When does Convergence of Asset Price Processes Imply Convergence of Option Prices?

Transcription

1 Whe does Covergece of Asset Price Processes Imply Covergece of Optio Prices? Friedrich Hubalek Walter Schachermayer Report No. 3 Istitut für Statistik, Uiversität Wie, Brüerstr.7, A-0 Wie, Austria. Friedrich.Hubalek@uivie.ac.at

2 Istitut für Statistik, Uiversität Wie, Brüerstr.7, A-0 Wie, Austria. SFB Adaptive Iformatio Systems ad Modellig i Ecoomics ad Maagemet Sciece Viea Uiversity of Ecoomics ad Busiess Admiistratio Augasse 6, 090 Wie, Austria i cooperatio with Uiversity of Viea Viea Uiversity of Techology Papers published i this report series are prelimiary versios of joural articles ad ot for quotatios. This paper was accepted for publicatio i: Mathematical Fiace This piece of research was supported by the Austria Sciece Foudatio (FWF) uder grat SFB#00 ( Adaptive Iformatio Systems ad Modellig i Ecoomics ad Maagemet Sciece ).

3 Abstract We cosider weak covergece of a sequece of asset price models (S ) to a limitig asset price model S. A typical case for this situatio is the covergece of a sequece of biomial models to the Black-Scholes model, as studied by Cox, Ross, ad Rubistei. We put emphasis o two dieret aspects of this covergece: rstly we cosider covergece with respect to the give "physical" probability measures (P ) ad secodly with respect to the "risk-eutral" measures (Q ) for the asset price processes (S ). (I the case of o-uiqueess of the risk-eutral measures also the questio of the "good choice" of (Q ) arises.) I particular we ivestigate uder which coditios the weak covergece of (P ) to P implies the weak covergece of (Q ) to Q ad thus the covergece of prices of derivative securities. The mai theorem of the preset paper exhibits a itimate relatio of this questio with cotiguity properties of the sequeces of measures (P ) with respect to (Q ) which i tur is closely coected to asymptotic arbitrage properties of the sequece (S ) of security price processes. We illustrate these results with geeral homogeeous biomial ad some special triomial models. Key Words: weak covergece, optio prices, asymptotic arbitrage, cotiguity, biomial models Itroductio Sice the semial paper by Cox, Ross, ad Rubistei (979) the approximatio of cotiuous time asset price models by discrete time models is a well-kow theme i Mathematical Fiace. Rachev ad Ruschedorf (994) ivestigated i a systematic way the questio which cotiuous time models may occur as limits of biomial models. We shall address the followig issue: there are two dieret aspects of weak covergece of a sequece (S ) of discrete time models which are relevat i the cotext of Mathematical Fiace: rstly it is the usual questio of covergece with respect to the origial, sometimes called "physical", probability measures (P ); but secodly there is also the questio of covergece with respect to the so-called "risk-eutral" measures (Q ), which, e.g., i the case of biomial models are uique (provided they exist). This sequece of probability measures is relevat for the covergece of the prices of derivative securities o the uderlyig stock price process (S ). The geeral theme of this paper is: How is covergece of (P ) related to covergece of (Q )? More precisely: Uder what coditios does the covergece of (P ) imply the covergece of (Q )? Bearig i mid that the equivalet martigale measures Q dee (via takig discouted expectatios) a pricig rule for derivatives (such as optios) a rough reformulatio of this questio is formulated i the title of the preset paper: Whe does covergece of asset price processes imply covergece of optio prices? Our aim is to make these ituitively formulated questios mathematically precise ad to provide suciet coditios for a armative aswer. (We cosider here covergece of processes i distributio, or equivaletly with respect to weak covergece of the laws of the processes, exclusively. Related results ad further refereces ca be foud i Jakubowski, Memi, ad Pages (989), Due ad Protter (99), Kurtz ad Protter (99), Cutlad, Kopp, ad Williger (993), Kurtz (99), Mada, Mile, ad Shefri (989), Nelso ad Ramaswamy (990), He (990), Avram (988), Memi ad Slomiski (99), Stricker (984/85).

