Regularization for the Inverse Problem of Finding the Purely Time-Dependent Volatility Dang Due TVong', Dinh Ngoc Thaiifa=. Nguyen Nhu Lan=

Regularization for the Inverse Problem of Finding the Purely Time-Dependent Volatility

Dang Due TVong', Dinh Ngoc Thaiifa=.

Nguyen Nhu Lan=

Received: 29 lano«j2014/Actepled:IOApnl20l5/Pi,bliil,edonbne: 7 September'015 0 Vienrn Ae^lem, ol Science md Technnlog, (VAST! i d Sponge, Scienc.tBasmes, Medi. Singapore

Abstract We consider die inverse problem of finding die volatility o e i ' (0 7) such that llBstX. R.r I. f o-(T)dT) = ullt.O < I s T. where O . j is die Black-Scholes formula and «(,, IS die observable fair pnce of an European call option. The problem is ill-posed Using die residual mediod, wc shall regulanze die problem An csplicn enor estimate is given

Keywords Calibration • Voladhty • Ill-posed Regularization Mathenmtics Subject Classification (2010, 35R30 . 65,20 91B24

1 Introduction

In die classical stochastic finance dieory. die stock price X ,s assumed to saUsfy an lio process (see [19, p. 13],

dX = pXdn-oA:dW.

where „ is die stock drift.o =oU. X)i, the stock volatility. Wis, Wiener process Denote Ik , V..'' 'c ' ' " " °^"" E"">P™ rail "Pn™. where X is die current price of

stock. K IS die smke pnce. r is the cominuously compounded Interest nite, / is time As

: Nguyen Nhu Lan nnl.ncanlho@gm.ll ct

fcpaoneni of M.lhem.ncs ai Compnier Science. Ho Ch, M,nl, C „ , U n , . ™ , of Scence Vielnm N.non.1 U n , . . ™ , . 227 Nguyen V „ Cn. D,.«cl 5. Ho C h Min" C™, V i . ™ Tay Do Umversuy. Can Tho. Vieuin

^

i!Z DJ7. Trong etil, shown by die Black-Scholes dieory (21, p. 49), if die stock price X satisfies an Ito process dien die fau- price C satisfies die (generalized) Black-Scholes et,uation

a , + ' ' * 3 5 + i - * " ' < ' ' - * ^ ' 8 5 2 = ' ' ' :;., In die latter equadon, die quantity o (I, , t ) plays a cnicial role m opdon pricing. The volatfl*'.

ity IS used to measure die uncertainty of die fumre remni of die stocks [20, p. 377]. If die: ' funcdonisknown, wecancdculatedieoptionprice C(Ar, AT, r,r,,T).TheproblemiscaBed ••

the direct problem of option pricing.

However, die vo,aohty cr(j. X) is a market parameter and not direcdy observable Hence, rt IS also called die imphed (or impUcit) voladlity. In die primitive Black-Scholes model, the volatilities are assumed to be constants. However, ui practice, die observadon shows that diey are not constant over dme Therefore, die market is concerned widi die problem ot idennfying of non-consMnI volatility [20, p. 741]. We have die problem of calibrating th, implied volatility oit.X) ham the fair price of die opdon. It is called the inverse problem of opnon pricing. The problem is ill-po.sed and smdied m many papers [2-15 18 19 241 Among these papetii, die mediod of mtegral equadons (Green funcdon) is used'in [ W 19 ' The moMcanon mediod is used m [12]. In odier papers [7, 15, 18], die audlot, usL die I ikhonov mediod (or mimmizadon mediod) to regulanze die problem

In the presem paper, we deal widi the case <r = cr(/). We put o(,) = o\t) and call il the

dX ^ti-Xdt+o{t)XdW.

We also note that similar purely ,-dcpendem models can be seen in many papers. For example, the value of an annually callable bond saosfies die Hull-'White model

•Ir = Ml) - fl.t))rlt) + olDdW.

f o l n t i o r , ? i h ™ T ^ ' ° S ' ' " " ' - '•' "•" ^ ° " " " ' ' S""™""^ ' ^ " • ™ » "O'i™ *™ »>=

sohit on of fte pmial differenual equadon of C can be represented dnough die Black- Scholesfunction[17,pp 71-72]

UBSIX. K.r.t.s) = I - * * « > ' - "'-"fldi). s > 0,

\lX-Ke-)+. s = 0.

where u = ,n (^) and

''' " ^ ^ ' ' * = ''I - •^. *'=) = ; ^ / ' '-''"dx.

Now, wc can give a precise form of die inverse problem.

:^^™:^s^r-:js-sr::(;i;s:i;:;s:sr-'""*'

UBS ( X . K. r. ,,£a(z)dr^ = ult). 0 < r < 7, (1) If we put for short

Noia)il) = VBS (X. K. r. I. / " a ( r W r ) then Ml is an operator NQ : a \-* Naia).

