Two methods for optimal investment with trading strategies of finite variation

Transcription

1 IMA Joural of Maagemet Mathematics Page of 24 doi:0.093/imama/dpxxx Two methods for optimal ivestmet with tradig strategies of fiite variatio BUJAR GASHI AND PARESH DATE Departmet of Mathematical Scieces, The Uiversity of Liverpool, Liverpool, L69 7ZL, UK Departmet of Mathematical Scieces, Bruel Uiversity, Uxbridge, UB8 3PH, UK. Two methods for desigig optimal portfolios are proposed. I order to reduce the variatio i the asset holdigs ad hece the evetual proportioal trasactio costs, the tradig strategies of these portfolios are costraied to be of a fiite variatio. The first method miimizes a upper boud o the discretetime logarithmic error betwee a referece portfolio ad the oe with a costraied tradig strategy, ad thus pealizes the shortfall oly. A quadratic pealty o the logarithmic variatio of the tradig strategy is also icluded i the objective fuctioal. The secod method miimizes a sum of the discrete-time log-quadratic errors betwee the asset holdig values of the costraied portfolio ad a certai referece portfolio, which results i trackig the referece portfolio. The optimal tradig strategy is obtaied i a explicit closed form for both methods. Simulatio examples with the log-optimal ad the Black- Scholes replicatig portfolios as refereces, show smoother tradig strategies for the ew portfolios ad a sigificat reductio i the evetual proportioal trasactio cost. The performace of the ew portfolios are very close to their refereces i both cases. Keywords: Sigle-period dyamic optimizatio, differetiable tradig strategies, evetual proportioal trasactio cost.. Itroductio Oe of the mai problems of mathematical fiace is achievig some pre-specified objective via the portfolio selectio, i. e. the tradig of the assets. A example of such a objective is the optimal wealth growth for the ivestor. The best criterio to maximize i this case is the logarithm of the termial wealth, ad the obtaied portfolio is termed log-optimal; see Merto (969), Kor (997), Lueberger (998). Aother example of the objective is the replicatio of a cotiget claim; see Black & Scholes (973), Merto (973), Wilmott (998), Bigham & Kiesel (999). The miimal iitial wealth required to achieve such a replicatio represets the price of the claim ad the replicatig portfolio is said to hedge the claim. May of the kow portfolio selectio methods, icludig the above examples, assume a idealized market where there is o trasactio cost. This simplifies greatly the aalysis ad i special cases gives explicit solutios. But whe applied i practice, trasactio costs are always icurred ad such tradig strategies may lead to a very large trasactio cost. There are three mai existig approaches to dealig with the problem of trasactio cost. The first approach icludes the trasactio cost explicitly i the model; see Davis & Norma (990), Davis et al. (993), Shreve & Soer (994), Wilmott (998), Kor (997), ad the refereces therei. Replicatio Some of the results i this paper were preseted i Gashi & Date (2005). Correspodig author. Bujar.Gashi@liverpool.ac.uk Paresh.Date@bruel.ac.uk c The authors Published by Oxford Uiversity Press o behalf of the Istitute of Mathematics ad its Applicatios. All rights reserved.

2 2 of 24 B. GASHI ad P. DATE portfolio strategies for hedgig uder proportioal trasactio costs have bee discussed i Lai & Lim (2009). Whe the costs are small, perturbatio theory has bee employed i Mokkhavesa & Atkiso (2002) to derive a solutio to the portfolio selectio problem for a broad class of utility fuctios for a sigle risky asset case. I all these cases, the tradig strategy allows allows ifiite variatio. The secod approach also icludes the trasactio cost explicitly i the model ad further uses a tradig strategy of a fiite variatio; see Kabaov (999), Kabaov & Last (2002). The third approach is developed i the cotext of optio pricig, ad is the most recet oe; see Brodie et al. (998), Soer & Touzi (2000), Soer (2007), Cheridito et al. (2005). It assumes that there is o trasactio cost i the market, ad itroduces a costrait o the diffusio coefficiet of the tradig strategy. If the umber of shares held for the asset i is deoted by v i (t) ad its correspodig equatio is give as dv i (t) = v () i ( )dt + v (2) i ( )dw i, (.) where W i (t) is a stadard browia motio, the this approach imposes the costrait Γ ( ) v (2) i Γ ( ). Here the bouds Γ ( ) ad Γ ( ) ca deped o the stock price ad are user-specified. We iterpret the itroductio of this costrait as a attempt to reduce the variability of the tradig strategy, ad thus the evetual trasactio cost. A similar approach with purely determiistic volatility fuctio is followed i Compay et al. (200), where the authors solve a o-liear Black-Scholes equatio umerically. I this paper we propose a alterative approach to dealig with the problem of proportioal trasactio cost. We also assume, as i the third approach above, that there is o trasactio cost i the market, but costrai the tradig strategy to be of a fiite variatio. This meas that we costrai the diffusio term v (2) i to be idetically zero almost surely. Such a tradig strategy also has a fiite first variatio. Thus, this class of a tradig strategy ca be see as a subset of the tradig strategies of both the secod ad the third approach above. We further costrai the tradig strategy to be positive i this paper. A additioal elemet of our approach is to use criteria that pealize the logarithmic variatio of the tradig strategy. The proportioal trasactio cost is proportioal to the variatio of the asset holdigs, ad thus pealizig a proxy of such a variatio gives the ivestor the meas for trade off betwee a higher profit ad a lower evetual proportioal trasactio cost. It is the combiatio of a differetiable tradig strategy ad the pealizatio of the variatio that leads to a sigificat reductio i the evetual proportioal trasactio cost. This ca be see as a implicit approach to dealig with the problem of proportioal trasactio cost. I this respect, it is similar to the method of Gamma costraits as metioed above. Various differet objectives to be achieved ca be imposed o the costraied portfolio. I this paper, we propose to use a already desiged portfolio as a referece to our costraied portfolio. This is doe with a expectatio that the costraied portfolio will behave very closely or perhaps outperform such a referece portfolio, ad will have a lower evetual proportioal trasactio cost. I order to obtai explicit ad closed-form results ad to permit a broad class of market ad referece models, the proposed optimizatio is i discrete time ad over a sigle period. There are several distict advatages to this approach i compariso with the existig approaches. For example, the method of Davis & Norma (990), Akia et al. (995), ca oly be applied to 2 or 3 risky assets sice the computatioal effort for a large umber of assets is prohibitively high. I our methods, the solutios are i a explicit closed form ad are easily implemeted. Aother promiet class of methods of usig Gamma costraits as advocated i Brodie et al. (998), Soer & Touzi (2000), Soer (2007), Cheridito et al. (2005), ofte leads to a higher iitial optio price tha the Black Scholes price, whereas our method is demostrated to give a lower evetual proportioal trasactio costs for the exact iitial (replicatio) price.

