Stochastic Portfolio Theory: An Overview

Transcription

1 Stochastic Portfolio Theory: An Overview ROBERT FERNHOLZ INTECH One Palmer Square Princeton, NJ 8542, USA IOANNIS KARATZAS Departments of Mathematics and Statistics Columbia University New York, NY 27, USA November 24, 26 Abstract Stochastic Portfolio Theory is a flexible framework for analyzing portfolio behavior and equity market structure. This theory was introduced by E.R. Fernholz in the papers Journal of Mathematical Economics, 999; Finance & Stochastics, 2) and in the monograph Stochastic Portfolio Theory Springer 22). It was further developed in the papers Fernholz, Karatzas & Kardaras Finance & Stochastics, 25), Fernholz & Karatzas Annals of Finance, 25), Banner, Fernholz & Karatzas Annals of Applied Probability, 25), and Karatzas & Kardaras 26). It is a descriptive theory, as opposed to a normative one; it is consistent with observable characteristics of actual portfolios and markets; and it provides a theoretical tool which is useful for practical applications. As a theoretical tool, this framework offers fresh insights into questions of market structure and arbitrage, and can be used to construct portfolios with controlled behavior. As a practical tool, Stochastic Portfolio Theory has been applied to the analysis and optimization of portfolio performance and has been the basis of successful investment strategies for over a decade. To appear in Mathematical Modelling and Numerical Methods in Finance Alain Bensoussan and Qiang Zhang, Editors), Special Volume of the Handbook of Numerical Analysis.

2 Table of Contents I Basics 3 Markets and Portfolios 3. General Trading Strategies The Market Portfolio 6 3 Some Useful Properties 8 4 Portfolio Optimization II Diversity & Arbitrage 2 5 Diversity 2 6 Relative Arbitrage and Its Consequences 3 6. Strict Local Martingales On Beating the Market Proof of 6.2) Diversity leads to Arbitrage 7 8 Mirror Portfolios, Short-Horizon Arbitrage 9 8. A Seed Portfolio Relative Arbitrage on Arbitrary Time Horizons A Diverse Market Model 22 Hedging and Optimization without EMM 23. Completeness without an EMM Ramifications and Open Problems Utility Maximization in the Absence of EMM III Functionally Generated Portfolios 26 Portfolio generating functions 27. Sufficient Intrinsic Volatility leads to Arbitrage Rank, Leakage, and the Size Effect Proof of the Master Equation.2) IV Abstract Markets 34 2 Volatility-Stabilized Markets Bessel Processes

3 3 Ranked-Based Models Ranked Price Processes Some Asymptotics The Steady-State Capital Distribution Curve Stability of the Capital Distribution Some Concluding Remarks 45 5 Acknowledgements 45 6 References 46 Introduction Stochastic Portfolio Theory SPT), as we currently think of it, began in 995 with the manuscript On the diversity of equity markets, which eventually appeared as Fernholz 999) in the Journal of Mathematical Economics. Since then SPT has evolved into a flexible framework for analyzing portfolio behavior and equity market structure that has both theoretical and practical applications. As a theoretical methodology, this framework provides insight into questions of market behavior and arbitrage, and can be used to construct portfolios with controlled behavior under quite general conditions. As a practical tool, SPT has been applied to the analysis and optimization of portfolio performance and has been the basis of successful equity investment strategies for over a decade. SPT is a descriptive theory, which studies and attempts to explain observable phenomena that take place in equity markets. This orientation is quite different from that of the well-known Modern Portfolio Theory of Dynamic Asset Pricing DAP), in which market structure is analyzed under strong normative assumptions regarding the behavior of market participants. It has long been suggested that the distinction between descriptive and normative theories separates the natural sciences from the social sciences, and if this dichotomy is valid, then SPT resides with the natural sciences. SPT descends from the classical portfolio theory of Harry Markowitz 952), as is the rest of mathematical finance. At the same time, it represents a rather significant departure from some important aspects of the current theory of Dynamic Asset Pricing. DAP is a normative theory that grew out of the general equilibrium model of mathematical economics for financial markets, evolved through the capital asset pricing models, and is currently predicated on the absence of arbitrage and on the existence of equivalent martingale measures). SPT, by contrast, is applicable under a wide range of assumptions and conditions that may hold in actual equity markets. Unlike dynamic asset pricing, it is consistent with either equilibrium or disequilibrium, with either arbitrage or no-arbitrage, and remains valid regardless of the existence of equivalent martingale measures). This survey reviews the central ideas of SPT and presents examples of portfolios and markets with a wide variety of different properties. SPT is a fast-evolving field, so we also present a number of research problems that remain open, at least at the time of this writing. Proofs for some of the results are included here, but at other times simply a reference is given. The survey is divided into four parts. Part I, Basics, introduces the concepts of markets and portfolios, in particular, the market portfolio, that most important portfolio of them all. In this first part we also encounter the excess growth rate process, a quantity that pervades SPT. Part II, Diversity & Arbitrage, introduces market diversity and shows how diversity can lead to relative arbitrage in an equity market. Historically, these were among the first phenomena analyzed using SPT. Portfolio generating functions are versatile tools for constructing portfolios with particular properties, and these functions are discussed in Part III, Functionally Generated Portfolios. Here we also consider stocks identified by rank, as opposed to by name, and discuss implications regarding 2