0 Spnnger

R^olarizalioa for ihe [overse Problem of Finding the Purely..

The case of purely r-dependent volatiliiy a ~ a{t) is studied only in a few papers as [10. 14. 15. 23J. especially, m [14, 23]. Since the problem is iU-posed. these papers studied the regularization of the problem in many common spaces C[0. 7"], Z,-(0. 7"). Z.P(0, 7") {p > 1), We will discuss the details later. From now on, we shall denote by u^, the fair price data, which is a fiinction in the range of the operator A'o- Since (Vo(fl)(() is mcrt;asing with respect to r, the operator N^ is not smjective on the set of positive functions W = C+[0, n C C[0. 7-j or W = t ^ + ( 0 . 7-) = {r e L^(0, T) : vix) > Ofor.r G (0. T)l Let V be a normed space such that NoiW) c V. The problem of finding conditions of a datum u e V such that u e NQIW) is called the problem of solvability: Moreover, in practical situations, the exact datum M„ is the result of a measurement. Hence, we often do not have the exact value of the datum u „ . We only get a noise u^ such that

The nojse a, is often not in A'o(W), i.e., the problem Noia) = «, possibly has no solution.

Therefore, the unsolvabihty often happens for real data. Even if u^ e NoiW). correspond- ing solutions might have large error (see the next section). Hence, the problem of stability seems to be more important than the one of solvability. In the present paper, we only consider the regularization for the problem

We recall the definition of a regularizanon scheme m the present context. Since our problem is nonlinear, we choose the classical definition in [22, Ch 2. p 43]. In fact, let (i/- II • llu) and (y. II • llj^) be two normed spaces and A is an operator from U into y with A : AiU) -> U not continuous. Let u)„ e AiU) be the exact datum, tv, e W be the corresponding exact solution satisfying Aiv„) = w „ and let u;, e y be an inexact (but real) datum of ui„. From w^ e y. we have to find a "stable solution'- v, e U (called a regulanzation solution) for the problem. To this end. we consider some problems.

Lei i, > 0 and C be a neighborhood of u;„ (in the topology of y). The first problem IS the one of convergence analysis (or consistency problem) which is of finding an operator P . O X iO.Si] -> U. called a regularizanon scheme such that there exists a positive fiinction S = S(E). hmj-^n* S(E) = 0. satisfying

||fi(Wf.a(c))^K„||„ < £ (otai\w,eO.\\w„-wAySS(E).

As shown in [22, pp. 44-t5], the latter property is fulfilled if two following properties (i) (eonunuous property) for each S e (0, ij], fl(-. S) is continuous on O (ii) (inverse property) RiAv. S)-> v ssS ^ 0+for every Av e O.

The second problem is the one of convergence rate which is of establishing error estimate between the regulanzation solution and the exact solution. As known, the problem is not ea.sy. Readers can see [16, 23] for some nontrivial estimates.

We turn to discuss our problem. In [14]. the audiors studied three versions the C-soace casewiihW = i ' = C[O,r],theL/'-casewithi/^L/'(O.7-),3^ = /.'?(0 T) and t h e V space case with U = y= L-(0. T). As shown in the paper, in all cases, the problems are itl-posed in the Hadamard sense 11, p. 17]. A regulanzation for the case i^ = V = C[0 T"]

was no. established in [14]. In [23], using an a priori information, the authors consinicted a regulanzation scheme for the continuous case. Moreover, they also considered the problem

•

WWfJ>' ^ H

!!1 D J l T t a u g e t i S

-_ W

of conveigence rate for die C-space case and obtained die rate (? (in ( i ) ] ~ . The rate is ' optimal m die sense that diere exists a Co > 0 satisfying

sup l l « ( i » , « ( e ) ) - i , „ | | u > C o ( ' l n ( ' i " ) ' )

I" 1"..—b-<«l V ycJJ

for every regularizanon « and every function He) 0 which has die propettv

In die IC-case, the audiors of [,4] considered only convergence analysis of regnlar- ization problem. Moreover, die regularization is descnpbve. Hence, to get die consistency ZTcl:','"'" "•^°™''"'"'» I ' solution was used The audiors smdied die outer operator' A'(«(l) = U,s(X. K.r.l. SID) widi S in die set

fl?:=|S6(."=(0.7'):0<S(„)<S(r2)<,. „ . , , 6 [ 0 , r ] , „<,2J.

