產權市場短期套利的探討

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應用數學系

數學建模與科學計算研究所

碩 士 論 文

產權市場短期套利的探討

Some remarks on short-term relative arbitrage

in equity markets

研 究 生 : 吳國禎

指導教授 : 許元春 教授

產權市場短期套利的探討

Some remarks on short-term relative arbitrage in

equity markets

研 究 生 : 吳國禎 Student : Guo-Jhen Wu

指導教授 : 許元春 Advisor : Yuan-Chung Sheu

國 立 交 通 大 學

應用數學系數學建模與科學計算碩士班 碩 士 論 文

A Thesis

Submitted to Department of Applied Mathematics College of Science, Institute of Mathematical Modeling and Scientific Computing

National Chiao Tung University in Partial Fulfillment of the Requirements

for the Degree of Master

in

Applied Mathematics June 2011

Hsinchu, Taiwan, Republic of China

產權市場短期套利的探討

學生 : 吳國禎 指導教授 : 許元春

國立交通大學應用數學系數學建模與科學計算碩士班

摘 要

我們給予了一個在任意短期時間內，以及任意產權市場

會存在套利的充分條件。相較於 Banner 和 Fernholz 在 2008

年所提出的充分條件，我們給的條件更具一般性，然而，另

一方面，我們給的條件也比較難去驗證。儘管如此，我們給

出了一類的財務模型，它們可能不會滿足 Banner 和 Fernholz

所給出的充分條件，可是卻滿足我們這篇論文所給出的條

件。

Some Remarks on Short-term Relative Arbitrage in

Equity Markets

Student : Guo-Jhen Wu Advisor : Yuan-Chung Sheu

Institute of Mathematical and Scientific Computing

National Chiao Tung University

Hsinchu, Taiwan, R.O.C.

Abstract

We provide a sufficient condition for the existence of relative

arbitrage over arbitrarily short time horizon in equity markets.

Compared with the sufficient condition given in Banner &

Fernholz(2008), our condition is much general, but, on the other

hand, it is more difficult to check. However, we give a family of

abstract models which do not satisfy the criteria in Banner &

Fernholz (2008) but satisfy the criteria here.

誌

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謝

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首先我要感謝的是我的指導教授許元春老師，在他的不

斷的鼓勵之下，讓我能夠完成這篇論文。也感謝他這兩年來，

給了我很多有用的建議，不論是在課程的選擇上或是生涯的

規劃上，讓我能在疑惑中走出來。此外，也感謝他一直以來

對我的肯定與栽培，期望將來能有所作為，不要辜負老師的

期待。

其次，我要感謝我的同學們，感謝你們陪伴我一起修課，

一起討論課業，讓研究生活增添了許多樂趣。也感謝眾多陪

我踢球的球友們，讓我能夠有適當的運動，在作問題遇到瓶

頸的時候能有抒發的管道。

最後，我最想感謝的是我的家人，特別是我的爸爸媽媽，

感謝你們多年來的培育，教導我許多做人處世的道理，總是

默默的付出，全年無休的工作，只為讓我們受到好的教育，

成為一個有用的人，真的是辛苦你們了，我一定不會白費你

們這些年來的努力。

Contents

1

Introduction………. 1

2 Preliminaries………... 2

2.1 The Model……… 2

2.2 Investment Strategies and Portfolios……….. 3

2.3 Relative Arbitrage……… 5

3 Some Useful Properties…..……….. 6

3.1 Relative Return Process……….. 6

3.2 Portfolio Generating Functions.……….. 7

3.3 Concave Functions………..……….. 9

4 Main Results……… 11

5 Generalized Volatility-Stabilized Market Model 17

6 Some Related Results……….. 18

6.1 Diffusion Models……… 18

6.2 Arbitrage and Diversity……….. 19

1

Introduction

In a financial market, there are three major topics for study, that is arbitrage, pricing and hedging. Arbitrage, in some sense, is to make money without any cost. Simply speaking, if there are two objects in the market with the same price today, and assume we are very sure about that which one of them will be cheaper than the other some specific day in the future, then we can sell the will-be-cheaper one and buy another one today, also, buy the cheaper one and sell another in the future, in this way, we will earn the price difference between these two objects without any cost today. Obviously, many people are really interesting about finding arbitrages in equity markets. Therefore, we may ask that do there exist arbitrages in any equity market? If not, then under what kind of equity markets do arbitrages exist? If arbitrages really exist in some equity markets, then we want to know more about how long can we achieve these arbitrages? a month? or a year? and how can we make these arbitrages? that is, by what combination of stocks in equity markets?