4 A overview regardig various covergece cocepts used i Mathematical Fiace is Williger ad Taqqu (99). The reader iterested i pathwise approximatio should also cosult Eberlei (99), Delbae ad Schachermayer (996).) The startig poit is a puzzlig ad at rst sight amazig example, which is due to Th. Schlumprecht ad, idepedetly, to K. Potzelberger.. Example For R, > 0, ^ > 0 there is a sequece of biomial asset price models with discouted asset price processes (S ), physical probabilities (P ), ad risk-eutral probabilities (Q ), such that (i) the sequece (S jp ) coverges weakly to (SjP ), which is geometric Browia motio with parameters ad, (ii) the sequece (S jq ) coverges to (SjQ), which is geometric Browia motio with parameters?^ = ad ^. As a cosequece the (uique arbitrage-free) price of a Europea optio o S may be dieret (ad, ideed, very dieret!) from the limit of the (uique arbitrage-free) prices of the same optio o S. The costructio of such a example (see Sectio 3 below) is actually quite simple: the trick is to use a sequece of biomial models with dieret behavior for odd ad eve icremets. I order to aalyze the pheomeo arisig i this example let us recall the basic idea behid the etire theory of pricig derivatives by o-arbitrage-argumets: if a derivative is "wrogly priced" there should be a possibility for arbitrage. The situatio described by Example. ca loosely be described by sayig that the optio is "asymptotically wrogly priced". This leads to the idea that this "asymptotically wrog price" should be related to some kid of "asymptotic arbitrage". The otio of asymptotic arbitrage was itroduced (i several variats) ad studied i the work of Kabaov ad Kramkov (994, 998), Klei ad Schachermayer (996a, 996b), ad Klei (996) ad is itimately related to cotiguity properties of the sequeces of measures (P ) with respect to the sequece of measures (Q ) ad vice versa. It turs out that there is ideed a close coectio alog these lies:. Theorem Let (S jp ) be a sequece of (ot ecessarily complete) asset price models that coverges weakly to the complete asset price model (SjP ). Let (Q ) be a sequece of equivalet martigale measures for (S jp ), such that the sequece of termial values (S T ) is uiformly (Q )-itegrable. If the sequece (Q ) is cotiguous with respect to (P ), the (S jq ) coverges weakly to (SjQ), where Q is the uique equivalet martigale measure for (SjP ). Let us put the message of Theorem. (precise deitios of the used terms will be give below) ito a more iformal laguage: by assumptio we x, for each (S jp ), a equivalet martigale measure Q, which we cosider (by takig expected values) as a pricig rule for derivatives. Note that we did ot assume that each Q is uique, i.e., that each S uder P is a complete market; we oly assume that the limitig model S uder P is complete. Uder a techical uiform itegrability assumptio the cotiguity of (Q ) with respect to (P ) the implies the covergece of (Q ) to Q. I particular this implies covergece of prices of Europea optios o S to the prices of the correspodig Europea optios o S. (We

5 do ot address covergece of America optio prices here. Related questios ad further refereces o America optios ca be foud i Lamberto ad Pages (990), Muliacci ad Pratelli (996), Ami ad Khaa (994), Lamberto (993).) The cotiguity of (Q ) with respect to (P ) is closely coected to the idea of asymptotic arbitrage: for example, if we make the additioal assumptio that each Q is the uique equivalet martigale measure for (S jp ), e.g., i the case of biomial models, the (Q ) is cotiguous with respect to (P ) i there is o asymptotic arbitrage of secod kid as was show by Kabaov ad Kramkov (994) (compare also Klei ad Schachermayer (996a, 996b), ad Kabaov ad Kramkov (998) for related ad more geeral results). Usig this relatio betwee cotiguity ad asymptotic arbitrage we obtai from Theorem.:.3 Corollary I the settig of Theorem. suppose i additio that each Q is the uique equivalet martigale measure for (S jp ). If (S jp ) permits o asymptotic arbitrage of secod kid the (S jq ) weakly coverges to (SjQ). The paper is orgaized as follows: i Sectio we x otatio ad deitios, ad i Sectio 3 we do the costructio of the "odd-eve" Example.. I Sectio 4 we give the proof of Theorem.. We also provide examples showig oe of the assumptios ca be dropped for the theorem to hold true. O the other had, we also give a example showig that the reverse implicatio of the theorem does ot hold true, i.e., covergece of (S jq ) to (SjQ) does ot imply cotiguity of (Q ) with respect to (P ), see however, Propositio 3.9 below for a partial reverse result. O the other had we show (Theorem 3.8) that for homogeeous biomial models (where, i particular, the distictio betwee the odd ad eve icremets caot be made) the pheomeo of Example. caot occur: loosely speakig, i the case of homogeeous biomial models covergece of stock prices implies covergece of optio prices. This result seems to be wellkow ad of folklore type but we have bee uable to trace a precise referece i the literature ad so we provide a proof. I Sectio 5 we apply Theorem. to a homogeeous triomial model (similar results ca be obtaied for more geeral block multiomial models). I this settig, for each N the process S does ot dee a complete market ad there is a wide variety of possible choices of equivalet martigale measures Q ; o the other had, i our settig the limitig model S is just geometric Browia motio (with drift) ad therefore dees a complete market. So we d ourselves precisely i the situatio of Theorem.. For certai homogeeous triomial models we give explicit ecessary ad suciet coditios characterizig those sequeces (Q ) of martigale measures such that (S jq ) coverges to (SjQ). Ackowledgemets We thak Thomas Schlumprecht for brigig to our attetio Example. which was the startig poit of this paper as well as Klaus Potzelberger, who costructed idepedetly this example ad preseted it i the Semiar o Stochastic Processes ad Mathematical Statistics i the summer term 997 i Viea. 3