For a datum „< e L'm. 7-), „•(,) > 0 (0 < , < T), die audiors found a mmimizer j ' € D+

Ot the extremal problem « " c iv,

||A'(S) - u |[i.«(o,r) -* min, subject to 5 e Z)+.

ll.^'lZ T t r o t , " V ° " " ' • "•' " " " "•= " " " " ^ " " ' " • ' * « ''• - S in i n o , T)

" ,, . ™ ' " " f ' ™ "f convergence rate was not smdied in diis case

Finally, m the Z, -case, W„ is an operator in die Hilbert space i,=(0, T) 'in [141 usin. die

^^zr^otr™*"""""-''"'^"''""^™™'"^™"^"—"^^

o ( ' = ) - " llt,,(Oj-,+ff||n-a''||=j|Pj.j-> min, subject to a e 7)()Vo), where a' e 0(«„) is a fixed initial guess damm and

D(«o) .= jci e i=(0. 7) ; ess^ M a > c„ > 0 j .

solution proccL^umci;?. r:;'A^i?,?°-°?;%7r;s't,r rt%*'

derivative of «„ at „ 6 O(«o) such dial ' ' ' '^ '"• '^^ ' » " " ' ^ * ' » '

l»(a-A-.(n)-G(»-„„|„,.„<£||,_„||J^^__^^

-»^n::s^?;^pSJ°;"j?^=:^r'^'"'"---°'

there exists a function »; e /.^(O, T) satisfying o - «- = c ' l ^ and the closeness condition lliwll.^ ^ ' i.. .u

condition implies the condMoli ' ' " ' I ' ' " *•= " " ' = « "f -"r problem, die soume

" - " • . « ' ( 0 , r , , » ( 7 ' ) - „ . ( 7 ' ) = 0 , l f ^ , , z , „ ^ , ^ ' a. . and the closeness condition has the form

' " li!|o.r, L

® Spnnger

Regularization for the Inverse fti*lem of Finding the Purely..

Since the ftmction a is unknown, it is not easy to find an initiai guess damm a' such that a{T) ~ a'iT) = 0 and that the other conditions hold.

In the present paper, we shall consider the problem of consistency and the problem of convergence rate for the case W = i^(0, 7-), 3^ = L''(0.7-)(1 <X < p < oc). We note that, in this case, £.''(0. T) (p / 2) is not a Hilbert space, hence the standard Tikhonov method (as in [14,15]) could not be applied easUy to solve the problem. Moreover, we also give explicitly error estimates of order O (bi ( i ) ) '' ( / > 0. see Corollaiy D in which we do not use the strict conditions as ait) > CQ > 0, the source coiuiition and the closeness condition. In die present paper, we do not obtain an optmial result for rate convergence. But, from the optimal result in [23], the logarithmic rate in the present work is reasonable. We will study the optimal rate convergence in a fumre work.

The remaining part of the paper is divided into five sections. In Section 2, we decompose die problem into two problems called the outer and the inner ones, give some properties of Black-Scholes fonnula and prove the instabihty of the inverse problems. Section 3 gives the existence and the regularization of the outer problem The regularization of whole problem and rate of convergence are presented in Section 4. In Section 5, we give some numerical ejiperiments. Finally, we give conclusion of our paper in Section 6,

2 Preliminary Results

2.1 Some Properties of the Black-Scholes Formula

The following properties are presented in [23], but for convenience, we recall it here without proofs Since X. K. r are market observable constants, we can denote (for short)

kit.s) = UBsiX. K.r.t.s).

The ftinction Ugs is defined in the domain X > 0, K > 0, r > 0. / > 0. i > 0. By duect calculation, one has (see its proof, e.g., in [14])

^Utn^UBsiX. K.r.t.s) = X.

iitn^UBsiX.K.r.t..s) = {X- Ke-''')+.

From the latter limit, weshalIpuH(,.0) = f/B5(X,/i:,ri,0)'=(A--A:e-")+ Weeasilv

gel (see [23] for proofs) ^ Lemma! Fort.suS2.s > 0,

aj kit. j | ) < kit. S2) If and only ifO < Si ^ i,. Hence. k{t. 5,) = k(t. s^) if and only if S\=S2. - / J b) (X-Ke-")+ <k{t.s)<X

c) If an a satisfies (X - Ke'^')^ < „ < X then there exists uniquely a ^(/) > 0 such that i(f. ? ( / ) ) = a.

D.D. Tiling et.sfc Now. we consider die problem of finding a funcdon S (Lebesgue, measurable on 10 Tl such diat t ( , . 5) = ,(,. The solution will be denoted by S = S(i^).

From Lemma 1, we have Proposition 1 Put

O = lu e i°=(0, T):IX- Ke-")* s f(() <X ae.\.

Ul t : [0, 7] -~ Rbe a measurable function The problem k(t, S) = f has a imiau, 'neasurablesolutionS = S(.^r)l.t)tfandonlyifft:D.