Pricing is another topic about how to price a derivative financial product with some sort of function. For example, futures contracts, if I have a deal with you that I will give you 1000 ipad2s next month but you want to pay me right now for some reasons, then how much should I charge for those ipad2s? Or from your point of view, how much money should you pay for those ipad2s? Such questions are highly related to another topic — hedging. Since no matter much money I get from you today, I still have the risk to lose money next month. For a relative conservative person as me, we don’t want to take a chance on it. Hence, we would like to ask that whether there some products which are highly correlated to ipad2 so that we can buy an appropriate combination of them to reduce the possible-loss. In other words, use the money I have today to make enough money at least to cover the possible-loss in the future. Furthermore, we may like to know that what are the best combination among all of them?

There are many theories for the three topics above, the most well-known one is Dynamic Asset Pricing Theory. In this theory, the market structure is analyzed under strong norma-tive assumptions with respect to the behavior of market participants. This theory is based on a market model with existence of equivalent martingale measure(s) and the absence of arbitrage, so we can only use this theory to answer the topics about pricing and hedging. Black-Scholes Option Pricing Formula is one of the most famous results among Dynamic Asset Pricing Theory.

Another recently-developed theory is Stochastic Portfolio Theory. This is a theory con-structed on a rather general setting. It uses the class of semi-martingales to describe the stocks in the real markets, and by making some descriptive assumptions to study arbi-trage, pricing and hedging. The differences between Dynamic Asset Pricing Theory and Stochastic Portfolio Theory is that the assumptions made in Stochastic Portfolio Theory are usually some observable properties of the markets, while the assumptions in Dynamic Asset Pricing Theory are usually made for technical concerns. One of the most powerful tools of Stochastic Portfolio Theory is portfolio generating function, it is a tool developed by Fernholz, R.(1999). With this tool and another powerful tool in stochastic analysis — Itˆo’s formula, we can construct a class of bounded portfolios whose returns can be decom-posed into several components with proper characteristics. Also, we can use these portfolios to investigate the issue of arbitrage for time large enough, or even better, for arbitrarily time-horizon. Moreover, we can use such tools to understand that what kind of portfolios can achieve arbitrage. Recently, there are a few works which concerns about arbitrage, and the conditions from those works may be present in actual market and the corresponding

portfolio can be implemented in practical. This theory has been the basis of successful equity investment strategies for a decade.

The most significant concern of this paper is that under what conditions, we can have strong arbitrage over any time-horizon. In section 2, we proceed with an introduction to the standard equity market model that we use, and then we introduce the concepts of investment strategies, portfolios and arbitrage in an equity market. Also, we mention a few important quantities corresponding to portfolios which are related to existence of arbitrages. At the end of this section, we also give some examples which exist an arbitrage opportunity for time large enough. Then, we introduce a powerful tool for Stochastic Portfolio Theory— portfolio generating functions, in section 3. With portfolio generating function, we can derive the master formula, we also give a simple example to show how to use the master formula to achieve arbitrage. Moreover, we mention some properties of concave functions in the same section for the later use. Section 4 is the main work of this paper, we gives a sufficient condition for arbitrarily strong arbitrage, and weaken the sufficient condition in Banner and Fernholz (2008). As for section 5, we propose an abstract model which may satisfy the condition for arbitrage given in section 4 but may not satisfy the sufficient condition in Banner and Fernholz (2008) .

