Research Paper Number 197 From Discrete to Cotiuous Time Fiace: Weak Covergece of the Fiacial Gai Process Darrell Duffie ad Philip Protter November, 1988 Revised: September, 1991 Forthcomig: Mathematical Fiace Abstract: Coditios, suitable for applicatios i fiace, are give for the weak covergece (or covergece i probability) of stochastic itegrals. For example, cosider a sequece S of security price processes covergig i distributio to S ad a sequece θ of tradig strategies covergig i distributio to θ. We survey coditios uder which the fiacial gai process θ ds coverges i distributio to θ ds. Examples iclude covergece from discrete to cotiuous time settigs, ad i particular, geeralizatios of the covergece of biomial optio replicatio models to the Black-Scholes model. Couterexamples are also provided. Duffie is with the Graduate School of Busiess, Staford Uiversity, Staford, CA 9435, ad ackowledges the fiacial support of Batterymarch Fiacial Maagemet. Protter is with the Departmets of Mathematics ad Statistics, Purdue Uiversity, West Lafeyette, IN 4797, ad is supported i part by NSF grat #DMS- 885595. We thak Jaime Sa Marti, Jea Jacod, Tom Kurtz, Erst Eberlei, ad a uusually attetive referee for helpful commets.
1. Itroductio Although a large part of fiacial ecoomic theory is based o models with cotiuoustime security tradig, it is widely felt that these models are relevat isofar as they characterize the behavior of models i which trades occur discretely i time. It seems atural to check that the limit of discrete-time security market models, as the legths of periods betwee trades shrik to zero, produces the effect of cotiuoustime tradig. That is oe of the pricipal aims of this paper. If {(S, θ )} is a sequece of security price processes ad tradig strategies covergig i distributio to some such pair (S, θ), we are cocered with additioal coditios uder which the sequece { θt dst } of stochastic itegrals defiig the gais from trade coverges i distributio to the stochastic itegral θ t ds t. Coditios recetly developed by Jakubowski, Mémi, ad Pagès (1989) ad Kurtz ad Protter (1991a, 1991b) are re-stated here i a maer suitable for easy applicatios i fiace, ad several such examples are worked out i this paper. The paper gives parallel coditios for covergece of gais i probability. I short, this paper is more of a user s guide tha a set of ew covergece results. A good motivatig example is Cox, Ross, ad Rubistei s (1979) proof that the Black-Scholes (1973) Optio Pricig Formula is the limit of a discrete-time biomial optio pricig formula (due to William Sharpe) as the umber of time periods per uit of real time goes to ifiity. Aside from providig a simple iterpretatio of the Black-Scholes formula, this coectio betwee discrete ad cotiuous time fiacial models led to a stadard techique for estimatig cotiuous-time derivative asset prices by usig umerical methods based o discrete-time reasoig. Oe of the examples of this paper is of the followig related form. Suppose that {S } is a sequece of security price processes covergig i distributio to the geometric Browia motio price process S of the Black-Scholes model, with {S } satisfyig a basic techical coditio. (For this, we show that it is eough that the cumulative retur process R for S coverges i distributio to the Browia motio cumulative retur process R uderlyig S, plus the same techical coditio o the sequece {R }. For example, S could be a price process that is adjusted oly at discrete-time itervals of legth 1/, with i.i.d. or α-mixig returs satisfyig a regularity coditio.) We show that if a ivestor, i igorace of the distictio 2
betwee S ad S, or perhaps at a loss for what else to do, attempts to replicate a call optio payoff by followig the associated Black-Scholes stock hedgig strategy C x (St, T t), the the ivestor will be successful i the limit (i the sese that the fial payoff of the hedgig strategy coverges i distributio to the optio payoff as ), ad the required iitial ivestmet coverges to the Black-Scholes call optio value. (This ca be compared with the o-stadard proof of the Black-Scholes formula give by Cutlad, Kopp, ad Williger (1991), which draws a differet sort of coectio betwee the discrete ad cotiuous models.) While this kid of stability result is to be expected, we feel