2,2 The m-posedness of the CalibraHon Problem

We shall consider die ill-posedness of «„ as an operator from t ^ O , T) to £,'(0 T) Ut

» . e L (0, T) u, . 7,-.(0, T, satisfy ,V„(.„) = « . 7„en k (,, fiaMd,) = ^fr,; We put u.(,) = t ( , , / „ ' „ . ( „ J j ) „ i , „ »,(I) = o„(r) + ft.(,) and

«.(» = ("; ''^',^'--;-

I " • T - i < t < 7 •

? UsTnVlh" f-P- •>" " ^ l ' ™ — " . ! ' ) = « ( » . From Lemma ,, one has 0 < «.(„ <

JT. Using die Lebesgue dominated convergence Uieorem, one has - "• l ' ) <

.'i'So I I " " - " » " " = "• and „lim^|o„-oolli = ||/i,[|» = „=A = oo.

This follows diat die calibration problem is unstable at „„. Hence, it is ill-posed.

2,3 The Outer and the Inner Problems

Ti^^olerpi^bLmis " '" ' " ' "'^ "" "•"""- =«" "= "'•"•>'•' - ' » ""• P'oWem"

Ht'ZZmtZn """•'" ""'" "•""'"• ^"" 1°' ^1 - [».») - » •<•-

and die inner problem is

3 The Outer Problem

.'^tirdttoi„?irc:r;rf"fr*™'"*""'"p'°"™'^"-i"

Sections 3 2 3 3 we shITro^e * "='^'"'>"^«"" •"=*o<ls in [14, 23]. ,n diTtivn .ion 3.2, we ha,Yprovet ' Z ]? T ^ T " " " ' W " " ™ ' " " ' ' " fa", in See- .o ». m die d c n n Z of gul':i;a ion : * m e ° m T T S ' " ' ' ' « • - " ' — - 1 = - P - IS investigated in Section 3 3 M e "Introduction. The inverse properiy (ii) convergence rate ""'• '" ' " " ° " ' ' * ' >•« ^ ^ l ""»Wer die problem of

^ Springer

in for [be Inverse Problem of Rndmg die Purely.

3,1 DeGnition of Regularization Scheme for the Outer Problem Let u € L^-*(0, r , = [n £ t ' ( 0 , T) . » > 0). 'Wc pm

J(v)m = kit, v(t)).

Let O be as in Proposition I. As shown in Lemma 1, J : L^-+(0, T) -. D c LP(Q T) is injectivc. As noted in Section 1, for 5 > 0, we shall construct «'(. S) • D ^ Z.*+'(0 T) such diat limj^o. R(J(n), S) = v. In die section, we shall use the mediod of residuals [I, p. 31) to regulanze die outer problem. Hence, our regularization for (CP) can be called a modified mediod of residuals. Precisely, let 5 > 0 and let u, e D. Put

ri.,j = (C E i*-+(0, T) r, t ~ ( 0 , T). Ilu, - /(iiJII^ < Sf.

From w. we construct a (rainimizer) taction Riw, i) .= y such dial r e S , , , and

*'(')< ii(r) foreveiyi e [ 0 . r ] , C e C u a . (2) The defmition defines uniquely die function y. We shall prove die existence of the min-

imizen Before going to deads of the construction, we have some comments. In [14]

the authors gave a result of convergence analysis for the outer problem in i'-sp.ces In fact, widi the datum », diey used die approximation Riut) as minimizer of die problem nJiv) — tu||£«[o,5-) —> min subject to

" e D ; = lu e t » ( 0 . T) : 0 < »(,,, < „(,j) < ^ for 0 < ,, < i; < T].

The a,gondirii depends on die constant , which is an available a pnori information As mentioned, die regularization is descriptive. Similarly, in [23], we used die formula Rlw.S) = mmiK, max(S(u,),«)] From die mathematical point of view, n might be bener If we have not to use the a priori constant,. This is die reason of our algoridim

Now, we give die closed-fonn of fl(u,, S). Put

ui^P) = mtailX - Ke-")*. «,(,) _ s\.

H T h l f i T ' c f + { , ™ ' " ' ? ' " ' * " O'Fn'd'P'oPraliou 1, wecan get die ftinction S(u,;) such that J(r, S(a,+)) = ,„,+. Wc shall prove diat « ( » , S) = Slnit) in the following result Lemma 2 ier 5 > 0 and ietujs D. Then

'* iHl " '^'/' ? " ' * " " - "''" " " ' ' ' " '"• '•I '"' " <^ " . ,. Hence the function Stwl) satisfies (2);

ii. Therefore. i(u,J)(/) > St.w*)U)forallt a (0, T) andO < « , < « . Pwof We first prove S («,,+) e S..,. For every, 6 [0. T]. we have 0 < u>(,) - ,„,+ („ < s

y(S(«,j+))(,) = kit. S(n,,+)(,), =: „+(,) So, Ilu, - y(J(u,+),||^ < 5, „ f o „ „ „ ,^„ 5,,^+) ^ g^ ^

, . r o ^ n l e T "l""'"','" -J -°' " ' " " " ' ' - " * " " = ' C ' SW) i u,(,) - J for ah e (0. T). Lemma I implies kit. „(,)) >IX- Ke-)*. Using both inequalities gives

* ( I , i i ( l ) ) > u , + ( r ) = t ( , . S ( „ + ) ( , ) ,

for^every , MO, T) By die monotonicity of <.(,, ,, we get 0 < « » / , ( , ) < 0(„ f„, „ , USr<7'.TbisimpliesOiati(u,+)isasolutionof(2)

Finally, we venfy Pari ii). In fact, we have l | u , - J ( S ( u , + ) , | | „ < j , .

