portfolio optimization in discrete time with proportional transaction costs under stochastic...

Ann Finance (2012) 8:405–425DOI 10.1007/s10436-010-0149-3

SYMPOSIUM

Portfolio optimization in discrete time with proportionaltransaction costs under stochastic volatility

Ha-Young Kim · Frederi G. Viens

Received: 10 December 2009 / Accepted: 12 February 2010 / Published online: 4 March 2010© Springer-Verlag 2010

Abstract This paper is devoted to evaluating the optimal self-financing strategy andthe optimal trading frequency for a portfolio with a risky asset and a risk-free asset.The objective is to maximize the expected future utility of the terminal wealth in astochastic volatility setting, when transaction costs are incurred at each discrete trad-ing time. A HARA utility function is used, allowing a simple approximation of theoptimization problem, which is implementable forward in time. For each of varioustransaction cost rates, we find the optimal trading frequency, i.e. the one that attainsthe maximum of the expected utility at time zero. We study the relation between trans-action cost rate and optimal trading frequency. The numerical method used is basedon a stochastic volatility particle filtering algorithm, combined with a Monte-Carlomethod. The filtering algorithm updates the estimate of the volatility distribution for-ward in time, as new stock observations arrive; these updates are used at each of thesediscrete times to compute the new portfolio allocation.

Keywords Portfolio optimization · Stochastic volatility · Particle filtering ·Monte-Carlo method · Discrete trading · Transaction costs

JEL Classifications C6 · C4 · G1

H.-Y. KimDepartment of Mathematics, Purdue University, 150 N. University St.,West Lafayette, IN 47907-2067, USAe-mail: [email protected]

F. G. Viens (B)Department of Statistics and Department of Mathematics, Purdue University,150 N. University St., West Lafayette, IN 47907-2067, USAe-mail: [email protected]

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406 H.-Y. Kim, F. G. Viens

1 Introduction

The Black-Scholes model is an essential tool to understand stock-market movements,and rightly continues to be celebrated by many authors. It is equally well-acceptedthat its principal drawback is the constance of its coefficients, most notably its vola-tility. There are many ways of taking into account the fact that stock market volatilityis far from being constant. This article situates itself in the popular continuous-timeframework of stochastic volatility (SV). Such models are mathematically convenientbecause, at the cost of an analysis of multidimensional stochastic differential equa-tions even for single stocks, many of the stochastic calculus tools from the standardBlack-Scholes theory for option pricing and portfolio optimization can be extendedto handle SV. See the excellent treatment in Fouque et al. (2000) along these lines.The book edited by Shepard (2005) contains a collection of recent articles with thesame motivations; one can also consult the book by Fornari and Mele (2000) for apresentation of how discrete and continuous-time modeling of SV are related.

1.1 Stochastic volatility framework and estimation

For a stock price diffusion process model S given in continuous time by d S (t) =αS (t) dt + V (t) S (t) dW (t), where V is the unobserved volatility process, and Wis a Brownian motion, continuous-time observation would imply that one can observethe quadratic variation process [S] of S; since stochastic calculus tells us that [S] (t) =∫ t

0 V (r)2 S (r)2 dr , one immediately has access, in continuous time, to the squaredvolatility V (t)2 = S (t)−2 d [S] (t) /dt . However, the main practical problem withSV models is that volatility itself is not directly observable, and must be somehowestimated. This proves that, even if one believes or assumes from a modeling point ofview (as we do) that stock prices are continuous-time stochastic processes, we can onlyassume that they are observed in discrete time. Even high-frequency (e.g. tick-by-tick)data for highly traded assets and indices cannot be considered as continuous-time datafor the purpose of understanding volatility.

In this paper, we estimate stochastic volatility in what one might call a Bayesianstatistical framework. More specifically, in the language of stochastic calculus, weconsider the pair of processes (S, V ) as above, and seek the stochastic filter of theunobserved process V given the observed process S; both processes are defined incontinuous time, but S is observed in discrete time only, and therefore, it is consistentto only require an estimation of V at the same discrete instants, given the informationcontained in all past observations. For simplicity of notation in this introduction, andoften in the remainder of this paper, observation times are denoted by the set of integersi = 0, 1, 2, . . .. In other words, we seek the conditional probability law

P [Vi ∈ dy|S1 = s1; S2 = s2; . . . ; Si = si ] (1)

where s1, s2, . . . , si are observed values of S up to time i . We adopt an approach whichwas introduced in Viens (2002) to actually compute an approximation of this prob-ability distribution, by way of a so-called stochastic volatility particle filter, which

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Portfolio optimization in discrete time 407

is adapted from the generic method of Del Moral et al. (2001). More details on thismethod are given in Sect. 2

1.2 Motivations

With this framework being established, one can ask a variety of quantitative financequestions, such as option pricing, or portfolio optimization, under our discretelyobserved stochastic volatility stock price.

The former topic was considered using a special quadrinomial recombining tree inFlorescu and Viens (2008); part of the issue there was to handle the fact that the marketis incomplete under an SV model. See the references therein for other approaches tothe problem, not based on the Bayesian estimation of volatility. Recently, asset pricingquestions in continuous time using transaction costs have appeared in the literature(Dempster et al. 2006, and Guasoni et al. 2010).

In this article, we concentrate on the latter topic: portfolio selection in order to opti-mize a given utility function; in this case, incompleteness of the model is not relevant,since we do not attempt to hedge any positions. This optimization was the topic origi-nally studied in Viens (2002), but the numerical method therein was too cumbersometo be implemented beyond a highly simplified binomial version of the model in Desaiet al. (2003). A practical breakthrough was achieved in the paper of Batalova et al.(2006), where it was noticed that in the case of power utility, the cumbersome portionof the algorithm, which requires numerically solving a discretized Bellman problemwith high-dimensional state space in reverse time, can be short-circuited in practice ina purely forward-time algorithm which nonetheless updates its Bayesian estimationof the stochastic volatility distribution at every time step. None of the above worksallow transaction costs.