4 the size effect. Roughly speaking, these first three parts of the survey outline the techniques that historically have comprised SPT; the fourth part looks toward the future. Part IV, Abstract Markets, is devoted to the area of much of the current research in SPT. Abstract markets are models of equity markets that exhibit certain characteristics of real stock markets, but for which the precise mathematical structure is known since we can define them as we wish!). Here we see volatility-stabilized markets that are not diverse but nevertheless allow arbitrage, and we also look at rank-based markets that have stability properties similar to those of real stock markets. Several problems regarding these abstract markets are proposed. Part I Basics SPT uses the logarithmic representation for stocks and portfolios rather than the arithmetic representation used in classical mathematical finance. In the logarithmic representation, the classical rate of return is replaced by the growth rate, sometimes referred to as the geometric rate of return or the logarithmic rate of return. The logarithmic and arithmetic representations are equivalent, but nevertheless the different perspectives bring to light distinct aspects of portfolio behavior. The use of the logarithmic representation in no way implies the use of a logarithmic utility function: indeed, SPT is not concerned with expected utility maximization at all. We introduce here the basic structures of SPT, stocks and portfolios, and discuss that most important portfolio of them all: the market portfolio. We show that the growth rate of a portfolio depends not only on the growth rates of the component stocks, but also on the excess growth rate, which is determined by the stocks variances and covariances. Finally, we consider a few optimization problems in the logarithmic setting. Most of the material in this part can be found in Fernholz 22). Markets and Portfolios We shall place ourselves in a model M for a financial market of the form dx i t) = X i t) b i t)dt+ dbt) = Bt)rt) dt, B) =, d ν= ) σ iν t) dw ν t), X i ) = x i >, i =,..., n, consisting of a money-market B ) and of n stocks, whose prices X ),, X n ) are driven by the d dimensional Brownian motion W ) = W ),, W d ) ) with d n. Contrary to a usual assumption imposed on such models, here it is not crucial that the filtration F = {Ft)} t T be the one generated by the Brownian motion itself. Thus, and until further notice, we shall take F to contain possibly strictly) this Brownian filtration F W = {F W t)} t T, where F W t) := σw s), s t). We shall assume that the interest-rate process r ), the vector-valued process b ) = b ),..., b n ) ) of rates of return, and the m d) matrix-valued process σ ) = σ iν ) ) of volatilities, i n, ν d are all F progressively measurable, and satisfy for every T, ) the integrability conditions T rt) dt + T b i t) +.) d σiν t) ) ) 2 dt <, a.s..2) This setting admits a rich class of continuous-path Itô processes, with very general distributions: no Markovian or Gaussian assumption is imposed. In fact, it is possible to extend the scope of the theory to general semimartingale settings; see Kardaras 23) for details. ν= 3

5 With the notation a ij t) := d σ iν t)σ jν t) = σt)σ t) ) ij = d dt log X i, log X j t).3) ν= for the non-negative definite matrix-valued variance/covariance process a ) = a ij ) ) of the i,j n stocks in the market, as well as we may use Itô s rule to solve.), in the form or equivalently: γ i t) := b i t) 2 a iit), i =,..., n,.4) d log X i t) = γ i t) dt + { t X i t) = x i exp γ i u) du + d σ iν t) dw ν t), i =,..., n,.5) ν= d ν= t } σ iν u) dw ν u), t <. We shall refer to the quantity of.4) as the growth rate of the i th stock, because of the a.s. relationship T ) lim log X i T ) γ i t) dt =, i =,, n..6) T This is valid when the individual stock variances a ii ) do not increase too quickly, e.g., if we have lim t t a ii t) log log t =, a.s.; then.6) follows from the law of the iterated logarithm and from the representation of local) martingales as time-changed Brownian motions.. Definition. A portfolio π ) = π ),..., π n ) ) is an F progressively measurable process, bounded uniformly in t, ω), with values in the set { π,..., π n ) R n π + + π n = }. A long-only portfolio π ) = π ),..., π n ) ) is a portfolio that takes values in the set n := { π,..., π n ) R n π,..., π n and π + + π n = }. For future reference, we shall introduce also the notation n + := { π,..., π n ) n π >,..., π n > }. Thus, a portfolio can sell one or more stocks short though certainly not all) but is never allowed to borrow from, or invest in, the money market. A long-only portfolio, of course, sells no stocks short at all. The interpretation is that π i t) represents the proportion of wealth V t) V w,π t) invested at time t in the i th stock, so the quantities h i t) = π i t)v w,π t), i =,, n.7) are the amounts invested at any given time t in the individual stocks. The wealth process V w,π ), that corresponds to a portfolio π ) and initial capital w >, satisfies the stochastic equation dv w,π t) V w,π t) = = b π t) dt + π i t) dx it) X i t) = π t) [ bt)dt + σt) dw t) ] d σ πν t) dw ν t), V w,π ) = w, ν= 4.8)

6 and b π t) := π i t)b i t), σ πν t) := π i t)σ iν t) for ν =,..., d..9) These quantities are, respectively, the rate-of-return and the volatility coëfficients that correspond to the portfolio π ). By analogy with.5) we can write the solution of the equation.9) as or equivalently d log V w,π t) = γ π t) dt + d σ πν t) dw ν t), V w,π ) = w,.) ν= { t V w,π t) = w exp γ π u) du + d ν= t } σ πν u) dw ν u), t <. Here is the growth rate, and γ π t) := π i t)γ i t) + γπt).) γπt) := n π i t)a ii t) 2 j= ) π i t)a ij t)π j t).2) the excess growth rate, of the portfolio π ). As we shall see in Lemma 3.3 below, for a long-only portfolio this excess growth rate is always non-negative and is strictly positive for such portfolios that do not concentrate their holdings in just one stock. Again, the terminology growth rate is justified by the a.s. property lim T ) log V w,π T ) γ π t) dt =,.3) T valid when all eigenvalues of the variance/covariance matrix process a ) of.3) are bounded: i.e., when ξ at)ξ = ξ σt)σ t)ξ K ξ 2, t [, ) and ξ R n.4) holds almost surely, for some constant K, ). We shall refer to.4) as the uniform boundedness condition on the volatility structure of M. Without further comment we shall write V π ) V,π ) for initial wealth w = $. Let us also note the following analogue of.), namely d log V π t) = γ πt) dt + π i t) d log X i t)..5).2 Definition. We shall use the reverse-order-statistics notation for the weights of a portfolio π ), ranked at time t in decreasing order, from largest down to smallest: max π it) =: π ) t) π 2) t)... π n ) t) π n) t) := min π it)..6) i n i n 5