520 r. I-. T Wl D.D. Trong i » ^ ^ But 0 < 5, < 5. hence Siw^) e fi„,^. From Part (i) of the lemma, we have 5 ( K ; / ) { ( ) <

S(Wg^ )(r) which gives the last desired inequality. This completes the proof of Lemma 2. Q 3.2 TheContiniiityof ff(ii),3)

For a fixed 5. to prove the continuity of Riw. S) with respect toweD. we state and prove some lemmas.

Lemmas Let

Vl>,n:=maxJ2|v|.2[u + r r | , , + m » t j H ^ ^ ± £ . , j j .

Tha them e.vlsis a posiiive number 0 < «„ < , such that if the inegtialtn klt.s)<X-t, holdsforaSelQ.taJ.then . i , 1 - A a

I V s)

Proof We can rewrite the inequality k(t.s) < X - S tis

Put ft = max[|i.|. Ill + , - r | | . We have for s > m= > 4^=

y £ _ A y£_ 1

2 y/;- 2 2- This implies

So, we have ^ 2

Putting ; = # - ^ , we have

f'-'^-^-r^-^^'

It follows dial

X+Ke-''n x + K [" ,„

v S ^ T ^ - ^ ^ / e - ' - * > J 2(A• + ^)e-'^^-l'Vs

- " - ' + " " I ^ B T ' ' )• ™ •*''n 8=t c-'.A-l)-/» > s. This implies

™:c™S'th; "prr:fTeTm;r''"' ^"' *- ^--^ '"-'-•"•^ <-»"»< * ^ 'g

fl Springer

R^nlarizailmi for the Inverse Problem of finding itie Purely,.

L e m m a 4 Leia.L satisfy 0 < or < /,. f o r every S ] , S2 e [0, L], one has

\s\ ~S2\^a + M(a,L.t)\kit.Si)~kit.S2)\.

where

Ml I ,\ 2yfhis /[v-\-ri)^ (v + rr) j \ M{a.L.i)= naax — - — e x p -f — i l - i - J 1 '•<s<L X ^\ 2s ^ 2 ^'&j W v = In(f).

Proof We consider three cases.

Case 1 0 < 51, J2 < o

Is, - S2| < a < a + M(o. i . r,|t(t. s,) - kU. si)\.

Case2 a < si..!2 < L.

We h a v e i ( r . s , ) - « , , j 2 ) = | i ( , . c ) ( s i - j ; , , where a < min(s|.S2l Sc < maxl.ti.s,) < L.

Hence, 0 < ( f ( c ) ) " ' < Mia. L. I). This gives

|si - « l < o + Mlat. L. t)\kU. s,) - kli. S2)|.

Cases 0 < i i < a < J2 < L.

One has

ISI-..2I = « - i | + s 2 - o

< t, + Mla.L.t)\kli.s2)-klt.a)\

But, from Uimma 1. one has * ( , , „ , < ku.a) < i,,,,,,). THis implies dial |i(/.o) - ' ( ' . 52)1 < lid. . t i ) - J O . S2)|. Hence,

|5i - .•2I < o + M(«. L. t)\klt. I,) -1-(,. S2)|

This completes die proof of Lemma 4. r-.

Proposition 2 Utabe as w Umma 4 and let J > 0 For two function, u.w,= Do„e has

\Sl.ut)V)l < X, ^ (l-y fi^^'.

\Sliit)lt) - Slw*)U)f < o + Mia. AT,, ,)!«(„ - u,(,)|

and

\ISIu*) - S(u,+)|, < a y l " +^mi«^ M(«, it,. , ) r ' i ? |[„ - u-II,.

L' (0. T, (.1 defined in Proposiiion I

Proof For,,, c 6 ^ ' ( 0 . T) and r 6 (0. T) fixed, we give die estimate of l«(n, nO) - R(».i)(,)| = |S(u;)(,) - j ( „ ; , „ ) j

^ D D- TVoBgaat^

One has i(r. S(»+)(()) - kO. S(v+)(t)) = u+(t) - u+(r). For S small enough, we obtain 0 < u^ it). Vg it) < X - S From Lemma 3, we shall get

(

, — • • V 2

l + JSInM .

Using Lemma 3, one has

\Siut)it) - S(i,.+)(0| < a + Mia. Ks. t)\u+(l) - v+it)\.

On the other hand, |u+(/) - i,+(,)| < [«(,) - i,(f )|. Hence.

\Siuj)ii) - S(v+)(t}\ < « + Mia. Kg. /)!«(/) - v(t)\.