6 Deitios ad Notatios. Deitio A asset price model is a ltered probability space (; F; (F t ) t[0;t ] ; P ) with a R d -valued semi-martigale (S t ) t[0;t ] deotig the discouted price processes of d assets. A probability measure Q o F will be called a equivalet martigale measure for the asset price model, if Q is equivalet to P ad S is a Q-martigale. If we do ot specify the ltratio the term martigale pertais to the (augmeted, right-cotiuous) ltratio geerated by S. We shall sometimes write briey S for the asset price model, or (SjP ) ad (SjQ) if we wat to stress that we cosider S relative to P or Q respectively. We say that a sequece (S jp ) of asset price models coverges weakly to a asset price model (SjP ), if the sequece of probability measures deed by (S jp ) o the space D d [0; T ] of cadlag trajectories equipped with the Skorokhod topology coverges to the probability measure deed by (SjP ), with respect to the weak covergece of probability measures, cf. Jacod ad Shiryaev (987) or Billigsley (968). A remark o the above deitio seems appropriate: usually a asset price model is deed as a R d+ -valued semimartigale (S t ) t[0;t ], where the rst coordiate S 0;t plays the role of the bod or riskless asset, which is assumed to be a strictly positive process. The term equivalet martigale measure the pertais to a measure Q P, uder which the discouted processes S ;t =S 0;t ; : : :; S d;t =S 0;t are martigales. I the preset paper we are ot really iterested i the covergece of the bod price processes, i.e., of the 0-th coordiate (S 0 ) of the asset price models; our iterest rather focuses o the covergece of the d stock-price processes. Maily i order to simplify the otatio we therefore cosider from the very begiig the d-dimesioal process S = (S ; : : :; S d ) of discouted stock price processes; i other words we choose the popular approach to use the bod as umeraire (compare, e.g., Delbae ad Schachermayer 995). It is the process of discouted stock prices which is relevat for the pricig of derivative securities ad therefore this settig allows to give more compact formulatios; we remark, however, that it is also possible mutatis mutadis ad ivolvig more cumbersome formulatios to preset our results i the laguage of (d + )-dimesioal processes takig also explicitly ito accout the covergece of the bod price processes. To prove weak covergece i our examples we ofte use the followig fuctioal versio of the Lideberg-Levy cetral limit theorem Jacod ad Shiryaev (987, VII.5.4). Without loss of geerality we set the time horizo T =.. Theorem Assume ( k ) k=;:::; is a rowwise idepedet triagular array uder (P ) satisfyig the coditio (.) for all " > 0 as!. Let X t = (.) X k= X E P [j kj fj k j>"g ]! 0 [t] k k= [t] X k= deote the partial sum process. If E P [ k ]! t 4

7 uiformly i t [0; ] as! ad (.3) [t] X k= V P [ k ]! t for ay t [0; ] as! the (XjP )! (XjP ), which is Browia motio with drift ad variace o [0; ]. We also shall use the followig lemma..3 Lemma If (X jp )! (XjP ) the (exp(x )jp )! (exp(x)jp ). This follows from the fact that exp is uiformly cotiuous o compact itervals, ad therefore the mappig! exp is cotiuous from D to D Jacod ad Shiryaev (987, VI..5 ad 3.8). The subsequet otios of cotiguity ad etire separatio ca be foud i Jacod ad Shiryaev (987), Wittig ad Muller-Fuk (995), Strasser (985), Roussas (97), Greewood ad Shiryaev (985), Shiryaev (984)..4 Deitio (i) A sequece of probability measures (Q ) is cotiguous to the sequece (P ), both deed o measure spaces ( ; F ), if P (A )! 0 implies Q (A )! 0 as! for all A F. We shall deote this by writig (Q ) (P ). If (Q ) (P ) ad (P ) (Q ) we say the sequeces are mutually cotiguous. (ii) The sequeces (Q ) ad (P ) are etirely separated, if there is a subsequece k! ad for each k a set A k, such that P k (A k)! ad Q k(a k)! 0 as k!. We shall deote this by writig (Q ) 4 (P ). A useful criterio for cotiguity ad etire separatio is the followig, which we adapt from Jacod ad Shiryaev (987, V..3) for our applicatios..5 Theorem Assume ( k ) k=;:::; is a row wise idepedet triagular array uder (P ) as well as uder (Q ). Let p k ad q k deote the law of k uder P resp. Q, ad let (.4) h () = X k= [? H(; p k ; q k )] ; where H(; p ; k q k ) is the Helliger itegral of order (0; ). (i) We have (Q ) (P ) i (.5) lim!0 lim sup h () = 0:! (ii) We have (Q ) 4 (P ) i there is (0; ) such that (.6) lim sup h () = or lim if if H(; p!! k; q k ) = 0; k=;:::; ad i this case (:6) holds for all (0; ). Note: We write p ad k q k sometimes to deote probability distributios, sometimes to deote related probabilities, but we prefer this to itroducig a further otatio. 5