7 . General Trading Strategies For completeness of exposition and for later use in this subsection, let us go briefly beyond portfolios and recall the notion of trading strategies: these are allowed to invest in or borrow from) the money market. Formally, they are F progressively measurable, R n valued processes h ) = h ), h n ) ) that satisfy the integrability condition T h i t) b i t) rt) + h 2 i t)a ii t)) dt <, for every T, ). The interpretation is that the real-valued random variable h i t) stands for the dollar amount invested by h ) at time t in the i th stock. If we denote by V w,h t) the wealth at time t corresponding to this strategy h ) and to an initial capital w >, then V w,h t) n h it) is the amount invested in the money market, and we have the dynamics dv w,h t) = or equivalently, V w,h t) Bt) V w,h t) ) h i t) rt) dt + h i t) b i t) dt + d ν= a.s. ) σ iν t) dw ν t), t h s) bs) ) ) = w + rs)i ds + σs) dw s), t T..7) Bs) Here I =,, ) is the n dimensional column vector with in all entries. Again without further comment, we shall write V h ) V,h ) for initial wealth w = $. As mentioned already, all quantities h i ), i n and V w,h t) h )I are allowed to take negative values. This possibility opens the door to the notorious doubling strategies of martingale theory e.g. [KS] 998), Chapter ). In order to rule these out, we shall confine ourselves here to trading strategies h ) that satisfy P V w,h t), t T ) =..8) Such strategies will be called admissible for the initial capital w > on the time horizon [,T ]; their collection will be denoted Hw; T ). We shall also find useful to look at the collection H + w; T ) Hw; T ) of strongly admissible strategies, with P V w,h t) >, t T ) =. Each portfolio π ) generates, via.7), a trading strategy h ) H + w) := T > H + w; T ); and we have V w,h ) V w,π ). It is not difficult to see from.8) that the trading strategy generated by a portfolio π ) is self-financing see, e.g., Duffie 992) for a discussion). 2 The Market Portfolio Suppose we normalize so that each stock has always just one share outstanding; then the stock price X i t) can be interpreted as the capitalization of the i th company at time t, and the quantities Xt) := X t) X n t) and µ i t) := X it), i =,..., n 2.) Xt) as the total capitalization of the market and the relative capitalizations of the individual companies, respectively. Clearly < µ i t) <, i =,..., n and n µ it) =, so we may think of the vector process µ ) = µ ),..., µ n ) ) as a portfolio that invests the proportion µi t) of current wealth in the i th asset at all times. The resulting wealth process V w,µ ) satisfies dv w,µ t) V w,µ t) = µ i t) dx it) X i t) 6 = dx i t) Xt) = dxt) Xt),

8 in accordance with 2.) and.8). In other words, V w,µ ) w Xt) ; 2.2) X) investing in the portfolio µ ) is tantamount to ownership of the entire market, in proportion of course to the initial investment. For this reason, we shall call µ ) of 2.) the market portfolio. By analogy with.) we have d log V w,µ t) = γ µ t) dt + d σ µν t) dw ν t), V w,µ ) = w, 2.3) ν= and comparison of this last equation 2.3) with.5) gives the dynamics of the market-weights d log µ i t) = γ i t) γ µ t) ) dt + d σiν t) σ µν t) ) dw ν t) 2.4) in 2.) for all stocks i =,..., n in the notation of.9),.); equivalently, ν= dµ i t) µ i t) = γ i t) γ µ t) + 2 τ µ ii t) ) dt + d σiν t) σ µν t) ) dw ν t). 2.5) ν= Here we introduce, for an arbitrary portfolio π ) and with e i denoting the i th unit vector in R n, the quantities τ π ijt) := for i, j n, and set a πi t) := d σiν t) σ πν t) ) σ jν t) σ πν t) ) 2.6) ν= = πt) e i ) at) πt) ej ) = aij t) a πi t) a πj t) + a ππ t) π j t)a ij t), a ππ t) := j= j= π i t)a ij t)π j t) = It is seen from.) that this quantity is the variance of the portfolio π ). d σπν t) ) ) We shall call the matrix-valued process τ π ) = τij π )) of 2.6) the variance/covariance i,j n process of individual stocks relative to the portfolio π ). It satisfies the elementary property The corresponding quantities d τ µ ij t) := ν= ν= τijt)π π j t) =, i =,, n. 2.8) j= σiν t) σ µν t) ) σ jν t) σ µν t) ) = d µ i, µ j t), i, j n 2.9) µ i t)µ j t)dt of 2.6) for the market portfolio π ) µ ), are the variances/covariances of the individual stocks relative to the entire market. For the second equality in 2.9), we have used the semimartingale decomposition of 2.5).) 7

9 3 Some Useful Properties In this section we collect together some useful properties of the relative variance/covariance process in 2.6), for ease of reference in future usage. For any given stock i and portfolio π ), the relative return process of the i th stock versus π ) is the process Ri π Xi t) t) := log V t)) w=xi w,π ), t <. 3.) 3. Lemma. For any portfolio π ), and for all i, j n and t [, ), we have, almost surely: τ π ijt) = d dt Rπ i, R π j t), in particular, τ π iit) = d dt Rπ i t), 3.2) and the matrix τ π t) = τij πt)) is a.s. nonnegative definite. Furthermore, if the variance/covariance matrix at) is positive definite, then the matrix τ π t) has rank n, and its i,j n null space is spanned by the vector πt), almost surely. Proof: Comparing.5) with.) we get the analogue dr π i t) = γ i t) γ π t) ) dt + d σiν t) σ πν t) ) dw ν t), of 2.4), from which the first two claims follow. Now suppose that at) is positive definite. For any x R n \ {} and with η := n x i, we compute from 2.4): ν= x τ π t)x = x at)x 2ηx at)πt) + η 2 π t)at)πt). If n x i =, then x τ π t)x = x at)x >. If on the other hand η := n x i, we consider the vector y := x/η that satisfies n y i =, and observe that η 2 x τ π t)x is equal to y τ π t)y = y at)y 2y at)πt) + π t)at)πt) = y πt) ) at) y πt) ), thus zero if and only if y = πt), or equivalently x = ηπt). 3.2 Lemma. For any two portfolios π ), ρ ), we have d log V π ) t) V ρ = γ t) πt) dt + ) Xi t) π i t) d log V ρ. 3.3) t) In particular, we get the dynamics V π ) t) d log V µ = γ t) πt) dt + π i t) d log µ i t) 3.4) = γπt) γµt) ) dt + πi t) µ i t) ) d log µ i t) for the relative return of an arbitrary portfolio π ) with respect to the market. Proof: The equation 3.3) follows from.5), and the first equality in 3.4) is the special case of 3.3) with ρ ) µ ). The second equality in 3.4) follows upon observing from 2.4), that µ i t) d log µ i t) = µ i t)γ i t) γ µ t)) dt = γµt) dt. 8

