This question is easily answered by the assumptions on F , espe iallythe weak losedness.

Introdu tion 18� stability for �xed � > 0:i.e. we show that solutions to the approximate problem (with y sub-stituted by yÆ in (1.15)) depend in a stable way on the data, that isfor yk ! yÆ the minimizers xk� have at least a onvergent subsequen e, onverging to a minimizer of (1.15) and every onvergent subsequen e onverges to a minimizer.� onvergen e as �, Æ ! 0:It an be shown that a hoi e of � independent of Æ annot, in gen-eral, lead to a onvergent regularization. If we hoose �(Æ) ! 0 asÆ ! 0 with �2(Æ)=Æ ! 0, then every sequen e of solution xÆ� to (1.15)has a onvergent subsequen e and every limit is an x�-minimum-normsolution.By substitution of the term kF (x)�yÆk2 in (1.15) by (kF (x)�yÆk�Æn)2, one an look upon (1.15) also as a kind of Lagrange fun tional to the problemof minimizing kx � x�k subje t to the restri tion kF (x)� yÆk = Æn. 1� thenplays the role of a Lagrange multiplier.By the spe ial hoi e of a �xed value of � one an for e a minimizer of (1.15)to be loser to x� or to satisfy F (x) ' yÆ with more a ura y.In the above statement of onvergen e, � was hosen in dependen e of Æa-priori, without knowledge of the solutions xÆ�. It is also possible to useknowledge about xÆ� to determine regularization parameters, whi h is referredto as a-posteriori parameter hoi e rules. One method to do this is thedis repan y prin iple. For some � > 1 the value of �(Æ) is determined by�(Æ; yÆ) = sup f� > 0 : kF (xÆ�)� yÆk � � ÆgA motivation for this strategy is very intuitive: it does not make sense tosolve F (x) = yÆ with more a ura y than is provided by the data yÆ.A fourth question is for the� rate of onvergen e:How fast do the approximate solutions xÆ� onverge to xy if the noise

Introdu tion 19level Æ ! 0? This question an be answered under additional assump-tions on the mapping F :1. F is Fre het-di�erentiable2. there exists > 0 su h that kF 0(xy)� F 0(x)kY � kxy � xkX3. there exists a w 2 Y su h that xy�x� satis�es the sour e ondition(F 0(xy))� w = xy � x�where (F 0)� denotes the adjoint of F 0.4. w satis�es the smallness ondition kwkY < 1Usually, the assumptions 1 and 2 an be veri�ed quite easily. We will do thisin Chapter 4 for our problem. The assumptions 1� 4 are so restri tive that,even if the x�-minimum-norm solution is not unique, there an only be onesatisfying 1� 4 (see [9℄).In [12℄ a di�erent proof of onvergen e rates is formulated for a di�usionproblem under mu h weaker assumptions. Espe ially the sour e ondition(3) is better interpretable and no smallness ondition (4) is needed. The fa tthat fxÆ�g always onverge to the right x�-minimum-norm solution may bean indi ation for uniqueness for the identi� ation problem in the dis ussedparaboli di�usion equation.Of ourse in pra ti e onvergen e rates will never (really) be observed, sin eusually the data noise (and also noise added by numeri al simulation) annotbe redu ed to zero and the results are valid only asymptoti ally. But theresults about onvergen e and rates guarantee to use the right method andgive a theoreti al ba king for the appli ation of a spe ial pro edure.We turn now to a general dis ussion of the theory of linear paraboli equa-tions.

Chapter 2Paraboli EquationsIn this se tion we study linear paraboli equations in a general ontext andsummarize some basi results about existen e and uniqueness of a solutionto equations of the form �ut + Lu = f (2.1)u(x; 0) = u0(x) (2.2)The main results an be derived with two di�erent approa hes:1. Fundamental solution: For smooth parameters, it is possible to derivean expli it representation of the solution in dependen e of the initialvalue and a sour e term of the formu(x; t) = Z �(x; t; �; 0) u0(�) d� + Z t0 Z �(x; t; �; �)f(�; �) d� d�2. Variational formulation: Multiplying by a test fun tion �, integratingover the domain and integration by parts give an (under appropriate onditions) equivalent equation for (2.1)(ut; �) + a(u; �) = (f; �) 8� 2 Vwith an appropriate bilinear form a(�; �) and a spa e of test fun tionsV . In many physi al appli ations (e.g. heat-transfer) this formula-tion seems to be the natural one. The oeÆ ients of the PDE may20

Paraboli Equations 21be well interpreted (e.g. heat ondu tivity) and it is possible to getexisten e and uniqueness results under mu h weaker onditions on the oeÆ ients, initial value and sour e term.2.1 Classi al TheoryThis se tion is on erned with solutions of (2.1) in spa es of ontinuousfun tions. We sket h the on ept of fundamental solutions, whi h an beused to derive existen e results for the Cau hy problem.A basi property of fundamental solutions is, that they also satisfy a dualequation. This property an be used to derive a uniqueness result and willbe used in the next se tion to derive a dual equation to the Bla k-S holesequation.Finally, we present a maximum prin iple, whi h guarantees uniqueness underweaker assumptions on the oeÆ ients.2.1.1 Cau hy ProblemConsider the partial di�erential operatorLu = nXi;j=1 aij(x; t) �2u�xi�xj + nXi=1 bi(x; t) �u�xi + (x; t) u� �u�t (2.3)where the oeÆ ients aij, bi, are de�ned on a ylinder Q = [0; T ℄� Rn .De�nition 2.1. The di�erential operator L is alled (uniformly) paraboli ,if there exist positive onstants a0 and a1 su h that, for every ve tor � 2 Rna0 j�j2 � nXi;j=1 aij(x; t) �i �j � a1 j�j2uniformly for all (x; t) 2 Q.For the moment let us make the following assumptions

Paraboli Equations 22(A1) L is uniformly paraboli in Q(A2) the oeÆ ients of L are ontinuous fun tions in Q and satisfy theH�older onditionsjaij(x; t)� aij(�; �)j � A �jx� �j� + jt� � j�=2�jbi(x; t)� bi(�; t)j � Ajx� �j�j (x; t)� (�; t)j � Ajx� �j� (2.4)for all (x; t); (�; �) 2 Q, 1 � i; j � n.Given a fun tion f(x; t) in Q = [0; T ℄� Rn and a fun tion u0(x) in Rn , theproblem of �nding a fun tion u(x; t) satisfying the paraboli equationLu = f in Q0 = (0; T ℄� Rn (2.5)where L is de�ned in (2.3), and the initial onditionu(x; 0) = u0(x) on Rn (2.6)is alled a Cau hy problem (in the strip 0 � t � T ). The solution is al-ways required to be ontinuous in Q. Additionally f(x; t) and u0(x) will beassumed to satisfy the boundedness onditionsjf(x; t)j � eh jxj2 (2.7)ju0(x)j � eh jxj2 (2.8)where h is any positive onstant satisfying h < a0=(4T ) and > 0.2.1.2 Fundamental SolutionsThe on ept of fundamental solutions is very intuitive. One tries to har-a terize the a tion of the solution operator on initial data and right-handside expli itly (e.g. the di�usion of a given heat distribution) by �nding afun tion � su h that the solution u an always be written in the formu(x; t) = Z �(x; t; �; 0) u0(�) d� + Z t0 Z �(x; t; �; �)f(�; �) d� d�

Paraboli Equations 23The so alled parametrix method yields a way to onstru t fundamentalsolutions to the Cau hy problem (2.5) - (2.8) and to prove their existen eand spe ial properties. We only sket h the main ideas. For details and proofswe refer to [15℄.De�nition 2.2. A fundamental solution of Lu = 0 is a fun tion �(x; t; �; �)de�ned for all (x; t); (�; �) 2 Q = Rn � [0; T ℄, t > � , whi h satis�es thefollowing onditions(i) for �xed (�; �) 2 Q it satis�es, as a fun tion of (x; t), (x 2 Rn ,� < t � T ) the equation Lu = 0.(ii) for every ontinuous fun tion f(x) in Rn satisfying jf(x)j � exp(hjxj2) for some positive onstant h < a0=(4T )limt&� ZRn �(x; t; �; �)f(�) d� = f(x) (2.9)A fundamental solution to the heat equation�ut + uxx = 0is given by �(x; t; �; �) = 1p4�(t� �)e (x��)24 (t��)whi h an be derived using Fourier transformation and an be veri�ed bysubstituting into the di�erential equation. A fundamental solution for themulti dimensional ase an be derived in the same way. The main idea of theparametrix method is now to generalize for the ase of variable oeÆ ients.1. For L paraboli let aij denote the inverse matrix to aij and set�y;�(x; �) = nXi;j=1 aij(y; �)(xi � �i)(xj � �j)For � > t we introdu e the fun tionswy;�(x; t; �; �) = (t� �)�n=2 exp ���y;�(x; �)4(t� �) � (2.10)

Paraboli Equations 24Z(x; t; �; �) = �2p���n �det �aij(x; t)��1=2 w�;�(x; t; �; �) (2.11)For ea h �xed (�; �) the fun tion Z(x; t; �; �) solves the equation with onstant oeÆ ientsL0u(x; t) = nXi;j=1 aij(�; �) �2u�xi�xj � �u�t = 0whi h may be veri�ed by straightforward al ulations. Additionally,Z(x; t; �; �) satis�es (2.9). Hen e Z = � is a fundamental solution toL0u = 0.2. In order to onstru t a fundamental solution for Lu = 0 one looksupon L0 as the �rst approximation to L and tries to �nd a fundamentalsolution � to Lu = 0 of the form�(x; t; �; �) = Z(x; t; �; �)+Z t0 ZRZ(x; t; �; �) �(�; �; �; �) d� d� (2.12)where � is to be determined in a way that � satis�es L� = 0. This anbe guaranteed, if � solves the Volterra integral equation�(x; t; �; �) = LZ(x; t; �; �) + Z t0 ZRn LZ(x; t; y; �) �(y; �; �; �) dy d�(2.13)Now one an �nd a solution of the form�(x; t; �; �) = 1Xm=1 (LZ)m (x; t; �; �) (2.14)where (LZ)1 = LZ and(LZ)m+1 (x; t; �; �) = Z t� ZRn [LZ(x; t; y; �)℄ (LZ)m (y; �; �; �) dy d�(2.15)3. Under assumptions (A1) and (A2) there exists a fundamental solutionto the Cau hy problem (2.5),(2.6). For the proof see [15℄.

Paraboli Equations 252.1.3 Existen e and Uniqueness for the Cau hy Prob-lemTheorem 2.1. Suppose that L satis�es (A1), (A2) and let f(x; t), u0(x)be ontinuous fun tions in Q and Rn respe tively, satisfying (2.7), (2.8).Assume also that f(x; t) is lo ally H�older ontinuous (exponent �) in x 2 Rn ,uniformly with respe t to t. Then the fun tionu(x; t) = ZRn �(x; t; �; 0) u0(�) d� � Z t0 ZRn �(x; t; �; �) f(�; �) d�; d� (2.16)is a solution to the Cau hy problem (2.5), (2.6) andju(x; t)j � ek jxj2 (x; t) 2 Qwhere k depends only on h, a0, T .2.1.4 The Adjoint Equation and a Uniqueness ResultOne method to prove uniqueness of the solution u to the Cau hy problem(2.5), (2.6) uses the adjoint operator to (2.3), whi h is by de�nitionL�v = nXi;j=1 �2�xi�xj (aij(x; t)u) + nXi=1 ��xi (bi(x; t)u) + (x; t) u+ �u�t (2.17)In order to justify this de�nition, we will have to make further assumptionson the oeÆ ients.aij; �aij�xh ; �2aij�xh�xk ; bi; �bi�xj ; (2.18)(A3) Let the fun tions in (2.18) be bounded ontinuous fun tions in Qand satisfy the uniform H�older ondition (exponent �) in x 2 Rn ,uniformly with respe t to t, and letjaij(x; t)� aij(�; �)j � A �jx� �j� + jt� � j�=2�hold throughout.

Paraboli Equations 26Proposition 2.1. Let (A1) - (A3) hold. Then L and L� satisfyZQ(vLu� uL�v) dx dt = 0for all suÆ iently smooth u, v.Under assumption (A1), (A3) one an again onstru t a fundamental solution�� to the problem L�v = 0and prove (see [15℄) �(x; t; �; �) = ��(�; � ; x; t) t > � (2.19)The proof of this identity is based on Green's theorem and the fa t L� = 0,L��� = 0.In the next se tion we use this identity to establish an adjoint problem tothe Bla k-S holes PDE.The onsiderations about the adjoint equation yield a uniqueness result:Theorem 2.2. Let L satisfy the assumptions (A1),(A3) in the domain Q =[0; T ℄ � Rn . Then there exists at most one solution to the Cau hy problem(2.5) - (2.8) satisfying the boundedness onditionZ T0 ZRn ju(x; t)j e�k jxj2 dx dt <1 (2.20)for some positive number k.2.1.5 Maximum Prin iple and Appli ation to the Cau hyProblemA widely used tool to prove uniqueness and spe ial properties (e.g. positivity,boundedness,...) of solutions to paraboli or ellipti equations are maximumprin iples. We present here a very general formulation of a maximum prin i-ple for paraboli equations, whi h is valid also on unbounded domains, and

Paraboli Equations 27use it to derive a uniqueness result for the solution to the Cau hy problem(2.5)-(2.8).Let Q0 = � (0; T ) and Q = � [0; T ℄. We onsider the Operator L of (2.3)on the domain Q0.Theorem 2.3. (Strong Maximum Prin iple)Let (A1), (A2) hold on Q0 and (x; t) � 0. If Lu � 0 in Q0 and if u hasa positive maximum whi h is attained at a point (x; t) 2 Q0 then u(x; t) =u(x; t) for all x 2 Q0.A more general version of this statement and proof an be found in [15℄.The ondition (x; t) � 0 may even be repla ed by (x; t) � 1. To seethat, we de�ne v(x; t) := u(x; t) e� 1 t (this te hnique is often referred to as"exponential shift"). Obviously v(x; t) solves Lv � 1 v = 0. Now we applyTheorem 2.3 to (L� 1) v = 0. Re-substituting gives the orresponding resultfor u. If is a bounded domain, then the theorem indu es that a maximummust be attained at the boundary � � [0; T ℄ [ � f0g. To establish asimilar result for unbounded , one has to assume a ondition on ju(x; t)j forjxj ! 1. One possible way would be the additional assumption ju(x; t)j ! 0for jxj ! 1. In fa t, a mu h weaker growth ondition is suÆ ient:Theorem 2.4. Let L be paraboli in Q0 and let the oeÆ ients of L be ontinuous fun tions satisfyingjaij(x; t)j �M; jbi(x; t)j �M(jxj+1); (x; t) �M(jxj2+1) (x; t) 2 Q0Then there exists at most one solution to the Cau hy problem (2.5), (2.6)satisfying ju(x; t)j � B e� jxj2 (x; t) 2 Qfor some positive onstants B, �.For the proof we refer again to [15℄. Note that this uniqueness result isderived under mu h weaker assumptions on the oeÆ ients aij, bi than thoseof the orresponding result of Theorem 2.2. The only additional assumptionneeded is ju(x; t)j � B exp(� jxj2) (x; t) 2 Q.

