On he Timing Opion in a Fuures Conrac Francesca Biagini, Mahemaics Insiue Universiy of Munich Theresiensr. 39 D-80333 Munich, Germany phone: +39-051-2094459 Francesca.Biagini@mahemaik.uni-muenchen.de Tomas Björk Deparmen of Finance Sockholm School of Economics Box 6501, SE-113 83 Sockholm SWEDEN phone: +46-8-7369162 omas.bjork@hhs.se (This version November 2005, firs version June 2003) To appear in Mahemaical Finance Absrac The iming opion embedded in a fuures conrac allows he shor posiion o decide when o deliver he underlying asse during he las monh of he conrac period. In his paper we derive, wihin a very general incomplee marke framework, an explici model independen formula for he fuures price process in he presence of a iming opion. We also provide a characerizaion of he opimal delivery sraegy, and we analyze some concree examples. Key words: Fuures conrac, iming opion, opimal sopping Corresponding auhor. Suppor from he Tom Hedelius and Jan Wallander Foundaion is graefully acknowledged. Boh auhors are graeful o B. Näslund, J. Kallsen, C. Kuehn, an anonymous associae edior, and an anonymous referee for a number of very helpful commens and suggesions. 1
1 Inroducion In sandard exbook reamens, a fuures conrac is ypically defined by he properies of zero spo price and coninuous (or discree) reselemen, plus a simple no arbirage condiion a he las delivery day. If he underlying price process is denoed by X and he fuures price process for delivery a T is denoed by F (, T ) his leads o he well known formula F (, T )=E Q [X T F, 0 T (1) where Q denoes he (no necessarily unique) risk neural maringale measure. In pracice, however, here are a number of complicaing facors which are ignored in he exbook reamen, and in paricular i is ypically he case ha a sandard fuures conrac has several embedded opion elemens. The mos common of hese opions are he iming opion, and he end-of-he-monh opion, he qualiy opion, and he wild card opion. All hese opions are opions for he shor end of he conrac, and hey work roughly as follows. The iming opion is he opion o deliver a any ime during he las monh of he conrac. The end of he monh opion is he opion o deliver a any day during he las week of he conrac, despie he fac ha he fuures price for he las week is fixed on he firs day of ha week and hen held consan. The qualiy opion is he opion o choose, ou of a prespecified baske of asses, which asse o deliver. The wild card opion is, for example for bond fuures, he opion o iniiae delivery beween 2 p.m. and 8 p.m. in he afernoon during he delivery monh of he conrac. The poin here is ha he fuures price is seled a 2 p.m. bu he rade in he underlying bonds goes on unil 8 p.m. The purpose of he presen paper is o sudy he iming opion wihin a very general framework, allowing for incomplee markes, and our goal is o invesigae how he general formula (1) has o be modified when we inroduce a iming opion elemen. Our main resul is given in Theorems 4.2 and 4.3 where i is shown ha, independenly of any model assumpions, he fuures price in he presence of he iming opion is given by he formula F (, T ) = inf τ T EQ [X τ F. (2) where τ varies over he class of opional sopping imes, and inf denoes he essenial infimum. This formula is of course very similar o he pricing formula for an American opion. Noe, however, ha (2) does no follow direcly from sandard heory for American conracs, he reason being ha he fuures price is no a price in he echnical sense. The fuures price process insead plays he role of he cumulaive dividend process for he fuures conrac, which in urn can be viewed as a price-dividend pair, wih spo price idenically equal o zero. 2