Studies of portfolio optimization under transaction costs are well known, andsome can now be considered classical: the first treatment of the problem was byDavis and Norman (1990), where a continuous-time framework is considered for theclassical Black-Scholes model; Shreve and Soner (1994) considerably weakened theconditions of Davis and Norman (1990), also in the continuous time setting; Framstadand Øksendal (2001), in which a jump diffusion model is considered, again in contin-uous time, showed the existence of a cone of non-activity, with trading given via localtime on the boundary of this cone. In the context of stochastic volatility, Pham andQuenez (2001) gave a continuous-time resolution of the portfolio optimization prob-lem assuming that one only observes the stock prices. In this last reference, volatilityis driven by a stochastic process whose dynamics are non-autonomous (meaning thatthey depend also on the stock process), making the problem only partially observed,even though stock observations and trading in Pham and Quenez (2001) are allowedin continuous time. This incomplete observation is of a different nature than the onedescribed in Sect. 1.1 above, where only discrete-time observations are considered.

With discrete information and trading being our motivation, in this paper, we pro-pose to take up the program in the paper of Batalova et al. (2006), and study itsimplementation and analyze its performance under the “real-market” assumption thatthere are transaction costs in discrete time, each time a stock allocation is changed

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(each time an individual traders buys or sells a stock). Our work contains propor-tional transaction cost in each time interval. Even if transaction costs are proportionalto the trades, which is the case in this article, they cannot be approximated by thecontinuous-time method of reducing the stock’s mean rate of return by a continuouslycompounded transaction rate. Indeed, in practice, trades incur costs which are notproportional to the frequency of trading, unlike the use of a continuously compoundedconstant trading rate.

Having transaction costs which are proportional to the trade sizes regardless ofthe trading frequency is more challenging to implement in practice, and is the mainsubject of this article. More specifically, for a given risk aversion parameter (for agiven power in our Hyperbolic Absolute Risk Averse (HARA) power utility) we findthe optimal self-financing portfolio that maximize the wealth’s expected future util-ity at time zero, with proportional transaction costs at each trading time i , under ourstochastic volatility model, assuming that S is observed only at the integer tradingtimes. The optimal portfolio selection at time i is required to only depend on thevalues of S at times 0, 1, 2, . . . , i .

The main quantitative issue at hand then becomes that of trading frequency. If notransaction costs are incurred, the practitioners may buy or sell stocks as many timesas they choose, and theirs becomes only an issue of gathering information. In thiscase, the gain increases as the number of transactions per unit of time increases. Inthe best of cases, where they can handle trading at the tick-by-tick frequency, theirproblem is still discrete, but they take advantage of all the available information.However, when discrete transaction costs are considered, we have the limitation of notletting the transaction costs destroy all our profits just for the sake of taking advantageof as much information as possible, and adjusting our portfolio every time the stockmoves. In other words, while increasing the frequency of information usage causes theexpected utility to increase, increases the trading frequency also increases the numberof transactions and their costs, causing the expected utility to decrease.

1.3 Summary of results

The main goal of this paper is to understand the trade-off between these two opposingforces by determining the trading frequency which maximizes the expected utility ofthe terminal wealth. We call this the optimal trading frequency and denote it by N∗.Evidently, N∗ will also vary depending on the size of the transaction costs, which inour case is measured by their proportionality constant λ to the trade size; the relationbetween N∗ and λ is also studied in this paper. For illustration purposes, we choose aone-year time horizon, and let N be the number of trades, so that N also denotes thefrequency, in trades per year, and N∗ is also the optimal number of trades per year.We take the point of view of an individual “day trader”, who is not likely to trademore frequently than once a day, i.e. likely to have a trading frequency that is boundedabove by 250. We will see that typical parameters for day trades, this frequency isusually far higher than N∗, and we will see that our results can be reinterpreted forthe case of fixed (non-proportional) transaction costs once per trading day, yieldingrecommendations for such day traders.

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We adapt the solution of the portfolio optimization problem proposed by Batalovaet al. (2006), using their time-forward algorithm with HARA utility function, andincorporate fixed positive transaction costs whether trades are buys or sells.

Specifically, in this article, we show the following. At any time i , we allow allo-cations of our wealth w into arbitrary quantities of SV stock Si and risk-free assetBi = eri . Let si = (s1, s2, . . . , si ) be the sequence of observed stock prices up to timei , and let U (i, si , w) be the approximate expected HARA terminal utility (at time N ),given the observations si up to time i , with wealth at time i equal to w.

1. We show how to take advantage of the fact that the HARA utility structure canbe preserved in the utility function: if for a fixed risk aversion parameter p, weassume U (N , sN , w) = w p/p, then the expected utility of the terminal wealth attime i, U (i, si , wi ) can be written as the product of the HARA functionw p

i /p anda time-dependent function � the observed stock prices up to time i only, whichwe then exploit to express � and an optimal portfolio allocation at time i . This isexplained in full detail in Sect. 3.2.

2. Our mathematical analysis also yields a forward-time algorithm. We describe thisin Sect. 3.3. Let ξi be the quantity of stock in our portfolio at time i ; this ξi isallowed to depend only on the initial wealthw0 and the observations si up to timei . By a self-financing condition, the sequence ξ determines the entire portfoliostrategy. An approximation from Batalova et al. (2006) is used to produce theforward-time algorithm for ξi = wi erκ∗, where r is the rate of interest and wi isthe current wealth at time i before the trade takes place. Specifically, κ∗ uniquelysolves the algebraic equation

n∑

k=1

(κ∗βk(si )+ 1)p−1βk(si ) = 0,

where βk(si ) := Si+1,k − (1 + λsign (ξi−1 − ξi ))si er , where n is the number ofsimulation particles, and Si+1,k is the k-th particle of the simulated future stockprices given the observed si . We simulate the future stock prices Si+1,k by usingthe stochastic volatility particle filter to initialize the volatility distribution at timei . This is explained in Sect. 2.2. We state the above method in formulas for com-puting κ∗ in Sect. 3.3.

3. In Sect. 4, we give a detailed algorithm based on this method. We run the algorithmon sets of simulated data s for many different trading frequencies N and severalvalues of the transaction cost rate λ. We identify the optimal trading frequency N∗for each λ, and find that it increases as λ decreases. This is given in Figs. 1, 2 andTable 2 in Sect. 5. We also provide recommendations for day traders who incurfixed transaction costs.