that it is importat to have precise ad easily verifiable mathematical coditios that are sufficiet for this kid of covergece result. As we show i couterexamples, there are coditios that are ot obviously pathological uder which covergece fails. Our geeral goal is to provide a useful set of tools for explorig the boudaries betwee discrete ad cotiuous time fiacial models, as well as the stability of the fiacial gai process θ ds with respect to simultaeous perturbatios of the price process S ad tradig strategy θ. 2. Prelimiaries This sectio sets out some of the basic defiitios ad otatio. We let ID d deote the space of IR d -valued càdlàg 1 sample paths o a fixed time iterval T = [, T ]. There are atural extesios of our results i each case to T = [, ). The Skorohod topology 2 o ID d is used throughout, uless otherwise oted. A càdlàg process is a radom variable S o some probability space valued i ID d. A sequece {S } of càdlàg processes (which may be defied o differet probability spaces) coverges i distributio to a càdlàg process S, deoted S = S, if E[h(S )] E[h(S)] for ay bouded cotiuous real-valued fuctio h o ID d. 1 That is, f ID meas that f : T IR has a limit f(t ) = lim s t f(s) from the left for all t, ad that the limit from the right f(t+) exists ad is equal to f(t) for all t. By covetio, f( ) = f(). The expressio RCLL (right cotiuous with left limits) is also used i place of càdlàg (cotiu à droite, limites à gauche). 2 The Skorohod topology is defied by the covergece of x to x i ID d if ad oly if there is a sequece λ : T T of strictly icreasig cotiuous fuctios ( time chages ) such that, for each t T, sup t t λ (t) t ad sup t t x [λ (t)] x(t). 3
A famous example is Dosker s Theorem, whereby a ormalized coi toss radom walk coverges i distributio to Browia Motio. That is, let {Y k } be a sequece of idepedet radom variables with equally likely outcomes +1 ad 1, ad let R t = (Y 1 + + Y [t] )/ for ay time t, where [t] deotes the smallest iteger less tha or equal to t. The R = B, where B is a Stadard Browia Motio. Dosker s Theorem applies to more geeral forms of radom walk ad to a class of martigales; Billigsley (1968), Ethier ad Kurtz (1986), or Jacod ad Shiryayev (1988) are good geeral refereces. I fiacial models, we are more likely to thik of {Y k } as a discrete-time retur process, so that R is the ormalized cumulative retur process. The correspodig price process S is defied by S t = S E(R ) t, for some iitial price S >, where the stochastic expoetial E(R ) of R is give i this case by E(R ) t = [t] k=1 ( 1 + Y ) k. The geeral defiitio of the stochastic expoetial, itroduced ito this fiacial cotext by Harriso ad Pliska (1981), is give i Sectio 3. It is well kow that S = S, where S t = S e Bt t/2. That is, with returs geerated by a coi toss radom walk, the asset price process coverges i distributio to the solutio of the stochastic differetial equatio ds t = S t db t. This is the classical Black-Scholes example (leavig out, for simplicity, costats for the iterest rate ad the mea ad variace of stock returs). We retur later i the paper to exted this example, showig that the Black-Scholes formula ca be foud as the limit of discrete-time models with a geeral class of cumulative retur processes R covergig i distributio to Browia Motio. A process X is a semimartigale if there exists a decompositio X = M + A where M is a local martigale ad A is a adapted càdlàg process with paths of fiite variatio o compact time itervals. Semimartigales are the most geeral processes havig stochastic differetials. Protter (199) is a itroductory treatmet of stochastic itegratio ad stochastic differetial equatios; Dellacherie-Meyer (1982) is a comprehesive treatmet of semimartigales ad stochastic itegratio. 4