Furhermore we prove ha he opimal delivery policy ˆτ(), for a shor conrac enered a, is given by ˆτ = inf { 0; F (, T )=X }. (3) We also sudy some special cases and show he following. If he underlying X is he price of a raded financial asse wihou dividends, hen i is opimal o deliver immediaely, so ˆτ() = and hus F (, T )=X. (4) If he underlying X has a convenience yield which is greaer han he shor rae, hen he opimal delivery sraegy is o wai unil he las day. In his case we hus have ˆτ() =T and F (, T )=E Q [X T F, (5) which we recognize from (1) as he classical formula for fuures conracs wihou a iming opion. Opion elemens of fuures conrac have also been sudied earlier. The qualiy opion is discussed in deail in Gay and Manaser (1984), and he wild card opion is analyzed in Cohen (1995) and Gay and Manaser (1986). The iming opion is (among oher opics) reaed in Boyle (1989) bu heoreical resuls are only obained for he special case when X is he price process of a raded underlying asse. In his seing, and under he added assumpion of a consan shor rae, he formula (4) is derived. The organizaion of he paper is as follows. In Secion 2 we se he scene for he financial marke. Noe ha we make no specific model assumpions a all abou marke compleeness or he naure of he underlying process, and our seup allows for discree as well as coninuous ime models. In Secion 3 we derive a fundamenal equaion, he soluion of which will deermine he fuures price process. We aack he fundamenal equaion by firs sudying he discree ime case in Secion 4.1, and prove he main formula (2). In Secion 4.2 we prove he parallel resul in he echnically more demanding coninuous ime case. We finish he main paper by some concree financial applicaions, and in paricular we clarify compleely under which condiions he fuures price process, including an embedded iming opion, coincides wih he classical formula (1). A he oher end of he specrum, we also invesigae under which condiions immediae delivery is opimal. 2 Seup We consider a financial marke living on a sochasic basis (Ω, F, F,Q), where he filraion F = {F } 0 T saisfies he usual condiions. We allow for boh discree and coninuous ime, so he conrac period is eiher he inerval [0,T or 3
he se {0, 1,...,T}. To se he financial scene we need some basic assumpions, so for he res of he paper we assume ha here exiss a predicable shor rae process r, and a corresponding money accoun process B. In coninuous ime B has he dynamics db = r B d. (6) In he discree ime case, he shor rae a ime will be denoed by r +1 so he bank accoun B has he dynamics B +1 =(1+r +1 ) B. (7) In his case he shor rae is assumed o be predicable, i.e. r is F 1 -measurable (r is known already a 1) for all, wih he convenion F 1 = F 0. The marke is assumed o be free of arbirage in he sense he measure Q above is a maringale measure w.r. he money accoun B for he given ime horizon. Noe ha we do no assume marke compleeness. Obviously; if he marke is incomplee, he maringale measure Q will no be unique, so in an incomplee seing he pricing formulas derived below will depend upon he paricular maringale measure chosen. We discuss his in more deail in Secion 5. We will need a weak boundedness assumpion on he shor rae. Assumpion 2.1 For he res of he paper we assume he following. In he coninuous ime case we assume ha he ineres rae process is predicable, and ha here exiss a posiive real number c such ha r c, (8) wih probabiliy one, for all. Defining he money accoun as usual by B by B ( ) = exp 0 r sds we assume ha E Q [B T <. (9) In he discree ime case we assume he ineres rae process is predicable, and ha here exiss a posiive real number c such ha wih probabiliy one, for all n. Remark 2.1 We noe ha he if we define C by 1+r n c, (10) C = sup τ T E Q [B τ F (11) where τ varies over he class of sopping imes, hen he inequaliy E Q [B T < easily implies C < (12) Q a.s. for all [0,T. 4
Wihin his framework we now wan o consider a fuures conrac wih an embedded iming opion. Assumpion 2.2 We assume he exisence of an exogenously specified nonnegaive adaped cadlag process X. The process X will henceforh be referred o as he index process, and we assume ha E Q [X <, 0 T (13) The inerpreaion of his assumpion is ha he index process X is he underlying process on which he fuures conrac is wrien. For obvious reasons we wan o include conracs like commodiy fuures, index fuures, fuures wih an embedded qualiy opion, and also fuures on a non-financial index like a weaher fuures conrac. For his reason we do no assume ha X is he price process of a raded financial asse in an idealized fricionless marke. Typical choices of X could hus