1.4 Further directions: trading versus observation frequency

This article’s scope is deliberately restricted to the results summarized above. A num-ber of other questions emerge from our work, which we do not attempt to solve,

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Fig. 1 Expected utility of WN

Fig. 2 Expected utility of WN

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because they entail significant new lines of research which we hope to be the subjectof future articles.

Our article assumes that trading and observations occur at the same frequency.This is a simplifying assumption: a more realistic approach would have trading occurat a lower frequency, and possibly a much lower frequency for small portfolios forwhom high-frequency transactions are clearly suboptimal. Stated in a different way,our conclusions of how often to trade, given in Sect. 5, ignore the fact that observationscan occur much more frequently than trading; by including more information fromthe observations at a high frequency, one should be able to achieve a higher expectedutility, even if trading occurs at a low frequency. A naive way to deal with this issueis to simply posit that trading can occur only every mth observation, and to repeat theanalysis of optimal trading frequency based now on the parameter m rather than theobservation frequency which could then be considered as fixed. This technique doesnot require any other change to our methodology.

Another approach, which can be combined with the above, would be to ignore anytrading for which the amplitude of the change in portfolio value is too close to theactual transaction cost to be deemed worthwhile. This would have the advantage oftaking into account periods of high volatility, when frequent trading may be desirableto capture rapid changes in the market. In both suggestions above, one might formu-late the problem mathematically by searching for values of the parameters (m, or thethreshold level below which trading is not worth it) which yield optimal values thatremain approximately constant at those parameter values.

However, these “fixes” do not provide an optimal way of dealing with the issueof trading frequency against observation frequency. The true optimization problemshould be framed as an optimal exercise problem: the portfolio manager needs todecide when the best time is to trade, given the constraint of having to pay for eachstock transaction, and given high-frequency data. This problem can be phrased in dis-crete or in continuous time, and both problems require the introduction of non-trivialtools from the theory of optimal stopping and its free-boundary interpretation, whichwe do not discuss here.

Other directions of research based on our exploratory work include model vali-dation and parameter estimation questions for our model, based on real data. Theparameter values used in Sect. 5 are typical of stock and index behavior in the currenteconomy. More information on parameter estimation for mean-reverting stochasticvolatility models can be found in the book of Fouque et al. (2000), which contains evi-dence for the validity of the mean-reverting Ornstein-Uhlenbeck process, as a modelof volatility for prices processes such as the S&P 500. This process, which we adopt inthis article, has become a common choice for such modeling, because it is the simplestmean-reverting diffusion process. The methodology in our article can be applied toa number of other popular stochastic volatility models without any difficulties, how-ever. In all cases the question of model validation and estimation is far from beingcompletely resolved. We also mention an interesting new direction, in Chronopoulouand Viens (2010), to validate or invalidate the use of long memory in volatility mod-eling.

We wish to thank an anonymous referee for a number of comments that helped usgreatly improve the presentation from an earlier version of this paper.

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2 Model and framework

2.1 Stochastic volatility model

Let P be an “objective market” probability measure, under which the stock priceprocess S, the risk-free asset process B, and a stochastic volatility driving process Yhave the following stochastic dynamics: for all t ≥ 0

⎧⎨

⎩

d Bt = r Bt dt, B0 = 1d St = αSt dt + σ(Yt )St dWt

dYt = μ(ν − Yt )dt + √μd Zt

where r is the deterministic constant short rate of interest, α is the deterministic con-stant mean rate of return of the stock price S, and σ is a deterministic function of thestochastic process Yt . As is typically done in SV models (see Fouque et al. 2000 orBatalova et al. 2006 for instance), we assume σ(x) = exp(x), and our Y above is themean-reverting Ornstein-Uhlenbeck process with a large mean-reversion rate param-eter μ (Y is often known as a fast mean-reverting process); the positive constant ν isthe mean level around which the process Y tends to revert. Note that our entire studycan be repeated with any number of distributions for the diffusion process Y , such asdYt = μ(Yt )dt + θ(Yt )d Zt , where μ and θ satisfy typical Lipshitz and boundednessconditions; we make no further comments on such extensions. In our study W andZ are Brownian motions which may be correlated in order to account for complexleverage effects: we denote ρw,z ∈ (−1, 1) their correlation coefficient.

2.2 Stochastic volatility filtering method

We refer to the articles of Florescu and Viens (2008) and Del Moral et al. (2001) fordetails of the interacting particle algorithm, including convergence results.

The main task of the stochastic volatility particle filter is to find the distribution ofthe volatility process Y when the discrete time observations of stock prices are given,and to do this dynamically in time, as the observations become available. This theo-retical problem, which is to estimate the probability distribution of the volatility giveninformation of the stock prices up to time i , as stated for instance in the introductionin (1), can be rewritten here specifically for the driving process Y as

psi (dy) := P[Yi ∈ dy|S0 = s0, S1 = s1, . . . , Si = si ], (2)

where s0, s1, . . . , si are the observed stock prices up to time i . This conditional time-dependent probability law, which we call the stochastic volatility filter given discreteobservations, cannot be computed explicitly, and its numerical approximation is non-trivial. it is unlikely to know the exact distribution of Y .

To estimate psi (dy), we adopt the particle method inspired by Del Moral et al.

(2001), introduced in Viens (2002), and explained in detail in Florescu and Viens(2008). We now briefly explain this procedure.

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It uses n particles(Yi,k

)nk=1 which evolve in discrete time. At time 0, we initialize

the Y0,k in order to approximate our best unconditional “guess” for the distributionof Y0. This “guess” can be seeded systematically by starting the filtering procedurein the distant past, long before trading occurs, which will yield an initial empiricaldistribution of the system

(Y0,k

)nk=0 which is close to the stationary measure of Y . We

do not give further details on this point, assuming only that(Y0,k

)nk=0 are chosen.

At any given time i −1, the past stock prices up to this time are given: S0 = s0,

S1 = s1, . . . , Si−1 = si−1. Assume by induction that(Yi−1,k

)nk=1 have been com-

puted. Then, as soon as the i th observation Si = si becomes available, we implementa “mutation” step by running an m-sub-time-step Euler scheme to give, for eachk = 1, . . . , n, a simulated value of the pair of particles (Si,k, Yi,k), started at time i −1at

(si−1,Yi−1,k

).