3. Two Couterexamples This sectio presets two couterexamples. I each case, ad obviously for differet reasos, eve though a tradig strategy θ coverges i distributio to a tradig strategy θ ad a price process S coverges i distributio to a price process S, it is ot true that the fiacial gai process θ ds coverges i distributio to the fiacial gai process θ ds. Example 1. Our first example is determiistic ad well kow. It is essetially the same as Example 1.1 of Kurtz ad Protter (1991a). Let there be d = 1 security ad cosider the tradig strategies θ = θ = 1 (T/2,T ], all of which hold oe uit of the security after time T/2. Let S = 1 [T/2+1/,T ] for > 2/T ad let S = 1 [T/2,T ]. Although θ = θ ad S = S, it is ot the case that (θ, S ) = (θ, S) i the sese explaied i Sectio 2. O the other had, t θ ds = 1 for all > 2/T ad all t > T/2 + 1/, while t θ ds = for all t. Failure of weak covergece occurs for a rather obvious reaso that will be excluded by our mai covergece coditios. Example 2. Our secod example is more subtle. Let B be a stadard Browia motio ad let R = σb describe the ideal cumulative retur o a particular ivestmet, for some costat σ. Suppose, however, that returs are oly credited with a lag, o a movig average basis, with Rt = t R(s) ds, so that we are t 1/ dealig istead with the stale returs. Suppose a ivestor chooses to ivest total wealth, X t at time t, by placig a fractio g(x t ) i this risky ivestmet, with the remaider ivested risklessly (ad for simplicity, at a zero iterest rate). We assume for regularity that g is bouded with a bouded derivative. I the ideal case, the wealth process is give by X t = x + t g(x s )X s dr s, where x is iitial wealth. With stale returs, likewise, the wealth process X is give by X t = x + t X s g(x s ) dr s. 5
It ca be show that the stale cumulative retur process R coverges i distributio to R. Is it true, as oe might hope, that the correspodig wealth process X coverges i distributio to X? The aswer is typically No. I fact, we show i a appedix that X = Y, where Y t = X t + 1 2 t g(x s )X 2 s [1 + g (X s )] ds. (1) For istace, with g(x) = k, a costat ivestmet strategy, for all x, we have Y t = e kt X t, which ca represet a substatial discrepacy betwee the limit of the gais ad the gai of the limit strategy ad returs. I particular, the price process S = S E(R) correspodig to the limit retur process is ot the same as the limit of the price processes S = S E(R ). Agai, the sufficiet coditios i our covergece results to follow would preclude this example. At the least, however, the example shows that care must be take. 4. Weak Covergece Results for Stochastic Itegrals This sectio presets recetly demostrated coditios for weak covergece of stochastic itegrals, i a form simplified for applicatios i fiacial ecoomic models. The followig setup is fixed for this sectio. For each, there is a probability space (Ω, F, P ) ad a filtratio {Ft : t T } of sub-σ-fields of F (satisfyig the usual coditios) o which X ad H are càdlàg adapted processes valued respectively i IR m ad IM km (the space of k m matrices). (We ca, ad do, always fix a càdlàg versio of ay semimartigale.) We let E deote expectatio with respect to (Ω, F, P ). There is also a probability space ad filtratio o which the correspodig properties hold for X ad H, respectively. Moreover, (H, X ) = (H, X). For (H, X ) = (H, X), we emphasize that the defiitio requires that there exists oe (ad ot two) sequece λ of time chages such that λ (s) coverges to s uiformly, ad (Hλ (s), X λ (s)) coverges i law uiformly i s to (H, X). We assume throughout that X is a semimartigale for each, which implies the existece of H s dxs. 6
4.1. Good Sequeces of Semimartigales The followig property of {X } is obviously the key to our goal. Defiitio. A sequece {X } of semimartigales is good if, for ay {H }, the covergece of (H, X ) to (H, X) i distributio (respectively, i probability) implies that X is a semimartigale ad also implies the covergece of (H, X, H dx ) to (H, X, H dx) i distributio (respectively, i probability). The followig result, showig that goodess is closed uder appropriate stochastic itegratio, is due to Kurtz ad Protter (1991b), who also provide ecessary ad sufficiet coditios for goodess. Propositio 1. If (X ) is good ad (H, X ) coverges i distributio, the ( H dx ) is also good. For our purposes, it remais to establish some simple coditios for a sequece {X } of semimartigales to be good. We will start with a relatively simple coditio for goodess, ad the exted i geerality. Before statig the coditio, recall that each semimartigale X is defied by the fact that it ca be writte as the sum M + A of a local martigale M with M = ad a adapted process A of fiite variatio. The total variatio of A at time t is deoted A t, ad the quadratic variatio of M is deoted [M, M]. (If M is