be one of he following. X is he price a ime of a commodiy, wih a non rivial convenience yield. X is he price a ime of a, possibly dividend paying, financial asse. X = min { } S 1,...,Sn where S 1,...,S n are price processes of financial asses (for example socks or bonds). This seup would be naural if we have an embedded qualiy opion. X is a non financial process, like he emperaure a some prespecified locaion. We now wan o define a fuures conrac, wih an embedded iming opion, on he underlying index process X over he ime inerval [0,T. If, for example, we are considering a US ineres rae fuure, his means ha he inerval [0,T corresponds o he las monh of he conrac period. Noe ha we hus assume ha he iming opion is valid for he enire inerval [0,T. The analysis of he fuures price process for imes prior o he iming opion period, is rivial and given by sandard heory. If, for example, we le he iming opion be acive only in he inerval [T 0,T, hen we immediaely obain F (, T )=E Q [F (T 0,T) F, 0 T 0, (14) where F (T 0,T) is given by he heory developed in he presen paper. We can now give formal definiion of he (coninuous ime) conrac. See below for he discree ime modificaion. Definiion 2.1 A fuures conrac on X wih final delivery dae T, including an embedded iming opion, on he inerval [0,T, wih coninuous reselemen, is a financial conrac saisfying he following clauses. 5
A each ime [0,T here exiss on he marke a fuures price quoaion denoed by F (, T ). Furhermore, for each fixed T, he process F (, T ) is a semimaringale w.r.. he filraion F. Since T will be fixed in he discussion below, we will ofen denoe F (, T ) by F. The holder of he shor end of he fuures conrac can, a any ime [0,T, decide wheher o deliver or no. The decision wheher o deliver a or no is allowed o be based upon he informaion conained in F. If he holder of shor end decides o deliver a ime, she will pay he amoun X and receive he quoed fuures price F (, T ). If delivery has no been made prior o he final delivery dae T, he holder of he shor end will pay X T and receive F (T,T). During he enire inerval [0,T here is coninuous reselemen as for a sandard fuures conrac. More precisely; over he infiniesimal inerval [, + d he holder of he shor end will pay he amoun df (, T )=F ( + d, T ) F (, T ). The spo price of he fuures conrac is always equal o zero, i.e. you can a any ime ener or leave he conrac a zero cos. The cash flow for he holder of he long end is he negaive of he cash flow for he shor end. The imporan poin o noice here is ha he iming opion is only an opion for he holder of he shor end of he conrac. For discree ime models, he only difference is he reselemen clause which hen says ha if you hold a shor fuure beween and + 1, you will pay he amoun F ( +1,T) F (, T ) a ime +1. Our main problem is he following. Problem 2.1 Given an exogenous specificaion of he index process X, wha can be said abou he exisence and srucure of he fuures price process F (, T )? 3 The Fundamenal Pricing Equaion We now go on o reformulae Problem 2.1 in more precise mahemaical erms, and his will lead us o a fairly complicaed infinie dimensional sysem of equaions for he deerminaion of he fuures price process (if ha objec exiss). We focus on he coninuous ime case, he discree ime case being very similar. 6
3.1 The Pricing Equaion in Coninuous Time For he given final delivery dae T, le us consider a fixed poin in ime T and discuss he (coninuous ime) fuures conrac from he poin of view of he shor end of he conrac. From he definiion above, i is obvious ha he holder of he shor end has o decide on a delivery sraegy, and we formalize such a sraegy as a sopping ime τ, wih τ T, Q a.s. If he holder of he shor end uses he paricular delivery sraegy τ, hen he arbirage free value of her cash flows is given by he expression [ E Q e τ rsds (F τ X τ ) τ e u rsds df u F. (15) The firs erm in he expecaion corresponds o he cash flow for he acual delivery, i.e. he shor end delivers X τ and receives he quoed fuures price F τ, and he inegral erm corresponds o he cash flow of he coninuous reselemen. Since he iming opion is an opion for he holder of he shor end, she will ry