3 Two methods for optimal ivestmet with tradig strategies of fiite variatio 3 of 24 We propose two ew criteria for the optimal ivestmet, which result i two differet methods. I the first method, a upper boud o the discrete-time logarithmic error is miimized, which pealizes oly the shortfall with respect to the referece portfolio. This is achieved by first makig its variace zero, ad the miimizig its mea. The optimal tradig strategy is derived i a explicit closed form. A simulatio example for the log-optimal portfolio as a referece shows a sigificat reductio i the evetual proportioal trasactio cost. I the secod method, a sum of discrete-time log-quadratic errors betwee asset holdig values of the costraied portfolio ad the referece portfolio is miimized. This meas that we view each asset holdig value as a referece ad try to track it, rather tha trackig the portfolio value. Thus, this kid of a portfolio ca be see as a replicatig portfolio. The optimal tradig strategy is derived i a explicit closed form for this case as well. A modified versio of this method is used to track the Black-Scholes replicatig portfolio. A simulatio example shows that the performace of the ew portfolio is very close to that of the Black-Scholes replicatig portfolio, while havig a idetical iitial value ad a sigificatly lower evetual proportioal trasactio cost. The rest of the paper is orgaized as follows. I Sec. 2 a geeral model of asset prices is used to derive the dyamics of a self-fiacig portfolio with a positive differetiable tradig strategy. Such a model is liear i cotrol variables, which i this case are the logarithmic variatios of the tradig strategies. There are also o explicit costraits o either the state or the cotrol variables. The first method for desigig optimal portfolios is preseted i Sec. 3. Here a upper boud of the discretetime logarithmic error betwee the referece ad the costraied portfolios is derived, ad used as a criterio for optimal ivestmet. The optimizatio task is formulated as a cotrol problem with a additioal quadratic pealty o the cotrols, ad solved i a explicit closed form. A simulatio example illustrates a sigificat reductio i the evetual proportioal trasactio cost as compared to the logoptimal portfolio. I Sec. 4 the secod method for optimal ivestmet is proposed. The optimality criterio is a sum of the discrete-time log-quadratic errors of the asset holdig values. The dyamics of such errors are derived for the geeral refereces. The cotrol problem also icludes a quadratic cotrol pealty ad is solved i a explicit closed form. The optimal tradig strategy ca the be obtaied from such cotrols. A modified versio of this approach that allows borrowig is used to track the Black-Scholes replicatig portfolio for a Europea Call optio. Simulatio results show that for almost the same performace of wealth ad a idetical iitial value, the ew portfolio has a lower evetual proportioal trasactio cost. 2. Market model ad the portfolio with a tradig strategy of a fiite variatio. We study a market cosistig of a sigle risk-free asset S 0 (t), ad risky assets S i (t), i =,2,...,, the prices of which are give i the followig form; see, e. g. Björk (2004): ds 0 (t) = r(t)s 0 (t)dt, (2.) [ ] ds i (t) = S i (t) µ i (t,s(t))dt + m j= σ i j (t,s(t))dw j (t) = S i (t)[µ i (t,s(t))dt + σ i (t,s(t))dw] (2.2) where S(t) = [S (t),...,s (t)], S i (0) > 0, i = 0,,...,. The risk-free iterest rate r(t) is a cotiuous ad determiistic fuctio of time, the drift µ i (t,s) ad the volatility σ i j (t,s) are assumed to satisfy the coditios that esure the strict positivity of asset prices; see, e. g. Cvitaiić & Ma (996). The The results of this paper will ot chage eve if r(t) = r(t,s 0,S).

4 4 of 24 B. GASHI ad P. DATE volatility matrix σ(t,s) is of order ( m) ad has vectors σ i (t,s) as rows. We do ot assume that the matrix σ(t,s)σ (t,s) is positive defiite, a assumptio ecoutered i all of previous work o trasactio cost; see Kor (997) ad the refereces therei. The ucertaity is due to a m-dimesioal stadard Browia motio W(t). For simplicity of otatio, we shall frequetly write µ i ad σ i j rather that idicatig explicitly their depedecies o the time ad asset prices. Equatios for asset price logarithms l[s i (t)] are foud by applyig Ito s lemma to (2.) ad (2.2) to obtai dl[s 0 (t)] = r(t)dt, (2.3) dl[s i (t)] = [µ i (t,s) (/2)σ i (t,s)σ i (t,s)]dt + σ i (t,s)dw(t). (2.4) The tradig strategy is defied as a adapted real-valued process [v 0 (t),...,v (t)], that satisfies the stadard itegrability coditios; see e. g.( Bigham & Kiesel (999), Sec. 6.). Here v i (t) deotes the umber of shares of asset i held by the ivestor. The portfolio value (ivestors total wealth) y(t) is give by y(t) = v i (t)s i (t) = y i (t). (2.5) Here y i (t), i = 0,,...,, deotes the value of the holdigs per asset. A portfolio is self-fiacig if the chage i its value occurs oly due to price chages, ad is described by dy(t) = v i (t)ds i (t) (2.6) We shall costrai the tradig strategy to be positive ad thus make the followig assumptio. Assumptio A. The borrowig ad the short-sellig is ot permitted, i. e. v i (t) > 0, a.s, i = 0,,2,...,. Most of the kow portfolio selectio methods give a tradig strategy of ifiite first variatio due to the dw term i their equatio; see, e. g. ( Bigham & Kiesel (999), Sec ). Whe applied to a real-world situatio, where there is always some trasactio cost, a discrete-time approximatios of such a strategy may lead to a very large evetual trasactio cost. Hece we costrai the tradig strategy further to be differetiable ad thus of fiite variatio as follows. Assumptio A2. The elemets of the tradig strategy v(t) = [v 0 (t),...,v (t)] are differetiable ad defied as d l[v i (t)] = u i ( )dt, (2.7) where i = 0,,2,...,, ad the scalars u i ( ) are adapted ad cotiuous fuctios. We ext develop the cotiuous-time ad discrete-time models of the self-fiacig portfolio with a tradig strategy that satisfies the above assumptios. LEMMA 2. A portfolio is self-fiacig uder assumptio (A2) if Proof. Applyig Ito s lemma to (2.5) uder assumptio (A2) oe obtais dy(t) = y i (t)u i (t)dt = 0. (2.8) v i (t)ds i (t) + S i (t)dv i (t). (2.9)