2

Preliminaries

2.1 The Model

On a filtered probability space (Ω, F , P, F), F = {F(t)}0≤t<∞with the usual conditions, that

is, right-continuity and augmentation by P-negligible sets. Also, we assume F(0) = {∅, Ω} modulo P. Consider the following market model :

d log Xi(t) = γi(t)dt + n

X

k=1

σik(t)dWk(t) , i = 1, · · · , n . (1)

Here the vector-valued process γ(t) = (γ1(t), ..., γn(t)) is the growth rates process for the

stocks, and the matrix-valued process σ(t) = [σik(t)]1≤i,k≤n is called the volatility of the

stocks in the market. The covariance process of the stocks in the market is the matrix-valued process α(t) = σ(t)σ0(t) with elements

αij(t) = n X k=1 σik(t)σjk(t) = d dthlog Xi, log Xji (t), 1 ≤ i, j ≤ n. (2) By Itˆo’s formula, the market model can be formulated as

dXi(t) = Xi(t) βi(t)dt + n X k=1 σik(t)dWk(t) ! , i = 1, · · · , n (3) where βi(t) = γi(t) + 1 2αii(t), for all t > 0 (4) is the mean rate of return for the stock i, for each i = 1, · · · , n.

Remark 1. (A)A special case of (1) is the so-called volatility-stabilized market given by d log Xi(t) = δ 2µi(t) dt + 1 pµi(t) dWi(t) , i = 1, · · · , n (5)

where δ ≥ 0 is a constant and µi(t) = Xi(t)/Pn_j=1Xj(t), 1 ≤ i ≤ n.

This model captures the property in real markets that the smaller stocks are more likely to have greater growth rates and volatilities than the larger stocks. Therefore, it is not surprising that each stock in such market fluctuates heavily. See Fernholz and Karatzas (2005) for details.

(B)Another case of (1) is called Atlas model given by

d log Xi(t) = ng1{Xi(t)=X_pt(n)(t)}+ σdWi(t) , i = 1, · · · , n (6)

where g > 0 and σ are constants. (Here pk(t) is the name of stock which rank kth position

of all stocks at time t; If there are stocks with the same value, then sort them by their indexes.) This model captures the characteristic in the real market that smaller stocks should have higher growth rates, by assigning zero growth rate to all the stocks except the smallest one.(For details, see Banner, Fernholz and Karatzas(2005). )

2.2 Investment Strategies and Portfolios

Now, consider a model with a money-market dB(t) = B(t)r(t)dt, B(0) = 1, and consider a small investor whose actions in the market cannot affect market prices. This investor decides, at each time t, that proportion πi(t) of current wealth Z(t) to invest in the ith

stock, i = 1, · · · , n; the proportion π0(t) := 1 −Pni=1πi(t) gets invested in the money

market. Thus, given a strategy π(·) and initial capital z ∈ (0, ∞), the corresponding wealth process Zz,π(·) for this strategy satisfies :

         dZz,π(t) Zz,π(t) = n X i=1 πi(t) dXi(t) Xi(t) + π0(t) dB(t) B(t) = π 0 (t)[(β(t) + r(t))dt + σ(t)dW (t)] . Zz,π(0) = z (7)

We shall say π(·) is an investment strategy, and write π(·) ∈ H, if π : [0, ∞) × Ω → Rn is a F-progressively measurable process which satisfies for each T ∈ (0, ∞)

Z T

0

|π0(t)(β(t) + r(t))| + π0(t)α(t)π(t)
dt < ∞ , a.s. (8) An investment strategy π(·) ∈ H with Pn

i=1πi(t, ω) = 1 for all (t, ω) ∈ [0, ∞) × Ω will

be called a portfolio. A portfolio never invests in the money market and never borrows from it. And we shall say a portfolio π(·) is bounded if there exist a constant M > 0 such that ||π(t)|| ≤ M , for all (t, ω) ∈ [0, ∞) × Ω. We shall call a portfolio long-only, if it never sells any stock short. Clearly, a long-only portfolio is also bounded.

Remark 2. We will write Zπ(t) for Zz,π(t), if Zz,π(0) = 1.