o choose he sopping ime τ so as o maximize he arbirage free value. Thus, he value of he shor end of he fuures conrac a ime is given by [ sup E Q e τ τ rsds (F τ X τ ) e u rsds df u F, (16) τ T where, for shor sup denoes he essenial supremum. We now recall ha, by definiion, he spo price of he fuures conrac is always equal o zero, and we have hus derived our fundamenal pricing equaion, which is in fac an equilibrium condiion for each. Proposiion 3.1 The fuures price process F, if i exiss, will saisfy for each [0,T, he fundamenal pricing equaion [ sup E Q e τ τ rsds (F τ X τ ) e u rsds df u F =0, (17) τ T where τ varies over he class of sopping imes. Some remarks are now in order. Remark 3.1 A firs sigh, equaion (17) may look like a sandard opimal sopping problem, bu i is in fac more complicaed han ha. Obviously: for a given fuures price process F, he lef hand side of (17) represens a sandard opimal sopping problem, bu he poin here is ha he fuures process F is no an a priori given objec. Insead we have o find a process F such ha he opimal sopping problem defined by he lef hand side of (3) has he opimal value zero for each T. Thus, our formalized problem is as follows. 7
I is no a all obvious ha here exiss a soluion process F o he fundamenal equaion 3.1, and i is even less obvious ha a soluion will be unique. These quesions will be reaed below. I may seem ha we are only considering he fuures price process from he perspecive of he seller of he conrac. However; he oal cash flows sum o zero, so if he fundamenal pricing equaion above is saisfied, he (spo) value of he conrac is zero also o he buyer (and if exercised in a non opimal fashion, he value would be posiive for he buyer and negaive for he seller). The main problems o be sudied are he following. Problem 3.1 Consider an exogenously given index process X. Our primary problem is o find a process {F ; 0 T } such ha (17) is saisfied for all [0,T. If we manage o find a process F wih he above properies, we would also like o find, for each fixed [0,T, he opimal sopping ime ˆτ realizing he supremum in [ sup E Q τ T e τ τ rsds (F τ X τ ) e u rsds df u F. (18) We also noe ha even if we manage o prove he exisence of a soluion process F, here is no guaranee of he exisence of an opimal sopping ime ˆτ, since in he general case we can (as usual) only be sure of he exisence of ɛ-opimal sopping imes. 3.2 Some Preliminary Observaions A complee reamen of he pricing equaion will be given in he nex wo secions, bu we may already a his sage draw some preliminary conclusions. Lemma 3.1 The fuures price process has o saisfy he condiion F (, T ) X, T, (19) F (T,T) = X T. (20) Proof. The economic reason for (19) is obvious. If, for some, we have F (, T ) >X hen we ener ino a shor posiion (a zero cos) and immediaely decide o deliver. We pay X and receive F (, T ) hus making an arbirage profi, and immediaely close he posiion (again a zero cos). A more formal proof is obained by noing ha he fundamenal equaion (17) implies ha [ E Q e τ τ rsds (F τ X τ ) e u rsds df u F 0, (21) 8
for all sopping imes τ wih τ T. In paricular, he (21) holds for τ = which gives us E Q [F X F 0, (22) and since boh F and X are adaped, he inequaliy (19) follows. The boundary condiion (20) is an immediae consequence of no arbirage. We finish his secion by proving ha for he very special case of zero shor rae, we can easily obain an explici formula for he fuures price process. Noe ha, for simpliciy of noaion, he symbol inf henceforh denoes he essenial infimum. Proposiion 3.2 If r 0, hen F (, T ) = inf τ T EQ [X τ F. (23) Proof. Wih zero shor rae he fundamenal equaion reads as sup E [(F Q τ X τ ) df u F =0. (24) τ T Using he fac ha τ df u = F τ F we hus obain sup E Q [F X τ F =0. (25) τ T Since F is adaped his implies τ F = sup E Q [ X τ F = inf τ T τ T EQ [X τ F. (26) In he Secion 4 we will prove ha he formula (23) is in fac valid also in he general case wihou he assumpion of zero shor rae. 3.3 The Pricing Equaion in Discree Time Case By going hrough a compleely parallel argumen as above, i is easy o see ha in a discree ime model he fundamenal equaion (17) will have he form [( τ ) ( sup E Q 1 τ n ) 1 (F τ X τ ) F n F =0, τ T 1+r n=+1 n 1+r n=+1 u=+1 u (27) where F n = F n F n 1. 4 Deermining he Fuures Price Process In his secion we will solve he fundamenal pricing equaions (17) and (27), hus obaining an explici represenaion for he fuures price process. We sar wih he discree ime case, since his is echnically less complicaed. 9