10 3.3 Lemma. For any two portfolios π ), ρ ) we have the numéraire-invariance property γπt) = n n π i t)τ ρ ii 2 t) π i t)π j t)τ ρ ij ). t) 3.5) j= In particular, recalling 2.8), we obtain the representation γ πt) = 2 π i t)τiit) π 3.6) for the excess growth rate, as a weighted average of the individual stocks variances τii π ) relative to π ), as in 2.6). Whereas, from 3.6), 3.2) and Definition., we get for any long-only portfolio π ) the property: γπt). 3.7) Proof: From 2.6) we get as well as π i t)τ ρ ii t) = n π i t)a ii t) 2 π i t)a ρi t) + a ρρ t) π i t)τ ρ ij t)π jt) = π i t)a ij t)π j t) 2 π i t)a ρi t) + a ρρ t), j= j= and 3.5) follows from.2). For the market portfolio, equation 3.6) becomes γ µt) = 2 µ i t)τ µ ii t) ; 3.8) the summation on the right-hand-side is the average, according to the market weights of individual stocks, of these stocks variances relative to the market. Thus, 3.8) gives an interpretation of the excess growth rate of the market portfolio, as a measure of the market s intrinsic volatility. 3.4 Remark. Note that 3.4), in conjunction with 2.4), 2.5) and the numéraire-invariance property 3.5), implies that for any portfolio π ) we have the relative return formula V π ) t) π i t) d log V µ = t) µ i t) dµ it) π i t)π j t)τ µ ij 2 t) dt. 3.9) j= 3.5 Lemma. Assume that the variance/covariance process a ) of.3) satisfies the following strong non-degeneracy condition: all its eigenvalues are bounded away from zero. To wit, assume there exists a constant ε, ) such that ξ at)ξ = ξ σt)σ t)ξ ε ξ 2, t [, ) and ξ R n 3.) holds almost surely. Then for every portfolio π ) and t <, we have in the notation of.6) the inequalities ε π i t) ) 2 τ π ii t), i =,, n, 3.) almost surely. If the portfolio π ) is long-only, we have also ε π) t) ) γ 2 πt). 3.2) 9

11 Proof: With e i denoting the i th unit vector in R n, we have τiit) π = πt) e i ) at)πt) e i ) ε πt) e i 2 = ε πi t) ) 2 ) + πj 2 t) from 2.6) and 3.), and 3.) follows. Back into 3.6), and with π i t) i =,, n, this lower estimate gives γ π t) ε 2 π i t) πi t) ) 2 ) + πj 2 t) j i = ε n π i t) π i t) ) 2 n + πj 2 t) π j t) )) 2 = ε 2 j= π i t) π i t) ) 2 ε π) t) ). 3.6 Lemma. Assume that the uniform boundedness condition.4) holds; then for every long-only portfolio π ), and for t <, we have in the notation of.6) the a.s. inequalities τ π iit) K π i t) ) 2 π i t) ), i =,, n 3.3) γ πt) 2K π ) t) ). 3.4) Proof: As in the previous proof, we get τiit) π K πi t) ) 2 ) + πj 2 t) j i K πi t) ) 2 ) + π j t) = K π i t))2 π i t)) as claimed in 3.3), and bringing this estimate into 3.6) leads to γ π t) K = K π i t) π i t) ) j i π ) t) π ) t) ) + K π) t) ) + 4 Portfolio Optimization π k) t) π k) t) )) k=2 j i ) π k) t) = 2K π ) t) ). We can already formulate some interesting optimization problems. k=2 Problem : Quadratic criterion, linear constraint Markowitz, 952). Minimize the portfolio variance a ππ t) = n n j= π it)a ij t)π j t), among all portfolios π ) with rate-of-return b π t) = n π it)b i t) b at least equal to a given constant. Problem 2: Quadratic criterion, quadratic constraint. Minimize the portfolio variance a ππ t) = j= π i t)a ij t)π j t)

12 among all portfolios π ) with growth-rate at least equal to a given constant γ : π i t) γ i t) + ) 2 a iit) γ + 2 j= π i t)a ij t)π j t). Problem 3: Maximize the probability of reaching a given ceiling c before reaching a given floor f, with < f < w < c <. More specifically, maximize P[ T c < T f ], with T ξ := inf{t V w,π t) = ξ} for ξ, ). In the case of constant coëfficients γ i and a ij, the solution to this problem comes in the following simple form: one looks at the mean-variance, or signal-to-noise, ratio γ π a ππ = n π iγ i + 2 a ii) n n j= π ia ij π j 2, and finds a portfolio π that maximizes it Pestien & Sudderth, 985). Problem 4: Minimize the expected time ET c ) until a given ceiling c w, ) is reached. Again with constant coëfficients, it turns out that it is enough to try and maximize the drift in the equation for log V w,π ), namely γ π = n π i γi + 2 a ) ii n n 2 j= π ia ij π j, the portfolio growth-rate Heath, Orey, Pestien & Sudderth, 987). Problem 5: Maximize the probability P[T c < T T f ] of reaching a given ceiling c before reaching a given floor f with < f < w < c <, by a given deadline T, ). Always with constant coëfficients, suppose there is a portfolio ˆπ = ˆπ,..., ˆπ n ) that maximizes both the signal-to-noise ratio and the variance, γ π a ππ = n π iγ i + 2 a ii) n n j= π ia ij π j 2 and a ππ = π i a ij π j, j= respectively, over all π,..., π n with n π i =. Then this portfolio ˆπ is optimal for the above criterion Sudderth & Weerasinghe, 989). This is a big assumption; it is satisfied, for instance, under the very stringent, and unnatural...) condition that, for some G >, we have γ i + 2 a ii = G, for all i =,..., n. As far as the authors are aware, nobody seems to have solved this problem when such simultaneous maximization is not possible.

13 Part II Diversity & Arbitrage Roughly speaking, a market is diverse if it avoids concentration of all its capital into a single stock, and the diversity of a market is a measure of how uniformly the capital is spread among the stocks. These concepts were introduced in Fernholz 999), and it was shown in Fernholz 22), Section 3.3, and in Fernholz, Karatzas & Kardaras 25) that market diversity gives rise to arbitrage. Unlike classical mathematical finance, SPT is not averse to the existence of arbitrage in markets, but rather studies market characteristics that imply the existence of arbitrage. Moreover, the existence of arbitrage does not preclude the development of option pricing theory or certain types of utility maximization. These and other related ideas are presented in this part of the survey. 5 Diversity The notion of diversity for a financial market corresponds to the intuitive and descriptive) idea, that no single company can ever be allowed to dominate the entire market in terms of relative capitalization. To make this notion precise, let us say that the model M of.),.2) is diverse on the time horizon [, T ], if there exists a number δ, ) such that the quantities of 2.) satisfy almost surely µ ) t) < δ, t T 5.) in the order-statistics notation of.6). In a similar vein, we say that M is weakly diverse on the time horizon [, T ], if for some δ, ) we have T µ ) t)dt < δ 5.2) T almost surely. We say that M is uniformly weakly diverse on [T, ), if there exists a δ, ) such that 5.2) holds a.s. for every T [T, ). It follows directly from 3.4) of Lemma 3.6 that, under the condition.4), the model M of.),.2) is diverse respectively, weakly diverse) on the time interval [, T ], if there exists a number ζ > such that γ µt) ζ, t T respectively, T ) γ T µt) dt ζ holds almost surely. And 3.2) of Lemma 3.5 shows that, under the condition 3.), the conditions of 5.3) are satisfied if diversity respectively, weak diversity) holds on the time interval [, T ]. As we shall see in section 9, diversity can be ensured by a strongly negative rate of growth for the largest stock, resulting in a sufficiently strong repelling drift e.g., a log-pole-type singularity) away from an appropriate boundary, as well as non-negative growth rates for all the other stocks. If all the stocks in M have the same growth rate γ i ) γ ), i n) and.4) holds, then we have almost surely: T lim γ T µt) dt =. 5.4) In particular, such an equal-growth-rate market M cannot be diverse, even weakly, for long time horizons, provided that 3.) is also satisfied. Here is a quick argument for these claims: recall that for X ) = X ) + + X n ) we have lim T ) log XT ) γ µ t) dt =, lim T 2 T ) log X i T ) γt) dt = T 5.3)