5 Two methods for optimal ivestmet with tradig strategies of fiite variatio 5 of 24 The equatios (2.7) give dv i (t) = v i (t)u i (t)dt, ad by comparig (2.9) to the self-fiacig equatio (2.6), we obtai S i dv i = S i (t)v i (t)u i (t)dt = y i (t)u i (t)dt = 0 (2.0) LEMMA 2.2 Let x i (t) = l[y i (t)], i = 0,,2...,. For a self-fiacig portfolio, uder the assumptios (A) ad (A2), the followig holds dx 0 (t) = i= e x i(t) x 0 (t) u i (t)dt + rdt (2.) dx i (t) = [u i (t) + µ i (t) (/2)σ i (t)σ i (t)]dt + σ i (t)dw(t) (2.2) Proof. First cosider the case whe i =,2,...,. Takig the logarithm of y i (t) = v i (t)s i (t), which is allowed due to the assumptio (A), we obtai l[y i (t)] = l[v i (t)] + l[s i (t)]. Its differetial is d l[y i (t)] = d l[v i (t)] + d l[s i (t)], which after substitutig (2.7) ad (2.4) gives equatios (2.2). Similarly we obtai the dyamics of l[y 0 (t)] as d l[y 0 (t)] = d l[v 0 (t)] + d l[s 0 (t)] = where we have used the self-fiacig costrait (2.8) i the form d l[v 0 (t)] = i= i= e l[y i(t)] l[y 0 (t)] u i (t)dt + rdt, e l[y i(t)] l[y 0 (t)] u i (t)dt. (2.3) REMARK 2. Equatios (2.) ad (2.2) represet the cotiuous-time state-space model of a selffiacig portfolio with a positive differetiable tradig strategy 2. Note that there are o explicit costraits o the state variables x i (t), i = 0,,...,, or o the cotrol variables u i (t), i =,2,...,. Usig the Euler approximatio 3 with a sufficietly small samplig time T, we obtai the discretetime form of (2.) ad (2.2) as x 0 (k + ) = x 0 (k) i= e x i(k) x 0 (k) u i (k)t + rt, (2.4) x i (k + ) = x i (k) + [u i (k) + µ i (k) (/2)σ i (k)σ i (k)]t + σ i (k)e(k) T, (2.5) where e(k) = [e (k),...,e m (k)] is a vector of zero mea, uit variace, i. i. d. Gaussia radom variables for each k. The model (2.4) ad (2.5) ca be writte i the followig more coveiet matrix form x(k + ) = x(k) + A(k,x(k))u(k)T + D(k)T + Σe(k) T (2.6) 2 I geeral, part of this model are also the risky asset price dyamics (2.2). 3 Similarly oe ca use other forward approximatio schemes; see, e. g. Klode & Plate (992).

6 6 of 24 B. GASHI ad P. DATE where x(k) = [x 0 (k),...,x (k)], u(k) = [u (k),...,u (k)]. Deotig by β(x,k) = [ e x (k) x 0 (k),..., e x (k) x 0 (k) ], 0 m a m-vector of zeros, I the -th order idetity matrix, we ca express the matrices i (2.6) as follows A(k,x(k)) = [ β(x,k) I ] [ ], Σ = 0 m, (2.7) σ D(k) = [ r(k), µ (/2)σ (k)σ (k),, µ (/2)σ (k)σ (k) ]. Equatio (2.6) will be used i the followig sectios as a model of a self-fiacig portfolio with positive tradig strategies of fiite variatio. REMARK 2.2 Note that the matrix A(k,x(k)) is of a full rak, ad thus its colums are liearly idepedet. This meas that for a positive defiite matrix R, the matrix quadratic form A RA will always be positive defiite. This fact will be used later i the proof of Theorem 4.. as The total wealth of the costraied portfolio y(k + ) ca be easily obtaied from (2.4) ad (2.5) y(k + ) = e x i(k+) (2.8) 3. The first method: a upper boud o the log-error. The aim to be achieved with the costraied portfolio of the previous sectio is to either track closely or outperform some already desiged referece portfolio that has a positive tradig strategy; see, e. g. Cvitaiić & Karatzas (992), Cvitaiić & Karatzas (993), Kor (997), Karatzas & Shreve (99), ad the refereces therei, for possible examples of such referece portfolios. This is doe with the aim of obtaiig a lower evetual proportioal trasactio cost due to the differetiable tradig strategy of the costraied portfolio. I this sectio the criterio of optimality will be a upper boud o the discrete-time logarithmic error e l (k + ) betwee the two portfolios e l (k + ) = l[y r (k + )] l[y(k + )]. (3.) Here y r (k + ) is the value of the self-fiacig referece portfolio at time T (k + ) with its logarithm defied below, ad y(k + ) is the value of the costraied trackig portfolio (2.8). Miimizatio of this criterio pealizes oly the shortfall of the costraied portfolio with respect to the referece portfolio. As we see later, by workig with a upper boud to e l (k + ) rather tha e l (k + ) itself, simple explicit closed-form solutios are obtaied. Deotig for the referece portfolio v r i (t)-the umber of shares held for the asset i, αr i (t)-the fractio of the wealth allocated to asset i, ad for which it holds α0 r +αr +...+αr =, we ca derive the dyamics

7 of its value y r (t) as Two methods for optimal ivestmet with tradig strategies of fiite variatio 7 of 24 dy r (t) = v r i (t)ds i (t) = v r 0rS 0 dt + = y r [α r 0rdt + = y r [( = y r [rdt + i= αi r i= i= i= α r i (µ i dt + σ i dw) ) rdt + i= α r i (µ i r)dt + v r i S i (µ i dt + σ i dw) ] α r i (µ i dt + σ i dw) i= α r i σ i dw ] ]. (3.2) Deotig by α = [α r,...,αr ], M = [µ r,..., µ r], ad applyig Ito s lemma to (3.2), we obtai the equatios for l[y r (t)] ad l[y r (k + )] as d l[y r (t)] = [r + α M 0.5α σσ α]dt + α σdw, l[y r (k + )] = l[y r (k)] + [r + α M 0.5α σσ α]t + α σe(k) T (3.3) A upper boud o e l (k+) ca be foud usig Jese s iequality (see e. g. Roberts & Varberg (973)), as follows. Let γ i (k + ), i = 0,,...,, be variables such that 0 γ i (k + ), ad γ 0 (k + ) γ (k + ) =. The the Jese s iequality gives the followig for each k l[y(k + )] γ i (k + )l[y i (k + )]. (3.4) A upper boud o the logarithmic error e u (k + ) e l (k + ) ca thus be expressed as e u (k + ) = ly r (k + ) γ i (k + )l[y i (k + )] = ly r (k + ) γ i (k + )x i (k + ) (3.5) The aim ow is to miimize this upper error boud. Oe possibility is to miimize its mea ad variace. We do so by first selectig the variables γ i (k + ), i = 0,,...,, such that the variace of e u (k + ) is zero, ad the miimize its mea. We stress that this is oly oe of may possibilities ad it is also supported by simulatio results. THEOREM 3. For k = 0,,..., the variace of Var[e u (k + ) x(k)] is equal to zero if for each i = 0,,...,. γ i (k + ) = α r i (k) (3.6) Proof. Substitutig l[y r (k + )] from (3.3) i (3.5), together with x 0 (k +) ad x i (k +), i =,2,...,, from (2.4) ad (2.5), ad takig the variace, we obtai Var[e u (k + ) x(k)] = m } 2 [αi j={ r (k) γ i (k + )]σ i j T. (3.7) i= It is clear that a sufficiet coditio for (3.7) to be equal to zero is for (3.6) to hold.