vector-valued, [M, M] is the matrixvalued process whose (i, j)-elemet is [M i, M j ].) Coditio A. A sequece {X = M +A } of semimartigales satisfies Coditio A if both {E ([M, M ] T )} ad {E ( A T )} are bouded. If M is a local martigale with E ([M, M] T ) <, the E ([M, M] T ) = var (M T ) [for example, see Protter (199), p. 66]. Thus the followig coditio is sufficiet for Coditio A, ad may be easier to check i practice. Coditio A. A sequece {X = M + A } of semimartigales satisfies Coditio A if M is a martigale for all, {var (MT )} is bouded, ad {E ( A T )} is bouded. Theorem 1. If {X } has uiformly bouded jumps ad satisfies Coditio A or A, the {X } is good. 7
This result ca be show as a easy corollary of results i Kurtz ad Protter (1991a). The assumptio of uiformly bouded jumps for {X } is strog, ad ot ofte satisfied i practice, but we also obtai covergece if, with jumps appropriately trucated, {X } satisfies Coditio A. Sice the obvious method of trucatig jumps is ot cotiuous i the Skorohod topology, we proceed as follows. For each δ [, ), let h δ : [, ) [, ) be defied by h δ (r) = (1 δ/r) +, ad let J δ (X) be the process defied by J δ (X) t = X t h δ ( X s ) X s, s t where X s = X s X s. Theorem 2. If, for some δ, the sequece {J δ (X )} satisfies Coditio A or A, the {X } is good. A proof is give i Kurtz ad Protter (1991a), ad also i Jakubowski, Mémi, ad Pagès (1989). The coditios here are desiged to be easy to verify i practice. I some cases, the restrictio o the quadratic variatio i Coditio A may be more difficult to verify tha the followig coditio. Coditio B. A sequece {X = M +A } of semimartigales satisfies Coditio ( B if {E supt T Mt ) } ad {E ( A T )} are bouded. Theorem 3. If {X } satisfies Coditio B, the {X } is good. This result is proved i a earlier versio of this paper, ad follows from a result by Jakubowski, Mémi, ad Pagès (1989), based o a applicatio of Davis iequality (Dellacherie ad Meyer (1982), VII.9). Commet: If the semimartigales X have uiformly bouded jumps, the they are special: that is, there exists a uique decompositio X = M + A, for which the fiite variatio process A is predictable with A =. Such a decompositio is called caoical. For the caoical decompositio, it ca be show that the jumps of M (ad hece of A ) are also bouded, ad therefore for the caoical decompositio i the case of bouded jumps, {X } is good if {E ( A T )} is bouded. I fact, i Theorem 3, it is eough to replace the restrictio that {E ( A T )} is 8
bouded with a weaker restrictio: that the measures iduced by {A } o [, T ] are tight. (See, for example, Billigsley (1968) for a defiitio of tightess of measures.) We also offer the followig help i verifyig this tightess coditio for special semimartigales. Lemma 1. Suppose {Z }, with Z = M + A, is a sequece of special semimartigales for which the measures iduced o [, T ] by (A ) are tight. The, for the caoical decompositio Z = M + Ã, the measures iduced by {à } are also tight. proof: Sice Z is special, A is locally of itegrable variatio [Dellacherie ad Meyer (1982), page 214]. Sice à is the predictable compesator of A, the result follows from the Corollary of Appedix B, Lemma B1. 4.2. Stochastic Differetial Equatios We ow address the case of stochastic differetial equatios of the form Z t = H t + Z t = H t + t t f (s, Z s ) dx s, f(s, Z s ) dx s, where f ad f are cotiuous real-valued fuctios o IR + IR k ito IM km such that: (i) x f (t, x) is Lipschitz (uiformly i t), each, (ii) t f (t, x) is LCRL (left cotiuous with right limits, or càglàd ) for each x, each, ad (iii) for ay sequece (x ) of càdlàg fuctios with x x i the Skorohod topology, (y, x ) coverges to (y, x) (Skorohod), where y (s) = f (s+, x (s)), y(s) = f(s+, x(s)). (If f = f for all, the coditio (iii) is automatically true.) The followig theorem is proved i more geerality i Kurtz ad Protter (1991a). See also S lomiński (1989). 9