Remark 3. An important long-only portfolio is the market portfolio; this invests in all stocks in proportion to their relative weights,

µi(t) =

Xi(t)

X(t), 1 ≤ i ≤ n, (9)

where X(t) := X1(t) + · · · + Xn(t).Clearly, we have Zz,µ(·) = zX(·)/X(0) and the resulting

vector process µ(·) = (µ1(·), · · · , µn(·)) of market weights takes values in the positive simplex

∆n= {x ∈ Rn: x1+ · · · + xn= 1; 0 < xi < 1, i = 1 · · · , n} .

For a portfolio π(·) with initial capital z > 0, since a portfolio never invests in the money market and never borrows from it( i.e. π0(·) ≡ 0 ), we can find by (7) that the

corresponding wealth process for this portfolio is the solution of the following SDE :          dZz,π(t) Zz,π(t) = n X i=1 πi(t) dXi(t) Xi(t) = π0(t)[β(t)dt + σ(t)dW (t)] . Zz,π(0) = z (10) or, equivalently, Zz,π(t) = z exp Z t 0  π0(s)β(s) − 1 2π 0_{(s)α(s)π(s)}  ds + Z t 0 π0(s)σ(s)dW (s)  . (11) By analogy with (1) we can write the SDE (10) as

d log Zz,π(t) = γπ(t)dt + n X k=1 σπk(t)dWk(t) , Zz,π(0) = z , (12) or, equivalently, Zz,π(t) = z exp ( Z t 0 γπ(s)ds + n X k=1 Z t 0 σπk(s)dWk(s) ) , 0 ≤ t < ∞. (13) Here σπk(t) := n X i=1 πi(t)σik(t) for k = 1, · · · , n (14)

are the volatility coefficients associated with the portfolio π(·), and γπ(t) :=

n

X

i=1

πi(t)γi(t) + γπ∗(t) (15)

is the growth rate of the portfolio π(·), where γ_π∗(t) := 1 2   n X i=1 πi(t)αii(t) − n X i=1 n X j=1 πi(t)αij(t)πj(t)   (16)

is the excess growth rate of the portfolio π(·).

For an arbitrary portfolio π(·), and with ei denoting the ith unit vector in Rn, let us

introduce the quantities τ_ijπ(t) := n X k=1 (σik(t) − σπk(t))(σjk(t) − σπk(t)) (17) = (π(t) − ei)0α(t)(π(t) − ej) = αij(t) − απi(t) − απj(t) + αππ(t) (18) 4

for 1 ≤ i, j ≤ n, and set απi(t) := n X j=1 αij(t)πj(t) , αππ(t) := n X i=1 n X j=1 αij(t)πi(t)πj(t) . (19)

We shall call the matrix-valued process τπ(·) = (τ_ijπ(·))1≤i,j≤n the process of individual

stocks’ covariance relative to the portfolio π(·). In fact, we have τ_ijπ(t) = d dt * log Xi Z_Xi(0),π, log Xj Z_Xj(0),π + (t), 1 ≤ i, j ≤ n. (20) It satisfies the equations

n X j=1 τ_ijπ(t)πj(t) = 0, i = 1, · · · , n. (21) Also we have γ_π∗(t) = 1 2   n X i=1 πi(t)τiiη(t) − n X i=1 n X j=1 πi(t)πj(t)τijη(t)  , (22)

for any portfolio η. In particular, when η = π, we have γ_π∗(t) = 1 2 n X i=1 πi(t)τiiπ(t). (23)

Remark 4. Since log(Xi/ZXi(0),π) (t) is a.s. nondecreasing,

τ_iiπ(t) ≥ 0, t ∈ [0, ∞), a.s.