Paraboli Equations 282.2 Variational FormulationThe on ept of weak solutions and variational formulations of di�erentialequations allows us to state results about existen e and uniqueness for so-lutions of linear paraboli problems under mu h weaker assumptions on the oeÆ ients. Additionally, it is possible to derive estimates for the solutions.The main result for linear ellipti problems is the lemma of Lax and Milgram.It may be generalized to over also paraboli equations.Following [30℄ we �rst develop the framework to treat linear evolution equa-tions as ordinary di�erential equations on Bana h spa es and then statethe main results about existen e, uniqueness and estimation of the solution.Some basi de�nitions of Lebesgue and Sobolev spa es an be found in theappendix. For further reading we refer to [1, 8, 37℄.2.2.1 Fun tions and Distributions Valued in Bana hSpa esLet B denote a Bana h spa e, S a bounded interval of real numbers (e.g.S=(0,T) for some T > 0). A fun tion f : S ! B is said to be di�erentiableat t if the di�eren e quotientÆhf(t) = h�1ff(t+ h)� f(t)g onverges in B as h 6= 0 goes to 0. The limit of Æhf(t) is the derivative of fat t and is denoted by f 0(t) or dfdt . If this limit exists for every point t of S,it de�nes a fun tion f 0 in S.2.2.2 IntegrationThe de�nition of the Riemann integralZS f(t)dt (2.21)of a ontinuous fun tion f : S ! B as the limit of the Riemann sums is imme-diate. The operation (2.21) extends, as Lebesgue integration, to measurablefun tions f .

Paraboli Equations 29The following is an overview of hapter 3 in [30℄. We present the main results,that will be needed later on. For more details and proofs, we refer to [30℄and the referen es therein.Let 1 � p � 1; we denote by Lp(S;B) the spa e of (equivalen e lasses of)measurable fun tions f : S ! B su h that kf(�)kB belongs to Lp(S;R) withthe respe tive normskfkLp = �ZS kf(t)kpB dt�1=p 1 � 1 <1kfkL1 = esssupfkf(t)kB j t 2 SgEa h Lp(S;B) is a Bana h spa e. If B is separable and 1 � p < 1, thenLp(S;B) is separable.Theorem 2.5. (Phillips). Let 1 < p < 1, 1=q + 1=p = 1 and B re exive.Then the dual (Lp(S;B))0 an be identi�ed with Lp(S;B0).If B is a Hilbert spa e, so is L2(S;B) with the inner produ t given by(f; g) = ZS(u(t); v(t))B dt; f; g 2 L2(S;B)Let W 1;p(0; T ;B) be the set of fun tions f : [0; T ℄ ! B su h that for someg 2 Lp(0; T ;B) f(t) = f(0) + Z t0 g(s) ds; t 2 [0; T ℄Let V be a Bana h spa e , H a Hilbert Spa e whi h is identi�ed with itsdual H �= H 0 and in whi h V is dense and ontinuously embedded. Then wehave V ,! H ,! V 0 with the equationf(v) = (f; v)H for f 2 H � V 0; v 2 VFor 1 � p <1, 1=p+ 1=q = 1, we de�ne the Bana h spa eWp(0; T ) = fu 2 Lp(0; T ;V ) j u0 2 Lq(0; T ;V 0)gwith norm kukWp = kukLp + ku0kLq .In analogy to the s alar ase, where a fun tion f 2 W 1;p(S) is ontinuous,we have

Paraboli Equations 30Proposition 2.2. The Bana h spa e Wp(0; T ) is ontained in C([0; T ℄;H).If a fun tion u 2 Wp(0; T ) then ju(�)j2H is absolutely ontinuous on [0; T ℄,ddt ju(t)j2H = 2u0(t)( u(t)) a:e: t 2 [0; T ℄and there is a onstant C for whi hkukC([0;T ℄;H) � C kukWp(0;T ); u 2 WpCorollary 2.1. If u; v 2 Wp(0; T ) then (u(�); v(�))H is absolutely ontinuouson [0; T ℄ andddt(u(t); v(t))H = u0(t)( v(t)) + v0(t)(u(t)) a:e: t 2 [0; T ℄As an analogon to the ompa t embeddings of Sobolev spa es in the s alar ase we getTheorem 2.6. (Lions-Aubin). Let B0; B; B1 be Bana h spa es with B0 �B � B1; assume B0 ,! B is ompa t and B ,! B1 is ontinuous. Let1 < p <1, 1 < q <1, let B0 and B1 be re exive and de�neW = fu 2 Lp(0; T ;B0) j u0 2 Lq(0; T ;B1)gThen the in lusion W ,! Lp(0; T ;B) is ompa t.2.3 The Abstra t Cau hy ProblemLet V be a separable Hilbert spa e with dual V 0. We de�ne the fun tionspa es V = L2 (0; T ;V ) V 0 = L2 (0; T ;V 0) (2.22)with the orresponding normskuk2V = Z T0 ku(�; t)k2V ; kuk2V 0 = Z T0 ku(�; t)kV 0 (2.23)

Paraboli Equations 31Assume that for every t 2 [0; T ℄ we are given a ontinuous bilinear forma(t; �; �) on V , or equivalently a linear operator A 2 L(V; V 0)(A(t)u) [v℄ = a(t; u; v) u; v 2 V; t 2 [0; T ℄su h that for ea h pair u; v 2 V a(�; u; v) 2 L1(0; T ;R). Thenja(t; u; v)j � K kukV kvkV u; v 2 V; t 2 [0; T ℄Assume that H is a Hilbert spa e identi�ed with its dual H 0 and that theembedding V ,! H is dense and ontinuous. Let u0 2 H and f 2 V 0 begiven.De�nition 2.3. The problem: Find u 2 V su h thatu0 + Au = f in V 0u(0) = u0 in His alled abstra t Cau hy problemThe following proposition gives a better understanding of what is meant byequality in V 0.Proposition 2.3. The following are equivalent1. u 2 V: u0 + Au = f in V 0, u(0) = u02. u 2 V: for every v 2 V with v 2 W 1;2(0; T ;H), v(T ) = 0� Z T0 (u(t); vt(t))H dt+Z T0 hA(t) u(t); v(t)iV dt = Z T0 hf(t); v(t)iV dt+(u0; v(0))H3. u 2 V: for ea h v 2 Vddt(u(t); v)H + hA(t) u(t); viV = hf(t); viV for a.e t 2 (0; T )and u(0) = u0.

Paraboli Equations 32As in the ellipti ase, the existen e and stability theory relies basi ally onthe lemma of Lax-Milgram. However, for the paraboli ase, a more generalversion due to J. L. Lions will be neededTheorem 2.7. (Lions) Let fH; j�jg be a Hilbert spa e and f�; k�kg a normedlinear spa e. Suppose E : H � � ! R is bilinear and E(�; �) 2 H0 for ea h� 2 �. Then the following are equivalent:1. infk�k=1 supjuj�1 jE(u; �)j � > 02. for ea h f 2 �0 there exists au 2 H : E(u; �) = f(�); � 2 �Note, in ontrast to the Lax-Milgram lemma, this theorem yields no unique-ness result. Nevertheless it indu es the following estimationCorollary 2.2. Let u be the solution of the above theorem. Then u satis�esthe estimate jujH � 1 kfk�2.3.1 Existen e, Uniqueness and EstimatesFirst we derive a uniqueness result:Let u 2 V be the solution to the abstra t Cau hy problemu0 + Au = f in V 0u(0) = u0 in Hwhere A(t) : V ! V 0 are bounded measurable operators as above. Byproposition 2.3 this implies12 ju(t)j2H + Z t0 hA(s) u(s); u(s)iV dt = Z t0 hf(s); u(s)iV dt+ 12 ju0j2H (2.24)with u(0) = u0. By use of the usual argument for linear equations we derive

Paraboli Equations 33Proposition 2.4. (Uniqueness) If ea h A(t) is monotone, that ishA(t) v; viV � 0; v 2 V; t 2 [0; T ℄then there is at most one solution to the Cau hy problem (2.24).To obtain existen e of the solution, we set H = L2(0; T ;V ) with the usuals alar produ t and � = f� 2 L2(0; T ;V ) j�0 2 L2(0; T ;H); �(T ) = 0gwith the norm given by k�k2� = j�j2H + j�(0)j2H. We de�neE(u; �) = Z T0 hA(t) u(t); �(t)iV � (u(t); �0(t))H dtand f(�) = Z T0 hf(t); �(t)iV dt+ (u0; �(0))Hand use proposition 2.3. The abstra t Cau hy problem (2.24) is equivalentto u 2 H : E(u; �) = f(�); � 2 �whi h is just a abbreviation for (2.24). The only remaining ondition toestablish in order to apply Theorem 2.7 isinfk�k=1 supjuj�1 jE(u; �)j � > 0A suÆ ient ondition is the uniform oer itivity of the operators A(t), thatis hA(t) v; viV � kvk2V ; v 2 V; t 2 [0; T ℄ThenE(�; �) = Z T0 hA(t)�(t); �(t)iV � (�(t); �0(t))H dt= Z T0 hA(t)�(t); �(t)iV dt+ j�(0)j2H � Z T0 hA(t)�(t); �(t)iV dt� k�k2 �

Paraboli Equations 34Proposition 2.5. (Existen e) Assume the operators A(t) are uniformly oer ive, i.e. hA(t) v; viV � kvk2V ; v 2 V; t 2 [0; T ℄for some > 0. Then there exists a unique solution to the Cau hy problem(2.24), and it satis�es the estimatekuk2L2(0;T ;V ) � (1= )2 (kfk2L2(0;T ;V 0) + ju0j2H)By "exponential shift" v(t) = e��tu(t)whi h is a standard te hnique for paraboli equations, it follows immediatelythat the result of proposition 2.5 remains valid if the oer itivity assumptionis repla ed byhA(t)v; vi+ � jvj2H � kvk2V ; v 2 V; t 2 [0; T ℄for some > 0, � 2 R.

Chapter 3Adjoint Equation and WeakFormulation; TheParameter-to-Solution MapWe return to the Bla k-S holes di�erential equation, whi h determines thepri e of a European Call option u(S; t) as a fun tion of sto k pri e S of theunderlying asset and time t. Real option pri es an be observed for di�erentmaturities T and di�erent strike pri es K. Thus we will formulate the Bla k-S holes di�erential equation in terms of these variables (u(K; T )) and sket hthe derivation of this adjoint equation, whi h an be found in [4℄ and is basedon the duality properties of fundamental solutions (see (2.19).The orresponding equation is of the same form as (1.6), i.e. it is in non-divergen e form. Nevertheless, we will transfer it into a variational formula-tion, whi h allows us to formulate the problem in Hilbert spa es and to showthat, under smoothness assumptions on the oeÆ ients, the unique solutionu to this problem has improved regularity and depends in a ontinuous anddi�erentiable way on the parameters (i.e. �) of the PDE. The problem ofdetermining the option values u for given parameters will be referred to asthe dire t problem.We on lude this hapter with basi properties of the parameter-to-solutionmap, whi h represents the fun tional relation of the volatility � and the35

Adjoint Problem - Parameter-to-Solution Map 36 orresponding option values u.Throughout this hapter, V and u will denote option values, t the time andT the expiry date. S and x will refer to the sto k values, whereas K and yrefer to the strikes. As we will see, t and T as well as S and K, respe tivelyx and y an be regarded as dual pairs of variables.3.1 An Adjoint ProblemRe all the extension of the Bla k-S holes PDE (1.8)Vt + 12S2�2(S; t)VSS + S (r(t)� q(t))VS � r(t)V = 0 (3.1)V (S; t = T ) = (S �K)+ (3.2)where VT (S) = V (S; t = T ;K). For the following derivations it will beimportant, that r and q depend only on time. Data (spot pri es) are availablefor di�erent strike values K, while the option value u is given dependent onthe sto k pri e S. In this setting the evaluation of n option pri es (the dire tproblem) would mean to solve n ba kwards paraboli equations with thesame oeÆ ients but di�erent terminal values (payo�s). Following [4℄, wetransform (3.1) into an equation in K and the expiry date T , whi h are theobservable quantities.We make the following assumptions on the oeÆ ients:(D1) 0 � r(t),q(t) 2 C([0; T ℄) do not depend on the state S(D2) �(S; t) 2 C�;�=2(R+ � [0; T ℄); 0 < �min � �for given 0 < � � 1. Equation (3.1) is degenerate as S2�2 is not boundedaway from 0, but by substituting S = K ex this degeneration an be resolved,leading tovt + 12�2(ex; t)vxx + ((r(t)� q(t) + 12�(ex; t)2)vx � r(t)v = 0 (3.3)v(x; T ) = K (ex � 1)+

Adjoint Problem - Parameter-to-Solution Map 37where v(x; t) = V (S; t).Re alling the results of the previous se tion, we know that the unique solutionof the Cau hy problem (3.3) is given byv(x; t) = ZR ~�(x; t; �; T ) vT (�) d�and has the regularity v 2 C2+�;1+�=2(Rn�℄0; T ℄) \ C(Rn � [0; T ℄). Resub-stituting for x = ln � SK � yields the orresponding existen e and uniquenessresults for V (S; t). Additionally, v(x; t) and hen e V (S; t) are positive by themaximum prin iple.Proposition 3.1. Let (D1), (D2) hold. Then there exists a unique solutionV (S; t) of the problem (3.1), (3.2) satisfying the growth onditionV (S; t) � e� ln2(S)for some ; � > 0.For the spe ial kind of terminal value (VT = (S � K)+), a fundamentalsolution � for equation (3.1) an be derived as follows (see [4℄):We build the di�eren e quotient V1(�;K; �) = 1� (V (�;K + �)� V (�;K)), whi hstill solves (3.1). Using the maximum prin iple, we get �1 < V1(�;K; �) < 0on R+ � (0; T ). It is now possible to pass the limit �! 0 (see [15℄)V1(�;K) = lim�!0V1(�;K; �)whi h still satis�es (3.1) with terminal data V1(s; T ;K) given by the negativeHeavyside step fun tion with jump at S = K. One an repeat this argumentfor the se ond di�eren e quotient. The limit � = VKK then solves (3.1) inR+ � (0; T ) and �(S; T ;K; T ) = Æ(s�K) in the sense of distributions. Theunique solution V (S; t) of proposition 3.1 an be represented in the formV (S; t) = Z 10 V (S; T ) �(S; t;K; T ) ds