4.1 The Discree Time Case We will no analyze equaion (27) direcly, bu raher use a sandard dynamic programming argumen as a way of aacking he problem. To do his we consider he decision problem of he holder of he shor end of he fuures conrac. Suppose ha a ime n you have enered ino he shor conrac. Then you have he following wo alernaives: 1. You can decide o deliver immediaely, in which case you will receive he amoun F n X n. (28) 2. You can decide o wai unil n + 1. This implies ha a ime n +1 you will obain he amoun F n F n+1. The arbirage free value, a n, of his cash flow is given by he expression E Q [ Fn F n+1 1+r n+1 F n = 1 1+r n+1 E Q [F n F n+1 F n, (29) where we have used he fac ha r is predicable. The value of your conrac, afer having received he cash flow above, is by definiion zero. Obviously you would like o make he bes possible decision, so he value a ime n of a shor posiion is given by [ 1 max (F n X n ), E Q [F n F n+1 F n. (30) 1+r n+1 On he oher hand, he spo price of he fuures conrac is by definiion always equal o zero, so we conclude ha [ 1 max (F n X n ), E Q [F n F n+1 F n =0, (31) 1+r n+1 for n =1,...T 1. We now recall he following basic resul from opimal sopping heory (see Snell 1952). Theorem 4.1 (Snell Envelope Theorem) Wih noaions as above, define he opimal value process V by V = inf τ T EQ [X τ F (32) where τ varies of he class of sopping imes. Then V is characerized by he propery of being he larges submaringale dominaed by X. The process V above is referred o as he (lower) Snell Envelope of X wih horizon T, and we may now sae and prove our main resul in discree ime. 10
Theorem 4.2 Given he index process X, and a final delivery dae T, he fuures price process F (, T ) exiss uniquely and coincides wih he lower Snell envelope of X wih horizon T, i.e. F (, T ) = inf τ T EQ [X τ F, (33) where τ varies over he se of sopping imes. Furhermore, if he shor posiion is enered a ime, hen he opimal delivery ime is given by ˆτ() =inf{k ; F k = X k }. (34) Proof. We will show ha here exiss a unique fuures price process F and ha i is in fac he larges submaringale dominaed by X. The resul hen follows direcly from he Snell Envelope Theorem. We sar by noing ha, since by Assumpion 2.1 1 + r n > 0, we can wrie (31) as max [ (F n X n ),E Q [F n F n+1 F n =0, (35) and since F is adaped his implies F n = max [ X n, E Q [F n+1 F n. (36) This gives us he recursive sysem F n = min [ X n,e Q [F n+1 F n, n =0,...,T 1 (37) F T = X T, (38) where he boundary condiions follows direcly from no arbirage. This recursive formula for F proves exisence and uniqueness. We now go on o prove ha F is a submaringale dominaed by X. From (37) we immediaely have F n E Q [F n+1 F n, which proves he submaringale propery, and we also have F n X n, which in fac was already proved in Lemma 3.1. I remains o prove he maximaliy propery of F and for his we use backwards inducion. Assume hus ha Z is a submaringale dominaed by X. In paricular his implies ha Z T X T, bu since F T = X T we obain Z T F T. For he inducion sep, assume ha Z n+1 F n+1. We hen wan o prove ha his implies he inequaliy Z n F n. To do his we observe ha he submaringale propery of Z ogeher wih he inducion assumpion implies Z n E Q [Z n+1 F n E Q [ F n+1 F n By assumpion we also have Z n X n, so we have in fac Z n min [ X n, E Q [F n+1 F n, and from his inequaliy and (37) we obain Z n F n. 11