14 a.s., from.3),.6) and γ i ) γ ) for all i n. But then we have also lim T ) log X ) T ) γt) dt =, a.s. T for the biggest stock X ) ) := max i n X i ), and note the inequalities X ) ) X ) nx ) ). Therefore, lim log X) T ) log XT ) ) =, thus lim T T γµ t) γt) ) dt =, T almost surely. But γ µ t) = n µ it)γt)+γ µt) = γt)+γ µt) because of the assumption of equal growth rates, and 5.4) follows. If 3.) also holds, then 3.2) and 5.4) imply lim T µ) t) ) dt = T almost surely, so weak diversity fails on long time horizons: once in a while a single stock dominates the entire market, then recedes; sooner or later another stock takes its place as absolutely dominant leader; and so on. 5. Remark. Coherence: We say that the market model M of.),.2) is coherent, if the relative capitalizations of 2.) satisfy lim T log µ it ) = almost surely, for each i =,, n 5.5) i.e., if none of the stocks declines too rapidly). Under the condition.4), it can be shown that coherence is equivalent to each of the following two conditions: lim lim T γi t) γ µ t) ) dt = a.s., for each i =,, n; 5.6) T T γi t) γ j t) ) dt = a.s., for each pair i, j n. 5.7) T 5.2 Remark. If all the stocks in the market M have constant though not necessarily the same) growth rates, and if.4), 3.) hold, then M cannot be diverse, even weakly, over long time horizons. 6 Relative Arbitrage and Its Consequences The notion of arbitrage is of paramount importance in mathematical finance. We present in this section an allied notion, that of relative arbitrage, and explore some of its consequences. In later sections we shall encounter specific, descriptive conditions on market structure, that lead to this form of arbitrage. 6. Definition. Given any two portfolios π ), ρ ) with the same initial capital V π ) = V ρ ) =, we shall say that π ) represents an arbitrage opportunity relative to ρ ) over the fixed, finite time horizon [, T ], if PV π T ) V ρ T )) = and PV π T ) > V ρ T )) >. 6.) We shall say that π ) represents a superior long-term growth opportunity, if L π,ρ V π ) := lim inf T log T ) V ρ > holds a.s. 6.2) T ) 3

15 6.2 Remark. The definition of relative arbitrage has historically included the condition that there exist a constant q = q π,ρ,t > such that P V π t) qv ρ t), t T ) =. 6.3) However, if one can find a portfolio π ) that satisfies the domination properties 6.) relative to some other portfolio ρ ), then there exists another portfolio π ) that satisfies both 6.3) and 6.) relative to the same ρ ). The construction involves a strategy of investing a portion w, ) of the initial capital $ in π, and the remaining proportion w in ρ ). This observation is due to C. Kardaras 26). 6. Strict Local Martingales Let us place ourselves now, and for the remainder of this section, within the market model M of.) and under the conditions.2),.4). We shall assume further that there exists a market price of risk or relative risk ) θ : [, ) Ω R d ; namely, an F progressively measurable process with σt)θt) = bt) rt)i, t T and T θt) 2 dt < 6.4) valid almost surely, for each T, ). If the volatility matrix σ ) has full rank, namely n, we can take, for instance, θt) = σ t) σt)σ t) ) [ bt) rt)i ] in 6.4).) In terms of this process θ ), we can define the exponential local martingale and supermartingale { Zt) := exp t θ s) dw s) 2 a martingale, if and only if EZT )) = ), and the shifted Brownian Motion Ŵ t) := W t) + t t } θs) 2 ds, t T 6.5) θs) ds, t T. 6.6) 6.3 Proposition. A Strict Local Martingale: Under the assumptions of this subsection, suppose there exists a time horizon T, ) such that relative arbitrage exists on [, T ]. Then the process Z ) of 6.5) is a strict local martingale: we have EZT )) <. Proof: Assume, by way of contradiction, that EZT )) =. Then from the Girsanov Theorem e.g. [KS], section 3.5) the recipe Q T A) := E[ZT ) A ] defines an probability measure on FT ), equivalent to P, under which the process Ŵ ) of 6.6) is Brownian motion. of Under Q T, the discounted stock prices X i )/B ), i =,, n are positive martingales, because d X i t)/bt) ) = X i t)/bt) ) d ν= σ iν t) dŵνt) and of the uniform boundedness.4), which is assumed to hold. As usual, we express this by saying that Q T is then an equivalent martingale measure EMM) for the model. Similarly, for any portfolio π ), we get then from 6.6) and.8): d V π t)/bt) ) = V π t)/bt) ) π t)σt) dŵ t), V π ) = ; 6.7) and because of condition.4), the discounted wealth process V π )/B ) is a positive martingale under Q T. Thus, the difference ) := V π ) V ρ ))/B ) ) is a martingale under Q T, for any other portfolio ρ ) with V ρ ) =. But this gives E Q T ) T ) = ) ) =, a conclusion inconsistent with 6.) which mandates Q T T ) = and QT T ) > >. 4