Adjoint Problem - Parameter-to-Solution Map 38For suÆ iently smooth oeÆ ients� 2 C2+�;1+�=2(R+ � [0; T ℄)the fundamental solution � also satis�es the dual equation��T + 12 �K2�2 ��KK � ((r � q)K �)K � r � = 0 (3.4)with initial ondition �(s; t;K; t) = Æ(s � K). We integrate this equationtwi e from K to 1 and integrate by parts using thatV; K VK K2 VKK ! 0 as K !1This leads to the following dual equation for V (K; T ):L�V = �VT + 12K2�2(K; T )VKK +K (q(T )� r(T ))VK � q(t)V = 0 (3.5)V (K; 0;S) = V0(K;S) = (S �K)+ (3.6)This result remains still valid for � 2 C�;�=2([0; T ℄�R), whi h may be shownby approximating � by �n 2 C2+�;1+�=2([0; T ℄� R).We will now derive weak formulations of (3.1) and (3.5). In order to resolvethe degenera y (for S;K ! 0), we apply the substitutions K = ey, S = exand denote a = 12�2 to getvt + a vxx + (r � q � a) vx � r v = 0 (3.7)�vT + a vyy + (q � r � a) vy � q v = 0 (3.8)with terminal value v(x; t = T ) = (ex�ey)+ respe tively initial value v(k; T =0) = (ex� ey)+. Note that the initial/terminal value (ex� ey)+ as a fun tionof x as well as of y is not in L2(R), whi h will be required in order to derivea weak formulation.Let us substitute v(x; t; y; T ) = e2x�y u(x; t; y; T ) to getut + a uxx + (r � q + 3a) ux + (r + 2a� 2q) u = 0 (3.9)

Adjoint Problem - Parameter-to-Solution Map 39�uT + a uyy + (q � r � 3a) uy + (2a� 2q + r) u = 0 (3.10)with terminal/initial value u(x; t = T ; y) = u(y; T = 0; x) = ey�2x(ex � ey)+.Under the assumptions(W1) a(y) 2 L1(R), 0 < a0 � a(y) � a1(W2) ay(y) 2 L1(R)(W3) b(t); (t) 2 L1(0; T )it is now possible to derive weak formulations, whi h read�(ut; �)0 + (a ux; �x)0 + ((r � q + 3a� ax) ux; �)0+ ((r + 2a� 2q) u; �)0 = 0 (3.11)for all � 2 H1(R), a.e. t � T and terminal value u(x; T ) = ey�2x(ex� ey)+ 2L2(R) and for the dual equation(uT ; )0 + (a uy; y)0 + ((q � r � 3a� ax) uy; )0+ ((2a� 2q + r) u; )0 = 0 (3.12)for all 2 H1(R), a.e. 0 � T and initial value u(y; 0) = ey�2x(ex � ey)+ 2L2(R). (�; )0 denotes the standard s alar produ t in L2(R).Remark 3.1. Theorem 2.4 yiedls existen e and uniqueness of lassi al so-lutions ud, uad to (3.9) and (3.10). Under smoothness assumptions on theparameters we have shown that ud = uad.By Proposition 2.5 the problems (3.11) and (3.12) have unique solutions ud,uad andkudkL2(0;T ;H1) � C kud(�; T )kL2 kuadkL2(0;T0;H1) � C kuad(�; 0)kL2We now show that the lassi al solution of the dual Problem (3.10) and theweak solution of (3.12) oin ide, if the parametes are smooth, that is we showthat the solution u of (3.9) is in L2(0; T0;H1(R)) \ L2(0; T0;H�1(R)):

Adjoint Problem - Parameter-to-Solution Map 40Let u denote the solution of (3.9). We investigate the following problem:�vT + a vyy + (q � r � 3a) vy + (2a� 2q + r) v = 0 T 2 [t0; T0℄ (3.13)with initial ondition v(y; t0) = u(y; t0) (3.14)Of ourse, the (unique) solution to this problem is given by v = uj[t0;T0℄.From Proposition 2.1 we know that u an be written in the formu(y; T ) = ZR �(y; T ; �; 0) u0(�) d�The derivatives uy, uyy and uT an be represented in the same way by sub-stituting � by ��y�, �2�y2� respe tively ��T � and following estimates hold (see[15℄, h. 1): j�(y; T ; �; 0)j � BpT e� �T (y��)2���� ��y�(y; T ; �; 0)���� � BT e� �T (y��)2and ���� �2�y2�(y; T ; �; 0)���� ; ���� ��T �(y; T ; �; 0)���� � BpT 3 e� �T (y��)2where � and B depend only on the parameters of the di�erential equationand the Hoelder oeÆ ient �. Additionally, we know 0 � u0(�) � C e j�j.We show that u(y; T ) has exponential de ay in y for any T � t0:ju(y; T )j = ����ZR �(y; T ; �; 0) u0(�) d������ ZR j�(y; T ; �; 0)j ju0(�)j d�� ZR BpT e� �T (y��)2 � C e� j�j d� = (�)

Adjoint Problem - Parameter-to-Solution Map 41The last term is independent of the sign of y, so we assume w.l.o.g. y � 0.For y � 4 T0� and e� j�j � 1 we get(�) � ZR B CpT s� T� � Gwith a onstant G independent of T . If now x � 4 T0� we get(�) = B CpT Z y=4�1 e� �T (y��)2 e� j�j d� + Z 1y=4 e� �T (y��)2 e� j�j d�! = B CpT (T1 + T2)The term (T2) may be estimated by(T2) � e� y2 Z 1y=4 e� �T (y��)2 d� � DpT e� 2 jyj(T1) an be further estimated by(T1) = Z y=4�1 e� �T (y��)2 e� j�j d�= Z 0�1 e� �T (y��)2� � d� +Z y=40 e� �T (y��)2+ � d� = Z 0�1 e� �T f(�) d� = + Z y=40 e� �T g(�) d�(�)Now, f(�) = (y � �)2 + T� �= (y � �)22 + T� y + �(y � �)22 + T� (� � y)�= (y � �)22 + T� y + (y � �)2 �y � � � 2 T� �Sin e y � 4 T0� , the term yx� � � 2 T� � 0 and hen ef(�) � (y � �)22 + T� y

Adjoint Problem - Parameter-to-Solution Map 42In the same way on derivesg(�) � (y � �)22 + T� y(�) � Z y=4�1 e� �2T (y��)2� y d� = e� y Z y=4�1 e� �2T (y��)2 d�� e� y ZR e� �2T (y��)2 d� = e� yr2t�= EpT e� jyjCombining the results, we haveju(y; T )j � K e� 2 jyjIt follows in the same way thatjuy(y; T )j � KpT e� 2 jyjand juyy(y; T )j � KT e� 2 jyj and juT (y; T )j � KT e� 2 jyjSo we get u 2 L2(t0; T0;H2(R)) \ H1(t0; T0;L2(R)). Hen e u is a solutionto the weak problem (3.13), whi h, by Proposition 2.5, is unique. By theestimates of Proposition 2.5 we get kukL2(t0;T0;H1(R)) � ku(t0)kL2(R) � Cwith C independent of t0. Hen e u 2 L2(0; T0;H1(R)).Proposition 3.2. If the parameters satisfy (W1) � (W3) and (D1) � (D3),than the lassi al solution of Problem 3.10 oin ieds with the weak solutionof Problem 3.12, and in the same way the solutions of the (3.9) and (3.11).3.2 Problem StatementIn the sequel we will on entrate on the derived weak formulation of the dualequation. We introdu e a suitable fun tion spa e:

Adjoint Problem - Parameter-to-Solution Map 43De�nition 3.1. Let the Hilbert spa e � be de�ned by� = fu(y; T ) 2 L2(0; T0;H1(R)) j uT 2 L2(0; T0;H�1(R))gwith the natural normkuk2� = Z T00 ku(�; T )k21 + kuT (�; T )k2�1 dTHere T0 is introdu ed for te hni al reasons. T0 an be interpreted as the lastdate of expiry, for whi h option pri es shall be al ulated.Problem 1. (Dire t Problem) Let (W1) - (W3) hold. Find a solutionu(y; T ) 2 L2(0; T0;H1(R)) whi h satis�esLu := uT +A u = 0 in L2(0; T0;H�1(R)) (3.15)with initial value u(�; 0) = ey�2x (ex� ey)+ in L2(R), where A(T ) : H1(R) !H�1(R) is de�ned byhA(T )w; vi = (awy; vy)0 + ((r � q + 3a + ay)wy; v)0 + ((2q � 2a� r)w; v)0(3.16)for 0 � T � T0 and all v; w 2 H1(R).Substituting into (3.15) gives immediatelyuT = �Au in L2(0; T0;H�1(R))and thus uT 2 L2(0; T0;H�1(R)).3.3 Existen e, Uniqueness and EstimatesNow we an apply the results about existen e and uniqueness of a solutionto the abstra t Cau hy problem derived in the previous se tion.

Adjoint Problem - Parameter-to-Solution Map 44Proposition 3.3. There is a unique solution u 2 L2(0; T0;H1(R)) to Prob-lem 1, whi h satis�es the estimateskukL2(0;T0;H1(R)) � C1 ku0k20; kuTkL2(0;T0;H�1(R)) � C2 ku0k20Proof. Under assumption W1�W3 proposition 2.5 and the following remarkyield the result.Plugging into the di�erential equation (3.15) yieldsCorollary 3.1. There is a unique solution u 2 � to Problem 1 andkuk� � C ku0k203.4 RegularityWe will show that for suÆ iently smooth oeÆ ients and initial data a solu-tion of problem 1 has improved regularity.Assume ayy 2 L2(R). The fun tion z := uy then solves the equationzT +A z = �Ay u (3.17)for every � 2 H1(R) and for a.e. T 2 (0; T0) with the initial valuez0 = �� yu0where Ay is de�ned byhAy(T )w; vi = (ay wy; vy)0 + (3ay + ayy)wy; v)0 � (2ay w; v)0for 0 � T � T0 and all v; w 2 H1(R).The right-hand side of (3.17) de�nes a linear fun tional in L2(0; T0;H�1(R)).Sin e the initial value z0 2 L2(R), the results of proposition 2.5 may beapplied and we get a unique solution z 2 L2(0; T0;H1(R)) with the normestimate kzk2L2(0;T0;H1(R)) � C (kz(�; 0)k2L2(R) + kuk2�)Hen e uy 2 � and u 2 L2(0; T0;H2(R)). Using the di�erential equation foru gives uT = �A u 2 L2(0; T0;L2(R)).

Adjoint Problem - Parameter-to-Solution Map 45Proposition 3.4. Let (W1) � (W3) hold and additionally ayy 2 L2(R). Ifu denotes the unique solution of Problem 1, then u 2 L2(0; T0;H2(R)) \H1(0; T0;L2(R)) and kukL2(0;T0;H2(R)) � C ku0k20where C depends only on the bounds for the oeÆ ients and T0.Re all that a was de�ned as a = �22 . Let(W4) 0 < �0 � � < �1(W5) �y 2 H1(R)hold. Sobolev's embedding theorems yield �y 2 L1(R). By di�erentiationwe get ay = � � �y and ayy = �2y + � � �yyand thus ay 2 H1(R). This yieldsCorollary 3.2. Let W3-W5 hold. Then there exists a unique solution u 2L2(0; T0;H2(R)) \H1(0; T0;L2(R)) of Problem 1 andkukL2(0;T0;H2(R)) � C ku0k20where C depends only on the bounds for the oeÆ ients and T0.Later on, we will need some properties ( ontinuity, di�erentiability) of theparameter-to-solution map F : a! u(a) (3.18)where a = �22 and u(a) again denote the parameter and the orrespondingsolution of problem 1.

Adjoint Problem - Parameter-to-Solution Map 463.5 The Parameter-To-Solution Map3.5.1 NotationDe�nition 3.2. We all the mappingF : K ! Wa 7! u(a) (3.19)parameter-to-solution map. The set of admissible parameters K and the spa eW still have to be spe i�ed.Let k � k0;� denote the norm on L2(T0 � �; T0;L2(R)), that iskuk20;� = Z T0T0�� ku(�; T )k20 dTDe�nition 3.3. In analogy to �, we de�ne the spa e = fv(y; T ) 2 L2(0; T0;H2(R)) j vT 2 L2(0; T0;L2(R))gand equip it with the normkvk2 = Z T00 �kv(�; T )k2H2(R) + kvT (�; T )k2L2(R)� dTDe�nition 3.4. Let the spa e X be de�ned asX = fq 2 L1(R) j qy 2 H1(R)gequipped with the norm kqkX = kqk1 + kqyk1.Sin e L1(R) and H1(R) are separable Bana h spa es, so is X .Proposition 3.5. The embedding X � C1=2(R) is ontinuous.