4.2 The Coninuous Time Case and Some Examples We now go on o find a formula for he fuures price process in coninuous ime and, based on he discree ime resuls of he previous secion, we of course conjecure ha also in coninuous ime we have he formula F (, T )= inf τ T E Q [X τ F. Happily enough, his also urns ou o be correc, bu a echnical problem is ha in coninuous ime i is impossible o jus mimic he discree ime argumens above, since we can no longer use inducion. Thus we have o use oher mehods, and we will rely on some very nonrivial resuls from coninuous ime opimal sopping heory. All hese resuls can be found in he highly readable Appendix D in Karazas and Shreve (1998). From now on, we assume he following furher inegrabiliy condiion on he underlying process X: [ E Q sup X < (39) 0 T Before proving our main resul, we need he following echnical resul. Lemma 4.1 Suppose ha he index process X saisfies condiion (39) and consider is Snell Envelope F (, T ) = inf τ T E Q [X τ F. Le H = e rudu 0, where r saisfies he weak boundness Assumpion 2.1. If τ T is a sopping ime such ha he sopped submaringale F τ is a maringale, hen he sochasic inegral s 0 e rudu 0 dfs τ is a maringale. Proof. We recall ha by Assumpion 2.2, he index process X is supposed o be a nonnegaive adaped càdlàg process. Hence if τ is a sopping ime such ha he sopped submaringale F τ is a maringale, i is indeed a càdlàg maringale and consequenly we need only o verify ha E Q ( T 0 H 2 s d[f τ s ) 1 2 < (40) in order o guaranee ha he sochasic inegral rudu 0 dfs τ is a maringale by using he Burkholder-Davis-Gundy inequaliies (see Revuz and Yor (1994), p.151, and Proer (2004) p.193). Here, he process [F τ is he quadraic variaion of F τ (for furher deails, see Proer (2004) p.66). Since H = e rudu 0 and r is uniformly bounded from below (Assumpion 2.1), we obain he following esimaes ( ) 1 ( T 2 T ) 1 E Q Hs 2 d[f τ s = E Q e 2 s 2 rudu 0 d[f τ s 0 E Q 0 ( T 0 0 e s ) 1 2 e 2cs d[f τ s 12
[ e ct E Q ([F τ T ) 1 2 Since he process F is given by he Snell Envelope of X, i is a nonnegaive submaringale dominaed by X, hence we can use he Burkholder-Davis-Gundy inequaliies and ge [ [ E Q ([F τ T ) 1 2 ke Q sup 0 T F τ ke Q [ sup F 0 T ke Q [ sup 0 T where k is a suiable consan. Since X saisfies (39), he las erm of he inequaliy is finie. Hence, we can conclude ha he sochasic inegral s 0 e rudu 0 dfs τ is a maringale. We may now sae our main resul in coninuous ime. Theorem 4.3 Under Assumpion 2.1 and if (39) holds, here exiss, for each fixed T, a unique fuures price process F (, T ) solving he he fundamenal equaion (17). The fuures price process is given by he expression F (, T ) = inf τ T EQ [X τ F. (41) Furhermore, if X has coninuous rajecories, hen he opimal delivery ime ˆτ(), for he holder of a shor posiion a ime is given by X ˆτ() =inf{u ; F (u, T )=X u }. (42) Proof. We firs show ha if we define F by (41) hen F solves he pricing equaion (17). Having proved his we will hen go on o prove ha if F solves (17), hen F mus necessarily have he form (41). We hus sar by defining a process F as he lower Snell envelope of X, i.e. F = inf τ T EQ [X τ F, (43) and we wan o show ha for his choice of F, he fundamenal pricing equaion (17) is saisfied. From he (coninuous ime version of) Snell Envelope Theorem, we know ha F is a submaringale. Thus (for fixed ) he sochasic differenial e u rsds df u, is a submaringale differenial, and since F X we see ha he inequaliy [ E Q e τ τ rsds (F τ X τ ) e u rsds df u F 0, (44) will hold for every sopping ime τ wih τ T. To show ha F defined as above saisfies (17) i is herefore enough o show ha for some sopping ime τ we have [ E Q e τ τ rsds (F τ X τ ) e u rsds df u F =0. (45) 13
For simpliciy of exposiion we now assume ha, for each, he infimum in he opimal sopping problem inf τ T EQ [X τ F, (46) is realized by some (no necessarily unique) sopping ime ˆτ. The proof of he general case is more complicaed and herefore relegaed o he appendix. From general heory (see Karazas and Shreve (1998) p. 355, Theorem D9) we cie he following facs. 1. Wih F defined by (43) we have 2. The sopped process F ˆτ defined by Fˆτ = Xˆτ. (47) F ˆτ s = F s ˆτ, (48) where denoes he minimum, is a maringale on he inerval [, T. Choosing τ =ˆτ, equaion (45) hus reduces o he equaion which we can wrie as E Q [ ˆτ E Q [ T e u rsds df u F =0, (49) e u rsds df ˆτ u F =0, (50) and since, by a small variaion of Lemma 4.1, he process s rvdv dfu ˆτ is a maringale, (50) is indeed saisfied. This proves exisence. In order o prove uniqueness le us assume ha, for a fixed T, a process F solves (17). We now wan o prove ha F is in fac he lower Snell envelope of X, i.e. we have o prove ha F is he larges submaringale dominaed by X. We firs noe ha, afer premuliplicaion wih he exponenial facor 0 0 e 0 rsds, u e he fundamenal equaion (17) can be rewrien as e u [ τ rsds 0 df u = inf τ T EQ e u rsds 0 df u + e τ rsds 0 (X τ F τ ) F Defining he process V by V = 0 (51) e u 0 rsds df u, (52) 14