Theorem 4. Suppose {X } is good ad let (f ) 1 ad f satisfy (i)-(iii) above. Suppose (H, X ) coverges to (H, X) i distributio (respectively, i probability). Let Z, Z be solutios 3 of respectively. Z t = H t + Z t = H t + t t f (s, Z s ) dx s f(s, Z s ) dx s, The (Z, H, X ) coverges to (Z, H, X) i distributio (respectively, i probability). Moreover, if H = Z ad H = Z, the {Z } is good. A importat special case is the stochastic differetial equatio Z t = 1 + t Z s dx s, which defies the stochastic expoetial 4 Z = E(X) of X. The solutio, extedig the special case of Sectio 2, is ( Z t = exp X t 1 ) 2 [X, X]c t <s t (1 + X s )e X s, where [X, X] c deotes the cotiuous part of the quadratic variatio [X, X] of X. With a Stadard Browia Motio B, for example, [B, B] c t = [B, B] t = t ad E(B) t = e B t t/2. 5. Covergece of Discrete-Time Strategies I order to apply our results to discrete-time tradig strategies θ ad correspodig price processes S, we eed coditios uder which (θ, S ) = (θ, S). We will cosider strategies that are discrete-time with respect to a grid, defied by times {t,..., t k } with = t < t 1 < < t k = T. The mesh size of the grid is sup k t k t k 1. The followig covergece result is sufficiet for may purposes. This result is trivial if f is uiformly cotiuous. The cotet of the lemma is to reduce it to that case. 3 Uique solutios exist. See Protter (199). 4 This is also kow as the Doléas-Dade expoetial. 1
Lemma 2. Let (S ) ad S be ID d -valued o the same probability space, S be cotiuous, ad S = S. For each, let the radom times {Tk } defie a grid o [, T ] with mesh size covergig with to almost surely. For some cotiuous f : IR d [, T ] IR, let Ht The (H, S ) = (H, S). = f [S (T k ), T k ], t [T k, T k+1 ), ad H t = f(s t, t). Sice the limit process S is cotiuous, covergece i the Skorohod topology is equivalet here to covergece i the uiform metric topology, so the proof is straightforward ad omitted. Corollary 1. Suppose, moreover, that {S } is good. The H t ds t = Ht ds t. The followig corollary allows the fuctio f defiig the tradig strategies to deped o. The proof ivolves oly a slight adjustmet. Corollary 2. Suppose f : IR d [, T ] IR is cotiuous for each such that: For ay ϵ >, there is some N large eough that, for ay (x, t) ad N, f (x, t) f(x, t) < ϵ. The, with Ht = f [S(Tk ), T k ], t [T k, T k+1 ), the coclusios of Lemma 2 ad Corollary 1 follow. 6. Example: Covergece to the Black-Scholes Model The objective of this sectio is to show that the weak covergece methods preseted i this paper are easy to apply to a stadard situatio: the Black-Scholes (1973) optio pricig formula. Uder stadard regularity coditios, the uique arbitragefree price of a call optio with time τ to expiratio ad exercise price K, whe the curret stock price is x, ad the cotiuously compoudig iterest rate is r, is C(x, τ) = Φ(h)x Ke rτ Φ(h σ τ), where Φ is the stadard ormal cumulative distributio fuctio ad h = log(x/k) + rτ + σ2 τ/2 σ, τ provided the stock price process S satisfies the stochastic differetial equatio ds t = µs t dt + σs t db t ; S = x >, (2) 11
for costats µ, r, ad σ >. 5 formula i two cases: We will show covergece to the Black-Scholes (a) A fixed stock-price process S satisfyig (2) ad a sequece of stock tradig strategies {θ } correspodig to discrete-time tradig with tradig frequecy icreasig i, with limit equal to the Black-Scholes stock tradig strategy θ t = C x (S t, T t), where T is the expiratio date of the optio ad C x (x, τ) = xc(x, τ). (b) A sequece of stock price processes {S } costructed as the stochastic expoetials of cumulative retur processes {X } covergig i distributio to a Browia Motio X, ad tradig strategies {θ } defied by θ (t) = C x (S t, T t) for discretely chose t. Case (a) hadles applicatios such as those of Lelad (1985); Case (b) hadles extesios of the Cox-Ross-Rubistei (1979) results. Case (a) Icreasig Tradig Frequecy. Let T > be fixed, ad let the set of stoppig times T = {T k } defie a sequece of grids (as i Lemma 2) with mesh size shrikig to zero almost surely. I the -th eviromet, the ivestor is able to trade oly at stoppig times i T. That is, the tradig strategy θ must be chose from the set Θ of square-itegrable predictable processes with θ (t) = θ (Tk 1 ) for t (T k 1, T k ]. For a simple case, let Tk = k/, or trades per uit of time, determiistically. We take the case r = for simplicity, sice this allows us to cosider stock gais aloe, bod tradig gais beig zero. For r >, a stadard trick of Harriso ad Kreps (1979) allows oe to ormalize to this case without loss of geerality. We cosider the stock tradig strategy θ Θ defied by θ () arbitrary ad θ (t) = C x [S(T k ), T T k ], t (T k, T k+1]. For riskless discout bods maturig after T, with a face value of oe dollar (the uit of accout) ad bearig zero iterest, we defie the bod tradig strategy 5 Note that S is the stochastic expoetial of the semimartigale X t = µt + σb t. 12