Adjoint Problem - Parameter-to-Solution Map 47Proof. Let v 2 X . For jy � xj � 1 we havejv(y)� v(x)j = j Z yx ��v(�) d�j �pjy � xjsZR j��v(�)j d�� pjy � xj kvkX :Hen e v is H�older ontinuous with exponent 1=2. If jy � xj > 1 we havesimply jv(y)� v(x)jjy � xj1=2 � jv(y)� v(x)j � 2kvk1Summarizing this yields kvk� � 3 kvkXDe�nition 3.5. Let K de�ne the set of admissible parametersK = fa 2 X j 0 < a0 � a � a1g (3.20)Proposition 3.6. The set K is losed in X and onvex.3.5.2 Basi Properties of the Parameter-To-SolutionMapProposition 3.7. Set W = �. Then F : a ! u(a) is lo ally Lips hitz ontinuous, i.e. let a 2 K, then for every d with ka� dkX �MkF (d)� F (a)k� = ku(d)� u(a)kV � L kd� akXwhere L does not depend on d.Proof. We denote the di�eren e u(d) � u(a) by wda and write in abstra tform uT + Au = 0

Adjoint Problem - Parameter-to-Solution Map 48where A = A(a) is de�ned by (3.16). The di�eren e wda then solveswdaT + A(a)wda = A(a� d) u(d)with homogeneous initial data. The existen e result of proposition 2.5 yieldskwdak� � C(a) kA(a)� A(d) u(d)kL2(0;T ;H�1) = fand we estimate the right hand side for p = a� d and �xed time t byjh(A(a)� A(d)) v; �ijH�1 = j ZR p vy �y + (3p+ py) vy �� 2p v �j� kpk1 kvk1 k�k1Integrating over (0; T ) and using ku(d)k� � C(a) for bounded d as abovegives the desired estimatekfkL2(0;T ;H�1) � C ka� dkXIn the same way we an derive the result also for the derivatives uy(a) anduy(d). This yieldsCorollary 3.3. Let W = . Then F : a! u(a) is lo ally Lips hitz ontin-uous.For dire tions p 2 X , � 2 H1(R) we formally derive equations for the dire -tional derivatives of the mapping F((u0p)t; �) + (a (u0p)y �y) + ((b+ 3a + ay) (u0p)y; �) + (( � 2a) u0p; �)= �(p uy; �y)� ((3p+ py) uy; �) + (2p u; �) (3.21)where u0p denotes F 0(a). Proposition 2.5 guarantees the existen e and bound-edness of a solution to this equation. The linearity is trivial and the ontin-uous dependen e on a follows as above. Hen e we get

Adjoint Problem - Parameter-to-Solution Map 49Proposition 3.8. Let W = �. Then the mapping F : a ! u(a) is Fre hetdi�erentiable.The same arguments applied to uy instead of u yieldu0p 2 L2(0; T ;H2(R)) \H1(0; T ;L2(R))andCorollary 3.4. Let W = . Then the mapping F : a ! u(a) is Fre hetdi�erentiable.The Lips hitz ondition for the Fre het derivative follows in the same way asthe Lips hitz ondition for F in proposition 3.7.Proposition 3.9. Let F : a ! u(a) be the mapping of proposition 3.7.ay 2 K. Then the Fre het derivative F 0 is lo ally Lips hitz ontinuous withkF 0(a)� F 0(ay)kL(X ;�) � L ka� aykXProof. We assume for brevity b = = 0 for the moment. Let w[p℄ and wy[p℄denote the derivatives of u in dire tion p at a respe tively ay. We know thatw and wy are de�ned as the unique solutions to the equations(w[p℄)t + A(a)w[p℄ = �A(p) u(a)with homogeneous initial value and with a and w repla ed by ay and wy forthe latter ase. Hen e the di�eren e z[p℄ = w[p℄� wy[p℄ solves(z[p℄)t + A(ay) z[p℄ = �A(p) (u(a)� u(ay))� (A(a)� A(ay))w[p℄ = fThe estimates of proposition 2.5 yieldkz[p℄k� � C(ay) kfkL2(0;T0;H�1)

Adjoint Problem - Parameter-to-Solution Map 50It remains to estimate f . By the boundedness and linearity of the operatorA we get for � 2 H1(R)hA(q) v; �i = (q vy; �y) + ((3q + qy) vy; �)� 2 (q v; �)� kqkX kvk1 k�k1Thus we getkfk2V 0 = Z T00 kpk2X ku(a)� u(ay)k2V + ka� ayk2X kw[p℄k2V dT = �Now (u(a) � u(ay)) an be further estimated by the Lips hitz ontinuity ofF (proposition 3.7) and we get� � ka� ayk2X (L kpk2X + kw[p℄k2V)whi h is the desired result.Equation (3.21) motivates the de�nition of an operator F : X ! W 0 withhF(a) p; �iW = � Z T0 (p uy; �y)0 + ((3p+ py) uy; �)0 dt� (2p u; �) dt (3.22)F an be seen as the orresponding operator to F 0 with respe t to a bilinearform indu ed by the di�erential equation (3.21). In our ase, F is ontinuousand its adjoint F� : W ! X 0 is given in the usual way by hF� �; piX =hF p; �iW, i.e.hF�(a)�; piX = � Z T0 (p uy; �y)0 + ((3p+ py) uy; �)0 dt� (2p u; �) dt (3.23)This adjoint operator will be a substitute for the adjoint of the Fre het deriva-tive in the proof of onvergen e rates (see [9, 12℄).

Chapter 4The Inverse Problem ofParameter Estimation - OutputLeast Squares Formulation withTikhonov RegularizationWe turn now to the problem of identifying the parameter a for a given so-lution u(a). Usually observations will be only available for ertain times tand strikes K and may ontain some noise. Therefore we use an output leastsquares formulation, i.e. try to minimize the output errorju(a)� zj2where a = �22 is the parameter to be identi�ed, u(a) is the solution of thedire t problem (problem 1), z are the observations and j � j is an appropriatenorm. As mentioned before, the inverse problem of parameter estimation isusually ill-posed (dependent on the onsidered norms) and we will thereforeapply Tikhonov regularization with a termkak2� = kayk20 + kayyk20 (4.1)multiplied by a regularization parameter and added to the output error term.51

Inverse Problem 52Thus we seek to minimize an obje tive fun tional of the formf(a) = ju(a)� zj2 + � ka� a�k2� (4.2)Although only �nitely many option pri es are available, we assume for themoment the ase of ontinuous observations. Generalizations of the Theo-rems 4.1, 4.2 and 4.3 for dis rete observations an be derived in the same wayas the stated theorems. Following [9, 12℄ we state and prove results aboutexisten e of a minimizer of (4.2) and stability of this solution with respe tto the data z. Additionally, we show onvergen e and onvergen e rates (seese tion 1.4).The results of the Theorems 4.1, 4.2 and 4.3 are quite similar to those derivedin [9℄ for abstra t problems. But our problem has some spe ial features, whi hpose some diÆ ulties in the proofs:1. The state spa e of the paraboli problem is unbounded. Sin e the valueof the obje tive fun tional has to be �nite, one an obviously gatheronly a �nite "amount" of information, that means one annot expe tto identify the parameter for very large or low state values. This willresult in some sort of lo al onvergen e.2. For numeri al al ulation, we will have to restri t ourselves anyway tobounded domains. One ould therefore ask to derive the results onlyfor this ase. We will see that we an treat solutions of the restri tedproblems as perturbations of the original one.3. The Bla k-S holes PDE (3.1) and its dual formulation (3.5) are in non-divergen e form. The parameter appears in the terma S2 uSS respe tively aK2 uKKIn the variational formulation, the parameter also appears with itsderivative in the onve tion term (see (3.12))(3 a+ ay) uy

Inverse Problem 53This will ause some diÆ ulties in the proofs and require additionalsmoothness assumptions on a. Nevertheless, the use of the variationalformulation allows us to treat the problem in Hilbert spa es and makesit easier to interpret an equation for the derivative of the solution u(a)with respe t to the parameter a, whi h in the ase ay 2 H1 exists as aFre het derivative.4. The use of a full norm term for regularization instead of (4.1) is possibleand straightforward.We just re all here the de�nition of the spa es� = fu(y; T ) 2 L2(0; T0;H1(R)) j uT 2 L2(0; T0;H�1(R))gX = fq 2 L1(R) j qy 2 H1(R)gand the set of admissible parametersK = fa 2 X j 0 < a0 � a � a1gFor many problems the state domain is bounded. The set of admissi-ble parameters is usually a onvex subset of a Hilbert spa e (for exampleL2(); H1()) and proofs are based on the re exivity of these spa es and ompa t embeddings on bounded domains (e.g. H1()! L2()). However,on the unbounded domain = R, the ellipti ity assumption 0 < a0 � a1 an obviously not be ful�lled by a 2 L2(R). Therefore the parameter spa eis repla ed here by X . Note that in a separable (not re exive) Bana h spa ebounded sequen es do not have weakly onvergent subsequen es in general.Our proofs will therefore rely on1. X is a subspa e of L1(R), whi h is the dual of L1(R). Bounded se-quen es in L1(R) have weakly� onvergent subsequen es , that isZR an g dy! ZR a g dy 8g 2 L1(R)On every ompa t subset K � R, L1(K) � L2(K) and hen e an *� ain L2(K). Sin e L2(K) is re exive, this is equivalent to an * a.

Inverse Problem 542. As shown above, X � C1=2(R), whi h is again not re exive. But on any ompa t subset ! � R the embedding C1=2(!) ! C�(!) is ompa t(Arzela-As oli) for 0 < � < 1=2, and therefore a bounded sequen e inC1=2(R) has a subsequen e whi h onverges uniformly on every ompa tsubset ! � R in the sense of C�(R).An immediate onsequen e of this observations isLemma 4.1. Let an be a bounded sequen e in K. Then there exist a 2 Kand a subsequen e am su h thatam *� a in L1(R) and amy * ay in H1(R)Proof. an is bounded in in X and hen e an, any are bounded in L1(R) re-spe tively H1(R). Hen e there exists a subsequen e ak of al and a 2 L1(R)su h that ak *� a in L1(R). aky is still bounded in H1(R). Again we an�nd a subsequen e am and ay 2 H1(R) su h that amy * ay. It remains toshow that ay = ay:8� 2 C10 (R) we haveZR a �x dy = limm!1ZR am �y dy (4.3)= limn!1ZR�amy � dy = ZR ay � dy (4.4)Hen e ay is the generalized derivative of a.4.1 Regularized Output Least Squares Prob-lemWe state now the optimization problem orresponding to the output leastsquares formulation with Tikhonov regularization with a spe ial hoi e ofj � j in (4.2). Minimizing (4.5) for �xed � > 0 orresponds to solving aneighbouring problem to F (a) = z, with F the parameter-to-solution mapdis ussed in se tion 3.5.

Inverse Problem 55Problem 2. For given � > 0 and �nal observation z 2 L2(T0��; T0;L2(R)),minimizeJz�(a) = R T0T0�� RR jua(y; T )� z(y; T )j2dy dT + � k(a� a�)yk21= kua(�; T )� zk20;� + � k(a� a�)k2� (4.5)over a 2 K, where ua(x; t) denotes the solution to Problem 1 orrespondingto the parameters a, T0.We assume here the ase of ontinuous observations on a �nite time interval[T0 � �; T0℄. If data are available only for dis rete points (in state or time),one has to modify the obje tive fun tional. Also the regularization termk � k� may be de�ned in another way. We make some remarks on possibleextensions at the end of this hapter.An important property to prove the existen e of a minimizer of (4.5) is theweak lower semi- ontinuity of Jz�(a):Proposition 4.1. Let an, a 2 K, any * ay in L2(R) and let un = u(an)denote the orresponding solution of problem 1. In addition let un * u =u(a) in �. Then Jz�(a) � lim infn!1Jz�(an)Proof. The result follows immediately by the weak lower semi- ontinuity ofnorms, that is an * a =) kak � lim infn!1 kank; f. e.g. [11, 32℄.4.2 Existen e of MinimizerIn order to show the existen e of a minimizer a of Problem 2 we will needa basi property (i.e. the weak losedness) of the nonlinear parameter-to-solution map F : a 7! u(a)). We make some onsiderations in advan e:

Inverse Problem 56Lemma 4.2. Let � be de�ned as�(an) = Z T00 ZR anuny �y + (3 an + any ) uny �� 2 anun � dy dTand fang denote a bounded sequen e in K. Then there is a 2 K su h that�(an)! �(a).Proof. Lemma 4.1 provides a 2 K and a subsequen e am su h thatam *� a in L1 and amy * ay in H1We denote the solutions a ording to a parameter am by um and an �nd asubsequen e uk of um and u 2 � su h that uk * u in �.We show the result only for the �rst termZ T00 ZR anuny �y dy dTThe rest follows in a similar way. First we split up the integralZ T00 ZR akuky �y dy dT = Z T00 ZR(ak � a) uky �y dy dT + Z T00 ZR a uky �y dy dT(4.6)For every k 2 N , the �rst term on the right side de�nes a linear fun tionalon H1(R) lk(�) = Z T00 ZR(ak � a) uky �y dy dTwhi h is bounded byjlk(�)j = ����Z T00 ZR(ak � a) uky �y dy dT ���� � 2 a1 k�ykL2(R) Z T00 kukykL2(R) dT� C k�kH1(R) kukk�Hen e, the family of fun tionals flkg is uniformly bounded in H�1(R). LetK denote the ompa t support of 2 C10 (R). We derive

Inverse Problem 57lk( ) = Z T00 ZR(ak � a) uky dy dT = Z T00 ZK(ak � a) uky dy dT� kak � ak1;K kuy(ak)kC(0;T0;L2(K)) k k1 T0� C T0 kak � ak1;K kuy(ak)k� k k1Sin e ak 2 C1=2(R), ak ! a for 0 < � < 1=2 and thus lk( )! 0. The se ondterm of (4.6) again de�nes a bounded linear fun tional a ting on uk and for 2 C10 (R) with ompa t support K we havefa; (uk) = Z T00 ZR a uky y dy dT = Z T00 ZK a uky y dy dTk!1! Z T00 ZK a uy y dy dT = fa; (a)Sin e C10 (R) is dense in H1(R) this result holds for every 2 H1(R) by theBana h-Steinhaus Theorem.Lemma 4.3. Let am 2 K be bounded in X and let um denote the orre-sponding solutions to Problem 1, whi h are uniformly bounded a ording toproposition 2.5. Then there exist a subsequen e an and a 2 X , u 2 � su hthat an *� a; any * ay and un * u in �Additionally, the limit of every su h onvergent subsequen e satis�es u(a) =u.Proof. Lemma 4.1 and the re exivity of � yield the existen e of a 2 K, u 2 �and of the orresponding subsequen es. It remains to show u(a) = u:We have 8T 2 (0; T0)ZR un0 � dy = ZR un(T )� dy � Z T0 dd� ZR un(�)� dy d�= ZR un(T )� dy +�(an) + Z T0 ZR(b uny + un)� dy d� = �

Inverse Problem 58The �rst and third term on the right hand side are ontinuous linear fun -tionals on �. Together with the previous lemma we obtain� = ZR un(T )� dy +�(a) + Z T0 ZR �uny + un �� dy d�= ZR u(T )� dy � Z T0 dd� ZR u(�)� dy d�Hen e u solves Problem 1 with parameter a. By the uniqueness of thissolution (see proposition 2.5) we have u = u(a).Theorem 4.1. Let an 2 K denote a minimizing sequen e of (4.5). Thenthere exists a subsequen e am of an and a 2 K su h thatam *� a in L1 and amy * ay in H1and the limit of every su h onvergent subsequen e is a minimizer of (4.5).Proof. In view of lemma 4.3 it remains to establish that a is a minimizer of(4.5), whi h follows by the weak lower semi- ontinuity of the fun tional (4.5)J�(a) � lim infm!1J�(am)Let Jinf = infq2KfJ�(q)g, and w.l.o.g. am su h that J�(am) � Jinf + 1m .Then 8q 2 K J�(a) � lim infm!1J�(am) � J�(q)4.3 StabilityBy stability we mean that for a �xed parameter � the minimizer of Problem 2depends ontinuously on the data zÆ, i.e. that the in uen e of perturbationsin the data zÆ on the minimizers aÆ of Problem 2 an be ontrolled by thenoise level Æ. Sin e a minimizer may not be unique, the result holds in aset-valued sense ( f. e.g. [11℄).