8 .6 Deitio (Harriso ad Pliska (98),Delbae ad Schachermayer (994)) A predictable R d -valued process H is called admissible for S, if the stochastic itegral with respect to the process S, deoted by (H S) t[0;t ] is well-deed ad there is a costat C > 0 such that (H S) t?c for all t [0; T ]..7 Deitio (Kabaov ad Kramkov (994, 998)) (i) A sequece (H ) of admissible tradig strategies realizes asymptotic arbitrage of rst kid (AA ), if there are umbers C! such that (H S ) t?, for t [0; T ], ad lim sup P [(H S ) T C ] > 0 as!. (ii) A sequece (H ) of admissible tradig strategies realizes asymptotic arbitrage of the secod kid (AA ), if there is a c > 0, such that (H S ) t?, for t [0; T ], ad lim sup P [(H S ) T c] = as!. (iii) A sequece (H ) of admissible tradig strategies realizes strog asymptotic arbitrage, if (H S ) t?, for t [0; T ], ad lim sup P [(H S ) T C] = for ay C > 0 as!. If there are o subsequeces permittig asymptotic arbitrage possibilities of rst, secod, or strog kid, we say there is o asymptotic arbitrage (NAA) of rst, secod, or strog kid, respectively. To come to the last (formally) udeed cocept appearig i Theorem.: the uiform Q -itegrability coditio of the sequece (S T ) with respect to the measures (Q ) meas (.7) E Q h js TjI fjs T j>cg i! 0 uiformly i N as C!. This implies i the preset cotext the uiform boudedess i L (Q ) ad uiform itegrability coditio as deed i Meyer ad Zheg (984). This coditio holds, for example, if (S ) is L p (Q )-bouded for some p >, i.e., (.8) sup N E Q [js Tj p ] < :. Notatio We write (.9) f = g + O(h); resp. f = g + (h); if there exists C > 0 (resp. c > 0 ad C > 0) such that (.0) jf? gj Cjhj; resp. cjhj jf? gj Cjhj 3 The odd-eve biomial model 3. Deitio A sequece of asset price models (S ) is called a biomial model if each discouted asset price process (S ) evolves as follows: For t [0; T ] (3.) S t = S 0 exp 6 0 [t] k k= A ;

9 where S P 0 > 0 is a costat ad the icremets ( k ) of the logarithmic discouted returs Xt [t] = k= form a row wise idepedet triagular array. The radom variables k k assume two values U ad k D with positive probabilities k p ad? k p, k (3.) P [ k = U k ] = p k; P [ k = D k ] =? p k for k = ; : : :;. To avoid trivial complicatios we always assume D < 0 < U. k k The model is called homogeeous, if (U ; k D; k p k ) deped o but ot o k. It is called a odd-eve biomial model if these parameters deped oly o ad the parity of k. For later usage we recall the followig lemma. 3. Lemma A biomial model has a uique martigale measure Q, which ca be characterized by the probabilities (3.3) Q [ k = U k ] = q k ; Q [ k = D k ] =? q k ; which are give by the familiar formula (Cox, Ross, ad Rubistei 979; Rachev ad Ruschedorf 994; Shiryaev, Kabaov, Kramkov, ad Mel'ikov 994; Pliska 997) (3.4) q k =? ed k? e D k e U k : 3.3 Deitio A asset price model S is called a Black-Scholes model with parameters (; ) if the discouted asset price process S evolves as follows: For t [0; T ] (3.5) S t = S 0 exp(x t ); where S 0 > 0 ad the logarithmic discouted returs satisfy X t = (? =)t + W t with a stadard Browia motio W. For the costructio of Example. we dee (3.6) ad (3.7) U j? = p + ; D j? =? p + ; U j = p + ; D j =? p + ; p j? = p p j =? p for j = ; : : :; [=], with > 0, > 0 ad 0 < p < to be xed later. We claim that the discouted logarithmic returs (X jp ) coverge i distributio to (XjP ), where the limit X is uder P a Browia motio with drift ad volatility = ( + ) p p(? p). Ideed, deotig by E P ad V P expectatio ad variace with respect to P, a easy calculatio shows that (3.8) E P [ ] = p k (?)k?? (? p) p + ; V P [] = ( + ) k p(? p); 7