2.3 Relative Arbitrage

Given any two investment strategies π and ρ, we shall say that π is an arbitrage relative to ρ over [0, T ], if we have

P (Z1,π(T ) ≥ Z1,ρ(T )) = 1 and P (Z1,π(T ) > Z1,ρ(T )) > 0. (24)

We call such relative arbitrage strong, if

P (Z1,π(T ) > Z1,ρ(T )) = 1. (25)

Existence of Arbitrage Relative to the Market Portfolio. (A) If there exists a real constant h > 0 such that

γ_µ∗(t) = 1 2   n X i=1 µi(t)αii(t) − n X i=1 n X j=1 µi(t)µj(t)αij(t)  ≥ h, ∀ 0 ≤ t < ∞ (26) holds almost surely, it can be shown that, for a sufficiently large constant c > 0, the long-only portfolio πi(t) = µi(t)(c − log µi(t)) Pn j=1µj(t)(c − log µj(t)) , i = 1, · · · , n (27) 5

is a strong arbitrage relative to the market portfolio µ over any time-horizon [0, T ] with T > (2 log n)/h. (See Fernholz and Karatzas (2009) Example 11.1 for a proof.) Note also that if the market is nondegenerate and diverse, then (26) holds. (Proposition 2.2.2 of Fernholz (2002).)

(B) If there exists a real constant h > 0 such that (µ1(t) · · · µn(t))1/n   n X i=1 αii(t) − 1 n n X i=1 n X j=1 αij(t)  ≥ h, ∀ 0 ≤ t < ∞, (28) holds almost surely, then for a sufficiently large constant c > 0, the long-only portfolio

πi(t) = c c + (µ1(t) · · · µn(t))1/n 1 n + (µ1(t) · · · µn(t))1/n c + (µ1(t) · · · µn(t))1/n µi(t) (29)

is a strong arbitrage relative to the market portfolio µ over any time-horizon [0, T ] with T > 2n1−(1/n)/h. (See Fernholz and Karatzas (2009) Example 11.2 for a proof, we will also give a proof in latter context.)

(C) Suppose there exists a continuous, strictly increasing function Γ : [0, ∞) → [0, ∞) with Γ(0) = 0, Γ(∞) = ∞ and such that

Γ(t) ≤ Z t

0

γ_µ∗(s)ds < ∞, for all 0 ≤ t ≤ ∞ (30) holds almost surely, then, for a sufficiently large constant c > 0, the portfolio

πi(t) =

cµi(t) − µi(t) log µi(t)

c −Pn

j=1µj(t) log µj(t)

, i = 1, · · · , n (31) is a strong arbitrage relative to the market portfolio µ over any time-horizon [0, T ] with T > T∗, where T∗ := Γ−1  − n X j=1 µj(0) log µj(0)  . (32)

(See Fernholz and Karatzas (2009) Remark 11.4 .)

Open Question: Does any one of the sufficient condition above guarantee the existence of strong relative arbitrage opportunities over arbitrary time-horizons?

3

Some Useful Properties

3.1 Relative Return Process

It is frequently of interest to measure the performance of stocks or portfolios relative to a given benchmark in the market portfolio, consisting of all the shares of all the stocks in the market.

For any two portfolios π and η, the relative return process of π versus η is defined by log Zπ(t)

Zη(t)



, t ∈ [0, ∞). 6

Then, by (1) and (12), we have d log Zπ(t) = n X i=1 πi(t)d log Xi(t) + γπ∗(t)dt, a.s., (33)

for t ∈ [0, ∞), so we also have d log Zπ(t) Zη(t)  = n X i=1 πi(t)d log  Xi(t) Zη(t)  + γ_π∗(t)dt, a.s., (34) for t ∈ [0, ∞).

In particular, when η = µ, the market portfolio, this equation can be expressed in an especially useful form :

d log Zπ(t) Zµ(t)  = n X i=1 πi(t)d log µi(t) + γπ∗(t)dt, a.s., (35) for t ∈ [0, ∞).

3.2 Portfolio Generating Functions

Functionally generated portfolios were introduced by Fernholz, in Fernholz (1999). For this class of portfolios one can derive a decomposition of their relative return which proves useful in the construction and study of arbitrages relative to the market. This decomposition is so powerful because it does not involve stochastic integrals, and opens the possibility for making probability-one comparisons over given fixed time-horizon.