Then using the function ψ(x) := 1 − |x |,−1 < x < 1, we let

ψn(x) := 3√

nψ(x 3√

n), (3)

and measure how close each of our simulated particles Si,k is to the actual observedvalue si by calculating for each k

Fi,k := ψn(Si,k − si ). (4)

If the simulated particle Si,k is close to the observed value si , then Fi,k will be verylarge. If Si,k is not close to si , then Fi,k will be close to zero. This Fi,k is the “weight”or “fitness” of Si,k . We normalize all particle fitnesses by computing Ci = ∑n

k=1 Fi,k .

Then pi,k := Fi,kCi

represents the approximate probability Si,k is a good approxima-

tion to si , and we transfer this likelihood concept to the Y particles as follows. Werearrange the particles Yi,k according to these probabilities, by picking n new parti-cles {Yi,k : k = 1, . . . , n} independently of each other according to the distributionP[Y = Yi,k] = pi,k : k = 1, . . . , n. This is the “selection” step: it thus results in nparticles (Yi,k) with the estimated probabilities pi,k ; their empirical distribution is theapproximate stochastic volatility particle filter at time i .

2.3 Self-financing portfolio strategies with proportional transaction costs

A portfolio strategy is a pair

{(ξi , ηi ), i = 0, . . . , N }, (5)

which is an adapted stochastic process such that (ξi Si , ηi ) in P-square-integrable forall i = 0, . . . , N . The ξi and ηi represent the number of units of stock S and thenumber of units of risk-free asset B held at time i , respectively.

When we sell or buy the risky assets such as stocks, we pay transaction costs. Hav-ing proportional transaction costs in discrete time means that cost that are proportionalto the dollar amount of stock that is traded at each specific time. Algebraically, this cost

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is the absolute value of risky asset that exchanges hands at time i , times the transactioncost rate λ for trading (λ ∈ R+): thus the proportional transaction cost at time i is

λ|ξi − ξi−1|Si , (6)

where |ξi − ξi−1| is the number of units of the traded stock at time i . In real markets,the transaction cost rates for purchasing are different from the ones for selling. Butwe use a common transaction cost rates for both purchasing and selling cases for thesake of simplicity.

In order to manage our portfolio, we consider the simple situation where the initialdollar amount w0 is determined, and thereafter allocation changes are financed onlyby stock movements and accrued interest; in other words we assume our portfoliois self-financing with no consumption: the wealth right before the transaction occursequals the one right after the transaction occurs. We also assume allocations at time iare solely determined by observed stock values at times 0, 1, 2, . . . , i .

The initial wealth W0 = w0 is given. The wealth at time i is:

Wi = ξi Si + ηi Bi − λ|ξi − ξi−1|Si .

For t ∈ [i, i + 1], Wt = ξi St + ηi Bt − λ|ξi − ξi−1|Si . Therefore, elementary algebrayields ηi = (Wi − ξi Si + λ|ξi − ξi−1|Si )e−ri , that is to say, we can eliminate the useof the risk-free account allocation variable by keeping track of the wealth. We callW(i)− the wealth right before the transaction occurs; calculated using the old portfolioallocation, this is

W(i)− = ξi−1Si + ηi−1 Bi .

For our portfolio strategy (ξ, η) to be self-financing, this W(i)− has to agree withthe value Wi of the portfolio under the new allocation at time i . Therefore, theself-financing condition for all i = 1, . . . , N − 1 reads as

Wi = ξi−1Si + (Wi−1 − ξi−1Si−1 + λ|ξi−1 − ξi−2|Si−1)er . (7)

We notice that for any t ∈ [i, i + 1), i.e. before the next transaction,

Wt = ξi−1St + (Wi−1 − ξi−1Si−1 + λ|ξi−1 − ξi−2|Si−1)er(t−i). (8)

3 Theoretical results

3.1 Problem formulation

The main computational goal of this study is to find a predictable self-financing port-folio to maximize the expected utility of terminal wealth for a given initial wealthwhen proportional transaction costs are incurred. For a portfolio to be admissi-ble, for each time i , ξi has to depend only on the initial wealth w0 and the past

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observed stock values s1, s2, . . . , si . We should find an admissible self-financingportfolio ξ∗ = (ξ∗

1 , . . . , ξ∗N ) that attains the following supremum over the set of

all admissible self-financing portfolios

U (0) = U (0, s0, w0) := supξ

EP[u(W ξ

N )|S0 = s0,W0 = w0

], (9)

where W ξ is the wealth process following strategy ξ , and u is a nondecreasing concaveutility function. We use u(x) = x p/p, for some positive constant p < 1, which iscalled the power utility function for a risk-averse investor or HARA utility function.Our study can also handle u(x) = log x , the log-utility function, but this is left tothe reader to check. Recall that η is determined by the value of ξ from the equationηi = (Wi − ξi Si + λ|ξi − ξi−1|Si )e−ri , and we use the notation si := {s0, . . . , si }.Then we can write ξi = ξi (si , w0).

3.2 Quantitative analysis

3.2.1 Setup

We immerse the portfolio optimization problem (9) in the following time-dependentproblem:

U (i, si , wi ) := supξ

EP[u(W ξ

N )|Si = si ,Wi = wi

]. (10)

This could be solved by an iteration of HJB equations backwards in time as in Viens(2002), but this algorithm is far too cumbersome to be implemented: its state spaceincludes the trajectorial variable si , whose dimension increases unwieldily in time.

To overcome this problem, we adopt the HARA utility function, and a Monte-Car-lo method, to estimate the problem (10) via a time-forward recursion, as in Batalovaet al. (2006). To state things in a more transparent way than was done in Batalova et al.(2006), the first idea is to imitate the standard structure of HJB equations under HARAutility, and more specifically to seek a solution which can be written, at time i , as afunction of time and past information only, multiplied by the HARA utility functionw

pi /p wherewi is the wealth at that time; this helps to obtain a forward-time algorithm.

The second idea is to replace the expectation in (10) by a Monte-Carlo average; thisis needed because the SV model under unobserved volatility in the interval (i, i + 1)does not allow for an exact resolution to the HJB equations in that interval.