Inverse Problem 59Theorem 4.2. For any � > 0 let fzng be a sequen e su h that zn ! zÆ inL2(T0 � �; T0;L2(R)). Let fang denote the minimizers of Problem 2 with zrepla ed by zn. Then there exists a subsequen e of famg and aÆ 2 K su hthat amy ! aÆy in H1(R); am ! aÆ in L1(R)and for every ompa t subset ! � Ram ! aÆ in H2(!)The limit of every su h onvergent subsequen e is a minimizer ofJzÆ� (a) = kua � zÆk20;� + �ka� a�k2� (4.7)Proof. Let again un denote the orresponding solution to an of problem 1. Byde�nition Jzn� (an) � Jzn� (a) for every a 2 K, espe ially for a � a0. Thereforkan � a�k� is uniformly bounded byk(an � a�)yk20 � 1�Jzn� (a) � 1�ku� � zÆk2� + ka�k2�and lemma 4.3 yields the existen e of aÆ 2 K, uÆ 2 � and the orrespondingsubsequen es am, um. Proposition 4.1 givesJzÆ� (aÆ) � lim infm!1JzÆ� (am) � limm!1 kzm � zÆk20;� + lim infm!1Jzm� (am)� lim infm!1Jzm� (a) � limm!1 kzm � zÆk20;� + JzÆ� (a) = JzÆ� (a)for all a 2 K. Hen e aÆ is a minimizer of (4.7).Assume amy 9 ay in H1(R), then there is a subsequen e ak and � > 0 su hthat kak � a�k2� � ka� a�k2� ! �Otherwiseka� akk2� = ka� a�k2� + kak � a�k2� � 2ha� a�; ak � a�i�! 0

Inverse Problem 60for k !1. We obtainkuÆ � zÆk20;� � lim infk!1 kuk � zkk20;�= lim infk!1(Jzk� (ak)� kak � a�k2�) � JzÆ� (aÆ)� lim infk!1 kak � a�k2�= kuÆ � zÆk20;� � lim infk!1(kak � a�k2� � kaÆ � a�k2�)� kuÆ � zÆk20;� � �2whi h is a ontradi tion. For ompa t sets ! � R the embedding L1(!)!L2(!) is ontinuous, thus we have am * a in L2. and by the ompa tembedding H1(!)! L2(!) that am ! a in H2(!).4.4 Convergen eSo far we ould �nd a minimizer to Problem 2 for a given noise level Æ and�xed regularization parameter � and show, that the solution aÆ depends ina stable way on the noisy data zÆ. Now we study the behaviour as � and Æboth tend to zero under the assumption that the data z are attainable, thatmeans that an a�-MNS ay with orresponding solution uy = z exists. Notethat the onvergen e rate of � and Æ an not be hosen independently, buta ording to an a-priori parameter hoi e rule.Theorem 4.3. Let zÆ 2 L2(T0 � �; T0;L2(R)) with kuy � zÆk0;� � Æ and�(Æ) su h that �(Æ) ! 0 and Æ2=�(Æ) ! 0 as Æ ! 0. Then every sequen efaÆk�kg, where Æk ! 0, �k = �(Æk) and aÆk�k is a solution to Problem 2, has a onvergent subsequen e famg in the sense amy ! ay in H1(R) and am ! a inH2(!) for every ompa t ! � R.Proof. By the de�nition of ak it is lear that Jzk�k(ak) � Jzk�k(ay), whi h indu esJzk�k (ak) = kuk � zkk20;� + �k kak � a�k2� (4.8)� kuy � zkk20;� + �k kay � a�k2� � Æ2k + �k kay � a�k2� (4.9)

Inverse Problem 61�k; Æk ! 0 imply kuk�zkk0;� ! 0 as k!1. By use of the triangle inequalitywe see that kuk � uyk20;� � kuk � zkk20;� + kzk � uyk20;� ! 0 (4.10)and thus uk ! uy in L2(T0 � �; T0;L2(R)). Skipping the �rst term in (4.8)and dividing by �k yields immediatelykak � a�k2� � Æ2k�k + kay � a�k2�and hen e lim supk!1 kak � a�k2� � kay � a�k2� (4.11)The sequen es fakyg and fukg are bounded and lemma 4.3 yields the existen eof a, u and the orresponding subsequen es and u = u(a). By the weak lowersemi- ontinuity of normsku� uyk0;� � lim infk!1 kuk � uyk0;�and (4.10) we know that u = uy on the interval (T0 � �; T0). Now ay is aminimum-norm-solution, and we getkay � a�k� � ka� a�k� � lim infn!1 kan � a�k�� lim supn!1 kan � a�)k� � kay � a�k� (4.12)and thus ka� a�k� = kay � a�k�. Hen e a is an a�-minimum-norm-solution.4.5 Convergen e RatesWe already know, that the estimated solution aÆ of the inverse problem onverges to ay in H2(!) on ompa t ! � R, if the noise level does so andthe regularization parameter �(Æ) is hosen in the right way. We will shownow that under additional assumptions on the true parameter ay, we havekuÆ � zk0;� = O(Æ) and kaÆ � a�k� = O(pÆ)

Inverse Problem 62In [9℄ these results are shown for abstra t problems with full norm regular-ization. We use the ideas of [9℄ and [12℄ to derive the same results (for thesemi-norm regularization) under weaker assumptions.We want to re all that a is just de�ned by a = �22 and thus all the resultsare valid analogously for �.1. The H2- onvergen e for a in Theorem 4.3 is only valid on boundeddomains, sin e only a semi-norm is used for regularization. It seemsadequate to laim that the parameter is known outside of a ompa tdomain !. Thus we ould adjust the set of admissible parameters toK�! = fq 2 K j q � a� = 0 in R � !gfor a given a-priori hoi e a�. The previous results arry over, if werepla e K by K�!. Moreover, the onvergen e of the subsequen es inTheorem 4.2, 4.3 is now strong in H2(!) and thus strong in X orH2(R) respe tively.2. For a 2 K�! we have a � a� 2 H20 (!) and a � a� = 0 outside !. Nowby Friedri hs' inequality, the norm kqkX an be estimated by the semi-norm kqyk0 ka� a�k� � ka� a�k2 � C ka� a�kXThe last inequality follows by the ontinuous embedding H2(R) !C1(R). The proof of the following theorem does not need su h anassumption, but, as we will see later, ay � a� must have very rapidde ay, the true parameter ay must be almost perfe tly known for jyjlarge.3. We will use the preliminary onsiderations of se tion 3.5 on the parameter-to-solution map F . Re all also the de�nition of F and its adjoint F�(see (3.22), (3.23)):hF�(a)�; pi = Z T0T0�� ZR�p u(a)y �y� (3p+py) u(a)y �+2p u(a)� dy dT(4.13)

Inverse Problem 63In the sequel we use the notation(a; b)� := (ay; by)0 + (ayy; byy)1 and as before kak� =p(a; a)�Theorem 4.4. Let ay 2 K and assume that there exists a fun tion� 2 H10 (T0 � �; T0;L2(R)) \ L2(T0 � �; T0;H2(R))su h that hF(ay)��; pi = (ay � a�; p)� (4.14)for all p 2 X . Then, with � � Æ, we havekaÆ� � ayk� = O(pÆ)Z T0T0�� kuÆ� � uyk20 dT = O(Æ2)Proof. For onvenien e we skip the � in the notation of aÆ�. By de�nition ofaÆ we have JzÆ� (aÆ) � JzÆ� (ay), whi h isZ T0T0�� ku(aÆ)� uyk20 dT + � kay � a�k2� � Æ2 + � kaÆ � a�k2�adding the term � (�kaÆ � a�k2� + kaÆ � ayk2�) on both sides yieldsZ T0T0�� ku(aÆ)� uyk20 dT + � kaÆ � ayk2�� Æ2 + � (kay � a�k2� � kaÆ � a�k2� + kaÆ � ayk2�)= Æ2 + 2 �(ay � a�; ay � aÆ)�We use the de�nition of the operator F� (see (4.13)) and the sour e ondition(4.14) to rewrite the last term as� (ay � a�; ay � aÆ)� = � hF(ay)� �; (ay � aÆ)i= � Z T0T0�� ZR�(ay � aÆ) uyy �y � (3(ay � aÆ) + (ay � aÆ)y) uyy �+2(ay � aÆ) uy � dy dT = �

Inverse Problem 64Now using the di�erential equation for uy and uÆ to eliminate the terms ontaining ay we get� = � Z T0T0�� ZR aÆ(uyy � uÆ)y �y + (3aÆ + aÆy) (uy � uÆ)y ��2aÆ(uy � uÆ)� dy dT � � Z T0T0�� ZR(uy � uÆ)�t+b ((uy � uÆ))y �+ ( � 2q) (uy � uÆ)� dy dT = �By integration by parts we eliminate terms ontaining derivatives of (uÆ�uy)� = � Z T0T0�� ZR(uy � uÆ) [��t + (aÆ �)yy+ ((b + 3aÆ + aÆy)�)y + ( � 2aÆ)�℄ dy dT = �Young's inequality yields� � � kuy � uÆk20;�+�24� k � �t + (aÆ �)yy + ((b+ 3aÆ + aÆy)�)y + ( � 2aÆ)�k20;� = �For � < 1 the �rst term an be shifted to the left hand side. The se ond termhas a leading �2, whi h is of the order of Æ2 by our hoi e.Note that by assumption � and aÆ are smooth enough su h that all termsexist in L2(T0 � �; T0;L2(R))and are bounded. Summarizing, this yieldsZ T0T0�� ku(aÆ)� uyk20 dT � 11� � Æ24�C(�)where 0 < � < 1, and C depends on � and the uniform bounds for theparameters aÆ only.

Inverse Problem 65Remark 4.1. By proposition 3.4 we already know that u(a) 2 C([0; T0℄;H1(R)),hen e even pointwise observation (in state and time) makes sense. The re-sults of Theorem 4.1, 4.2, 4.3 arry over and their proofs may be applied, ifwe repla e the �rst term in the obje tive fun tional byju�zj20;� = mXi=1 wi Z T0T0�� u(yi; T )�zi(T )j2 dT or ju�zj0 = mXi=1 ju(xi; T0)�zij2The following theorem provides onvergen e rates also for the ase of obser-vations dis rete in time and ontinuous in state . We therefore onsider themapping F : K ! L2(R)a ! u(a;T0)and regularize by a full norm term� ka� a�k2Note that F = P F with the (by proposition 2.2 ontinuous) mappingP : � ! L2(R)a 7! ua(�; T0)F 0 again satis�es a Lips hitz ondition of the formkF 0(ay)� F 0(a)kL(X ;L2(R)) � kay � ak2X (4.15)Sin e F 0(a) is linear and ontinuous, so is its adjoint (with respe t to L2(R)and X ) F 0(a)�.Theorem 4.5. Let zÆ 2 L2(R) with kuy(T0)� zÆk0 � Æ and let ay 2 K be ana�-minimum norm solution. Assume that there exists w 2 L2(R) su h thatay � a� = [F 0(ay)℄�

Inverse Problem 66satisfying the smallness ondition kwk0 < 1. Then for the hoi e � � Æ wehave kaÆ� � ayk2 = O(pÆ)and ku(aÆ�)(T0)� uy(T0)k0 = O(Æ)The result follows dire tly from Theorem 10.4 in [9℄.For the sake of ompleteness, we mention on e again that a was simplyde�ned as a = �22 , y = ln(K), x = ln(S), V (K; T ) = S2K u(ln(K); T ) is theBla k-S holes option value of the European Call with expiry T , strike K andspot pri e S of the underlying. The analogous results for V (K; T ) and �2follow by introdu ing new norms:Let ZÆ(K) be the noisy option values for di�erent strikes K. ZÆ(K) is relatedto zÆ(y) again by ZÆ(K) = S2K zÆ(ln(K)). The ondition zÆ 2 L2 is then agrowth ondition on ZÆ. Note, that U(K; T ) is always bounded (U(K; T =0) = (S � K)+ � S) for K ! 0 and has exponential de ay for K ! 1, sothat this assumption will always be satis�ed by reasonable data. The errorlevel of the data has simply to be measured in a spe ial norm:kU(�; T0 � ZÆ)k2+ = ZR+ K2S4 (ZÆ(ln(K)� U y(ln(K); T0)2 1K dKApplying the substitution and hain rule also for the term kak2 giveskak22 = ZR a2 + a2y + a2yy dy= ZR+((�2)2 + (2K � �K)2 + (2K � �K + 2K2 �2K + 2K2 � �KK)2 12K dK=: k�2k2�The rates orresponding to Theorem 4.5 for �2 and U Æ are thenk(�Æ�)2 � (�y)2k2� = (O)(Æ)and kU(�Æ�)(T0)� U y(T0)k+ = O(Æ)

Inverse Problem 674.6 Remarks on the Convergen e Rate Re-sultsWe dis uss here very brie y some aspe ts on erning an interpretation of thesour e ondition of Theorem 4.4. If we rewrite the sour e ondition in itsstrong formulation, we get feeling whi h onditions may have in uen e onthe solvability of this equation.1. We rewrite� Z T0T0��(puyy); �y)0 + ((3 p+ py) uyy; �) dT = Z T0T0��((p (uyy � 3uy); �)0 dTwhi h is possible, sin e uy 2 L2(0; T0;H2(R)) by proposition 3.4.2. Assume, that ay is known outside a bounded interval, i.e. ay � a� 2H20 (!). Then by integration by parts we have(ay � a�; p)� = �(�2y (ay � a�); p)0;! + (�4y (ay � a�); p)0;!= ((�4y � �2y) (ay � a�); p)0;!where (u; v)0;! = R! u v dy. This equality holds, if also p 2 H20 (!).Note that p will be of the form aÆ � a�, su h that this orresponds toa setting a 2 K�! as des ribed above.3. We note, that by the improved regularity of uy (see proposition 3.4)and � the produ t Z T0T0��(uyyy � 3 uy)� dTis in L2(R), thus the equalityZ T0T0��(uyyy � 3 uy)� dT = (�4y � �2y) (ay � a�)makes sense in L2(R).

Inverse Problem 684. Obviously, one has to guarantee, that uyyy � 3 uy does not vanish for agiven y for all t 2 [T0��; T ℄. In the original equation (3.1) this amountsto ukk not vanishing for a given k on [T0��; T ℄, whi h is quite natural,for then the solution of the di�erential equation does not depend onthe value of the parameter at k.5. In the proof of Proposition 3.2 we have shown that the derivatives ofu(a) tend exponentially to zero as jyj ! 1. Hen e the sour e ondition an be ful�lled by a bounded � only, if also ay�a� (i.e. their derivatives)show this behaviour.6. In Theorem 4.5 a full norm regularization is needed. In the ase, whereay�a� vanishes outside a bounded domain, k(ay�a�)yk1 and kay�a�k2are equivalent by Friedri hs' inequality.7. Theorem 4.4 does not need a smallness ondition. Sin e in general aminimum norm solution for a nonlinear problem may not be unique, thesmallness ondition plays the role of a sele tion riterion and guarantees onvergen e to the right solution. The fa t, that this is not ne essary forTheorem 4.4 suggests, that a minimum norm solution for this paraboli problem is in fa t unique, if the sour e ondition is ful�lled. In [27℄the authors prove a uniqueness result for this identi� ation problem instrong formulation for the use of a spe ial norm.8. In [12℄ the authors an even onstru t a solution � satisfying the sour e ondition for the similar problem�ut +r(aru) = f on for the ase � R1 , where they use krak0 for regularization.