Specifically speaking, given S a positive C2 function defined on some open neighborhood U of ∆n _{such that for all i = 1, · · · , n, x 7→ x}

iDilog S(x) is bounded on U . Consider also

the portfolio π(·) with weights πi(t) =  Dilog S(µ(t)) + 1 − n X j=1 µj(t)Djlog S(µ(t))  µ_i(t), 1 ≤ i ≤ n. (36) We call this the portfolio generated by S(·). It can be shown that the relative wealth process of the portfolio, with respect to the market, is given by the master formula

log Zπ(T ) Zµ(T )  = log S(µ(T )) S(µ(0))  + Z T 0 Θ(t)dt 0 ≤ T < ∞, (37) where the so-called drift process Θ(·) is given by

Θ(t) = −1 2S(µ(t)) n X i,j=1 DijS(µ(t))µi(t)µj(t)τ_ijµ(t). (38)

(For a proof, see Fernholz(2002) p.46)

Remark 5. The generated portfolio weights depend only on the market weights µ1(t), · · · , µn(t),

not on the covariance structure of the market. Hence, such portfolio can be implemented easily.

Remark 6. Suppose the function S(·) is concave, or, more precisely, its Hessian D2S(x) = (D2

ijS(x))1≤i,j≤n has at most one positive eigenvalue for each x ∈ U and, if a positive

eigenvalue exists, the corresponding eigenvector is orthogonal to ∆n. Then the generated portfolio corresponding to this S is long-only with the drift term Θ(·) being non-negative; if rank(D2S(x)) > 1 holds for each x ∈ U , then Θ(·) is positive.

Here are a few examples of simple generating functions and the portfolios they generate. 1. S(x) ≡ w, a positive constant, generates the market portfolio with Θ(·) ≡ 0;

2. S(x) = w1x1+ · · · + wnxngenerates the passive portfolio that buys at time t = 0, and

holds up until time t = T , a fixed number of shares wi in each stock i = 1, · · · , n( the

market portfolio corresponds to the special case w1 = · · · = wn= w of equal numbers

of shares across assets ); 3. S(x) = xp1

1 · · · x pn

n , where p1, · · · , pnare constants and p1+ · · · + pn= 1, generates the

constant-weighted portfolio with weights πi(t) = pi and dΘ(t) = γπ∗(t)dt. Indeed, for

this generating function, we have log S(x) = n X i=1 pilog xi⇒ Dilog S(x) = pi xi ⇒ n X j=1 µj(t)Djlog S(µ(t)) = n X j=1 pj = 1 and DiS(x) = pi xi S(x), DiiS(x) = pi(pi− 1) x2_i S(x), DijS(x) = pipj xixj S(x), for i 6= j. Therefore, πi(t) =  pi µi(t) + 1 − 1  µi(t) = pi, and dΘ(t) = −1 2S(µ(t))   n X i=1 pi(pi− 1)S(µ(t))τiiµ(t) + X 1≤i,j≤n,i6=j pipjS(µ(t))τijµ(t)  dt = −1 2   n X i=1 n X j=1 pipjτijµ(t) − n X i=1 piτiiµ(t)  dt = 1 2   n X i=1 πiτiiµ(t) − n X i=1 n X j=1 πiπjτijµ(t)  dt = γ_π∗(t)dt. (In the last line, we use equation (22).)

4. The modified entropy function Hc(x) = c −Pni=1xilog xi generates the portfolio in

(31). The drift process for this portfolio is dΘ(t) = γ_µ∗(t)/Hc(µ(t))dt. Since c <

Hc(x) ≤ c + log n, for x ∈ ∆n, similar argument as in 3 show that the portfolio is a

relative arbitrage to the market portfolio over sufficiently large time.

Proof of (B) in section 2.3 : Consider the equally weighted portfolio ϕi(·) ≡ 1/n,

i = 1 · · · , n, then condition (28) can be written as (µi(t) · · · µn(t))1/nγ∗ϕ(t) ≥

h 2n.

From 3 above, we know this portfolio is generated by S(x) = (x1. . . xn)1/n. For this S(·)

and any c > 0, consider another portfolio ϕc_i(t) = S(µ(t)) c + S(µ(t))· 1 n+ c c + S(µ(t))· µi(t), i = 1, · · · , n.