We call the resulting approximation U , i.e. we use the fact established in theMonte-Carlo method of Viens (2002) that, for i = 1, . . . , N , and k = 1, . . . , n

U (i, si , wi ) = maxξ∈R

1

n

n∑

k=1

U (i + 1, (si , Si+1,k), W ξi+1,k), (11)

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where the simulated value Si+1,k is computed using an Euler approximation as in theSV particle filter of Sect. 2.2. We set, in agreement with Eq. (8),

W ξi+1,k = ξi Si+1,k + (wi − ξi si )e

r + λ|ξi − ξi−1|si er . (12)

We notice that the maximum in the definition of U in (11) is over constant reals only,since this iteration is over a single time interval, during which stock allocation changesare not allowed.

3.2.2 Determination of an optimal strategy

In order to guarantee that the computation of U in (11) allows a forward time recursion,which includes an algorithm for a forward-time determination of an optimal portfolio(ξ∗

i )ni=1, we incorporate the typical HARA structure into the expression for U at each

integer time, via the following analysis. We make the following proposition.

Proposition 3.1 Let U be defined by (11). For all i = 1, . . . , N − 1, we can expressU as

U (i + 1, si+1, wi ) = wpi

p�(i + 1, si+1). (13)

where � (i, ·) is a function depending only on i, si , and the simulated values usedin (12).

The remainder of this section establishes this proposition. It is instructive to learnhow to express the corresponding function �: such an analysis yields the method forcomputing ξ∗, as we now show [below we drop the ∗ in ξ to lighten the notation].

When i = N − 1, since U (N , sN , w) = w p/p, the proposition is obviously truewith � = 1. Now writing

U (i + 1, si+1, wi ) = wpi

p�(i + 1, si+1), (14)

as in the proposition, assume that an optimal strategy ξ has been determined up to timei , with ξ j depending only on s j for all j ≤ i . And let bi−1 = λξi−1si er

W ξi+1,k

. This expression

depends only on i, si , and the simulated values used in (12). First we investigate thecase where ξi − ξi−1 ≥ 0. As we substitute all these into Eq. (12), we have

W ξi+1,k = ξi (Si+1,k − (1 − λ)si e

r )+ wi er − λξi−1si e

r . (15)

Now we set the following notation:

β+k (si ) := Si+1,k − (1 − λ) si e

r . (16)

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Then finally we have

W ξi+1,k = 1

(1 + bi−1)(ξi β

+k (si )+ wi e

r ). (17)

Note that by the self-financing principle (see (7)), wi is determined by the values si

only on the wealth of the portfolio at the end of the interval [i − 1, i). Then by theexpression (14), we have


1

n

n∑

k=1

(ξ β+

k (si )+ wi er)p

p(1 + bi−1)p�(i + 1, si , Si+1,k). (18)

By the definition of bi−1, we can rewrite �(i+1,si ,Si+1,k )

(1+bi−1)p := �(i, si ) where we see that

the function � (i, ·) depends only si and the simulated values used in (12).We now have


1

n

n∑

k=1

(ξ β+

k (si )+ wi er)p

p�(i, si ). (19)

To evaluate the maximum and its argmax, it is enough to find the zeros of the deriva-tive of the above function with respect to ξ . In other words the optimal ξi solves thefollowing equation

n∑

k=1

(ξi β

+k (si )+ wi e

r)p−1

β+k (si )�(i, si ) = 0. (20)

To simplify the notation let κi = ξiwi er . Then κi solves

n∑

k=1

(κi β

+k (si )+ 1

)p−1β+

k (si )�(i, si ) = 0. (21)

Calculating the derivative of the above Eq. (21) with respect to κi , we see that since0 < p < 1,

n∑

k=1

(p − 1)(κi β

+k (si )+ 1

)p−2(β+

k (si ))2�(i, si ) < 0. (22)

This negativity relies on the fact that 1 + bi−1 > 0. This expression can be calculated

and equals the quantityξi

(Si+1,k−(1−λ)si er

)+wi er

W ξi+1,k

. We know that Si+1,k is chosen to be

very close to si er at the selection step in the particle filtering algorithm and λ is lessthan one. There is a very small chance, depending on the Monte-Carlo samples, that

123


this expression would be negative. Our algorithm uses an approximate maximizationprocedure that is not sensitive to such events. This proves that any extremum κi is themaximum in Eq. (19); we denote any such value by κ∗

i [in theory it is possible to haveseveral global maxima, but this occurs with probability 0 w.r.t. the distribution of theMonte-Carlo randomness]. So the maximum in the expression in (19) is attained atξi := κ∗

i wi er . Substitute this ξi into Eq. (19), we get an explicit expression for thefunction in the proposition above:

U (i, si , wi ) = wpi

p

1

n

n∑

k=1

er p(κ∗

i β+k (si )+ 1

)p�(i, si ) (23)

=: wpi

p�(i, si ). (24)

Similarly, if it turns out that ξi − ξi−1 < 0, then

W ξi+1,k = 1

(1 − bi−1)(ξi β

−k (si )+ wi e

r ), (25)

where

β−k (si ) := Si+1,k − (1 + λ) si e

r . (26)

Then the rest of the proof is the same as the proof of the case ξi − ξi−1 ≥ 0 exceptsimply replacing β+

k (si ) by β−k (si ).

It is theoretically possible to solve the equation for κ∗i corresponding to the case

ξi − ξi−1 > 0 (increase in stock allocation), and find that the optimal ξi is actuallylower than ξi−1; and similarly if one chooses the equation for the other case. Thisoccurs with probability zero w.r.t. the Monte-Carlo distribution; because of roundingerrors, we have found that the occurrence is extremely low but non-zero; this can betaken care of by rejecting any selection with this feature, and repeating the selectionstep with a fresh Monte-Carlo sample. We include this procedure in the Algorithm inthe next section.