8 8 of 24 B. GASHI ad P. DATE LEMMA 3. If the volatility matrix σ(t) is square ad o-sigular, the the coditio (3.6) is also ecessary for Var[e u (k + ) x(k)] = 0. Proof. The ecessary coditios for (3.7) to be zero are i= for every j =,2,...,. This system of equatios ca also be writte as [α r i (k) γ i (k + )]σ i j = 0 (3.8) σ N = 0, (3.9) where N = [α r (k) γ (k + ),...,α r (k) γ (k + )]. Sice the volatility matrix σ is assumed square ad osigular, the the system of equatios (3.9) has a uique solutio give by N = 0, which gives (3.6) for each k. The expected value of e u (k + ) will be miimized if we maximize the followig E{ γ i (k + )l[y i (k + )] x(k)}=e{ where we have substituted the relatios (3.6). 3. Optimal tradig strategies. α r i (k)x i (k + ) x(k)} =E[α (k)x(k + ) x(k)], (3.0) I order to give the ivestor the meas for trade off betwee a lower evetual trasactio cost ad a higher profit, ad have a well defied optimizatio problem, we exted the criterio (3.0) to iclude a quadratic pealty o the logarithmic rates of chage of tradig strategies u i (k). The pealty matrix for cotrols i geeral ca be a fuctio of the state x(k) to reflect a wealth depedet pealizatio. The resultig optimizatio problem ca be stated as follows. Portfolio cotrol problem I (PCP-I). Let B(k,x(k)) R be a give positive defiite matrix, possibly state depedet. Fid the cotrol u(k), k = 0,,..., that miimizes the followig objective [ ] V () (k,x(k),u(k)) = E 2 u (k)b(k,x(k))u(k)t α (k)x(k + ) x(k), (3.) where x(k + ) is give i (2.6) ad α(k) = [α r (k),...,αr (k)]. THEOREM 3.2 The solutio to the PCP-I always exists, is uique ad for every k = 0,,2,..., is give by u (k) = B (k,x(k))a (k,x(k))α(k), (3.2) where A(k,x(k)) is defied i (2.7). Proof. Substitutig x(k + ) from (2.6) ito (3.) ad takig the expectatio coditioal o x(k), we obtai V () = 2 u (k)b(k,x(k))u(k)t α (k)[x(k) + A(k,x(k))u(k)T + D(k)T ]. (3.3) Differetiatig with respect to u(k) ad equatig to zero gives V () u(k) = B(k,x(k))u(k)T A (k,x(k))α(k)t = 0 u (k) = B (k,x(k))a (k,x(k))α(k). (3.4)

9 Two methods for optimal ivestmet with tradig strategies of fiite variatio 9 of 24 Due to the assumed positive defiite property of B(k, x(k)), the uique cotrol law (3.4) always exists ad represets the required optimum sice the Hessia of V () is positive defiite. REMARK 3. Optimal cotrols (3.2) cotai the fractios of wealth α(k) = [α r (k),...,αr (k)] of the referece portfolio. This meas that oe eeds to first desig the referece portfolio uder the o shortsellig costraits before implemetig (3.2). I the special case of the cotrol pealizatio matrix B(k, x(k) beig of diagoal form, the optimal cotrols (3.2) reduce to u i (k) = b i [α r i (k) α r 0(k)e x i(k) x 0 (k) ],i =,2,...,. (3.5) where b i, i =,2,...,, are the diagoal elemets of B. The optimal tradig strategies v i (k + ), i = 0,,...,, for k = 0,,...,, are foud by applyig the Euler s approximatio to (2.7) ad (2.3), ad usig the optimal cotrols u i (k) from (3.2) to obtai v i (k + ) = v i (k)e u i (k)t, i =,2,...,. (3.6) v 0(k + ) = v 0(k)e u 0 (k)t, u 0(k) = i= e x i(k) x 0 (k) u i (k). (3.7) Equatios (3.6) ad (3.7), do ot give a aswer o how to make the iitial optimal selectio v i (0), i = 0,,...,, which is uderstadable sice the optimizatio has bee carried out with respect to the logarithmic rates of chage rather tha the quatities themselves. Thus, we make the iitial selectio idetical to the referece portfolio, which gives e l (0) = l[y r (0)] l[y(0)] = 0. I this case we have v i (0) = αr i y(0),i = 0,,...,. (3.8) S i (0) REMARK 3.2 Note that the cotrols i (3.2) will have the same form for ay value of T. I particular, as T 0, they will represet a cotiuous cotrol with k replaced by t. The cotiuous-time optimal tradig strategies will have a fiite variatio ad are derived by solvig equatios i (2.7) ad (2.3) with the iitial coditios give by (3.8). The optimal variatio (3.2) does ot deped explicitly o the market parameters. Such a iformatio is cotaied i the fractios of wealth α(k) of the referece portfolio. This meas that this approach is also applicable to the market with ucertai parameters, i. e. the parameters are iterval umbers of the type r(t) [r (t),r + (t)], µ i (t,s) [µ i (t,s), µ i + (t,s)], ad σ i j (t,s) [σi j (t,s),σ+ i j (t,s)]; see, e. g. Ah et al. (997), Ah et al. (999), Wilmott (998), Wilmott & Oztukel (998), for the time-varyig ucertai parameters. I this case, the desig of the referece portfolio deals with the parameter ucertaity. 3.. Costraied portfolio cotrol. I some applicatios it could be required to have a certai guaratee o the quality of performace. Oe approach to achievig this is to place a costrait o the upper boud of the logarithmic error, i. e. to require that e u (k + ) ε(k), k = 0,,2,..., a. s., where ε(k) is some pre-specified positive variable. If we select the coefficiets γ(k + ) as i (3.6), tha the upper boud becomes e u (k + ) = l[y r (k)] + [r + α (k)m(k) 0.5α (k)σ(k)σ (k)α(k)]t α (k)[x(k) + A(k,x(k))u(k)T + D(k)T ]. (3.9)

10 0 of 24 B. GASHI ad P. DATE The costraied optimizatio is stated as follows: { } mi u(k) 2 u (k)b(k,x(k))u(k)t α (k)[x(k)+a(k,x(k))u(k)t +D(k)T ] s.t. : α (k)a(k,x(k))u(k)t ε(k) [r + α (k)m(k) 0.5α (k)σ(k)σ (k)α(k)]t (3.20) l[y r (k)] + α (k)[x(k) + D(k)T ] (3.2) This is a quadratic programmig problem that ca be solved umerically for each k. 3.2 Bouds o the tradig strategies ad the problem of restrictig the umber of shares. Usig the explicit optimal cotrols (3.5) for the case of a diagoal cotrol pealizatio, we ca derive bouds o the logarithmic variatios of the optimal tradig strategies lv i (k) = u i (k), i = 0,,2,...,,, as follows. LEMMA 3.2 Lower ad upper bouds o the optimal logarithmic chages lv i (k), i = 0,,2,...,, are max[α r i (k)]y(k) mi(b i )y 0 (k) α r 0 (k) b i y i (k) y 0 (k) lv 0(k) αr 0 (k) y 2 (k) mi(b i ) y 2 0 (k) (3.22) lv i (k) αr i (k) b i (3.23) where max[α r i (k)] ad mi(b i) represet the maximum α r i (k) ad miimum b i for i =,2,...,, respectively. Proof. We first prove (3.22). The lower boud is foud by startig from the discrete form of (2.0) ad makig use of (3.5) as 0 = max(αr i ) mi(b i ) y i (k) lv i (k) i= i= y i (k) αr i b i + y 0 (k) lv 0(k) y i (k)+y 0 (k) lv 0(k)= max(αr i ) mi(b i ) [y(k) y 0(k)]+y 0 (k) lv 0(k). lv 0(k) max(αr i ) mi(b i ) Similarly, we fid the upper boud as [ y(k) ] max(αr i )y(k) y 0 (k) mi(b i )y 0 (k). 0 = y i (k) lv i (k) α r 0 mi(b i )y 0 (k) i= i= α r 0 y2 i (k) b i y 0 (k) y 2 i (k) + y 0 (k) lv 0(k) + y 0 (k) lv 0(k) lv 0(k) αr 0 y 2 (k) mi(b i ) y 2 0 (k)