4.7 GeneralizationsSo far, we restri ted ourselves to the ase of a volatility onstant in timeand one maturity date. A generalization to several expiries and a volatilitypie ewise onstant in time is intuitive:

Inverse Problem 69Let T1 � ::: � Tn be given. For a1; :::; an 2 K let u1; :::; un denote thesolutions of problem 1, where ui(a) is the solution with expiry T0 substitutedby Ti and the parameter a is given bya(y; t)j(Ti�1;Ti) = ai(y)Let �n = nNi=1�(Ti), with �(Ti) de�ned like � on the time interval (0; Ti).The properties of � and X arry over dire tly to the produ t spa es �n andKn, if they are provided with the natural normskuk2�n = nXi=1 kuik2�; kak2Kn = nXi=1 kaik2K;Let u(a) = (u1; :::; un) 2 �n denote the solution ve tor to the parameterve tor a = (a1; :::; an) 2 Kn and the expiries (T1; :::; Tn). In analogy to the ase of one maturity, we de�ne the normskjukj20;� = nXi=0 Z TiTi�� kui(�; T )k20 dTfor u = (u1; :::; un) 2 �n and kjakj20 = nXi=1 kaik20for a = (a1; :::; an) 2 Kn.For given � > 0 and zi 2 L2(Ti � �; Ti;L2(R)), z = (z1; :::; zn) and a-priori hoi e a� = (a�1; :::; a�n), we de�ne the obje tive fun tionalJz�(a) = kju(a)� zkj20;� + � kj�y (a� a�)kj20as before. By repla ing the norms k�k0;� of Theorems 4.1, 4.2, 4.3 by kj�kj0;�,the proofs arry over verbatim. The analogon to lemma 4.3 an be shownfor every omponent of u = (u1; :::; un) seperately in the same way.

Chapter 5Numeri al RealizationIn this se tion we onsider the problem of numeri ally minimizingf(a) = ju(a)� zj2 + � ja� a�j2� (5.1)for a given regularization parameter � and given (noisy) data z, where u(a)denotes the solution to �uT +A(a)u = 0 (5.2)with initial value u0. We restri t ourselves here to the hoi ejuj2 = Z ju(y; T0)j2 dy or juj2 =Xi ju(yi; T0)j2and hoose for the regularizationkqk2� = �0 kqk20 + �1 jqj21 + �2 jqj22with �0 � 0, �1;2 > 0.Usually only dis rete data zi (the spot pri es of options with di�erent strikesand maturities) will be available, but if many observations are possible, onemight onsider interpolation or approximation of the dis rete data and thentreat the data as ontinuously given. The error introdu ed by this approxi-mation must be onsidered additionally.70

Numeri al Tests 71Solving (5.2) for given parameter a = �22 orresponds to al ulating optionvalues for a given volatility. In �nan e, several methods to do this numeri allyare known (e.g. binomial or polynomial trees, FEM and FDM,...). We willuse here a FEM s heme for dis retization of the state and a Crank-Ni holsons heme for time integration as proposed in [20℄. For an overview over severalfrequently used methods we refer to [28℄.As proposed in [20, 28℄ we restri t the state domain to a �nite interval, hen ewe approximate the dire t problem (5.2) by�uT + A(a) u = 0 on (m0; m1)� (0; T0) (5.3)and introdu e the boundary onditionsu(m0; T ) = g0(T ) and u(m1; T ) = g1(T ) 0 � T � T0The hoi e of the boundary values g0 and g1 has to be made su h that theerror introdu ed by this approximation is suÆ iently small. We will showlater, how this an be a hieved.For minimizing (5.1) we use a Quasi Newton method, in parti ular we on-sider a BFGS algorithm with line sear h as proposed in [14℄.The dis retization of the parameter a will be hosen independently of that ofthe PDE (5.3). We represent the parameter as a pie ewise ubi spline. Thisallows us to al ulate derivatives of the parameter expli itly. Additionally,we use only few degrees of freedom (dof's) for the parameter, that is a veryrough dis retization. This an be motivated by the following onsiderations:� option values ontain information about expe ted returns of portfo-lios, expe ted evolution of sto k values,... Thus the urrent values arein uen ed by the expe tation of a future volatility. Building the expe -tation is a smoothing pro ess, hen e it is reasonable to expe t assumea smooth volatility.

Numeri al Tests 72� It seems to be rational to expe t similar u tuations in the sto k valuesfor similar sto k values. Hen e high os illations in (the expe ted) �(S)seem to be unnatural.The restri tion to only a few dof's redu es the dimension of the optimizationproblem (5.1) and may be onsidered as regularization by itself.

5.1 Restri tion to Bounded DomainsIf we approximate the dual Bla k-S holes equation in weak formulation (3.12)by an equation on a bounded domain n � R (e.g. n = (�n; n)) withsuitable boundary onditions, for instan eu(�n; �) = 0; u(n; �) = 0the results of the previous se tion an be applied immediately to the re-stri ted problem. In this ase X is just repla ed by H2(n) and � by itsbounded analogon.Now, onsider equation 3.10uT + a uyy + (q � r � 3a) uy + (2a� 2q + r) u = 0 (5.4)with initial value u0(y) = ey�2x (ex � ey)+ (remember: ex = S, ey = K).In the proof of Proposition 3.2 we already showed that the solution to thisproblem has exponential de ay.We approximate this system by restri tion to a bounded interval n =(�n; n) . For n 2 N , let an; qn; rn be the restri tions of the parametersa, q, r to n and un0 = (u0 �mn)jn with a fun tion mn of the form

Numeri al Tests 73mn(y) = 8>>><>>>: y + n y 2 [�n;�n + 1℄1 y 2 (�n + 1; n� 1)y � n y 2 [n� 1; n℄0 elseLet un denote the solutions of the restri ted problem with homogeneousboundary onditions and initial values un0 . The di�eren e u � un obviouslysolves the problemvT + a vyy + (q � r � 3a) vy + (2a� 2q + r) v = 0 (5.5)for y 2 n, T 2 (0; T0) with boundary onditionsv(m0; T ) = u(m0; T ) v(m1; T ) = u(m1; T )and initial ondition v(y; 0) = [1�m(y)℄ u0(y)If the parameters are smooth, su h that the solution exists in the lassi alsense, it follows by the maximum prin iple that v � 0 and v � C max(y;T )2�fv(y; T )g,where � = f(y; 0) j y 2 [m0; m1℄g [ f(m0; T ); (m1; T ) jT 2 (0; T0)gand C depends only on the bounds of the oeÆ ients of the PDE and T0.Hen e v � 1 e� 2 n, that is un(T0)j ! u(T0)j in L2() for any bounded � R. The restri tion to a bounded domain an therefor be treated as errorin the data.5.2 Crank-Ni holson FEM S hemeWe onsider (3.8) restri ted to a bounded domain in variational formulation(uT ; �) + (a ux; �x) + ((a+ ax + r � q) ux; �) + (q u; �) = 08� 2 H10 (); 0 < T < T0 and for = (m0; m1) with given boundary andinitial data

Numeri al Tests 74u(m0; T ) = g0(T ) u(m1; T ) = g1(T ) u(x; 0) = u0(x)A semi-dis retization is performed using Galerkin's method for the state oordinate. The spa es V = H1() and V0 = H10 () are approximated bythe �nite dimensional subspa es Vn and Vn0 , where Vn is spanned by the Basisfun tions f�ni gi=1;::;2n+1. More pre isely, let fm0 = x0 < x1 < ::: < x2n =m1g be the spatial mesh. By linear transformation the ith element [y2i�2; yi℄ an be transformed to the standard element [�1; 1℄. In lo al oordinates �we de�ne the basis fun tionsN1 = �12(1� �) N2 = 1� �2 N3 = 12(1 + �)The orresponding basis f�igi=0;2n is onstru ted by retransformation andspans the 2n+1-dimensional subspa e Vn ofH1(). Vn0 := span(f�igi=1;2n�1)is the orresponding subspa e of H10 (). A fun tion u 2 Vn is representedby its oordinate ve tor u su h thatu = 2nXi=0 ui � �i = g0 � �0 + 2n�1Xi=1 ui � �i + g1 � �2nHere the inhomogeneous solution u is split up into u = v + g, with g =g0 �0 + g1 �2n and v =P2n�1i=1 ui �i 2 V0.The semi-dis retized equation yields a system of ODE'sMvT +K v = f (5.6)where M and K(T ) we denote the mass respe tively sti�ness matrixMij = (�i; �j)Kij = (a �jy; �iy) + ((a+ ay + r � q)�jy; �i) + (q �j; �i)for i = 1; ::; 2n� 1. (u; v) here denotes the s alar produ t of L2()The right-hand side f is de�ned byf = (gt; �i) + (a gy; �iy) + ((a+ ay + r � q) gy; �i)� (r g; �i)

Numeri al Tests 75The system (5.6) is now dis retized by a Runge-Kutta s heme. We introdu ethe temporal mesh f0 = t0 < t1 < ::: < tN = T0g and get(M + dtj � Kj+1)uj+1 = (M � dtj (1� �)Kj+1)uj+1+ dtj (� f j+1+ (1� �) f j)(5.7)For the hoi e � = 12 we get the orresponding Crank-Ni holson s heme whi his of se ond order a ura y in time. For � � 12 , the s hemes are absolutelystable in L2 and maximum norm. For details on this method and erroranalysis, we refer to [34℄.By the lo al support of the basis fun tions, the matrix (M + dtj � Kj+1)will be penta-diagonal and hen e the systems (5.7) an be solved with O(n)operations by LU -de omposition.5.3 Quasi-NewtonMethod and BFGS-AlgorithmQuasi-Newton methods provide a way of gaining super linear onvergen eproperties for well-posed optimization problems like (5.1) without having to al ulate the Hessian of the obje tive fun tional in ea h step. Together witha line sear h algorithm, global onvergen e an be proven under adequate onditions. To determine an approximation for the Hessian only gradientinformation (whi h must be provided for any des ent algorithm anyway) isneeded. There are di�erent ways to use this information for updating theapproximate Hessian. In general we have an algorithm of the following kind:while onvergen e not a hievedpk = �Bk�1gkxk+1 = xk + �kpk for an appropriate step length �sk = xk+1 � xk; yk = gk+1 � gkBk+1 = B(Bk; sk; yk);k = k + 1;end

Numeri al Tests 76where x denotes here the free variable. The version we use for our tests isthe BFGS method. An update is performed a ording toBk+1 = B � 1stB s B s stB + 1yt s y ytwith y = gk+1 � gk and s = xk+1 � xk. For details we refer to [14℄.To guarantee that the restri tion on the parameter (x � x0 > 0) is satis�ed,we proje t the gradients su h that xk is always in the onvex set fx �x0g. In our numeri al test, the restri tions were usually not met during theoptimization. To guarantee a global onvergen e of the algorithm, we resetthe guess for the Hessian Bk every n steps, where n is the number of degreesof freedom for the parameter (the problem dimension). Sin e we deal witha non quadrati problem, it annot be guaranteed, that the Hessian stayspositive de�nite during the iteration. If Bk is lose to singular or inde�nit,the reset is performed in a similar way.As a stopping rule, we limit the maximum number of iterations.5.4 Gradients and the Adjoint Approa hIn order to perform any des ent algorithm, one has to al ulate derivativesof the obje tive fun tional. In our ase, the dependen e of the obje tivefun tion f on the parameter a is given by the di�erential equation (5.2).Applying the hain rule formally gives the the dire tional derivative f 0[p℄ indire tion p dda(f(a; u(a))[p℄ = fu(a) u0(a)p+ fa(a)pwhere u0p is the dire tional derivative of u in dire tion p, given byu0(a)p = limt!0 1t (u(a+ tp)� u(a))Now there are several ways to ompute this derivative

Numeri al Tests 771. Finite di�eren es: Just approximateu0(a)p ' 1t0 (u(a+ t0p)� u(a))for a given t0 > 0. If u is in fa t Gateaux di�erentiable in dire tion p,we get an error estimateu0(a)p ' 1t0 (u(a+ t0p)� u(a)) + o(t0 kpk)2. Dire tional derivative: Let u be given as the solution to�uT + A(a) u = 0with a di�erential operator A(a) dependent on the parameter a andgiven initial and boundary values. Formally, the dire tional derivativeu0p then satis�es the following equation:�(u0(a)p)T + A(a) (u0(a)p) = �(A0(a)p)u(a) (5.8)with homogeneous initial and boundary onditions. This may be veri-�ed by plugging into the de�nition of the dire tional derivative.3. Adjoint approa h: Consider the ase, that we are not interested in u0pitself but in lw((u0p)), where lw is a given linear fun tional, e.g.lw(u0p) = (w; u0p(T ))with (�; �) denoting the standard L2 s alar produ t. Using the di�eren-tial equation (5.2), we get for a � 2 � with �(T0) = w(w; u0p(T0)) = (�(T0); u0p(T0)) = Z T00 ddT (�(T ); u0p(T )) dT + (�(0); u0p(0))= Z T00 h�T ; u0pi+ h�; (u0p)T i dT= Z T00 h�T ; u0pi+ h�;A(a)u0p+ [A0(a)p℄ u(a)i = (�)

Numeri al Tests 78Now let � be a solution to the equation�T + A�� = 0with terminal value w and A� the adjoint of A, then we have(�) = Z T00 hA0[p℄ u(a); �i dTHen e to al ulate the gradient of the obje tive fun tional f(a) we� solve the adjoint problem (ba kwards in time)�T + A�� = 0with terminal value �jT0 = (u(a)jT0 � z).� for every dire tion p al ulate the derivative byf 0(a)[p℄ = 2 (u(a;T0)� z; u0p(a;T0))= lw(u0p) = 2 Z T00 hA0[p℄ u(a); �i dTFor the �rst two possibilities of al ulating the derivatives, one solution ofthe PDE is ne essary for ea h dire tion p. The e�ort grows rapidly with thenumber of dof's for the parameter. The adjoint method, in this spe ial ase,only needs solving the di�erential equation one time for a gradient evaluation.We mention here, that derivatives may not always exist. Consider the ase ofthe Bla k S holes equation with given dis rete (in time and state) data. Theobje tive fun tional will then, despite of the regularization term, be given byf(a) =Xi ju(xi; T0)� zij2The derivative of f in dire tion p isf 0p = 2Xi (u(xi; T0)� zi) � (u0p)(xi; T0)

Numeri al Tests 79In our framework, we an only guarantee u0p 2 L2(0; T ;H1(R)). Hen eu0p(xi; T0) may not even make sense. But even if u0p is regular enough toallow pointwise evaluation, the adjoint problem may not be well-de�ned,sin e the initial value would beXi (u(xi; T0)� zi) � Æxiwhi h is not in L2.For our numeri al experiments we mostly use the adjoint method. Dis retedata are interpolated by ontinuous fun tions. Finite di�eren es are usedon e to he k the in uen e of data-interpolation and to test if the adjointmethod is properly working.5.5 Spline-Representation of the ParameterWe dis uss very brie y the representation of the parameter as a ubi spline.We divide the state domain into m equidistant intervals and get the dis- retization fm0 = y0 < y1 < ::: < ym = m1g. The parameter a will berepresented as pie ewise ubi interpolating spline. Hen e, a is de�ned by itsvalues fa(yi)gi=0;::;m = faigi in the following way:� a(y)j[yi�1;yi℄ is a ubi polynomial,i.e.a(y) = s(i)(y) = (i)0 + (i)1 (y � yi�1) + (i)2 (y � yi�1)2 + (i)3 (y � yi�1)3for y 2 [yi�1; yi℄ and i = 1; ::; m.� a(y) is required to interpolate at yis(i)(yi�1) = ai�1 and s(i)(yi) = ai i = 1; :::; m� a(y) is two times ontinuously di�erentiable, that iss(i)y (yi) = s(i+1)y (yi) s(i)yy(yi) = s(i+1)yy (yi)for i = 1; ::; m� 1.