16 Now let us consider the deflated stock-price and wealth processes X i t) := Zt) Bt) X it), i =,, n and Vw,h t) := Zt) Bt) Vw,h t) 6.8) for t T, for arbitrary admissible trading strategy h ) and initial capital w >. processes satisfy, respectively, the dynamics d X i t) = X i t) These d σiν t) θ ν t) ) dw ν t), Xi ) = x i, 6.9) ν= Zt)h d V w,h ) t) t) = σt) Bt) V w,h t)θ t) dw t), Vw,h ) = w 6.) in conjunction with.),.7) and 6.5). In particular, these processes are non-negative local martingales and supermartingales) under P. In other words, the ratio Z )/B ) continues to play its usual rôle as deflator of prices in such a market, even when Z ) is just a local martingale.) 6.4 Remark. Strict Local Martingales Galore: From 6.9), 6.) it can be shown that, in the setting of Proposition 6.3, the deflated stock-price processes Xi ) of 6.8) are all strict local martingales and strict) supermartingales: E Xi T ) ) < x i holds for every i =,, n. 6.) Actually, we shall prove in subsection 6.3 a considerably stronger result: In the setting of Proposition 6.3, for any given portfolio ρ ) the process V w,ρ ) = Z )V w,ρ )/B ) of 6.8) is a strict local martingale and a strict) supermartingale, namely E V w,ρ T ) ) < w. 6.2) 6.5 Proposition. Non-Existence of Equivalent Martingale Measure: In the context of Proposition 6.3, no Equivalent Martingale Measure can exist for the model M of.), if the filtration is generated by the driving Brownian Motion W ) : F = F W. Proof: If F = F W, and there exists some probability measure Q which is equivalent to P on FT ), then the martingale representation property of the Brownian filtration gives dq/dp) Ft) = Zt), t T for some process Z ) of the form 6.5) and some progressively measurable θ ) with T θt) 2 dt < a.s. Then Itô s rule leads to the extension d X ) i t) d d = b i t) rt) σ iν t)θ ν t) dt + σiν t) θ ν t) ) dw ν t) X i t) ν= of 6.9) for the deflated stock-prices of 6.8). But if all the X i )/B ) s are Q martingales that is, if Q is an equivalent martingale measure), then the X i ) s are all P martingales, and this leads to the first property σ ) θ ) = b ) r )I in 6.4). We repeat now the argument of Proposition 6.3 and arrive at a contradiction with 6.), the existence of relative arbitrage on [, T ]. 6.2 On Beating the Market Let us introduce now the decreasing function ft) := ) Zt) X) E Bt) Xt), t T. 6.3) If relative arbitrage exists on the time-horizon [, T ], then we know f) = > ft ) > from Remark 6.4. ν= 5

17 6.6 Remark. With Brownian filtration F = F W, n = d and an invertible volatility matrix σ ), consider the maximal relative return RT ) := sup { r > h ) H; T ) s.t. V h T )/V µ T ) ) r, a.s. } 6.4) in excess of the market, that can be obtained by trading strategies over the interval [, T ]. It can be shown easily that this quantity is computed in terms of the function of 6.3), as RT ) = /ft ). 6.7 Remark. The shortest time to beat the market by a given amount: Let us place ourselves again under the assumptions of Remark 6.6, but now assume that relative arbitrage exists on [, T ] for every T, ); see section 8 for elaboration. For a given exceedance level r >, consider the shortest length of time Tr) := inf { T > h ) H; T ) s.t. V h T )/V µ T ) ) r, a.s. } 6.5) required to guarantee a return of at least r times the market. It can be shown that this quantity is given by the number Tr) = inf { T > ft ) /r }, the inverse of the decreasing function f ) of 6.3) evaluated at /r. Details can be found in [FK] 25). Question: Can the counterparts of 6.4), 6.5) be computed when one is not allowed to use general strategies h ) H; T ), but rather long-only portfolios π )? 6.3 Proof of 6.2) First, some notation: we shall take w = for simplicity, then employ the usual notation V ρ ) V,ρ ), V ρ ) V,ρ ) for the wealth and the deflated wealth of our given portfolio ρ ). With h ) := V ρ )ρ ) and θ ρ ) := σ )ρ ) θ ), we notice that the equation 6.) takes the form d V ρ t) = V ρ t) θ ρ t) ) dw t), or equivalently { t V ρ t) = exp On the other hand, introducing the process we obtain W t) := W t) t V ρ t) ) = exp { θ ρ s) ) dw s) 2 t } θ ρ s) 2 ds. 6.6) t θ ρ s)ds = Ŵ t) σ s)ρs) ds, t T, 6.7) t θ ρ s) ) d W s) 2 t } θ ρ s) 2 ds. 6.8) We shall argue 6.2) by contradiction: let us assume that it fails, namely, that V ρ ) is a martingale. From the Girsanov theorem, the process W ) of 6.7) is then a Brownian motion under the equivalent probability measure P T A) := E ) V ρ T ) A on FT ). On the other hand, Itô s rule gives V π ) t) V π ) t) d d V ρ = t) V ρ πi t) ρ i t) ) σ iν t) d W ν t) 6.9) t) ν= for any portfolio π ), in conjunction with 6.7), 6.8) and 6.7). Then the uniform boundedness condition.4) implies that the ratio V π )/V ρ ) is a martingale under P T ; in particular, E P T V π T )/V ρ T ) ) =. Now consider any portfolio π ) that satisfies the conditions of 6.) on the time-horizon [, T ] ; such a portfolio exists, because we are operating in the setting of Proposition 6.3. This gives, in 6

18 particular, PT V π T ) V ρ T ) ) =. In conjunction with the equality E P T V π T )/V ρ T ) ) = just proved, we obtain P T V π T ) = V ρ T ) ) =, or equivalently: P V π T ) = V ρ T ) ) = for every portfolio π ) that satisfies 6.). But this contradicts the second condition P V π T ) > V ρ T ) ) > of 6.). 7 Diversity leads to Arbitrage We provide now examples which demonstrate the following principle: If the model M of.),.2) is weakly diverse over the time-interval [, T ], and if 3.) holds, then M contains arbitrage opportunities relative to the market portfolio, at least for sufficiently long time horizons T, ). The first such examples involve heavily the diversity-weighted portfolio µ p) ) = µ p) defined in terms of the market portfolio µ ) of 2.) by ),..., µp) n ) ) µ p) µi t) ) p i t) := n j= µj t) ) p, i =,..., n 7.) for some arbitrary but fixed p, ). Compared to µ ), the portfolio µ p) ) in 7.) decreases the proportions) held in the largest stocks) and increases those placed in the smallest stocks), while preserving the relative rankings of all stocks; see 7.8) below. The actual performance of this portfolio relative to the S&P 5 index over a 33-year period is discussed in detail in Fernholz 22), Chapter 7. We show below that if the model M is weakly diverse on a finite time horizon [, T ], then the value process V µp) ) of the diversity-weighted portfolio in 7.) satisfies V µp) T ) > V µ T ) n /p e εδt/2) p 7.2) almost surely. In particular, ) P V µp) T ) > V µ T ) =, provided that T 2 log n, 7.3) pεδ and µ p) ) is an arbitrage opportunity relative to the market µ ), in the sense of 6.3)-6.). The significance of such a result for practical long-term portfolio management cannot be overstated. What conditions on the coëfficients b ), σ ) of M are sufficient for guaranteeing diversity, as in 5.), over the time horizon [, T ]? For simplicity, assume that 3.) and.4) both hold. Then certainly M cannot be diverse if b ) r ),..., b n ) r ) are bounded uniformly in t, ω), or even if they satisfy a condition of the Novikov type { E exp 2 T bt) rt)i 2 dt} ) <. 7.4) The reason is that, under all these conditions 3.),.4) and 7.4), the process θ ) = σ ) σ )σ ) ) [ b ) r )I ] satisfies the requirements 6.4), and the resulting exponential local martingale Z ) of 6.5) is a true martingale contradicting Proposition 6.3, at least for sufficiently large T >. Proof of 7.3): Let us start by introducing the function n ) /p, G p x) := x p i x n +, 7.5) 7