10 for k = ; : : :;, therefore (3.9) [t] X k= E P [ t ]! t; [t] X k= V P [ t ]! ( + ) p(? p)t: Sice U! 0 ad k D k! 0 uiformly i k as! our claim follows from the Lideberg- Feller cetral limit theorem. ad.3. So far othig very surprisig. For optio pricig we are iterested i the behavior of the above markets uder the riskeutral probability measures. Isertig the special values of (3.6) ad (3.7) ito (3.4) we obtai asymptotically (3.0) q j? = +? + p + O q j = +? + p + O ; uiformly i j = ; : : :; [=] as!. For the expectatio ad variace uder Q, deoted by E Q ad V Q, we obtai (3.) E Q [ k ] =?? + O for k = ; : : :;. Thus uder Q (3.) [t] X k= E Q [ t ]!? t;?3= ; V Q [ k ] =? + O?3= [t] X k= V Q [ t ]! t: Cosequetly the (X jq ) coverges to (XjQ), where the limit X uder Q is a Browia motio with drift? = ad volatility ^ = p. A elemetary cosideratio shows that we ca produce ay combiatio of > 0 ad ^ > 0 by choosig appropriate values for > 0, > 0 ad 0 < p <. This ishes the costructio of Example Remark If we cosider the odd-eve models as (cotrolled) Markov chai approximatios to the limitig Browia motio, the these models are ot locally cosistet i the sese of Kusher (997), see also Kusher ad Dupuis (99). Let us cosider the cosequeces of Example. for optio prices. A priori we could thik of two ways to calculate the price of a optio o S: Either as limit of the prices of the correspodig optio o S, or alteratively as discouted expectatio uder the martigale measure Q. We will show, that for the Europea call optio ay pair of values withi the trivial bouds for arbitrage-free optio prices may occur i this way. 3.5 Propositio For ay R, > 0 there is a sequece of odd-eve biomial markets that coverge uder the origial measures P to a Black-Scholes market with parameters ;, but the price of a Europea call optio with strike price K R approaches the lower arbitrage boud, i.e., uder the risk-eutral measures Q (3.3) E Q [(S T? K) + ]! (S 0? K) + : 8

11 Also, for ay R, > 0 there is a sequece of odd-eve biomial markets that coverge uder the origial measures P to a Black-Scholes market with parameters (; ), but the price of a Europea call optio with strike price K R approaches the upper arbitrage boud, i.e. (3.4) E Q [(S T? K) + ]! S 0 : Proof: We x ad ad choose a arbitrary ^ > 0. We have see that there is a odd-eve model, such that the limit of (S jp ) ad (S jq ) are geometric Browia motios with volatility resp. ^. A direct ad wellkow calculatio shows that, for xed strike price K, the limit of the price of the Europea call optio (3.5) lim! E Q [(S? K) + ] = E Q [(S? K) + ] = f(^); where f(^) deotes the price of the optio i a Black-Scholes model with volatility ^. Sice (3.6) lim f(^) = (S 0? K) + ; ^!0 lim ^! f(^) = S 0; ad ^ > 0 was arbitrary ay price withi the trivial arbitrage bouds (S 0? K) + ad S 0 may occur as limitig optio price of a odd-eve biomial model, which approximates uder the origial measures a give Black-Scholes model. I the ext propositio we relate the "asymptotically wrog" optio price, which arises if we choose 6= ^ i Example. with the otio of asymptotic arbitrage. 3.6 Propositio If we have 6= ^ i Example. above, the there are strog asymptotic arbitrage possibilities, ad (P ) 4 (Q ): Proof: Let p k ad q k deote the distributio of k uder P ad Q respectively. The the Helliger itegral of order = is give by (3.7) H(p k; q k ) = p p k q k + q(? p k )(? q k ) For the odd-eve model uder discussio we get (3.8) lim! H(p k ; q) = p p k pq + (? p)(? q) uiformly i k = ; : : :; with q = =( + ). By assumptio 6= ^, so p 6= q, hece? p pq? p (? p)(? q) > 0. This implies (3.9) h ( ) = [? H(p k; q k )]! as!. By.5 this is equivalet to etire separatio, which is equivalet to strog asymptotic arbitrage Kabaov ad Kramkov (998, Prop.4).. Actually this proof leads to a example, which was poited out to us by K.Potzelberger, that shows, that cotiguity (or absece of asymptotic arbitrage) is ot a ecessary assumptio for the coclusio of Theorem Example There is a odd-eve model such that uder the physical probability measures (P ) the sequece (S ) coverges i distributio to geometric Browia motio with parameters ad, ad uder the risk eutral probability measures (Q ) the sequece of stock prices (S ) coverges to the correct limit, i.e. geometric Browia motio with parameters? = ad, although the sequece of biomial models permits strog asymptotic arbitrage. 9