Numeri al Tests 80This gives 2m + 2(m � 1) equations for the 4m oeÆ ients (i)j , i = 1; ::; m,j = 0; ::; 3.� As additional onditions we hoose (1)2 = (m)2 = 0 (5.9)This determines the oeÆ ients of the ubi spline fun tion in a unique way.A ubi spline satisfying (5.9) is alled natural ubi spline and possesses thevariation diminishing property, i.e. it minimizes the urvature R m1m0 s2(y) dyunder all ubi spline fun tions with the same interpolation points. Fordetails and approximation properties see [31℄.The dis retized parameter a therefore is equivalently represented by its nodalvalues ai.In [36℄ Wilmott et al. suggest parametrizing � by a low order polynomial,i.e. a quadrati parabola �(S) = 0 + 1 S + 2 S2As we will see, the option values (even far o� the urrent spot value of theunderlying) depend mainly on the volatility near the spot pri e. Hen e theshape of the parameter there will mostly determine the urvature of thisparabola and some information of option values with strikes far o� the spotwill be lost. The spline representation has lo al properties and will be ableto resolve this sort of shapes better.5.6 Numeri al TestsWe set up the Crank-Ni holson-FEM s heme, the BFGS optimization algo-rithm and the spline evaluation in MATLAB. The speed of the MATLABFEM- ode for a rough dis retization is omparable to an implementation inC++. The advantage of the MATLAB implementation is the possibility ofusing the various MATLAB vizualisation tools.

Numeri al Tests 81To he k the integrity of the FEM ode, we ompare the results for onstantvolatility with the values gained by the Bla k-S holes formula (1.7). Thenwe try to re onstru t a unknown (for the moment onstant) volatility usingexa t ontinuous data.We set up a test example, and try to re onstru t in the ase of� exa t ontinuous data� exa t dis rete data: here we also try to interpolate the dis rete dataand he k the in uen e of this interpolation on the result. This willgive a justi� ation for using data interpolation for the ase of noisydata.� noisy data: We he k the in uen e of the regularization term. As wewill see, a non-regularized identi� ation will lead to os illating param-eters, even in the ase of only few dof's.� real data: We on lude the numeri al examples with the identi� ationof the volatility smile for real data, option values of the S&P500 index.The estimated volatility suÆ es to reprodu e the market values verya urately. Additionally, due to the regularization the vloatility smileis very smooth.To make the results of the various test examples omparable we make thefollowing settings throughout� S = 1, whi h orresponds to x = ln(S) = 0� m0 = �1, m1 = 1, this takes into a ount possible strikes of K 2[0:37; 2:72℄� Additionally, we set for simpli ity r = q = 0� T0 = 1 year to expiry� Boundary values g0 and g1 for u are determined in the following way:We use the Bla k-S holes formula to al ulate Bla k-S holes implied

Numeri al Tests 82volatilities for u(m0) = z0 and u(m1) = z1, where z0 and z1 are thedata for the lowest and highest possible strike. Then we use thesevolatilities to al ulate option values g0 = u(m0; T ) and g1 = u(m1; T )for T 2 [0; T0℄ again by the Bla k-S holes formula.For the FEM- al ulation, we hoose the following dis retization� 50 equally sized elements in spa e� uniform temporal mesh with 100 elementsBoth, the temporal and the spatial mesh size ould be hosen smaller nearthe point y = 0; T = 0, where the initial value has some sort of singularity.The parameter is represented as� natural ubi spline with 11 equally spa ed interpolation points. Thevalues of the parameter at the interpolation points are used as degreesof freedom.The results are presented again in terms of V and �. We just re allK = ey; S = exand V (K; T ) = V (ey; T ) = u(y; T ); �(K) = �(ey) =p2 a(y)

Example 1. Constant VolatilityWe onsider the dual Bla k-S holes equation�uT + �22 uyy + (q � r � �22 ) uy � q u = 0 (5.10)with initial value u0(y) = (S� ey)+ whi h is, a ording to the onsiderationsin Chapter 3, equivalent to the Bla k S holes equation (3.1). For onstant

Numeri al Tests 83parameters �, r and q = 0, given S, K the solution is given by the Bla k-S holes formula~u(y; T ) = V (S; 0;K; T ) = S �(d1)�K e�rT �(d2) (5.11)with the umulative normal distribution �(x),d1 = 1�p(T ) (ln SK + (r + �22 )T )d2 = d1 � �pT .We hoose � = 0:5, S = 1 and r = q = 0. In this ase the boundary valuesare given by g0 = ~u(�1; T ; � = 0:5) and g0 = ~u(�1; T ; � = 0:5).First, we he k the FEM ode, and ompare the numeri al with the analyti alresults. We list here the value at every 10th grid point.strike exa t (v) numeri al (u) error(u� v)0.368 0.635 0.635 0.0000000.449 0.558 0.558 -0.0000110.549 0.472 0.472 -0.0000210.670 0.378 0.378 -0.0000260.819 0.284 0.284 -0.0000261.000 0.197 0.197 -0.0000311.221 0.125 0.125 -0.0000131.492 0.072 0.072 -0.0000081.822 0.037 0.037 -0.0000042.226 0.017 0.017 -0.0000012.718 0.007 0.007 0.000000Table 1: dire t solverThe maximum error is less than 3:06 � 10�5, the maximum relativ error is8:62� 10�4.

Numeri al Tests 84The se ond part for this �rst example onsists of identifying the (unknown) onstant volatility � = 0:5. As initial guess we hoose �0 = 0:8. We list theoutput error after 0 and 20 iterations and ompare the results to the outputerror obtained by the guess � = 0:51.strike exa t (v) error0 error20 error�=0:510.368 0.635 0.04550 -0.00005 0.001350.449 0.558 0.02138 0.00004 0.000600.549 0.472 0.08291 -0.00004 0.002470.670 0.378 0.07099 -0.00006 0.002280.819 0.284 0.09432 0.00000 0.003201.000 0.197 0.11054 -0.00005 0.003821.221 0.125 0.11521 -0.00002 0.003911.492 0.072 0.10591 -0.00002 0.003391.822 0.037 0.00000 0.00000 0.000002.226 0.017 0.04758 0.00005 0.001332.718 0.007 0.00000 0.00000 0.00000Table 2: output error with re onstru ted volatility after 0, 10 and 20 iterationsAfter only 20 iterations, the maximum output error is redu ed to about 0:1%of the starting value and is far less than the output error obtained by the"good" guess � = 0:51. The error made by the bad identi� ation near theboundary is negligible.Figure 1 shows the estimated parameters after 0 and 20 iterations. Near thespot of the sto k (1) the volatility an be identi�ed quite well. The values atthe boundary, in ontrast, are highly determined by the initial guess. We willmake this observation throughout all examples. In the ase of regularization,the value of the parameter at the boundary will be determined by by theregularization and the value of the parameter near the spot.

Numeri al Tests 85

0.5 1 1.5 2 2.50.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9goalstartp20

Figure 1: identi�ed volatility after 20 iterations, K vs. �Before we ontinue with the numeri al tests for the ase of dis rete and noisydata, we want to make some remarks on this result and the identi�ability ofthe parameter a:� Sin e very high or very low sto k values are rea hed only with lowprobability, the values of the volatility there does not in uen e theoption value very mu h, i.e. the sensitivity of the option value at timeT0 does depend only very little on volatilities for sto k values far o�the spot (in our ase 1).� We restri ted ourselves to a bounded domain with Diri hlet boundary onditions. As we have seen in (5.8), the derivative of the option valuewith respe t to the volatility solves a PDE with homogeneous boundary onditions, i.e. the option value does not depend on the value of thevolatility at the boundary at all.� The stability and onvergen e results of the last se tion are based on ompa tness arguments and hen e do not have quantitative on lu-sions. In the proof of onvergen e rates, we needed a non vanishing

Numeri al Tests 86 ondition for a uyy + 3 a uy, whi h orresponds to uKK in the originalform of the Bla k S holes equation. We already noted the expli itformula for � � =s2 uT + (q � r)K uK � q uK2 uKK (5.12)whi h we will not use to identify � for the given reasons. Nevertheless,this formula gives an insight on the kind of ill posedness of our problem.Building the derivatives in (5.12) is moderately ill posed. But evensmall errors in uKK may have a huge e�e t on the solution, where uKKis lose to zero. To visualize the problem, we plot the denominator of(5.12) for di�erent values of � in �gure 2.

0 50 100 150 200 250 300−100

0

100

200

300

400

500

600

700

800σ=0.05σ=0.2 σ=0.5

Figure 2: K vs. K2 ukkThis ondition orresponds to the non-vanishing ondition mentionedin se tion 4.6. Due to the spe ial stru ture of our problem, we know,that uKK has exponential de ay in K. For noisy data, the parameter �therefore will be mainly determined by the regularization term and/orthe initial guess for large jS�Kj. We will see that behaviour in all ourtest examples.

Numeri al Tests 87We demonstrate on a very simple example that the sensitivity of the solutionu to the parameter values near the boundary of the onsidered domain is verylow. Hen e one an not expe t a good identi�ability there:Consider equation (5.10) on [�5; 5℄ with boundary onditions and initialvalue as above. Let r = q = 0 and ~a = �22 = 0:55 + 0:01 y, with y = ln(K).We parameterize the guess for ~a bya = a0 (5 + x)10 + a1 (5� x)10We use the FEM ode to al ulate the solution with the exa t parameter,and then for several hoi es for a0 and a1, and plot the value of the obje tivefun tional f(a) = ku� zk2where u = u(a; �; T ) and z = u(~a; �; T ). Obviously, the minimum of f will beattained at a0 = 0:6 and a1 = 0:5.

0.5

0.55

0.60.5

0.550.6

0

0.005

0.01

0.015

0.02

0.025

0.03

Figure 3: a0, a1 vs. residualIn this simple example we see, that f (the residual) is quite sensitive to thevalue of a0+a12 , whi h is the value of a at y = 0, but not at all on the spe ial

Numeri al Tests 88 hoi e a0; a1 (the values of a at the boundary), as long as their mean is asabove.The immediate onsequen es of this behaviour are:� The volatility estimated out of option values for European Call optionsare not very reliable for strikes far o� the spot value of the underlying.� In order to identify the volatility for high/low sto k values, one mustuse additional information, e.g. some a-priori guess for the volatility oroption values for options with high sensitivity on the volatility in theregion of interest.The next test example is on erned with identifying a variable volatility and omparing the ases of ontinuous, dis rete and noisy data. We set� �y = 0:4� 0:1 ar tan(ln(K)).� The data z = u(T ) are onstru ted with the FEM solver on the domain[�5; 5℄ and with �ne grid of 500� 200 elements.� the initial guess: �0 = 0:3The following tests are performed:1. The ase of ontinuous data is treated the same as in the exampleabove.2. For the ase of dis rete data, we use observations at every 10th grid-point and use the obje tive fun tionalf(u; z) = 10Xi=1 (u(yi; T0)� zi)2Then we prolongate the dis rete observations to a ontinuous fun tion,i.e. we aproximate the observations zi by a ubi spline with 9 dof'sand minimize the l2 error. Then we identify the parameter a ordingto 1. In Table 3 we list the exa t data and the values obtained by thisapproximation:

Numeri al Tests 89strike uexa t uapprox error0.368 0.634 0.634 -0.000010.449 0.557 0.557 0.000020.549 0.468 0.468 -0.000050.670 0.368 0.368 0.000050.819 0.261 0.261 -0.000041.000 0.157 0.157 0.000031.221 0.072 0.072 -0.000041.492 0.023 0.023 0.000051.822 0.005 0.005 -0.000052.226 0.001 0.001 0.000032.718 0.000 0.000 -0.00001Table 3: data approximationThe error introdu ed by this approximation is quite small and is in theorder of the total a ura y of the FEM ode.3. For noisy data we use the strategy of point 2; we �rst interpolate thedata and then identify the orresponding parameter. We add somearti� ial noise to the original data, i.e. we reate normal distributedrandom variables dz = N(0; v) with varian e v and add them to thedis rete data zi. We ompare the results obtained for di�erent noiselevels, i.e. diferent vallues of v.Example 2. Continuous and Dis rete Data - No NoiseHere we present the results of test 1 and 2. In Figure 5.6 the identi�edparameters are plotted.The di�eren e between the exa t ontinuous data and the approximated data,based on dis rete observations, hardly dete table. This indi ates, that theerror introdu ed by data-approximation an be negle ted.For omparison we list the al ulated option values with the exa t data inTable 4.

Numeri al Tests 90

0.5 1 1.5 2 2.50.25

0.3

0.35

0.4

0.45

0.5

0.55

goals

0s

conts

disc

Figure 4: ontinuous and dis rete data: K vs. �strike exa t z u on error on udis errordis0.368 0.634 0.634 0.00000 0.634 -0.000020.449 0.557 0.557 -0.00006 0.557 -0.000080.549 0.468 0.468 0.00000 0.468 0.000020.670 0.368 0.368 0.00004 0.368 0.000070.819 0.261 0.261 -0.00002 0.261 -0.000071.000 0.157 0.157 0.00000 0.157 0.000051.221 0.072 0.072 -0.00001 0.072 -0.000011.492 0.023 0.023 0.00003 0.023 -0.000011.822 0.005 0.005 -0.00003 0.005 0.000002.226 0.001 0.001 0.00002 0.001 0.000032.718 0.000 0.000 0.00000 0.000 -0.00001Table 4: ontinuous and dis rete dataFor both identi�ed volatilities, the al ulated option values mat h the exa tdata very well. The di�eren e in the results for dis rete and ontinuous data an be negle ted.