α Θ by the self-fiacig restrictio α (t) = α () + T k θ t ds t θ (T k )S(T k ) + θ ()S(), t (T k, T k+1], (3) where α () = C(S, T ) θ ()S. The total iitial ivestmet α + θ S is the Black-Scholes optio price C(S, T ). (Note that, α Θ.) The total payoff of this self-fiacig strategy (α, θ ) at time T is C(S, T ) + T θ t ds t. For our purposes, it is therefore eough to show that C(S, T ) + T θ t ds t = (S T K) +, the payoff of the optio. This ca be doe by direct (tedious) calculatio (as i, say, Lelad (1985)), but our geeral weak covergece results are quite simple to apply here. It should be coceded, of course, that i simple cases such as that cosidered by Lelad (1985), oe could likely obtai 6 almost sure covergece. Propositio 1. I the limit, the discrete-time self-fiacig strategy θ pays off the optio. That is, C(S, T ) + T θ t ds t = (S T K) +. proof: For X = X = S, it is clear that {X } is good. With Ht = C x (S(Tk 1 ), T Tk 1 ), t [T k 1, T k ), the coditios of Corollary 1 of Lemma 2 are satisfied sice C x is cotiuous. Sice θt = Ht, it follows that C(S, T ) + T θ t ds t = C(S, T )+ T θ t ds t. By Black ad Scholes (1973), C(S, T )+ T θ t ds t = (S T K) + a.s. [For the details, see, for example, Duffie (1988), Sectio 22.] Thus C(S, T ) + T θ t ds t = (S T K) +. We ca geeralize the result as follows. We ca allow S to be ay diffusio process of the form ds t = µ(s t, t) dt + σ(s t, t) db t. The, subject to techical restrictios, for ay termial payoff g(s T ), there is a sequece of discrete-time tradig strategies whose termial payoff coverges i distributio to g(s T ). The followig techical 6 Lelad allows for trasactios costs that coverge to zero, ad (despite appearaces) actually makes a argumet for covergece i mea, which does ot ecessarily imply almost sure covergece. 13
regularity coditios are far i excess of the miimum kow sufficiet coditios. For weaker coditios, see, for example, the refereces cited i Sectio 21 of Duffie (1988). Coditio C. The fuctios σ : IR [, T ] IR ad g : IR IR together satisfy Coditio C if they are Lipschitz ad have Lipschitz first ad secod derivatives. Propositio 2. Let (σ, g) satisfy Coditio C. Suppose ds t = µ(s t, t) dt+σ(s t, t) db t, ad that {S } is good with S = S. The there exist (discrete-time self-fiacig) strategies (θ ) i Θ such that E [g(x T )] + T where X t = S + t σ(x s, s)db s, t [, T ]. θ t ds t = g(s T ), The result implies that oe obtais the usual risk-eutral valuatio ad exact replicatio of the derivative payoff g(s T ) i the limit, as S = S. proof: Let F (x, t) = E [ g(x x,t T )], where Xτ x,t = x + τ t σ (Xx,t s, s) db s, τ t. The, as i Duffie (1988) Sectio 22, the partial F x is a well-defied cotiuous fuctio ad θ t = F x (S t, t) satisfies E [g(x T )] + T θ t ds t = g(s T ) a.s. For the tradig strategies θt = f[s(tk ), T k ], t (T k, T k+1 ], the result the follows as i the proof of Propositio 1. Related results have bee obtaied idepedetly by He (199). Case (b) (Cumulative Returs that are approximately Browia Motio). The cumulative retur process X correspodig to the price process S of (2) is the Browia Motio X defied by X t = µt + σb t. (4) That is, S = S E(X), where E(X) is the stochastic expoetial of X as defied i Sectio 2. We ow cosider a sequece of cumulative retur processes {X } with X = X. 14
Example 1. (Biomial Returs) A classical example is the coi-toss walk with drift used by Cox, Ross, ad Rubistei (1979). That is, let Xt = 1 [t] k=1 Y k, (5) where, for each, {Yk } is a sequece of idepedet ad idetically distributed biomial trials with E(Y1 ) µ ad var(y1 ) σ 2. It is easy to show that X = X. (See, for example, Duffie (1988), Sectio 22.) Let us show that the assumptios of Theorem 3 (for example) are satisfied i this case. For ay umber t, recall that [t] deotes the largest iteger less tha or equal to t. Sice the (Y k ) k 1 are idepedet ad have fiite meas, we kow that Mt = 1 [t] k=1 [Y k E (Y k )] is a martigale, ad thus a decompositio of X is: Xt = 1 [t] [Y k=1 k E(Y 1 )] + 1 [t]e (Y 1 ) M + A. The jumps of M are uiformly bouded. I