12 Dee a odd-eve model as i Example. with p 6= = ad q =? p. The the variace is ot aected, ad the limitig measures P ad Q are equivalet, although by the.5 we have etire separatio.. I Rachev ad Ruschedorf (994, Theorem.) ecessary ad suciet coditios for covergece of a sequece of homogeeous biomial markets to a Black-Scholes market are give, as well as suciet coditios for the covergece of optio prices (see Rachev ad Ruschedorf 994, Theorem 3.). First we demostrate, that i the homogeeous situatio covergece of the stock prices implies i fact covergece of optio prices, or loosely speakig, 'homogeeous biomial models have automatically good covergece properties', cf. the discussio i Williger ad Taqqu (99, 5.). This theorem seems to be wellkow ad of folklore type, it is implicit i Rachev ad Ruschedorf (994), it was metioed to us by K. Potzelberger, but we have bee uable to trace a precise referece i the literature. 3.8 Theorem Suppose a sequece of homogeeous biomial models (S ) with U! 0, D! 0 coverges i distributio uder P to a Black-Scholes model with parameters ;.The uder the correspodig martigale measures (Q ) the sequece (S jq ) coverges to the Black-Scholes model with parameters? =;. Proof: Istead of applyig Theorem. we prefer to give a elemetary proof. The covergece assumptio is equivalet to (3.0) [U p + D (? p )]! ; (U? D ) p (? p )! : This follows from the cetral limit theorem, or may be deduced easily from the coditios give i Rachev ad Ruschedorf (994). We claim (3.) =? lim U D : By assumptio (U p + D (? p ))!, so U p + D (? p ) = O(=). We ca write (3.) (U? D )p =?D + O ; (U? D )(? p ) = U + O : Multiplyig these equatios yields (3.3) (U? D ) p (? p ) =?D U + O (U ) + O (D ) + O : We cosider here oly models with U! 0 ad D! 0, so the claim is proved. Now we calculate the asymptotic expasio of the risk eutral probabilities, (3.4) q =?D? U U? D + O (U? D ) : We d U? (U? D ) q (? q ) =?D U + O (D U (U? D D (3.5) )) + O showig (U? D ) q (? q )!. Fially : (3.6) U q + D (? q ) = D U 0 + O (U? D ) 3 ;

13 showig (U q + D (? q ))!? =. I the settig of homogeeous biomial models we also ca give a coverse to Theorem Propositio Uder the assumptios of Theorem 3.8 suppose that (S jp ) coverges to a o degeerate limit, i.e. > 0. The we have o asymptotic arbitrage (either of rst or secod kid) ad therefore (P ) ad (Q ) are mutually cotiguous. Proof: Accordig to the.5 (Q ) (P ) i (3.7) lim!0 lim sup! X k=? H(; k ; 0 k ) = 0 I the homogeeous biomial world this equatio becomes (3.8) lim lim sup? p q (?)? (? p ) (? q ) (?) = 0:!0! From equatio (3.0), applied to P ad Q, we kow (3.9) (U? D ) p (? p )! ; (U? D ) q (? q )! ; with > 0, thus (3.30) U? D = O = ; From equatio (3.), applied to P ad Q, we kow (3.3) p (U? D ) =?D + O Combiig this estimate with (3.30) gives (3.3) p (? p ) q (? q )! : ; q (U? D ) =?D + O p? q = O = : Equatios (3.3) ad (3.30) imply lim p =q =, or equivaletly (3.33) q = p? w p ; w = O(?= ): Pluggig this expressios ito (3.7) gives (3.34)? p q (?)? (? p )(? q ) (?) = O ((? )): :