It is not hard to see that this portfolio is generated by Sc(x) = c + S(x), and with n−1/n ≥

S(x) ≥ 0, we can derive the following equation log Zϕc(T ) Zµ(T )  = log c + S(µ(T )) c + S(µ(0))  + Z T 0 S(µ(t))γ_ϕ∗(t) c + S(µ(t)) dt ≥ log  c c + n−1/n  + hT 2n(c + n−1/n) Moreover, log  c c + n−1/n  + hT 2n(c + n−1/n₎ > 0 ⇐⇒ T > 2n h (c + n −1/n_{) log} c + n−1/n c ! , and lim c→∞(c + n −1/n_{) log} c + n−1/n c ! = lim c→∞ logc+n_c−1/n 1 c+n−1/n = lim c→∞ c c+n−1/n · ( −n−1/n c2 ) −1 (c+n−1/n₎2 = lim c→∞ n−1/n(c + n−1/n) c = n −1/n_.

Thus, if T > 2n1−(1/n)/h, then there exists a c > 0 large enough such that the corresponding portfolio ϕcis a strong arbitrage relative to the market.



3.3 Concave Functions

We list some important results here which will be used in the later context. Lemma 1. If f is concave on [A, B], and a1, · · · , am ∈ [A, B], then

m X i=1 f (ai) ≤ mf ( 1 m m X i=1 ai). (39)

Proof. Since f is concave on [A, B], for any λ ∈ (0, 1) and a1, a2 ∈ [A, B],

(1 − λ)f (a1) + λf (a2) ≤ f ((1 − λ)a1+ λa2) , (40)

by induction, we have m X i=1 wif (ai) ≤ f m X i=1 wiai ! , (41)

where ai∈ [A, B] for all i = 1, · · · , n and w = (w1, · · · , wn) ∈ ∆n.

Finally, replace wi with _m1 for each i, it follows the result.

Lemma 2. Suppose that f is concave on [A, B], and a1, · · · , am ∈ [A, B] are chosen such

thatPm

i=1ai− (m − 1)A ≤ B. Then

(m − 1)f (A) + f m X i=1 ai− (m − 1)A ! ≤ m X i=1 f (ai). (42)

Proof. For any a1, · · · , am∈ [A, B] withPmi=1ai− (m − 1)A ≤ B, define

λj = aj− A Pm i=1ai− mA = aj− A (a1− A) + · · · + (am− A)

It is straightforward to check that 0 ≤ λj ≤ 1 and that

(1 − λj)A + λj m X i=1 ai− (m − 1)A ! = aj . SincePm

i=1ai− (m − 1)A ∈ [A, B] and f is concave, we have

(1−λj)f (A)+λjf m X i=1 ai− (m − 1)A ! ≤ f (1 − λj)A + λj m X i=1 ai− (m − 1)A !! = f (aj).

Summing these inequalities form j = 1 to j = m gives (m − 1)f (A) + f m X i=1 ai− (m − 1)A ! ≤ m X i=1 f (ai).

Lemma 3. Suppose that f is concave on [0, 1]. For x = (x1, · · · , xn) ∈ ∆n, we have

(n − 1)f (0) + f (1) ≤ (n − 1)f x(n)
+ f 1 − (n − 1)x(n)
≤ n X i=1 f (xi), (43) and n X i=1 f (xi) ≤ f x(n)
+ (n − 1)f _{1 − x} (n) n − 1  ≤ nf 1 n  . (44) where x_(n)= min{x1, · · · , xn}. 10

Proof. For x = (x1, · · · , xn) ∈ ∆n, without loss of generality, we may assume x(n) = xn.