order to verify Coditio B for goodess, it is therefore eough to show that E( A T ) is bouded. This follows from the fact that A is determiistic ad A t µt. Thus X is good. With S = S E(X ) ad S S, Theorem 4 implies that {S } is good ad that S = S. We cosider the discrete-time stock-tradig strategy θ Θ defied by θ t = C x [S (T k ), T k ], t (T k, T k+1], where S = S E(X ). I order to show that Black-Scholes applies i the limit, we must show that C(S, ) + T θ t ds t = (S T K) +. [The self-fiacig bod tradig strategy α is defied by the obvious aalogue to (3), ad the iitial ivestmet is the Black-Scholes value of the optio, C(S, T ).] It is implicit i the followig statemet that all processes are defied o the same probability space uless the stoppig times {Tk } are determiistic. 15
Propositio 3. Suppose S S >, {X } is good, ad X = X, where X is the Black-Scholes cumulative retur process (4). The S = E(X )S = E(X)S = S ad C(S, ) + T θ t dst = (S T K) +. proof: To apply Corollary 1 of Lemma 2, we eed oly show that S = S ad that S is good. This is true by Theorem 4. Sice C(S, )+ T θ t ds t = (S T K) + a.s., we are doe. What examples, i additio to the coi-toss radom walks {X } satisfy the hypotheses of Propositio 3? Example 2. (iid Returs). defied by (5), where: (i) {Yk } are uiformly bouded, (ii) for each, {Yk } is i.i.d., (iii) E(Y k ) µ, ad (iv) var(y k ) σ 2. Suppose {X } is a sequece of stock retur processes The, usig Lideberg s Cetral Limit i the proof of Dosker s Theorem, we have X = X, where X is give by (4). Furthermore, {X } satisfies the hypotheses of Theorem 3. Thus, the hypotheses of Propositio 3 are satisfied. This eds Example 2. Example 3. (Mixig returs). Let the sequece {X } of cumulative retur processes be defied by (5), where the followig coditios apply: (i) {Yk } are uiformly bouded, IR-valued, ad statioary i k (for each ). (ii) For Fm = σ{yk ; k < m}, Gm ), where φ F l = σ{y k ; k m}, ad φ p (m) = φ p (G m+l p (A B) = sup P (A B) P (A) L p, A A C = m=1 [φ p (m)] α <, each, where p = 2+δ 1+δ, α = δ 1+δ, for some δ >. (iii) E (Y k ) µ ad, for U k = Y k E [Y k ], sup C U 1 L 2+δ <. 16
(iv) σ 2 = E [(U 1 ) 2 ] + 2 k=2 E (U 1 U k ) is well-defied ad σ2 σ 2. Uder (i)-(iv), for X defied by (5), we have X = X. [See Ethier-Kurtz (1986), pp. 35-353, for calculatios ot give here.] I order to ivoke goodess, we eed to fid suitable semimartigale decompositios of X. To this ed, followig Ethier-Kurtz (1986) (p. 35 ff), defie: M l = l Uk + k=1 ( E U l+m Fl ). m=1 The series o the right is coverget as a cosequece of the mixig hypotheses (see Ethier-Kurtz (1986), p. 351), ad Ml is a martigale (with jumps bouded by twice the boud o {Yk }) with respect to the filtratio (F l ) l 1. We have where with X t = 1 M [t] + A t, A t = 1 V [t] + 1 [t] E (Y k ), V l = k=1 E (Ul+m Fl ). m=1 Note that the total variatio of the paths of the process V are majorized i that ( l ) E ( V l ) 2E E (Uk+m Fk ), k=1 m=1 ad usig a stadard estimate [Ethier-Kurtz (1986), p. 351], it follows that ( l ) E ( V l ) 8φ δ/1+δ p (m) U1 L 2+δ 8lC U1 L 2+δ. The sup k=1 m=1 1 ( E V ) [t] sup δ t C U 1 L 2+δ < by hypothesis (iii). Sice E (Yk ) µ, it follows that sup E ( A T ) <. Sice M has bouded jumps, Coditio B is satisfied, so X is good, ad Propositio 3 applies oce agai. This eds Example 3. 17
Appedix A: No-Covergece with Stale Returs, This appedix explais the failure of weak covergece for Example 2 of Sectio 3, i which returs i model are give by the stale retur process R, which coverges to the Browia motio R. We eed to show that the wealth process X defied by the ivestmet policy g coverges to the process Y give by (1). This is really a extesio of the Wog-Zakai pathology that was pursued by Kurtz ad Protter (1991a). To this ed, cosider the followig calculatios. (Without loss of geerality, let σ = 1.) We have R (t) = R(t) + [R (t) R(t)] = V (t) + Z (t), where V (t) = R(t) for all ad where Z (t) = R (t) R(t). The, clearly, V = R ad Z =. The key is to look at H (t) = t Z (s) dz (s) = 1 2 Z (t) 2 1 2 [Z, Z ] t = 1 2 Z (t) 2 1 2 t, sice [Z, Z ] t = [R R, R R] t = [R, R] t = t. Sice Z =, so does Z 2 by the cotiuous mappig theorem, ad hece H (t) = t/2. Similarly, lettig K = [V, Z ], we have K (t) = t. The U = H K = U, where U t = t/2. Theorem 5.1 of Kurtz ad Protter (1991a) the implies that X = Y defied by (1). What is really goig o i this example is that the Browia motio R, which is a cotiuous martigale with paths of ifiite variatio o compacts, is beig approximated by cotiuous processes R of fiite variatio; moreover the processes have o martigale properties. Thus the calculus of the R processes is the classical path by path Riema-Stieltjes first order calculus; while the calculus of the Browia motio R is the Itô secod order calculus. This leads to a discotiuity (or lack of robustess) whe we approximate R by R. 18 This