11 Two methods for optimal ivestmet with tradig strategies of fiite variatio of 24 Bouds i (3.23) follow directly from (3.6) ad (3.5). A importat applicatio of the upper bouds is whe we restrict the umber of shares per asset, where for some determiistic M i (k + ) it is required that v i (k + ) M i(k + ), k = 0,,2,...,, ad i = 0,,2,...,. The pealty coefficiets b i (k), i =,2,...,, ca be selected as follows i order for such a costrait to hold. First ote that the upper bouds i (3.22) ad (3.23) ca be expressed as [ α v 0(k + ) v r 0 (k)y 2 ] (k) 0(k)exp mi(b i )y 2 0 (k) (3.24) ( ) α v i (k + ) v r i (k)exp i (k),i =,2,...,. (3.25) By comparig these with v i (k + ) M i(k + ), it ca be see that sufficiet coditios for b i (k), i =,2,...,, to satisfy for every k =,2,..., are b i α0 r [ ] (k)y2 (k) M0 (k + ) mi(b i )y 2 l 0 (k) v 0 (k) αi r(k) [ ] Mi (k + ) l b i v i (k) (3.26) (3.27) For the special case of costat fractios of wealth for the referece portfolio α r i (k) = αr i, costat pealty coefficiets b i, a urestricted umber of shares i the risk free asset (e. g. the bak accout), ad a costat restrictio o the remaiig assets M i (k + ) = M i, i =,2,...,, we have the followig LEMMA 3.3 Let the iitial selectio be such that v i (0) < M i for every i =,2,...,. The the upper costraits v i (k) M i are satisfied for every k =,2,..., if b i Proof. Referrig to (3.25), for k =,2,..., we have A sufficiet coditio for v i (k) < M i is α r i l[ Mi v i (0) v i (k) v i (0)exp( kα r i b i v i (0)exp( kα r i ) α r exp( i b i b i ] (3.28) ). ) M i [ Mi v i (0) Due to assumptio [M i /v i (0)] >, the above iequality yields ] k ) [ ] α r exp( i Mi b i v i (0)

12 2 of 24 B. GASHI ad P. DATE for every k =,2,..., ad hece the result i (3.28). Oe solutio to the problem of havig v i (0) < M i, i =,2,...,, is to desig a referece portfolio that satisfies the coditio 3.3 Example: pseudo-log-optimal portfolio. αi r < S i(0)m i, i =,2,...,. y(0) The referece portfolio i this example is selected to be the log-optimal portfolio. Such portfolios are the best to use whe the aim of ivestmet is optimal wealth growth. These were itroduced i Kelly (956) ad Latae (959) for the case of discrete-time static portfolios ad more fully developed i Breima (96). A similar optimizatio problem i a market with trasactio cost is give i Iyegar & Cover (998). The log-optimal portfolio i a cotiuous-time dyamic case was itroduced i Merto (969), ad its versios with covex costraits ad trasactio cost ca be foud i textbooks such as Karatzas & Shreve (99), Kor (997), ad the refereces therei. Let us cosider a market havig a bak accout S 0 (t) ad a sigle stock S (t) with the followig dyamics ds 0 (t) = rs 0 (t)dt, (3.29) ds (t) = S (t)(µdt + σdw ). (3.30) We assume that the parameters are costat ad have these umerical values: r = 0.04, µ = 0.05, ad σ = The iitial ivestors wealth ad the iitial asset prices are assumed as y(0) = S 0 (0) = S (0) =. The fractio of wealth ivested i the stock for the log-optimal portfolio α r (k) is give as Merto (969): α r (k) = α r = µ r σ 2 = 0.6, which clearly satisfies the o short-sellig costrait. The iitial selectio for both the portfolios (the log-optimal ad the costraied oe) will thus be v 0 (0) = 0.84, v (0) = 0.6. The cotrol law (3.5) with a samplig time of T = 0.004, becomes u (k) = [ v ] (k)s (k). (3.3) b v 0 (k)s 0 (k) Let us also have two differet values for the pealty coefficiet, b () = 0.05, b(2) = 0.5, ad deote the correspodig tradig strategies for the stock of the costraied portfolio as v () (k) ad v(2) (k). I a market with o trasactio cost, for oe realizatio of the stock price, the tradig strategies for the stock of the log-optimal v r (k) = [αr (k)yr (k)]/s (k)], ad the costraied portfolios v () (k), v(2) (k), are show i Fig.. The tradig takes place durig the iterval of time [0, 0]. The total portfolio wealth is show i Fig. 2, where oe ca otice a almost udistiguishable behavior of the portfolios. This is the reaso why we propose to call this costraied portfolio pseudo-log-optimal. I Fig. 3, the ed period portfolio wealth is elarged. The evetual trasactio costs that would have accumulated at time (k + )T for the log-optimal C l (k + ) ad pseudo-log-optimal C p (k + ) portfolios, are assumed to be: C l (k + ) = C l (k) + 0.0α r y r (k + ) y r (k)s (k + )/S (k) C p (k + ) = C p (k) v (k + ) v (k) S (k)

13 Two methods for optimal ivestmet with tradig strategies of fiite variatio 3 of log optimal b=0.05 b= Tradig strategies Time kt FIG.. Tradig strategies for the stock. with C l (0)=C p (0)=0.0v (0)S (0). This correspods to a charge of % of the total trasactio value of buyig or sellig the stock, ad o trasactio cost for the bak accout. At the ed of the tradig period, the wealth y l f, y() f, y (2) f, ad the evetual proportioal trasactio cost C l f, C() f, C (2) f, of the log-optimal, pseudo-log-optimal with b (), ad pseudo-log-optimal with b(2), respectively, are: Log optimal : y l f =.6983, Cl f = b () : y () f =.628, C () f = b (2) : y (2) f =.58572, C (2) f = This shows that for almost the same fial wealth, the evetual trasactio cost is more tha ad 22 times smaller for the pseudo-log-optimal portfolios i compariso with the log-optimal oe. Further the differeces betwee the fial wealth ad the total evetual trasactio cost is higher for the pseudo-logoptimal portfolios. The average results of several realizatios are similar to the above sigle realizatio. The average of differeces (y l f y() f ), (y l f y(2) f ), ad the average of ratios C l f /C() f, C l f /C(2) f, for 00 realizatios of the stock price, are give below average average (y l f y() f ) = , average C l f /C() f =.555, (y l f y(2) f ) = , average C l f /C(2) f = The secod method: trackig portfolios. I this sectio the aim is to track as closely as possible a already desiged referece portfolio with the portfolio itroduced i Sec. 2. The criterio for the quality of the trackig is a quadratic form i the discrete-time log-square errors of idividual asset holdig values, i. e. the square of the logarithmic differece λ i (k + ) = l[y i (k + )] l[y r i (k + )], i = 0,,...,, where l[yr i (k + )] = a i(k + ) is the logarithm of the value of the holdigs for asset i of the referece portfolio. Thus we view each asset