14 4 Proof of the mai theorem 4. Proof of Theorem.: To show covergece we use a method iitiated by Prokhorov (see Jacod ad Shiryaev 987, VI.3.8): We prove that (S jq ) is tight ad that (SjQ) is the oly possible limit poit. We assumed that the sequece (S jp ) coverges weakly, hece it is tight. Our cotiguity assumptio guaratees that (S jq ) is tight as well (see Jacod ad Shiryaev 987, X.3.). We cosider the models (S jp ) ad (S jq ) as probability measures o the space D d [0; T ] of R d -valued cadlag fuctios equipped with the Skorokhod topology. Sice D d [0; T ] is a Polish space, for ay weak accumulatio poit of (S jq ), say (SjQ 0 ), there is a subsequece ( k ) k= with (S kjq k )! (SjQ 0 ). The subsequece iherits uiform itegrability. We ow are i a positio to apply a theorem of Meyer-Zheg Meyer ad Zheg (984, Theorem ), which asserts the followig: if (S kjq k ) is a sequece of martigales covergig weakly to a process (SjQ 0 ) with respect to the so called Meyer-Zheg topology o D d [0; T ] (which is weaker tha the Skorokhod topology) satisfyig the uiform itegrability coditio give i the assumptios of Theorem., the the limit (SjQ 0 ) agai is a martigale (with respect to its atural ltratio). Hece we obtai that Q 0 is a martigale measure for S ad from our cotiguity assumptio we obtai that Q 0 is absolutely cotiuous with respect to P (see Wittig ad Muller-Fuk 995, 6.3). Usig the easy Lemma 4. below we coclude that Q 0 equals the uique equivalet martigale measure Q for S. Hece (S jq ) is a tight sequece with (SjQ) beig its uique weak accumulatio poit, which readily shows the weak covergece of (S jq ) to (SjQ). 4. Lemma Suppose S is a Q-martigale ad Q is the oly martigale measure equivalet to P. If Q 0 is a martigale measure for S, which is absolutely cotiuous with respect to P, the Q = Q 0. Proof: Q 00 := (Q+Q0 ) is also a martigale measure, ad it is equivalet to P, thus Q = Q Remark Let us aalyze the assumptios of Theorem. ad covice ourselves that they ideed are ecessary for the theorem to hold true. Firstly we deal with the assumptio that (SjP ) is a complete arbitrage-free market, i.e., that there is a uique equivalet martigale measure Q for S. Clearly this assumptio caot be dropped: ideed, if (SjP ) is such that the set M e (S) of equivalet martigale measures cosists of more tha oe elemet, we may choose (S jp ) (SjP ), for all N, ad may choose a sequece Q M e (S ) = M e (S) which veries the assumptios of uiform itegrability ad cotiguity ad does ot coverge: for example, x Q 0 6= Q 00 i M e (S) ad let, for j N, Q j? = Q 0 ad Q j = Q 00. This trivial example shows that i the cotext of o-complete limitig models (SjP ) the questio has to be posed dieretly: we have to restrict ourselves to special elemets Q of M e (S ) ad M e (S), such as the miimal (Follmer ad Schweizer 99), the variace optimal (Schweizer 996; Delbae ad Schachermayer 996), the Esscher measure (Gerber ad Shiu 994), the etropy miimizig measure (Frittelli 996; Gradits 998; Miyahara 995) etc. ad ask whether it is true that these special choices Q M e (S ) coverge to the correspodig special choice Q M e (S).

15 This questio seems to be a iterestig ad challegig topic for future research. Let us metio i this cotext related results for the case of the miimal (Ruggaldier ad Schweizer 995) ad the variace-optimal martigale measure (Priget 995). A result o approximatios of the variace-optimal martigale measure i L is cotaied i Delbae ad Schachermayer (996). We ow deal with the secod techical assumptio we had to impose i Theorem., the uiform (Q )-itegrability of (S ) =. The subsequet Example 4.4 illustrates i the preset cotext the well-kow pheomeo that i the absece of uiform itegrability the weak limit of a sequece of martigales eed ot to be a martigale (ot eve a local martigale). 4.4 Example Cosider a odd-eve model as above, with = 0 ad this time (4.) U j? = p ; D j? =? p ; ad (4.) U j = l ; D j =? a ; where j = ; : : :; [=] ad a > 0. For simplicity we set R = 0. Choose the probabilities P so, that S is a P -martigale, i.e., such that (3.4) holds true, so that P = Q. A easy calculatio shows (4.3) [t] X k= E P [ t ]!? 4 + a t; [t] X k= V P [ t ]! t: Sice a > 0 the limitig measure P is ot a martigale measure (ot eve a local martigale measure), thus useless for the purpose of optio pricig: it is ot the (uique) martigale measure Q associated to the limitig process (SjP ), which is geometric Browia motio with parameters? =4 + a= ad =. Fially let us discuss the questio whether there is a coverse to Theorem., i.e., whether we ca deduce from the covergece of (S jq ) to (SjQ) somethig about the cotiguity of (Q ) with respect to (P = )? = Ufortuately there is o hope for a geeral result i this directio (compare, however, Propositio 3.9 for a positive result i the case of homogeeous biomial models). It is a wellkow pheomeo i Mathematical Statistics (see, e.g., Strasser 985; Wittig ad Muller-Fuk 995) that i the case of weak covergece of (S jp ) to (SjP ) ad (S jq ) to (SjQ) the absolute cotiuity of Q with respect to P does ot imply the cotiguity of (Q ) = with respect to (P ) =. Example 3.7 illustrates this situatio. 5 Triomial models I this sectio we shall cosider homogeeous triomial models. These are obvious extesios of the biomial models 3.. The dierece is that the icremets k of the logarithmic returs X assume three values U ; M ; D with positive probabilities. The resultig markets are icomplete. For simplicity we choose (5.) U = p ; M = 0; D =? p ; 3