we hus see ha V is he lower Snell envelope of he process Z, defined by Z = 0 e u rsds 0 df u + e rsds 0 (X F ), (53) i.e. V = inf τ T EQ [Z τ F, (54) From he Snell Theorem i now follows ha V is a submaringale, and since he exponenial inegrand in (52) is posiive, his implies ha also F is a submaringale. We have already proved in Proposiion 3.1 ha F X so i only remains o prove maximaliy. To his end, le us assume ha G is a submaringale dominaed by X. We now wan o prove ha G F for every T. To his end we choose a fixed bu arbirary. For simpliciy of exposiion we now assume ha, for a fixed, here exiss and opimal sopping ime aaining he infimum in (54), and he denoe his sopping ime by τ. The proof in he general case is found in he appendix. We obain from (17) [ E Q e τ τ r sds (F τ X τ ) e u rsds df u F = 0 (55) Since F X and F is a submaringale, his implies ha and ha E Q [ τ F τ = X τ, (56) e u rsds df u F =0, (57) which in urn (afer premuliplicaion by an exponenial facor) implies ha E Q [V τ F =V. (58) Since V is a submaringale, his implies ha he sopped process V τ is in fac a maringale on he ime inerval [, T, which in urn implies ha he sopped process F τ is a maringale on [, T. In paricular we hen have F = E Q [F τ F =E Q [X τ F, (59) where we have used (56). On he oher hand, from he assumpions on G we have G E Q [G τ F E Q [X τ F =F, (60) which proves he maximaliy of F. The second saemen in he heorem formulaion follows direcly from Theorem D.12 in Karazas and Shreve (1998). As a more or less rivial consequence, we immediaely have he following resul for fuures on underlying sub- and supermaringales. 15
Proposiion 4.1 If X is a submaringale under Q, hen and i is always opimal o deliver a once, i.e. If X is a supermaringale under Q, hen and i is always opimal o wai, i.e. F (, T )=X, (61) ˆτ() =. (62) F (, T )=E Q [X T F, (63) Proof. Follows a once from he represenaion (41). ˆτ() =T. (64) From his resul we immediaely have some simple financial implicaions. Proposiion 4.2 Assume ha one of he following condiions hold 1. X is he price process of a raded financial asse wihou dividends, and he shor rae process r is nonnegaive wih probabiliy one. 2. X is he price process of a raded asse wih a coninuous dividend yield rae process δ such ha δ r for all wih probabiliy one. 3. X is an exchange rae process (quoed as unis of domesic currency per uni of foreign currency) and he foreign shor rae r f has he propery ha r f r for all wih probabiliy one. Then he fuures price is given by and i is always opimal o deliver a once, i.e. F (, T )=X, (65) ˆτ() =. (66) Proof. The Q dynamics of X are as follows in he hree cases above dx = r X d + dm, dx = X [r δ d + dm, dx = X [r r f d + dm, where M is he generic noaion for a maringale. The assumpions guaranee, in each case, ha X is a Q-submaringale and we may hus apply Proposiion 4.1. Wih an almos idenical proof we have he following parallel resul, which shows ha under cerain condiions he fuures price process is no changed by he inroducion of a iming opion. 16
Proposiion 4.3 Assume ha one of he following condiions hold 1. X is he price process of an asse wih a convenience yield rae process γ such ha γ r for all wih probabiliy one. 2. X is an exchange rae process (quoed as unis of domesic currency per uni of foreign currency) and he foreign shor rae r f has he propery ha r f r for all wih probabiliy one. Then he fuures price is given by F (, T )=E Q [X T F, (67) and i is always opimal o wai unil T o deliver, i.e. 5 Conclusions and Discussion ˆτ() =T. (68) The main resul of he presen paper is given in Theorems 4.2 and 4.3 where we provide he formula F (, T ) = inf τ T EQ [X τ F (69) which