3.3 Forward-time iteration

Now we use the same method in the article of the Batalova et al. (2006) to make ouralgorithm forward in time and tractable. This is done by using a further approximation,assuming that the quantity �(i, si ) := �(i + 1, si+1, Si+1,k) in the Eq. (21) does notdepend on k. See Batalova et al. (2006) for an explanation of what this approximationentails. We then see that if ξi − ξi−1 ≥ 0, then κ∗ = κ∗+ is the (unique) solution ofthe equation

∑nk=1(κ

∗+β+k (si ) + 1)p−1β+

k (si ) = 0. Similarly, if ξi − ξi−1 < 0, thenκ∗ = κ∗− is the (unique) solution of the equation

∑nk=1(κ

∗−β−k (si )+1)p−1β−

k (si ) = 0.Thus we see that these two equations can be computed forward in time thanks to the

123


values β+k (si ) and β−

k (si ), defined in (16) and (26). We summarize these considerationshere.

• [Formulas for computing κ∗i ]

• Let U be as given (11). The maximum in this expected utility for i = 1, . . . , N −1,is attained at the approximate value ξi = ξ∗

i (si ) := wi erκ∗i , where κ∗

i is the uniquesolution to the following algebraic equation.

(i) If ξi − ξi−1 ≥ 0, κ∗i solves equation

n∑

k=1

(κ∗i β

+k (si )+ 1)p−1β+

k (si ) = 0, (27)

where

β+k (si ) := Si+1,k − (1 − λ)si e

r . (28)

(ii) If ξi − ξi−1 < 0, κ∗i uniquely solves the equation

n∑

k=1

(κ∗i β

−k (si )+ 1)p−1β−

k (si ) = 0, (29)

where

β−k (si ) := Si+1,k − (1 + λ)si e

r . (30)

Remark 3.2 We notice that if the transaction cost rate λ is zero, in other words, if weassume that there is no the proportional transaction costs, then our results in Proposi-tion 3.1 and formulas for computing κ∗

i are exactly the same as the ones in (Batalovaet al. 2006, Sect. 4). We state these briefly with our notation here, to highlight thesimilarity: they show that the approximation of the expected utility at time i + 1 is

U (i + 1, si+1, w) = w p

pK (i + 1, si+1) (31)

for some function K which does not depend on w, and the optimal portfolio is givenby ξ∗ = werκ∗ where κ∗ is the unique solution of

n∑

k=1

βk(si )(κ∗ (

Si+1,k − si er)

+ 1)p−1

. (32)

4 Algorithm

For the practitioners’ benefit, we restate all the above considerations in the form ofa complete algorithm. It is similar to that which was used (but not fully stated) in

123


Batalova et al. (2006); in our case, the maximization step calculates κ differently thantheirs, depending on the sign of ξi − ξi−1: if we need to buy (resp. sell) stocks at timei , we use Eq. (27) (resp. Eq. (29)) to calculate κ .

Let us briefly summarize the algorithm below. To provide an approximate solutionto our portfolio optimization problem (9), based on past observed prices up to timei , using the Euler and Bayesian Monte-Carlo methods yielding stochastic volatilityparticles Yi,kand their corresponding probabilities ps

i,k , we simulate the future stock

prices Si+1,k one unit of time into the future, using the SV dynamics on [i, i + 1].Then with these, we calculate κ∗

i and the optimal portfolio ξ∗i = wi erκ∗

i . In addition,we calculate a Monte-Carlo version of the initial expected utility U (0) based on theevolution of the optimal portfolio ξ∗ for a number of different simulated scenarios s;this step is not part of the optimization scheme, but allows us to estimate U (0), whichwill be crucial to our analysis of the optimal trading rate in the next section.

Now we present our forward-in-time algorithm in more detail; to lighten the nota-tion, we omit writing the functional dependence of all quantities on the fixed sequenceof observations s.

1. Initialization : Let the k-th initial stock price in the Monte-Carlo step be S0,k = s0.Let Y0,k = y0, where Y0,k is the k-th particle of the filter ps

0(·) = 1n

∑nk=1 δY0,k (·).

Let the initial wealth W0,k = w0 for all k = 1, . . . , n. Let U (N , w) = w p/p.Decide on the number of steps in the Euler scheme m and the number of particlesin the stochastic volatility filter n. We choose n such that the particle filtering errororder n−1/2 (see Del Moral et al. 2001 and Viens 2002) is sufficiently small, andsimilarly for m and the Monte-Carlo error order m−1/2. Let h = �t = ti − ti−1.

2. Calculation of the stochastic volatility particle filter : it has two steps, a mutationstep and a selection step. For i = 1, . . . , N − 1.

(i) (Mutation Step)For k = 1, . . . , n, we start with (si−1,Yti−1,k)

For 0 ≤ j ≤ m − 1, calculate

Sti−1, j+1 = Sti−1, j + α Sti−1, jh

m+ exp(Yti−1, j )Sti−1, j

√h

mZi−1, j

Yti−1, j+1 = Yti−1, j + μ(ν − Yti−1, j )h

m+ √

μ

√h

mZ ′

i−1, j , (33)

where Zi−1, j and Z ′i−1, j are i.i.d. standard Normal random variables. This

is the Euler scheme. Then let Sti = Sti−1,m and Yti ,k = Yti−1,m and repeatthis procedure n times to have n particles {Sti ,k, Yti ,k}n

k=1.(ii) (Selection Step)

For k = 1, . . . , n, we start with (Yti ,k) and set

Ci =n∑

k=1

ψn(Sti ,k − si ), (34)

123


where ψn is the function given in (3). Then calculate :

�ni =

{1

Ci

∑nk=1 ψn(Sti ,k − si )δYti ,k

if Ci > 0

δ0 otherwise(35)

Then sample n IID particles from the law �ni (in other words, sample n

independent times from the discrete distribution with atoms Yti ,k and cor-responding weights pi,k := ψn(Sti ,k − si )/Ci ). The resulting IID sampleis {Yti ,k}n

k=1 and together with its corresponding probabilities pi,k , this isthe stochastic volatility particle filter at time i .

3. Calculation of optimal portfolio strategy ξi : Let the initial optimal strategy ξ∗0 = 0,

which signifies that before the portfolio allocation starts, all money is in the risk-free account. Let the initial wealth be w0, and specify the transaction cost rate λ.For each i = 1, 2, . . . , N , we assume that wi−1 and ξi−1 have been determined,and we perform the following.