19 which we shall interpret as a measure of diversity ; see below. An application of Itô s rule to the process {G p µt)), t < } leads after some computation, and in conjunction with 3.9) and the numéraire-invariance property 3.5), to the expression log V µp) T ) V µ T ) ) = log ) Gp µt )) T + p) γ µ t) dt 7.6) G p µ)) p) for the wealth V µp) ) of the diversity-weighted portfolio µ p) ) of 7.); see also Section below, particularly.2) and its proof in subsection.3. One big advantage of the expression 7.6) is that it is free of stochastic integrals, and thus lends itself to pathwise almost sure) comparisons. For the function of 7.5), we have the simple bounds = µ i t) µi t) ) p = Gp µt)) ) p n p minimum diversity occurs when the entire market is concentrated in one stock, and maximum diversity when all stocks have the same capitalization), so that the function of 7.5) satisfies ) Gp µt )) log p log n. 7.7) G p µ)) p This shows that V µp) )/V µ ) is bounded from below by the constant m p)/p, so 6.3) is satisfied for ρ ) µ ) and π ) µ p) ). On the other hand, we have already remarked that the biggest weight of the portfolio µ p) ) in 7.) does not exceed the largest market weight: µ p) µ) t) ) p )t) := max i n µp) i t) = n k= µk) t) ) p µ ) t). 7.8) The reverse inequality holds for the smallest weights: µ p) n) t) := min i n µ p) i t) µ n) t).) We have assumed that the market is weakly diverse over [, T ], namely, that there is some < δ < for which T µ) t) ) dt > δt holds almost surely. From 3.2) and 7.8), this implies T γ µ p) t) dt ε 2 T µ p) ) t) ) dt ε 2 T µ) t) ) dt > ε 2 δt a.s. In conjunction with 7.7), this leads to 7.2) and 7.3) via ) V µp) T ) εt log V µ > p) T ) 2 δ ) p log n. 7.9) If M is uniformly weakly diverse and strongly non-degenerate over an interval [T, ), then 7.9) implies that the market portfolio will lag rather significantly behind the diversity-weighted portfolio over long time horizons. To wit, that 6.2) will hold: L µp),µ = lim inf T log V µp) T ) / ) V µ T ) p)εδ/2 >, In Figure we see the cumulative changes in the diversity of the U.S. stock market over the period from 927 to 24, measured by G p ) with p = /2. The chart shows the cumulative changes in diversity due to capital gains and losses, rather than absolute diversity, which is affected by changes in market composition and corporate actions. Considering only capital gains and losses a.s. 8

20 % YEAR Figure : Cumulative change in market diversity, has the same effect as adjusting the divisor of an equity index. The values used in Figure have been normalized so that the average over the whole period is zero. We can observe from the chart that diversity appears to be mean-reverting over the long term, with intermediate trends of to 2 years. The extreme lows for diversity seem to accompany bubbles: the Great Depression, the nifty fifty era, and the irrational exuberance period. 7. Remark. Fernholz, 22): Under the conditions of this section, consider the portfolio with weights ) 2 µi t) π i t) = Gµt)) µ i t), i n, where Gx) := x 2 i 2 for x n. It can be shown that this portfolio leads to arbitrage relative to the market, over sufficiently long time horizons [, T ], namely with T 2n/εδ 2 ) log 2. In this case, we also have π i t) 3µ i t), for all t [, T ], a.s., so, with appropriate initial conditions, there is no risk that this π ) will hold more of a stock than the market holds. 8 Mirror Portfolios, Short-Horizon Arbitrage In the previous section we saw that in weakly diverse markets which satisfy the strict non-degeneracy condition 3.), one can construct explicitly simple portfolios that lead to arbitrages relative to the market over sufficiently long time horizons. The purpose of this section is to demonstrate that, under these same conditions, such arbitrages exist indeed over arbitrary time horizons, no matter how small. For any given portfolio π ) and real number q, define the q-mirror image of π ) with respect to the market portfolio, as π [q] ) := qπ ) + q)µ ). This is clearly a portfolio; and it is long-only if π ) itself is long-only and < q <. If q =, we call π [ ] ) = 2µ ) π ) the mirror image of π ) with respect to the market. 9

21 By analogy with 2.6), let us define the relative covariance of π ) with respect to the market, as τ π µµt) := πt) µt) ) at) πt) µt) ), t T. 8. Remark. Recall from 2.8) the fact τ µ t)µt), and establish the elementary properties τ π µµt) = π t)τ µ t)πt) = τ µ ππt) and τ µ π [q] π [q] t) = q 2 τ µ ππt). 8.2 Remark. The wealth of π [q] ) relative to the market, can be computed as ) V π[q] T ) V π ) T ) q q) T log V µ = q log T ) V µ + τ T ) 2 ππt) µ dt. Indeed, let us write the second equality in 3.4) with π ) replaced by π [q] ), and recall π [q] µ = qπ µ). From the resulting expression, let us subtract the second equality in 3.4), now multiplied by q ; the result is d log V π[q] t) dt V µ t) q log V π ) t) V µ = q )γ t) µt) + γ π t) qγµt) ). [q] But from the equalities of Remark 8. and Lemma 3.3, we obtain 2 γ π [q] t) qγ πt) ) = π [q] t) qπ i t) ) τ µ ii t) τ µ π [q] π t) + qτ ππt) µ [q] = q) The desired equality now follows. 8.3 Remark. Suppose that the portfolio π ) satisfies and µ i t)τ µ ii t) + qτ µ ππt) q 2 τ µ ππt) = q) 2γ µt) + qτ µ ππt) ). P V π T )/V µ T ) β ) = or P V π T )/V µ T ) /β ) = T ) P τ ππt) µ dt η = for some real numbers T >, η > and < β <. Then there exists another portfolio π ) with P V π T ) < V µ T ) ) =. To see this, suppose first that we have P V π T )/V µ T ) /β ) = ; then we can just take π ) π [q] ) with q > + 2/η) log/β), for then Remark 8.2 gives log V π[q] T ) V µ T ) ) q log /β ) + q ) 2 η <, a.s. If, on the other hand, P V π T )/V µ T ) β ) = holds, then similar reasoning shows that it suffices to take π ) π [q] ) with q, 2/η) log/β) ). 8. A Seed Portfolio Now let us consider π = e =,,, ) and the market portfolio µ ); we shall fix a real number q > in a moment, and define the portfolio πt) := π [q] t) = qe + q)µt), t < 8.) 2