14 4 of 24 B. GASHI ad P. DATE.8.7 log optimal b=0.05 b= Total portfolio wealth Time kt FIG. 2. Total portfolio wealth durig the tradig period..66 log optimal b=0.05 b= Total portfolio wealth Time kt FIG. 3. Total portfolio wealth at the ed of the tradig period.

15 Two methods for optimal ivestmet with tradig strategies of fiite variatio 5 of 24 holdig values as refereces ad try to track them rather tha track the total value of the referece portfolio. The resultig portfolios ca thus be see as replicatig portfolios, sice their tradig strategies are tryig to replicate those of the referece portfolio. The geeral form of the dyamics of l[y r i (t)] = a i(t) is give as da i (t) = g i (t,s)dt + h i (t,s)dw, i = 0,,...,, where the scalars g i (t,s) ad the row vectors h i (t,s) are kow fuctios of time ad asset prices. The Euler approximatio gives the followig discrete form a i (k + ) = a i (k) + g i (k)t + h i (k)e(k) T, i = 0,,...,. For simplicity, we have used the otatios g i (k) ad h i (k) rather tha g i (kt,s(kt )) ad h i (kt,s(kt )). The dyamics of the vector of the logarithmic errors λ(k) = [λ 0 (k),...,λ (k)] is obtaied by takig the differece betwee the costraied portfolio model (2.6) ad the above refereces, which gives λ(k+)=λ(k)+a(x,k)u(k)t +[D(k) G(k)]T +[Σ(k) H(k)]e(k) T, (4.) where G(k) is a colum vector with g i (k) as elemets, ad H(k) is a ( + ) m matrix with vectors h i (k) as rows. We ca ow give the formulatio of the optimal ivestmet problem. Portfolio cotrol problem II (PCP-II). Let B(k,x(k)) R ad Q(k,x(k)) R (+) (+) be two give symmetric positive semi-defiite matrices, possibly state depedet ad at least oe beig positive defiite. Fid the cotrols u(k), k = 0,,..., that miimize the followig objective V (2) (k,λ(k),u(k)) = 2 E[ u (k)b(k,x(k))u(k)t + λ (k + )Q(k,x(k))λ(k + ) λ(k) ], (4.2) where λ(k + ) is give by (4.). THEOREM 4. The solutio to the PCP-II always exists, is uique ad for every k = 0,,..., is give by u (k) = (B + A QAT ) A Q[λ(k) + (D G)T ] (4.3) Proof. obtai Substitutig the expressio for error dyamics (4.) i (4.2) ad takig the expectatio, we V (2) = 0.5{u BuT + λ Qλ +λ QAuT +λ Q(D G)T +u A QλT +u A QAuT 2 + u A Q(D G)T 2 + (D G )QλT + (D G )QAuT 2 + (D G )Q(D G)T 2 + tr[(σ H )Q(Σ H)]T }, (4.4) where tr( ) deotes the trace of a matrix. Differetiatig V (2) with respect to u(k) ad equatig it to zero gives V (2) u(k) = Bu(k)T + 0.5A QλT + 0.5A QλT + A QAT 2 u(k) + 0.5A Q(D G)T A Q(D G)T 2 = 0 u (k) = (B + A QAT ) A Q[λ(k) + (D G)T ].

16 6 of 24 B. GASHI ad P. DATE Due to our assumptio o matrices B ad Q, the matrix B + A QAT is always positive defiite ad so is the Hessia of V (2). Thus the obtaied cotrol law always exists, is uique, ad it represets the required optimum. For a positive defiite matrix B ad cosiderig the limitig case of T 0, the cotrol law (4.3) becomes u (t) = B A Qλ(t) ad the correspodig cotiuous-time tradig strategy has a fiite first variatio. This cotrol law does ot deped explicitly o the market parameters. I the special case of a market with a bak accout S 0 (t) ad a sigle stock S (t) give by (3.29) ad (3.30), ad pealty matrices selected as B = b, Q = diag(q 0,q ), the cotrol law (4.3) reduces to the followig form that will be useful later: u (k) = q 0e x x 0 [x 0 a 0 +rt g 0 T ] q [x a +(µ 0.5σ 2 g )T ] q 0 Te x x 0 (k) +q T + b. (4.5) The optimal tradig strategy is foud by substitutig the optimal cotrols u i (k), k = 0,,2,..., from (4.3) i (3.6) ad (3.7). The iitial selectio is made idetical to the referece portfolio. Similarly to the Sec. 3.., i order to have a certai guaratee o the quality of trackig, oe ca solve a costraied portfolio cotrol problem of a quadratic programmig type with (3.20) replaced by (4.4), ad the elemets of the vector α(k) = [α r(k),...,αr (k)] i (3.2) be give as α r i (k) = yr i (k) j=0 yr j (k) = eai(k) j=0 ea j(k) 4. Example: Black-Scholes replicatig portfolio as a referece. As a referece we will use the well-kow Black-Scholes replicatig portfolio for a Europea Call optio o a sigle stock as described i Black & Scholes (973), Merto (973). This is a iterestig example sice it shows how the origial formulatio of the optimal portfolio ca be modified to deal with the case whe borrowig 4 is allowed, ad also the resultig optimal portfolio is of a practical importace. I a market give by (3.29) ad (3.30), the price of a Europea call optio C(S,t), i. e. the value of the Black-Scholes replicatig portfolio, is give as Wilmott (998) where E is the exercise price, T e the expiry time, ad N(x) = C(S,t) = S N(d ) Ee r(t e t) N(d 2 ), (4.6) 2π x e 0.5y2 dy (4.7) d 2 (S,t) = l S E + (r 0.5σ 2 )(T e t) σ, d (S,t) = d 2 (S,t) + σ T e t (4.8) T e t The umber of shares i the cash v BS 0 are obtaied from (4.6) as ad i the stock vbs v BS (the tradig strategy) for this referece portfolio 0 = Ee r(te t) N(d 2 ), (4.9) S 0 (t) v BS = N(d ). (4.0) 4 Similarly we ca deal with the Europea Put optio, i which case the short sellig of the stock is allowed.