(i) Simulate Si+1,k by the Euler scheme with time step hm for the pair (S,Y )

starting from (si ,Yi,k).(ii) (Maximization step) Set β+

k = Si+1,k − (1 − λ)si er and β−k = Si+1,k −

(1 + λ)si er . Find the unique solution κ+i of the equation

n∑

k=1

(κ+i β

+k + 1)p−1β+

k = 0 (36)

and find the unique solution κ−i of the equation

n∑

k=1

(κ−i β

−k + 1)p−1β−

k = 0. (37)

Now using wi in the Eq. (7) in Sect. 2.3, we calculate wi :

wi = ξi−1si + (wi−1 − ξi−1si−1 + λ |ξi−1 − ξi−2| si−1) er ; (38)

this expression contains only previous data and si , it is readily calculated attime i . In the formula for computing κ∗

i in Sect. 3.3, we know ξi = κiwi er

and we have κ+i , κ

−i and wi . With these, we calculate

ξ+i = κ+

i wi er (39)

and

ξ−i = κ−

i wi er . (40)

Now we choose the optimal strategy ξ∗i depending on the sign of (ξ+

i −ξi−1)

and the sign of (ξ−i − ξi−1) by the following procedure.

123


Table 1 U (0) for λ = 0.1

N U (0) N U (0) N U (0) N U (0) N U (0)

2 245.85 22 247.39 42 131.95 62 72.05 82 33.99

4 247.62 24 234.76 44 93.71 64 52.69 84 35.60

6 257.05 26 238.81 46 119.28 66 57.34 86 32.05

8 253.99 28 205.41 48 90.74 68 55.62 88 34.46

10 260.23 30 178.01 50 82.66 70 39.76 90 33.63

12 258.67 32 213.55 52 103.45 72 52.09 92 32.09

14 258.43 34 210.21 54 87.13 74 48.73 94 34.20

16 256.78 36 196.76 56 74.71 76 48.24 96 31.85

18 253.26 38 132.58 58 59.49 78 39.04 98 36.06

20 248.21 40 122.38 60 71.73 80 35.84 100 31.58

(a) If ξ+i − ξi−1 ≥ 0, then the optimal strategy is ξ∗

i = ξ+i .

(b) If ξ−i − ξi−1 ≤ 0, then the optimal strategy is ξ∗

i = ξ−i .

(c) If ξ+i − ξi−1 < 0 and ξ−

i − ξi−1 > 0, then repeat (i) and (ii) in Step 3with a fresh Monte-Carlo simulation to compute β+

k and β−k .

Since the occurrence of case (c) is very rare, a second iteration of (i) and(ii) in step 3 will, in practice, never end up again in case (c), and we are leftwith a choice between (a) and (b) in this second iteration.

4. Calculation of U (0) : Now we calculate the initial maximal expected future utili-ties U (0, s0, w0). In Proposition 3.1, we saw that a good approximation for U (0)is the average of the terminal wealth utilities i.e.

1

n

n∑

k=1

(wN ,k)p

p, (41)

where n is the number of scenarios. First, we simulate a large number of scenarioss by using the algorithm in Step 1 and Step 2. Then we use the algorithm in Step 3to calculate optimal strategies for each scenario s. With these optimal strategies,we calculate the terminal wealth utilities for each scenario, and then compute theaverage of these utilities as in (41).

5 Optimization and recommendations

We run our algorithms above with the following typical parameter choices: m =50, n =5,000, p = 0.4, r = 0.02, the number of scenarios s = 300, the initial wealthw0 =$100,000, α = 0.05, μ = 5, σ (Y0) = 0.25 for a variety of N ’s and λ’s. Thetime horizon is 1 year.

The results are summarized in Figs. 1 and 2, Tables 1, 2, and 3. The vertical axis inFigs. 1 and 2 is U (0), the expected utility of the terminal wealth, and the horizontal

123


Table 2 N∗λ N∗ U (0) avg(WN ) N Return (%) N∗∗

0.1 10 260.96 111324 10 11.32 20

0.05 11 256.94 107084 12 7.08 24

0.03 11 255.18 105261 12 5.26 28

0.02 14 254.28 104338 14 4.34 30

0.01 16 253.17 103200 22 3.20 40

0.008 32 253.62 103663 24 3.66 54

0.005 32 253.74 103784 43 3.78 79

0.004 62 254.09 104135 62 4.14 122

0.003 116 254.66 104725 118 4.73 162

0.002 204 256.65 106778 234 6.78 358

axis is the trading frequency N . First, we notice that for each transaction cost rate, λ,each graph has optimal transaction numbers N∗, which maximizes the U (0) in Figs. 1and 2. More specifically, looking at Table 3, which corresponds to the case λ = 0.008,we see that U (0) is increasing from N = 2 to about N = 32, and decreases from thenon. This means that if the transaction cost rate is 0.8%, the optimal trading frequencyis about 32 trades per year, since this frequency yields the highest utility from theinvestment in this case. Second, the tail of each graph eventually becomes zero as Nincreases, as seen in Fig. 1 for all but the very smallest λ. In Table 1, which correspondsto 10% transaction cost rate, an unreasonably high λ, the expected future utility is seento get to 0 much faster than in the other cases, and the trader can evidently not makeany profit at all unless she trades very infrequently.

The optimal trading frequency N∗, estimated for each λ as the point where thecorresponding graph in Figs. 1 or 2 is maximal, is seen to increases as λ decreases, aneffect that is to be expected, and which we have quantified in Table 2: this table showsthe optimal trading frequency N∗ as a function of λ, which increases quickly as thetransaction cost rate decreases to 0.

The notation avg(WN ) in the tables is the average of the terminal wealth overthe number of scenarios. The corresponding return for each frequency, is calculatedover one year as (Terminal Wealth-Principal)/Principal, and is reported in Table 3 forλ = 0.008, as well as in Table 2 for all λ and their corresponding optimal rates. Wenotice the very important fact that there can be a wide range of N ’s which has approx-imately the same return as the return under the optimal N∗ frequency. For example,if λ = 0.008 then for 10 ≤ N ≤ 36, the returns in this range are around 3.5 ∼ 3.6%which is almost same as the return of optimal N∗ which is 3.66%. There is only aslight difference between the return for N∗ and for the midpoint N of the stated range.This can be interpreted as a kind of robustness result for our portfolio: the trader maychoose a frequency which is roughly of the same magnitude as N∗, and still hope toget an optimal return. Our tables also allow one to determine which frequencies willcause the portfolio to be less profitable on average than the risk free account: sincer is 2%, those N which have less than $102,000 as their terminal wealth are to beavoided. In the case of λ = 0.008, we see that this corresponds to N > N∗∗ = 64.