Now, we prove the continuity of R{-, 5). One has

|5(u+) - S(w^Xt)\ < a + M(a. Ks. l)\u(t) - w{t)\.

This implies

lK(u.5)-Rlw.S)h <c,Ti -^ mnx^M(a.X,.i)T'^\]u-wll,.

So.for«, e O c ffO.T') such dial lim„,„. I|i«„-,»||„ = 0, we get after ,etiin8n . * 00

°, " ° ' " " . " " ^ " P " - * II^C""- ') - «<•«• *)!U = 0 It follows dial Rl: S) is continuons

at u, e D This completes die proof of Proposition 2. • 3,3 The Inverse Property of ff (tu, 5)

We shall prove dial « ( » . i) is a regulanzation .scheme. In fact, we prove Proposition 3 Ut „,. 6 t ' . + (o. T). Then « ( J ( „ „ , , J)(,) ^ „ „ ( „ a, SI Band

J i i n | l « ( y ( u „ ) , J ) - u „ | | j = 0 , where we recall that Jlv)ft) = kit. u(0).

o K^we have « ( » „ , s, = i ( „ + ^ , . po, 0 < S, < J. Lemma 2 implies Slw* , )(,) >

« « , . (r) for a„ , ^ (0, T) Hence, ,ims,„ S(„+ , , exists for every , E (0, T) Putting VC) - , i m i ; o 5 ( i u ^ ) , we claim that ,1, = u,„ In fact Ilu, - /rc,„,+ „,i ^ i c Lemma 2. L e t . . a we get „ „ „ - y f ^ ^ l u ' l o ' S i i ^ t i e s . , u ^ ^ S f f n . ^ , im7es • " ° ' " " " = *• " ~ ' ""= ' • " ' » S « ™ " " t a = conveJience tbeorei!;

l™ | | « ( A u „ ) , S) - v,.h = lim | | i ( „ ; , , , - „ „ , , = 0.

This completes the proof of Proposition 3.

*1 Springer

Jr tbe Inveise Problem of Rnding the Purely..

3.4 Error Estimates

We have

Proposition4 LetL.s.S.a > Qandletv^^ e L'-+(Q, T). «.„ = J(LVV). Let W^ e D be such that \\wi - ui„ \\p < E. Then

mw,.6)-i,,A\). < 2aT^'^-\-(( \i>,At)\'-dt\

\j{Kcxi.l)^L\ J

+S ^rnax^ Mia. L. I)T'T' -[-e max Mia.Ks.t)T^.

So. if we choose S = 5(e), a = a(&). L = US) such that

\\Tn,^o£'nax(,<,<TMia.Ks.t) = 0, hmi,_o«(^) = 0, Ums^oHS) = oo and

\imi-,oSMia{S).LiS).t) = Qforte iO.T)then Jim||fl(u),.5(£))-i,„||^==0, Proof We have

II«(V(u„),S)-„„||^ < (f | » „ ( „ | V r ) ' "

VJ|,v,(ii>t} /

* ( / i l - , U.L l * ' " ' " > l " - ^ < < . . l H ' > l ' < ' ' )

IfO < » „ ( „ < L dien 0 < S(u,,+ ,)(,) < „ „ ( „ < L. Hence, Proposition 2 gives IS(>»,.)(') - S(u,*.j)\ < o + SMia, L.i).

So.

[j I S ( « i „ ) « ) - S ( « , ; , ) ( r ) | ' r f l ) " ' ' < „ ? ' " » + J max M»(a. L . O T ' T ' This implies

\\RUIv„).S)-v„h < if l«„{r)|»AV"

On the other hand,

ll«(«>.i)-«(,!(»„). S,|k = l | i ( i U i + , - « < , ) | | i

- " ' ' ' " + o 5 ? r " ' ° ' *^''"ll< - »'*• illi

" r s ^ ^

The above give

||ff(U'.^)-i',,|U < 2aT"^ + ( f iVeAt)\^di]

+5 max Mia.L.t)T P -^E max Mia.Ks.i)T f

1

This completes the proof of Proposition 4 [1p We need an estimate for M(a. L.I).In fact, we have

Lemmas Piir up = max[|ul. | y + r ; | , , X, = ^ ^ exp {I}). Then

0 < Mia. L.t)<K, yiexp f-!* + - V

\2a Sj

Proof We have Mia. L.I) = m a x „ , , i = i ^ e x p ( i l i g l ; + & « a + • ) , But,for0 <

t < T, one has |u + r ( | < max(iu|, |u + r7'|| = va. Hence,

\2a^ 2^ sj- This completes the proof of Lemma 5.