discotiuity is precisely computable i the above calculatios, applied to Theorem (5.1) of Kurtz ad Protter (1991a). Appedix B: A Aid to Checkig Goodess of Special Semimartigales Lemma B1. Suppose {Z }, with Z = M + A, is a sequece of special semimartigales for which the measures o [, T ] iduced by (A ) are tight. Suppose further that sup E (sup t T A t ) <. The, for the caoical decompositio Z = M + Ã, the measures iduced by à are also tight. proof: Let H t = dã t d à t, where à t = dã s deotes the total variatio of the paths of the process Ã. The H is predictable ad, for ay stoppig time τ, [ τ ] [ τ ] E Ht da t = E Ht [ à ] dã t = E τ. Sice H t = 1, [ τ ] E [ A τ ] E Ht da [ à ] t = E τ. By the Leglart Domiatio Theorem [Jacod ad Shiryaev (1987), Lemma 3.3 (b), page 35, with ϵ = b ad η = b], ( à ) lim sup P τ b lim sup b b { [ ( 1 b + E b )] sup A t + P ( A τ b) } =. t τ Sice the measures iduced by (A ) are tight, it follows that the measures iduced by à are tight. Corollary. Suppose the measures iduced by (A ) are tight ad A is locally of itegrable variatio for all. If à is the predictable compesator for A, the the measures iduced by (à ) are tight. proof: For give b, let T = if{t : A t b}. The stopped process (A ) T is of bouded total variatio. For give t > ad ϵ >, there exists b large eough that sup P (T t ) sup 19 P ( A t b) < ϵ, (6)
sice the measures iduced by (A ) are tight. It follows that ( à ) ({ lim sup P t b lim sup ) {P b b à t b} {T > t } { ) lim b sup P ( (à ) T b t + P (T t ) } + P (T t ) } by the lemma ad (6). Sice ϵ is arbitrary, the lim sup is, so the measures iduced by (à ) are tight. < ϵ, 2
Refereces Billigsley, P. (1968) Covergece of Probability Measures, New York: Wiley. Black, F. ad M. Scholes (1973) The Pricig of Optios ad Corporate Liabilities, Joural of Political Ecoomy 3, pp. 637-654. Cox, J., S. Ross, ad M. Rubistei (1979) Optio Pricig: A Simplified Approach, Joural of Fiacial Ecoomics 7, pp. 229-263. Cutlad, N., E. Kopp, ad W. Williger (1991) A Nostadard Approach to Optio Pricig, Mathematical Fiace (to appear). Dellacherie, C. ad P. A. Meyer (1982), Probabilities ad Potetial B, Theory of Martigales. New York: North-Hollad. Duffie, D. (1988) Security Markets: Stochastic Models. Bosto: Academic Press. Ethier, S. N. ad T. G. Kurtz (1986), Markov Processes: Characterizatio ad Covergece, New York: Wiley. Harriso, M. ad D. Kreps (1979) Martigales ad Arbitrage i Multiperiod Security Markets, Joural of Ecoomic Theory 2, pp. 381-48. Harriso, M. ad S. Pliska (1981) Martigales ad Stochastic Itegrals i the Theory of Cotiuous Tradig, Stochastic Processes ad Their Applicatios 11, pp. 215-26. He, H. (199) Covergece from Discrete- to Cotiuous- Time Cotiget Claims Prices, Review of Fiacial Studies 3, 523-546. Jacod, J. ad A. Shiryaev (1987), Limit Theorems for Stochastic Processes, New York: Spriger-Verlag. Jakubowski, A., J. Mémi, ad G. Pagès (1989), Covergece e loi des suites d itégrales stochastiques sur l espace D 1 de Skorohod, Probability Theory ad Related Fields, 81, 111-137. Kurtz, T. ad P. Protter (1991a) Weak Limit Theorems for Stochastic Itegrals ad Stochastic Differetial Equatios, Aals of Probability 19, 135-17. Kurtz, T. ad P. Protter (1991b) Characterizig The Weak Covergece of Stochastic Itegrals, to appear i The Proceedigs of the Durham Symposium o Stochastic Aalysis. Lelad, H. (1985) Optio Pricig ad Replicatio with Trasactios Costs, Joural of Fiace 4, pp. 1283-131. Protter, P. (199), Stochastic Itegratio ad Differetial Equatios: A New Approach, (New York: Spriger-Verlag). S lomiński, L. (1989) Stability of Strog Solutios of Stochastic Differetial Equatios, Stochastic Processes ad Their Applicatios, 31, 173-22. 21