Numeri al Tests 91In the next example, we onsider the ase of noisy data. As in the previousexample, we approximate the dis rete data by a ontinuous fun tion.Example 3. Dis rete Noisy DataThe dis rete observations zi are arti� ially perturbed by normally distributedrandom variables dzi with mean 0 and varian e v. The varian e v respe tivelythe standard deviation sd = pv here plays the role of the error level. Wetest for the hoi es sd1 = 0:5%, sd2 = 0:1% and plot the identi�ed volatilitiesin �gure 5.6.

0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6 goals

0.5 %s

0.1 %

Figure 5: 0:5%, 0:1% data noise : K vs. �For a noiselevel of 0:5%, the estimated parameter is os illating and wouldbe ome less than 0:001 whi h is the lower bound �xed a-priori. Note, thatour experiment is already regularized by keeping the number of dof's for thespline low (The Tikhonov term is added in the next example). Note, that theestimated parameters vary a lot. In Table 5, we list the noisy data and theerrors of the di�eren es of the re onstru ted option values to the noise-freedata.

Numeri al Tests 92strike exa t z data0:5% error0:5% data0:1% error0:1%0.368 0.634 0.629 -0.00517 0.634 -0.000400.449 0.557 0.553 -0.00446 0.557 0.000160.549 0.468 0.466 -0.00205 0.468 0.000030.670 0.368 0.371 0.00345 0.367 -0.000760.819 0.261 0.267 0.00592 0.261 -0.000021.000 0.157 0.159 0.00203 0.158 0.000751.221 0.072 0.067 -0.00429 0.072 0.000801.492 0.023 0.024 0.00159 0.024 0.001221.822 0.005 0.008 0.00310 0.006 0.001072.226 0.001 0.002 0.00143 0.001 -0.000052.718 0.000 0.001 0.00116 0.002 0.00175Table 5: output error - noisy dataAlthough the re onstru ted option values mat h the noise-free data quitewell, the orresponding volatilities are not reliable at all. In the ase of 0:5%noise, the volatility would even be negative for some values of K. We alreadysaw this behaviour in the onsideration of the expli it formula (5.12).In the following example we test the in uen e of Tikhonov regularization.Therefore we add a term �1 kayk20 + �2 kayyk20to the obje tive fun tional.Example 4. Tikhonov RegularizationAs in the previous example, we add 0:5% noise to the dis rete data zi andinterpolate the dis rete observations as before. We list the results for various hoi es of the regularization parameter �, i.e. we hoose in ase 1 �1 =�2 = 1:0 � 10�2, in ase 2 �1 = 1:0 � 10�3, �2 = 1:0 � 10�4 and in ase 3�1 = �2 = 1:0� 10�6. We ompare the estimated parameters in �gure 5.6.The in uen e of the size of the regularization term is ru ial. If � is hosen toosmall, the noisy data are approximated very well. Therefore the parameter

Numeri al Tests 93

0.5 1 1.5 2 2.50.1

0.2

0.3

0.4

0.5

0.6

goals

1s

2s

3

Figure 6: in uen e of regularization : K vs. �has to vary a lot. A large value of � on the other hand makes it impossibleto extra t all possible information out of the data.strike no reg. ase 1 ase 20.368 0.00000 0.00000 0.000000.449 0.00025 0.00480 0.002970.549 -0.00043 0.00047 -0.000660.670 -0.00115 -0.00548 -0.002670.819 -0.00086 -0.00539 -0.000551.000 0.00063 0.00088 0.001011.221 0.00108 0.00524 -0.000201.492 0.00165 -0.00006 0.000281.822 0.00051 -0.00373 -0.000672.226 0.00014 -0.00228 -0.000812.718 0.00000 0.00000 0.00000Table 6: in uen e of regularization; Output errors for di�erent values of theregularization parameter �.Table 6 lists the output errors (re onstru ted option values - noisy data).The values for the di�erent values of � are of the same order of magnitude

Numeri al Tests 94but the regularized volatilities are mu h smoother. For the small regular-ization parameter, the approximation of the data is best, but therefore there onstru ted volatility shows the same behaviour as in the non regularized ase.Example 5. Real dataFinally we test the ode on real market data. The following option val-ues (in $) on S&P 500 Index options with expiry June, 2002, were foundunder www.big harts. om on 06/20/2001. The spot pri e of the sto k was1; 216:51 $.We just re all that a European Put option (with strike K and expiry T )provides the owner the right to sell a sto k at time T for K (dollar) and thePut-Call-Parity (see se tion 1.1).C + e�r (T�t0)K = P + Swhere we assume the risk free interest rate to be onstant over the timeinterval (t0; T ) and t0 is today. One an exploit this equality in two ways:� estimate the risk free interest rate r by regression over the quoted optionvalues� use Put values to al ulate more a urate Call values for strikes far outof the money, that is for S � K.

Numeri al Tests 95OPTION CHAIN FOR S&P 500 INDEX JUNE 2002Quote as of 01:24:29 PM on 06/20/2001CALL PUT CALL PUTbid ask strike bid ask397.20 399.20 850.00 10.50 12.00312.80 312.80 950.00 20.30 22.30275.60 275.60 995.00 26.30 28.30230.70 232.70 1,050.00 36.30 38.30193.60 195.60 1,100.00 47.20 49.20176.40 178.40 1,125.00 54.00 56.00159.40 161.40 1,150.00 61.10 63.10128.00 130.00 1,200.00 77.70 79.70113.80 115.80 1,225.00 87.50 89.50100.30 102.30 1,250.00 98.00 100.0077.20 79.20 1,300.00 122.90 124.9066.90 68.90 1,325.00 136.60 138.6057.00 59.00 1,350.00 150.80 152.80

bid ask strike bid ask39.80 41.80 1,400.00 181.70 183.7033.50 35.50 1,425.00 199.30 201.3027.50 29.50 1,450.00 217.30 219.3018.30 19.80 1,500.00 256.00 258.0014.40 15.90 1,525.00 276.10 278.1011.40 12.90 1,550.00 297.10 299.107.10 8.10 1,600.00 340.60 342.604.70 5.40 1,650.00 386.10 388.102.60 3.30 1,700.00 432.30 434.301.50 1.95 1,750.00 479.20 481.200.65 1.10 1,800.00 526.40 528.400.25 0.70 1,850.00 574.10 576.100.10 0.55 1,900.00 622.60 624.00sour e: www.big harts. omWe use T = T0 = 1:0 (years) and t0 = 0 get by regression r = 3:02%. Forthe parameter estimation algorithm, the data are again interpolated. Figure5.6 shows the estimated volatility for a regularization as in the previousexample with �1 = �2 = 1:0 � 10�6. The minimization is stopped after 25iterations. We ompare the value of the estimated volatility to the values ofBla k-S holes implied volatilities (see [29℄ and se tion 1.2).Both estimates are quite smooth, but the regularized output least squaresmethod is able to reprodu e the market data with mu h more a ura y. Welist the option values for di�erent strikes, the errors of the re onstru tedvalues using the interpolated Bla k-S holes implied volatility (see [29℄) andthe errors of the values re onstru ted by our Tikhonov regularized methodin Table 7.

Numeri al Tests 96

800 1000 1200 1400 1600 1800 20000.1

0.15

0.2

0.25

0.3

0.35

regularizedimplied

Figure 7: estimated and implied volatility (Shimko)strike all implied regularized850.00 398.20 0.000 0.000950.00 312.80 -3.978 -0.652995.00 275.60 -4.673 -0.5251050.00 231.70 -4.426 0.0891100.00 194.60 -3.980 0.2221125.00 177.40 -3.888 -0.0711150.00 160.40 -3.130 0.1641200.00 129.00 -1.484 0.4291225.00 114.80 -0.784 0.3181250.00 101.30 0.177 0.4261300.00 78.20 1.002 -0.4711325.00 67.90 1.541 -0.7391350.00 58.00 2.564 -0.447

strike all implied regularized1400.00 40.80 4.552 0.4081425.00 34.50 4.435 -0.0761450.00 28.50 4.742 0.0041500.00 19.05 4.794 0.0001525.00 15.15 4.883 0.2311550.00 12.15 4.592 0.1681600.00 7.60 3.915 0.1211650.00 5.05 2.718 -0.3451700.00 2.95 2.193 -0.1461750.00 1.72 1.599 -0.0761800.00 0.88 1.178 0.0981850.00 0.48 0.641 0.1061900.00 0.32 0.000 0.000Table 7: option values for implied and regularized volatilityIn our numeri al test examples we have shown that an output least squaresmethod is working well for the problem of identifying volatility smiles. In the ase of data noise (whi h will always be ontained in real data) one has toadd a regularization term to get reasonable results. With the regularized andvery smooth volatility we ould reprodu e option values with high a ura y

Numeri al Tests 97and mu h better than with interpolating implied volatilites as suggested in[29℄.

Chapter 6Con lusionsThe problem of identifying the volatility smile stru ture by aid of given optionvalues for di�erent strikes is a nonlinear ill posed problem, i.e. small errors inthe observations an ause arbitrarily large errors in the estimated volatility.This behaviour an already be seen in the analyti al formula� =s2 uT � (r � q)K uK + q uK2 uKKHere, the following problems o ur:� Di�erentiation of a fun tion is an ill posed problem� If only �nitely many strikes are available, � annot be determined forall K uniquely� By the spatial stru ture of our equation and the initial value, we know(by maximum prin iple) the the se ond derivative uKK tends exponen-tially to zero. Hen e the re onstru tion of � will be highly sensible todata noise.The dire t problem of al ulating the option values as fun tions of the strikeK was onsidered in a weak formulation.98

Con lusions 99In order to re onstru t the volatility in a stable way, we used an output leastsquares formulation with Tikhonov regularization term. For the resultingobje tive fun tional, we proved the existen e of a minimizer.We showed that the estimated volatility depends in a stable way on the dataz (the observed option values for di�erent strikes).Additionally, we ould show onvergen e and onvergen e rates: If the datauy are attainable (there exists �2y su h that uy = u(�2y)), if the noise levelgoes to zero (kuy � zÆk � Æ, Æ ! 0) and the regularization parameter is hosen in the right way, then a sequen e of minimizers �2Æn has a onvergentsubsequen e and every onvergent subsequen e onverges to a �2�-minimum-norm solution. Under the assumptions of Theorem 4.4k�2�;Æ � �2yk� = O(pÆ)Z T0T0�� kuÆ� � uyk20 dT = O(Æ2)We set up a FEM ode for solving the dire t problem and a BFGS algorithmto minimize the Tikhonov fun tionalku� zk2 + � k�2k�for �xed � in MATLAB. As long as � > 0 the minimization is a well-posedproblem.The dire t problem was restri ted to a bounded domain and the simulatedvalues ompared to the values gained by the Bla k-S holes formula, whi hwere reprodu ed very well.For the al ulation of derivatives we used an adjoint method.We tested the algorithm for dis rete and ontinuous data. Dis rete data wereapproximated by ubi splines. In the ase of exa t data, the volatility ouldbe re onstru ted very well near the spot. For noisy data, a identi� ation withno Tikhonov regularization lead to os illating volatilities, in spite of the fa tthat the number of degrees of freedom for the parameter was restri ted to10, whi h is regularization by itself. For a good hoi e of the regularization

Con lusions 100parameter �, the output error an be redu ed almost as well as in the non-regularized ase but the orresponding volatilities are mu h smoother and asmooth given volatility ould be re onstru ted very well.A generalization of the proposed method for a pie ewise onstant volatility intime is straightforward. To over ome the problem of bad identi�ablility faro� the strike, we suggest to use options of di�erent type (e.g. barrier options),whi h have larger sensitivities to the volatility for large/small values S of theunderlying.

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Bibliography 102[11℄ H. W. Engl, K. Kunis h, A. Neubauer, Convergen e Rates for TikhonovRegularization of Nonlinear Ill-posed Problems, Inverse Problems 5, 1989,523 - 540[12℄ H. W. Engl, Jun Zou, Stability and Convergen e Analysis of TikhonovRegularisation for Parameter Identi� ation in a Paraboli Equation, In-verse Problems 16(6), 2000, 1907 - 1924[13℄ L. C. Evans, Partial Di�erential Equations, Ameri an Mathemati al So- iety, 1991[14℄ R. Flet her, Pra ti al Methods of Optimization, J. Wiley & Sons, 1980[15℄ Avner Friedman, Partial Di�erential Equations of Paraboli Type, Pren-ti e Hall, 1964[16℄ T. C. Gard, Introdu tion to Sto hasti Di�erential Equations, NewYork, Basel: Dekker, 1988[17℄ I.I. Gihman, A.V. Skorohod, Sto hasti Di�erential Equations, Springer,1972[18℄ J. Hull, Options, Futures and Other Derivatives, Third Edition, Prenti eHall, 1997[19℄ H. Heuser, Funktionalanalysis, B. G. Teubner Stuttgart, 1992[20℄ N. Ja kson, E. S�uli, S. Howison, Computation of Deterministi VolatilitySurfa es, Journal of Computational Finan e, Volume 2, 1998/99[21℄ J. Keller, Amer. Math. Monthly 83, 1976, 107 - 118[22℄ R. Lagnado, S. Osher, A Te hnique for Calibrating Derivative Se urityPri ing Models: Numeri al Solution of an Inverse Problem, JCF, 1(1),1997[23℄ D. Lamberton, B. Lapeyre, Introdu tion to Sto hasti Cal ulus Appliedto Finan e, Chapman & Hall, 1996

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Bibliography 104[37℄ William P. Ziemer, Weakly Di�erentiable Fun tions, Springer, 1989In addition to the ited referen es I made use of the following le ture notes,distributed at the J. K. Univers�at Linz:� H. Gfrerer, Optimierung I� U. Langer, Numerik I: Operatorglei hungen, Institut f�ur Analysis undNumerik, WS 1999/2000 1996� U. Langer, Numerik II: Numeris he Verfahren f�ur Randwertaufgaben,Institut f�ur Mathematik, SS 1996� U. Langer, Numerik III: Numeris he Verfahren f�ur Anfangs- und Randw-ertaufgaben, Institut f�ur Mathematik, WS 1996/1997� R. Mahringer, Mathematik derivativer Finanzinstrumente, Istitut f�urIndustriemathematik, SS 1999