Theorem 1 Ut the assumptions of Proposition 4 hold Ut 0 < p < 1/2 S = ef Put VIL) = | u „ > 1) = (, s (0, T) . u„(r) > L). Then there exist a constant Cj > 0 luii Z. = L(E) > 0 such that Ump^o L(e) = cxi and

l l « ( u , „ S ) - „ „ | , 5 ^ ( 5 ) , where

; j i J + (/vaM)l""<')l'<") i/V(L(,))^H,

;;;(i) i/i'(t(i!)) = 0.

Proof We choose „ = - i j , 1 , C2 In ( l ) , where die const«,ts C,, C2 > 0 saUsly

° < 2 C 7 + 2 r < l , 2 ^ + - g i < r . Using Proposition 4 and Lemma 5, we can compute dircctlytogetdiedesiredestimate.ThiscomplctesdieproofofTheorem I D Corollary 1 Ut the assumptions m Theorem I bold If there exist p. K2 > 0 such that 0 < i j t < I m d 0 < «„ (1) < jJfjy, then them exists a conslanl Q > 0 such that

lRiu,,.s)-v„hi , ,';•' I n 4 ' ( l )

Pmof The assumption on u„ implies dial L < ^^^ when u„(t) > t . So, we have

/ iu„(r)i'* < r^ ^„„ --^d, = _ £ L _ /-ftr"'*'

A,„u,2t J r - ( ? ) "^ - ' ) * " ->,P-H\.L j

ion for the Inverse Problem of Finding the Purely..,

Direct computation gives the desired conclusion. •

4 The Inner Problem

As mentioned, the inner problem is the one of recovering a derivative ftom its mtegral Precisely, we consider the problem of recovering the function aMD from the approximate value SAD = R(u,. S(e)) of 5 „ := 5(H,,.)(/). The problem is ill-posed and many text- books presented well-known regularization schemes. Here, we give only a simple version of recovenng the ftmction «„((). We recall that S^A') = /ofl„(i:)^r. We first have Lemma 6 LetO <: h < 2h < T and let d e [0. h]. For f e L^(0. T), put

fi.(t) = 1-^^' + ^) forQ<t<T-h.

[f(t-e) forT-h<t<T.

andiafih) = supo^^^;, ||/e - /||^. Then lim„_(]-^ w/ih) = 0.

Proof Let £ > 0, Since C'[0. T] is dense in L'iQ. T). there exists a conunuous ft.nc.ion g e C[0, T] such that | [ / - g||i < | , We note that

Wss-feh = / \gi< + e)-fit-\-Q)\>-dt+ / \git-g)- fii-6)Y-dt

•'" JT-h rT-t,y, r-,

~ l « « ) - / ( r ) r * + / l g ( / ) - / ( r ) | V , i 2 1 1 s - / l i t

Since A > 1, we have

l » - / » l l » £ 2 | | g - / | u . Using the latter inequality, we have

11/ - ft\W < \\f - g\i + llg - gelli + ||g, - / , | | i

<3|/-Sll» + ll«-S«lll

< -« + »,(*,.

Since hm»^„n,,(«) = 0. we can find an /,„ > 0 such that \\f - / , | | , < , for every

d < ( l < f i < ) i n . T h i s comp,etes the proof of Lemma 6, • ThMtem 2 Le, an € (0.,]. 1 < i < p and let Ihe assumptions of Theorem I hold In

addition., we assume that a„ 6 Z,»(0, T,. For h £ (0. T) such that 0 < )i < 2* < T, p«,

, lA -t-2<»i,(«»)°"l ''•here He) is as m Theorem I and He) = (^(c),"o. Moreover if a„ is a Holder function

satisfying '

|o„(t|) -d„(t2)l < K2\n - t.V for t,.t2 e [0, T].

\ihere 0 < )/ < I, then we can choose OQ = ^ry "^ S^'

l|a^"''-«„lk<3'-'/M4r p -F2S|7-j (/?(E))?Tr.

If. in addition, we have the assumptions of Corollary I then

~—"i£f

Proof We have for 0 < r < T -h

K^f)-fl«(OI < -j^i\SAt + h)-S,At + h)\-^\SAt)~S,Am + / \a,,it+hd)-aeAt)\de.

Ja

Using the inequality (2+|±£:)'^< o^+b^+c' (ora.b.c >0.k > 1. we get / !<(;) - a.AOrdt < /] + /2 -I- /3.

where Jo

'' " ~b^i \SAt-\-h)-S,Ai+h)]'di^t-T'9^^is).

3).-1 /•r-A -li-i '= = - p r | I S . W - i „ ( 0 [ * * < --^T'Tf)\e).

'^^^'"' j„ [J l<'«('-l-'i9)-ii,.,0)|r/8) *<3*-'tt,*(*) The estimates of Z,. /2. h imply that

f"l<.,*(r)-.„(,)|V,.3>-'(^i:i^+„i.,„J.

Similarly, for T - A < ( < r . we have

l«^') - « „ ( / ) [ < -(15,(7) - s,Ar}\ + |5,(r -^/,) - S,Ai~h)\) + -^\Sexii) - S,Ai -h)- ha,At)\.