gives us he arbirage free fuures price process in erms of he underlying index X and he maringale measure Q. In Secion 4.2 we also gave some immediae implicaions of he general formula, bu hese resuls are of secondary imporance. We now have a number of commens on he main resul (69). We see ha he formula (69) for he fuures price in he presence of a iming opion looks very much like he sandard pricing formula for an American opion. Therefore, one may perhaps expec ha (69) is a direc consequence of he well known pricing formula for American conracs. As far as we can undersand, his is no he case. As noed above, he fuures price process F (, T ) is no a price process a all, since is economic role is ha of a cumulaive dividend process for he fuures conrac (which always has spo price zero). From a more echnical poin of view, we also see ha he deerminaion of he F process is quie inricae, since F has o solve he infinie dimensional fundamenal equaion (17) (which is in fac an equilibrium condiion for each ) or he corresponding discree ime equaion (31). We assumed absence of arbirage bu we did no make any assumpions concerning marke compleeness. In an incomplee marke, he maringale measure Q is no unique, so in his case formula (69) does no provide us wih a unique arbirage free fuures price process. In an incomplee seing, he inerpreaion of Theorems 4.2 and 4.3 is hen ha, given absence of arbirage, he fuures price process has o be given by formula (69) for 17
some choice of a maringale measure Q. This is of course compleely parallel o he sandard risk neural pricing formula which, in he incomplee seing, gives us a price of a coningen claim which depends upon he maringale measure chosen. Noe however, ha some of he resuls above are independen of he choice of he maringale measure. In paricular his is rue for Proposiion 4.2. A A Proof of Theorem 4.3 in he General Case In his Appendix we provide he proof of Theorem 4.3 for he general case, i.e. wihou assuming ha he infima in (46) and (54) are aained. We sar wih he exisence proof and o his end we define he process F (as before) by F = inf τ T EQ [X τ F, (70) and we have o show ha F hus defined saisfies he fundamenal pricing equaion [ sup E Q e τ rsds (F τ X τ ) τ T τ e u rsds df u F =0. (71) As in he simplified proof above i is easy o see ha [ E Q e τ τ rsds (F τ X τ ) e u rsds df u F 0, (72) for all sopping imes τ wih τ T. Thus; o prove ha F saisfies (71) i is enough o prove ha here exiss a sequence of sopping imes {τ n } n=1 such ha [ E Q e τn τn r sds (F τn X τn ) e u rsds df u F 1 n, (73) for all n. To do his we consider a fixed and define τ n by τ n = inf {s ; F s X s (1 1/n)}. (74) Like in he earlier proof we can rewrie he sochasic inegral in (73) as τn e u T rsds df u = e u rsds dfu τn, and i can be shown (see Karazas and Shreve 1998) ha he sopped process F τn is a maringale. Thus, by Lemma 4.1 we ge ha he sochasic differenial e u rsds dfu τn 18
is a maringale differenial, and we obain [ E Q e τn r sds (F τn X τn ) = E Q [ e τn τn r sds (F τn X τn ) F. e u rsds df u F From he definiion of τ n we hen have [ E Q e τn [ r sds (F τn X τn ) F E Q e τn ([ r sds 1 1 ) X τn X τn F n = 1 [ n EQ e τn r sds X τn F Furhermore we have [ E Q e τn r sds X τn F e c(t ) E Q [X τn F 1 1 1/n EQ [F τn F = e c(t ) F 1 1/n. where we again have used he maringale propery of he sopped process F τn. We hus have [ E Q e τn r sds (F τn X τn ) τn e u rsds c(t ) F df u F e n(1 1/n) which ends o zero as n. We now urn o he uniquenesss proof and for his we consider again a fixed and define for each n he sopping ime τ n by τ n = inf { s ; V s Z s 1 n }. (75) Since V τn is a maringale on [, T and since V is given by (52) i now follows ha F τn is a maringale on he same inerval. By definiion of τ n i follows ha e τn r sds (X τn F τn ) 1/n, (76) so we have X τn F τn + 1 n Bτ n (77) B Now assume ha G is a submaringale dominaed by X. We hen obain G E Q [G τn F E Q [X τn F (78) E Q [F τn F + 1 [ n Bτn EQ B F (79) = F + 1 [ n Bτn EQ B F F + 1 n C (80) B where C is given by (11). Leing n gives us G F and we are done. 19
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