22 which takes a long position in the first stock and a short position in the market. In particular, π t) = q + q)µ t) and π i t) = q)µ i t) for i = 2,, n. Then we have log V π ) ) T ) µ T ) qq ) T V µ = q log τ µ t)dt 8.2) T ) µ ) 2 from Remark 8.2. But taking β := µ ) we have µ T )/µ )) /β; and if the market is weakly diverse on [, T ] and satisfies the strict non-degeneracy condition 3.), we obtain from 3.) and the Cauchy-Schwarz inequality T T τ µ t)dt ε ) 2dt µ) > εδ 2 T =: η. 8.3) Recalling Remark 8.3, we see that the market portfolio represents then an arbitrage opportunity with respect to the portfolio π ) of 8.), provided that for any given T, ) we select q > qt ) := + 2/εδ 2 T ) log /µ ) ). 8.4) The portfolio π ) of 8.) can be used as a seed, to create long-only portfolios that outperform the market portfolio µ ), over any given time horizon T, ). The idea is to immerse π ) in a sea of market portfolio, swamping the short positions while retaining the essential portfolio characteristics. Crucial in these constructions is the a.s. comparison, a consequence of 8.2): ) q V π µ t) t) V µ t), t <. 8.5) µ ) 8.2 Relative Arbitrage on Arbitrary Time Horizons To implement this idea, consider a strategy h ) that, at time t =, invests q/µ )) q dollars in the market portfolio, goes one dollar short in the portfolio π ) of 8.), and makes no change thereafter. The number q > is chosen again as in 8.4). The wealth generated by this strategy, with initial capital z := q/µ )) q >, is V z,h t) = qv µ t) µ )) q V π t) V µ t) µ )) q q µ t)) q ) >, t <, 8.6) thanks to 8.5) and q > > µ t)) q. This process V z,h ) coincides with the wealth V z,η ) generated by a portfolio η ) with weights ) qµi t) η i t) = V z,h t) µ )) q V µ t) π i t)v π t), i =,, n 8.7) that satisfy n η it) =. Now we have π i t) = q )µ i t) < for i = 2,, n, so the quantities η 2 ),..., η n ) are strictly positive. To check that η ) is a long-only portfolio, we have to verify η t) ; but the dollar amount invested by η ) in the first stock at time t, namely dominates qµ t) µ )) q V µ t) [ q q )µ t) ] V π t) qµ t) µ )) q V µ t) [ q q )µ t) ] µ t) µ )) q V µ t), or equivalently V µ t)µ t) µ )) q q )µ t)) q + q [ µ t)) q ]) >, again thanks to 8.5) and q > > µ t)) q. Thus η ) is indeed a long-only portfolio. 2

23 On the other hand, η ) outperforms at t = T a market portfolio that starts with the same initial capital; this is because η ) is long in the market µ ) and short in the portfolio π ), which underperforms the market at t = T. Indeed, from Remark 8.3 we have V z,η T ) = q µ )) q V µ T ) V π T ) > zv µ T ) = V z,µ T ), a.s. Note, however, that as T, the initial capital zt ) = qt )/µ )) qt ) required to do all of this, increases without bound: It may take a huge amount of initial investment to realize the extra basis point s worth of relative arbitrage over a short time horizon confirming of course, if confirmation is needed, that time is money... 9 A Diverse Market Model The careful reader might have been wondering whether the theory we have developed so far may turn out to be vacuous. Do there exist market models of the form.),.2) that are diverse, at least weakly? This is of course a very legitimate question. Let us mention then, rather briefly, an example of such a market model M which is diverse over any given time horizon [,T ] with < T <, and indeed satisfies the conditions of subsection 4. as well. For the details of this construction we refer to [FKK] 25). With given δ /2, ), equal numbers of stocks and driving Brownian motions that is, d = n), constant volatility matrix σ that satisfies 3.), and non-negative numbers g,..., g n, we take a model d log X i t) = γ i t) dt + σ iν dw ν t), t T 9.) ν= in the form.5) for the vector X ) = X ),, X n ) ) of stock prices. With the usual notation Xt) = n j= X jt), its growth rates are specified as γ i t) := g i Q c i Xt)) M δ Qi Xt)) log ). 9.2) δ)xt)/x i t) In other words, γ i t) = g i if Xt) / Q i the i th stock does not have the largest capitalization); and γ i t) = M δ log δ)/µ i t) ), if Xt) Q i 9.3) the i th stock does have the largest capitalization). We are setting here { Q := x, ) } { n x max x j, Q n := x, ) } n x n > max x j, 2 j n j m { Q i := x, ) } n x i > max x j, x i max x j for i = 2,..., n. j i i+ j n With this specification 9.2), 9.3), all stocks but the largest behave like geometric Brownian motions with growth rates g i and variances a ii = n ν= σ2 iν ), whereas the log-price of the largest stock is subjected to a log-pole-type singularity in its drift, away from an appropriate right boundary. One can then show that the resulting system of stochastic differential equations has a unique, strong solution so the filtration F is now the one generated by the driving n dimensional Brownian motion), and that the diversity requirement 5.) is satisfied on any given time horizon. Such models can be modified appropriately, to create ones that are weakly diverse but not diverse; see [FK] 25) for details. Slightly more generally, in order to guarantee diversity it is enough to require min γ k)t) γ ) t), 2 k n min γ k)t) γ ) t) + ε 2 k n 2 M δ F Qt)), 22