It follows that

jl^ !<.;(,) - a.At)fdt S 3 - ( H I ^ + ^ r t , ) .

We get m view of the above inequalities

fK*<0-«„(„,=*,3-'(li:^+.„i.(„j.

fl Spnnger

^ u l a r i z a l i m i fig the Inverse Problem of Finding die Purely-

By choosing h = A(e) = (P(E))"O, we get

J^ \a^,'''(l)-a„(nNt < 3 ^ - ' UT^mE))<^-'^i^-i-2a>^jfiie))'^)\ . Since (i„ is m /.*(0, 7"), Lenmia 6 gives

limtUfl (A) ~ u.

Noting thai Pie) ^ 0 as c ^ 0, we get the desired result.

Moreover, if a „ is a Holder function as in the statement of the theorem then k«((]) - a „ ( / 3 ) l < K2lti - t2\^ < Ksh". n.tie [0. T]. |/i -t2\ < h.

In this case, we have l(a„)9(r) - a«(f)| < K2\8\y.ll follows that / l(fl«)e(/) -a„(Ol^rff < K^T\e\^>'.

6J*„(/j)< sup / {aeit)-a{t)\'-di < K^Th^'' o^e<k Jo

So.ifao = ^ . h = ifi{e))y^ then

^ la^,"Ht)~a,At)\'-dt <3^-' UT'T' -\-2K^Tyfi(E))V-*

This imphes

„ J,(„i I I , , / £-A \ '/*• ,

I K " - a « I U < 3 ' - " M 4 r - +2*r2'7'j ((,(s))i*I.

Now, if wc have die assumptions of Corollaty 1, dien we can choose/,(«, = get die final result. This completes die proof of Theorem 2.

^ ( 0

•fable 1 Error t)etween exact solution and approximale solunon

£ = 10-'

( ;

0

Error

0,02703 0.03922 0-06152 O.0358S

E = 10--'

' .

0 0 16667 0.33333 0 50000 066667 083333

Error 0-00752 0.04076 0.0317!

0 03505 0.04647 0.03926

£ = lO--"

/,

0 0 12500 0,25000 0.37500 0 50000 0.62500 0 75000 0.87500

En.r

002725 002914 0-02785 002820 0.03222

5 Numerical Results

From the results of the previous sections, we have many ways to establish a computatrojn""

program. Here, we give a rough algorithm. The given data are K. X. r, T. the noise levels"' and the noise fair option price «(/). Since H e D (see Proposition 1). we can replace K by!*

inin{H. X]. The algorithm is divided into three steps. •'i Step I We can choose ft = ft(e) = ^{e)

the interval [0. T] into n subintervals by f;

"(%)"

. in Theorems 1 and 2. We divide.' l)T/n,i = l n-H,whens Step 2 We construct the function Riu,S)(t) for 7 = f„ 1 < / < H -|- I. Let 5 = S(£) =

£'' be as in Theorem 1. We can find 5; := fl{M,,5)(r,-) by solving the equation

*(/,,->:) = max|(X - Ke"-)+,«(/,) - 5} (/ = I n-h I).

Step S We denote

_ -Si-n-^f n

The output is an approximation of Ihe implied volatility o(/) at /, = (i - \)T/n I = 1. .,M + l,i,e..{a(/,) ail„+i)) == (a, «„+i).

For example, we choose T=[.X= 100, AT = 101. r = 0.05 and a volatility function

— ^ '

We use bisprice (Price, Strike, Rate, Time, Volatility, ,n MATLAB R20,3a m calculate

•1(1,). Testing with « = 10 =, , 0 " ' . 10-» and error al points /, is as Table I. In Figs. 1 and 2, we plot the exact solution and die approximate solution. In left panels, die graph of exact outer solution IS blue and the graph of approximate outer solution is green. Right panels are graphs of exact (red) and approximate (blue) volatiUty functions.

(a) Outer solution (b) Volatility function

F i g l « .

fl Springer

Regulariiiation for the Inverse Problem of Finding the Purely..

(a) O u t e r s o l u t i o n ( b ) V o l a t i l i t y function

6 Conclusion

The regulanzation problem of finding a purely time-dependent volatility is considered in a lot of papers [ 1 4 - 1 6 , 23], O u r paper is devoted lo the / . P - c a s e of the problem In [14], the authors considered the case and established the convergence analysis for a d e s c n p t i v e regularization with p ?t 2 Bui the regularization needed an available a p r i o n constant, and the rate of convergence was not considered. Now. m our paper, w e construct a regular- ization scheme in which w e do not use the a p n o r i constant. We also give an analysis for convergence rate of o u r regulanzation and s o m e numerical experiments.

Acknowledgments The authors ai

leading lo ihe iraprovemeni version < grateful lo Uiree anonymous referees for iheir precioi

1-L17(I997) e problem iha

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