17 Two methods for optimal ivestmet with tradig strategies of fiite variatio 7 of Model of the costraied portfolio with v 0 (t) < 0. It ca be see from (4.9) that the umber of shares i the cash is always egative. Thus we caot apply the method of this sectio directly. Oe approach to achievig a log-quadratic trackig is to first costrai the umber of shares i the cash of the trackig portfolio v 0 (t) to also be egative; v 0 (t) < 0. This meas that both the referece portfolio ad the replicatig portfolio will have egative values of the holdigs i the cash. Thus we propose to try ad match the logarithms of the egative of the values of the holdigs i the cash, which are well defied i this case. Uder the assumptio of v 0 (t) < 0, ad followig the basic steps of Sec. 2, we derive the dyamics of the portfolio with a differetiable tradig strategy, which will be oly slightly differet from the oe of Sec. 2. The value of the holdig i the cash is y 0 (t) = v 0 (t)s 0 (t), ad the logarithm of its egative value is well defied ad give as l[ y 0 (t)] = l[ v 0 (t)]+l[s 0 (t)] d l[ y 0 (t)]=d l[ v 0 (t)]+d l[s 0 (t)], (4.) l[y (t)] = l[v (t)] + l[s (t)] d l[y (t)] = d l[v (t)] + d l[s (t)]. (4.2) The differetiability costrait o the tradig strategy is d l[ v 0 (t)] = u 0 ( )dt, d l[v (t)] = u ( )dt. The self-fiacig costrait (2.0) ow gives S 0 dv 0 + S dv = 0 S dv = S 0 d( v 0 ) S v u dt = S 0 ( v 0 )u 0 dt u 0 = e x x 0 u, where x 0 = l( y 0 ) ad x = l(y ). The dyamics of a self-fiacig portfolio is ow obtaied from (4.) ad (4.2) as dx 0 (t) = u 0 dt + rdt = e x x 0 u dt + rdt, (4.3) dx (t) = u dt + (µ 0.5σ 2 )dt + σdw. The refereces i this case are the logarithm of the egative value of the holdigs i the cash a 0 = l( v BS 0 S 0) ad the logarithm of the value of the holdigs i the stock a = l(v BS S ). We select the objective to be miimized as V (3) = 2 E{b u 2 (k) + q 0 [x 0 (k + ) a 0 (k + )] 2 + q [x (k + ) a (k + )] 2 }. The oly differece betwee this optimizatio problem ad PCP-II is i the dyamics for x 0 (t). If (4.3) is compared to (2.), the oly differece is that (4.3) does ot have a mius sig i frot of the expoetial term. This meas we ca write the optimal cotrol law for this problem u (k) directly from (4.5) by oly placig a mius sig i frot of the expoetial i the umerator (sice the expoetial i deomiator is squared) to obtai u (k) = q 0e x x 0 [x 0 a 0 +rt g 0 T ] q [x a +(µ 0.5σ 2 g )T ] q 0 Te x x 0 (k) +q T + b. (4.4) 4..2 Derivig the dyamics of the refereces. Here we derive the discrete-time dyamics of the refereces a 0 ad a, which are defied as a 0 = l( v BS 0 S 0) = l[ee r(te t) N(d 2 )] = l(e) r(t e t)+l[n(d 2 )], (4.5) a = l(v BS S ) = l[s N(d )] = l(s ) + l[n(d )]. (4.6)

18 8 of 24 B. GASHI ad P. DATE I order to fid the differetials of these refereces we shall make frequet use of the Ito s lemma ad make may elemetary calculatios. To derive the dyamics of d 2 = d 2 (t,l[s]) we eed the followig partial derivatives: d 2 t = l( S E ) (r 0.5σ 2 )(T e t) 2σ(T e t).5, d 2 l(s) = σ T e t, 2 d 2 l(s) 2 = 0. Applyig Ito s lemma to (4.8) ad substitutig the above partial derivatives, we obtai [ d2 d(d 2 ) = + d ] 2 t l(s) (µ 0.5σ 2 ) dt + d 2 l(s) σdw [ ] l[ S = E ] (r 0.5σ 2 )(T e t) 2σ(T e t).5 + (µ 0.5σ 2 ) σ dt + dw T e t σ T e t = m 2 dt + s 2 dw. Next we eed the differetial of l[n(d 2 )] which, together with its partial derivatives, is give as [ l[n(d 2 )] = l l[n(d 2 )] d 2 = 2π d2 ] e 0.5y2 dy, e 0.5d2 2 N(d 2 ) 2π, 2 l[n(d 2 )] d2 2 = Applyig Ito s lemma for this case as well, we obtai [ l[n(d2 )] d l[n(d 2 )]= m l[n(d 2 )] d 2 d2 2 s 2 2 l[n(d 2 )] = 0, t The differetial ad differece equatios of the first referece (4.5) are da 0 (t) = rdt + d l[n(d 2 )], ] [ e 0.5d2 2 d 2 N(d 2 ) 2π + e d2 2 2π[N(d 2 )] 2 dt+ l[n(d 2)] d 2 s 2 dw (4.7) a 0 (k + ) = a 0 (k) + rt + l[n(d 2 )], (4.8) where l[n(d 2 )] is obtaied by applyig Euler s method to (4.7), ad i particular ote that the term (T e t) i m 2 ad s 2 becomes (T e kt ) i this case. To calculate the optimal cotrol law u i (4.4), we eed the drift term g 0 (k) of (4.8), which is g 0 (k) = r + l[n(d 2)] m l[n(d 2 )] d 2 σ 2 (T e kt ) The dyamics of the secod referece (4.6) requires the differetial of l[n(d )], which ca be obtaied from that of l[n(d 2 )] by substitutig d 2 with d, ad (r 0.5σ 2 ) with (r +0.5σ 2 ). The differetial ad differece equatios of the secod referece are da (t) = d l[s (t)] + d l[n(d )] = (µ 0.5σ 2 )dt + σdw + d l[n(d )], d 2 2 a (k + ) = a (k) + (µ 0.5σ 2 )T + σe (k) T + l[n(d )], (4.9) ]