123


Tabl

e3

U(0)

forλ

=0.

008

NU(0)

avg(

WN)

Ret

urn

(%)

NU(0)

avg(

WN)

Ret

urn

(%)

NU(0)

avg(

WN)

Ret

urn

(%)

NU(0)

avg(

WN)

Ret

urn

(%)

225

2.87

1028

94.9

62.

8928

253.

2610

3293.1

33.

2954

252.

4710

2489.2

52.

4880

249.

6199

610.

13−0.3

9

425

2.29

1023

06.3

82.

3130

253.

3610

3393.5

23.

3956

252.

3010

2311.8

72.

3182

249.

8299

824.

66−0.1

8

625

2.95

1029

73.8

12.

9732

253.

6210

3663.7

93.

6658

251.

9610

1971.6

71.

9784

249.

8499

840.

05−0.1

6

825

2.49

1025

10.9

82.

5134

253.

4910

3527.2

43.

5360

252.

2710

2289.1

32.

2986

249.

0899

086.

43−0.9

1

1025

3.48

1035

11.3

53.

5136

253.

5010

3541.1

93.

5462

252.

2610

2278.8

62.

2888

249.

2199

208.

66−0.7

9

1225

3.47

1035

08.1

93.

5138

252.

9410

2970.1

92.

9764

251.

6110

1618.7

81.

6290

249.

2099

201.

32−0.8

0

1425

3.50

1035

31.7

93.

5340

252.

7910

2810.3

62.

9766

251.

6810

1689.2

71.

6992

248.

5498

543.

61−1.4

6

1625

3.45

1034

90.6

73.

4942

253.

0210

3051.5

13.

0568

251 .

3810

1387.0

71.

3994

248.

8198

816.

69−1.1

8

1825

3.53

1035

71.4

53.

5744

252.

7610

2778.2

42.

7870

250.

7510

0755.6

30.

7696

247.

8397

848.

95−2.1

5

2025

3.49

1035

24.6

13.

5246

252.

8310

2854.0

22.

8572

250.

9810

0987.7

00.

9998

248.

5898

590.

53−1.4

1

2225

3.51

1035

46.5

53.

5548

252.

5410

2561.4

52.

5674

250.

6310

0637.2

20.

6410

024

3.57

9369

1.04

−6.3

1

2425

3.44

1034

79.9

23.

4850

252.

3210

2340.3

72.

3476

250.

9710

0975.5

50.

98

2625

3.50

1035

33.4

93.

5352

252.

7510

2773.6

32.

7778

250.

7510

0754.0

10.

75

123


Lastly, let us consider the interesting question of the online day trader with fixed$10 transaction cost per trade, and N = 252 trades per year (one trade per day). Inorder to get that 252 corresponds to an optimal trading frequency, our algorithm yieldsthat it should come from a proportional transaction cost rate of λ∗ = 0.0018. We canrephrase this information into fixed transaction costs. If each trade is approximatelyfor P := $10

λ∗ $5,500, then the proportional transaction cost corresponding to everysuch trade will be λ∗( 10

λ∗ ) = $10. Therefore, we conclude that a frequency of 252 pertrades year (once per day), with $10 transaction costs per trade, can be optimal, solong as each trade is sufficiently large, in this case at least $5,500.

References

Batalova, N., Maroussov, V., Viens, F.: Selection of an optimal portfolio with stochastic volatility anddiscrete observations. Trans Wessex Inst Model Simul 43, 317–380 (2006)

Chronopoulou, A., Viens, F.: Estimation and pricing under long-memory stochastic volatility. Submitted toAnn Finance, p. 27 (2010)

Davis, M.H.A., Norman, A.R.: Portfolio selection with transaction costs. Math Oper Res 15(4),676–713 (1990)

Del Moral, P., Jacod, J., Protter, P.: The Monte-Carlo method for filtering with discrete time observa-tions. Probab Theory Relat Fields 120, 346–368 (2001)

Dempster, M.A.H., Evstigneev, I.V., Taksar, M.I.: Asset pricing and hedging in financial markets with trans-action costs: an approach based on the Von Neumann–Gale model. Ann Finance 2, 327–355 (2006)

Desai, R, Lele, T., Viens, F.: A Monte-Carlo method for portfolio optimization under partially observedstochastic volatility. In: IEEE International Conference on Computational Intelligence for FinancialEngineering, 2003. Proceedings, pp. 257–263 (2003)

Florescu, I., Viens, F.: Stochastic volatility: option pricing using a multinomial recombining tree. ApplMath Finance 15(2), 151–181 (2008)

Fornari, F., Mele, A.: Stochastic Volatility in Financial Markets: Crossing the Bridge to Continuous Time,Kluwer: Amsterdam (2000)

Fouque, J.-P., Papanicolaou, G., Sircar, R.: Derivatives in Financial Markets with Stochastic Volatility,Cambridge: Cambridge university Press (2000)

Framstad, N.C., Øksendal, B., Sulem, A.: Optimal consumption and portfolio in a jump diffusion market withproportional transaction costs. Arbitrage and control problems in finance. J Math Econom 35(2), 233–257 (2001)

Guasoni, P., Rásonyi, M., Schachermayer, W.: The fundamental theorem of asset pricing for continuousprocesses under small transaction costs. Ann Finance 6(2), 157–191 (2010)

Pham, H., Quenez, M.-C.: Optimal portfolio in partially observed stochastic volatility models. Ann ApplProbab 11(1), 210–238 (2001)

Shepard, N. (ed.): Stochastic Volatility, Selected Readings, New York: Oxford University Press (2005)Shreve, S.E., Soner, H.M.: Optimal investment and consumption with transaction costs. Ann Appl Pro-

bab 4(3), 609–692 (1994)Viens, F.: Portfolio optimization under partially observed stochastic volatility. In: Wells, W. (ed.) COMCON

8, pp. 1–12. Optim Soft, Inc Pub Div (2002)

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