16 with some > 0 ad the probabilities (5.) p (U ) = p (M ) = p (D ) = 3 : We will see, that i cotrast to homogeeous biomial models (cf. Theorem 3.8) homogeeous triomial models do ot possess good covergece properties automatically. 5. Propositio (i) The sequece of triomial asset price models (S jp ) deed above coverges weakly to (SjP ), which is geometric Browia motio with parameters 0; =3. (ii) The family of equivalet martigale measures Q, uder which the process is agai a homogeeous triomial model ca be characterized by the probabilities q (U ) =? ed ; q e U? (M ) =? ; q (D ) = eu? (5.3) e D e U? e D with 0 < <. (iii) If! =3 as! the (S jq )! (SjQ), which is geometric Browia motio with parameters? =3; =3. So i this case P is equivalet to Q. If (5.4) = 3 + O p ; the (P ) ad (Q ) are mutually cotiguous, otherwise we have etire separatio. Proof: (i) We have (5.5) E P [ k ] = 0; V P [ k ] = 3 ad by Theorem. (X jp )! (XjP ), where the limit X is a Browia motio with zero drift ad variace =3. (ii) We are iterested i martigale measures Q preservig the idepedece of the icremets. From the martigale equatio u = (k? )= ad t = k= with k = ; : : :; (5.6) E Q [S t jf u ] = S ue Q [e k ] = S u we see that these (Q ) ca be characterized by (5.7) e U q (U ) + q (M ) + e D q (D ) = : The solutios (5:3) are covex combiatios of the measure igorig the icremets with value zero ad the measure assigig all mass to it. The (5.8) E Q [ k ] =? + O?3= ; V Q [ k ] = + O?3= : If! by the Lideberg-Feller theorem (X jq )! (XjQ), where the limit is Browia motio with drift? =3 ad variace. We have (5.9)! 3 () lim V P [X ] = lim t V Q [X t ] () P Q:!! 4

17 (iii) To study cotiguity with the criterio from Theorem.5 we must cosider (5.0)h () =? p (U ) q (U )?? p (M ) q (M )?? p (D ) q (D )? : First we will show that! =3 is ecessary for cotiguity, ext we ree the argumet ad get (5:4) as ecessary coditios. Fially it turs out that this is actually suciet for mutual cotiguity. (5.) h ( ) = "? r r r # 3 q (U )? 3 q (M )? 3 q (U ) If we take ay coverget subsequece ( k ) with limit as k!, the (5.) q k (U k )! ; q k (M k )!? ; q k (D k )! ; ad (5.3) h ( ) If the sequece does ot coverge we have r r r!? 6? 3 (? )? 6 : (5.4) lim sup h ( ) = ;! which meas etire separatio. From ow o we assume = ( + 3 ) with! 0. Usig the asymptotic expressio (5.5)? p + O we obtai (5.6) h ( ) Therefore lim sup show (5.7)! h () = q = =? p +? 3 3 h ( ) < implies =? 3 (?)? p? + O = ( ) + O : O p. With this estimate we ca actually 3 (? )? + O ((? )) uiformly i (0; ). A Taylor expasio of? (=3)?a? (=3) (? )? aroud = =3 reveals (5.8)??? 3 (? )? = (? )?! ; 3 3 uiformly i (0; ) ad (0; ). Theorem.5 implies mutual cotiguity for = =3 + O(?= ), etire separatio otherwise. 5

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