19 Two methods for optimal ivestmet with tradig strategies of fiite variatio 9 of 24 ad the drift term of (4.9) ca be obtaied as g (k) = (µ 0.5σ 2 ) + l[n(d )] m l[n(d )] d σ 2 (T e kt ), where m is the same as m 2 with (r 0.5σ 2 ) substituted with (r + 0.5σ 2 ). Note that u (k) i (4.4) depeds explicitly o the drift µ through the parameters m ad m 2, which makes the trackig portfolio ot idifferet to it as is the case with the replicatig portfolio. d Simulatio results. Let the market parameters, the samplig time, ad the trasactio cost structure be selected as i the previous sectio. Also let the optio parameters be T e = 0, E = 2, ad the trackig portfolio parameters be q 0 = q = 00, b = 50. For oe realizatio of the stock price, the value processes of the two portfolios are show i Fig. 4, ad it ca be oticed that these are almost idetical. The correspodig tradig strategies for the cash ad the stock are show i Fig. 5 ad 6, respectively. Similarly to the previous example, the tradig strategy of the trackig portfolio is smooth ad it is ituitively clear that this will result i a lower evetual trasactio cost. At the ed of the tradig period, the wealth y BS f, ytr f, ad the evetual proportioal trasactio cost C BS f, C Tr f, for the Black-Scholes ad the trackig portfolios, respectively, are Black Scholes : y BS f = , C BS f = , Trackig : y Tr f = , C Tr f = Thus this example illustrates that whe the trackig portfolio is used to hedge a optio, it will have a idetical iitial value to the Black-Scholes replicatig portfolio, almost the same total value throughout the tradig period, ad a lower evetual trasactio cost (more tha 7 times lower i this example). The average differece [y BS f y Tr f ], ad the average ratio CBS f /C Tr f, for 00 realizatio of the stock price, are obtaied as average [y BS f y Tr f ] = 0.08, average C BS f /C Tr f = Ope problems There are two importat ope questios regardig this approach to optio hedgig. Due to borrowig, the value of the trackig portfolio ca become egative ad thus oe should use this approach with care. This problem does ot occur if the referece optio satisfies the positivity costrait o the tradig strategy (see, e. g. Cvitaiić & Karatzas (993)), i which case the trackig portfolio that also satisfies such a costrait is used ad its value is positive. The secod problem is that of the termial wealth, sice there is o guaratee that the trackig portfolio will be idetical to the replicatig portfolio at the termial time, as the above example illustrates. Whe the value of the trackig portfolio is higher tha that of the referece portfolio just before the last tradig step, the a simple ad-hoc solutio is as follows: first equalize the portfolio values by cosumig the excess wealth of the trackig portfolio, ad the employ a idetical tradig strategy to that of the replicatig portfolio at the very last tradig step. A better solutio to this problem is required, ad also a solutio for the case of the trackig portfolio havig a lower value to that of the referece portfolio.

20 20 of 24 B. GASHI ad P. DATE 2.8 Black Scholes Trackig.6.4 Value processes Time kt FIG. 4. Value processes for the Black-Scholes ad the trackig portfolios. 0 Black Scholes Trackig Tradig of cash Time kt FIG. 5. The tradig strategies for the cash.

21 Two methods for optimal ivestmet with tradig strategies of fiite variatio 2 of Black Scholes Trackig Tradig of stock Time kt FIG. 6. The tradig strategies for the stock. 5. Coclusios Two simple ad very geeral methods for desigig optimal portfolios are proposed. I order to reduce the evetual proportioal trasactio cost the tradig strategy is costraied to be differetiable ad thus of fiite variatio. I the first method, a upper boud o the discrete-time log-error betwee the referece ad the costraied portfolios is miimized, by which the shortfall with respect to such a referece portfolio is pealized. The criterio also has a quadratic pealty of the logarithmic rates of chage of the tradig strategy. This gives the ivestor the meas for trade off betwee a lower evetual trasactio cost ad a higher profit, ad also esures the existece ad uiqueess of the optimal solutio. The tradig strategy is obtaied i a explicit closed-form, ad a simulatio example illustrates a sigificat reductio i the evetual proportioal trasactio cost as compared to the log-optimal portfolio. I the secod method, optimal trackig is achieved by usig a sum of discrete-time log-quadratic errors of the asset holdig values. The optimal tradig strategy is obtaied i a explicit closed-form for this approach as well, ad a simulatio example shows the use of a trackig portfolio as a hedgig strategy for optios. I terms of the performace, simulatio results show o sigificat differece betwee these two methods. O the other had, the first method ca also be applied to the markets with iterval parameters, whereas the preset form of the discrete-time versio of the secod method ca ot, sice the tradig strategy depeds explicitly o the market parameters. A modified versio of the secod method ca also be applied whe borrowig is allowed, i which case the value of the trackig portfolio ca be egative. This meas the first method ca ot be applied i this case sice the log-error ca ot be defied. I both of these methods the referece portfolio eeds to be desiged first, ad it is ot ecessary for the desig method beig used to iclude the trasactio cost explicitly i the model. This meas that oe should be ecouraged to develop such methods, sice our approach icreases their practical relevace. Compariso of the secod method as applied to optio replicatio, with results of Kor (998), Martellii & Priaulet (2002), ad Whalley & Wilmott (999), is a importat future work. Furthermore, the model of a self-fiacig portfolio with a positive differetiable tradig strategy proposed i this paper is of a geeral use, e. g. a differet objective from the oes used i this paper ca be employed with it. I particular, formulatio of objective fuctios for various portfolio selectio problems with costraits

22 22 of 24 REFERENCES (such as the costrait o Capital-at-Risk discussed i Atkiso & Papakokkiou (2005)), ad iclusio of more geeral asset price models i our framework are importat topics for further research. Ackowledgemet The authors are thakful to the reviewer for suggestios which improved the quality of this paper. Refereces H. AHN, A.MUNI, & G. SWINDLE (997) Misspecified asset price models ad robust hedgig, Applied Mathematical Fiace, 4, H. AHN, A. MUNI, & G. SWINDLE (999) Optimal hedgig strategies for misspecified asset price models, Applied Mathematical Fiace, 6, M. AKIAN, P. SÉQUIER, & A. SULEM (995) A fiite horizo multidimesioal portfolio problem with sigular trasactio cost, Proceedigs of the 34 th Coferece o Decisio ad Cotrol, C. ATKINSON & M. PAPAKOKKINOU (2005) Theory of optimal cosumptio ad portfolio selectio uder a Capital-at-Risk ad a Value-at-Risk costrait, IMA Joural of Maagemet Mathematics, 6, N. H. BINGHAM & R. KIESEL (2000) Risk-eutral valuatio: pricig ad hedgig of fiacial derivatives, Spriger-Verlag. F. BLACK & M. SCHOLES (973) The pricig of optios ad corporate liabilities, Joural of Political Ecoomy, 8, T. BJÖRK (2004) Arbirage theory i cotiuous time, Secod editio, Oxford Uiversity Press. L. BREIMAN (96) Optimal gamblig systems for favorable games, Fourth Berkeley Symposium, I, M. BRODIE, J. CVITANIĆ, & H. M. SONER (998) Optimal replicatio of cotiget claims uder portfolio costraits, Review of Fiacial Studies,, P. CHERIDITO, H. M. SONER, & N. TOUZI (2005) The multi-dimesioal super-replicatio problem uder gamma costraits, Aales de L Istitut Heri Poicaré Aalyse Noliéaire, 22, R. COMPANY, L. JÓDAR, J.R. PINTO & M.D. ROSELLÓ (200), Computers ad Mathematics with Applicatios, 59, J. CVITAINIĆ & I. KARATZAS (992) Covex duality i costraied portfolio optimizatio, Aals Appl. Probability, 2, J. CVITAINIĆ & I. KARATZAS (993) Hedgig cotiget claims with costraied portfolios, Aals Appl. Probability, 3, J. CVITAINIĆ & J. MA (996) Hedgig optios for a large ivestor ad forward-backward SDE s, The Aals of Applied Probability, 6,