By Lemma 1 with A = 0, B = 1, m = n − 1 and ai = xi for i = 1, · · · , m , we have n X i=1 f (xi) = f (xn)+ n−1 X i=1 f (xi) ≤ f (xn)+(n−1)f 1 n − 1 n−1 X i=1 xi ! = f (xn)+(n−1)f  1 − xn n − 1  . Reapplying Lemma 1, with m = n, a1 = · · · = an−1= 1−x_n−1n, and an= xn, then we have

f (xn) + (n − 1)f 1 n − 1 n−1 X i=1 xi ! ≤ nf 1 n " xn+ (n − 1) 1 n − 1 n−1 X i=1 xi #! = nf 1 n  . Hence, n X i=1 f (xi) ≤ f (xn) + (n − 1)f  1 − xn n − 1  ≤ nf 1 n  . (45)

On the other hand, by Lemma 2 with m = n, A = xn, B = 1, and ai = xi, for

i = 1, · · · , n, we find (n − 1)f (xn) + f (1 − (n − 1)xn) = (n − 1)f (xn) + f n X i=1 xi− (n − 1)xn ! ≤ n X i=1 f (xi).

Reapplying Lemma 2 with A = 0, B = 1, m = n, a1 = · · · = an−1 = xn and an =

1 − (n − 1)xn,then (n − 1)f (0) + f (1) ≤ (n − 1)f (xn) + f (1 − (n − 1)xn) . Hence, (n − 1)f (0) + f (1) ≤ (n − 1)f (xn) + f (1 − (n − 1)xn) ≤ n X i=1 f (xi). (46)

Remark 7. For any r ∈ (0, 1/n], obviously, 1 − (n − 1)r ≥ r, then choose x = (1 − (n − 1)r, r, · · · , r) ∈ ∆n, x(n)= r and by Lemma 3, we have

(n − 1)f (0) + f (1) ≤ (n − 1)f (r) + f (1 − (n − 1)r) ≤ n X i=1 f (xi). and n X i=1 f (xi) ≤ f (r) + (n − 1)f  1 − r n − 1  ≤ nf 1 n  .

4

Main Results

We say that a bounded portfolio π is a strong relative arbitrage opportunity over the time horizon [0, T ], if

P (Z1,π(T ) > Z1,µ(T )) = 1.

Proposition 1. There exists a strong relative arbitrage opportunity over any time horizon [0, T ] in any market model of the form (1) satisfying the following conditions :

(A) There exists a positive differentiable function l defined on [0,_n1], such that τ_m(t)m(t)µ (t) ≥ l(µm(t)(t)), for all t ≥ 0

almost surely, where m(t) denotes the index of the stock of minimum weights at time t, namely, µm(t)(t) = min{µ1(t), · · · , µn(t)}.

(B) For this l(·), there exists a family of C3 functions F = {fα}α∈I, where I is some

nonempty index set, such that

(a) fα(x) > 0, ∀x ∈ (0, 1] and fα(0) = 0 for any α ∈ I.

(b) xf_α0(x) is bounded on (0, 1) for each α ∈ I. (c) f_α0(x) ≥ 0 for all x ∈ (0, 1) and α ∈ I. (d) f_α00(x) < 0 for all x ∈ (0, 1) and α ∈ I.

(e) _dxd[−f_α00(x)x2l(x)] ≤ 0 for all x ∈ (0, 1/n) and α ∈ I. (f ) For each T > 0, there exist β ∈ I, such that

Z 1

n

0

f_β0(x)

−f_β00(x)x2_l(x)dx ≤ T.

Proof. Given any T > 0, we want to find a strong relative arbitrage to the market portfolio over the time horizon [0, T ], that is, find a bounded portfolio π, such that log

 Zπ(T ) Zµ(T )  > 0 almost surely.

Consider S(x) = S(x1, · · · , xn) = f (x1) + · · · + f (xn) with f ∈ F . Since f satisfies (B.a)

and (B.b) so that S satisfies the statements in section 3.2. Apply the master formula to this S, it follows that

log Zπ(T ) Zµ(T )  = log S(µ(T )) − log S(µ(0)) + Z T 0 1 2S(µ(s)) n X i=1 −f00(µi(s))µ2i(s)τ µ ii(s)ds (47)

The reason we consider S(x) = f (x1) + · · · + f (xn) is that with S being this form, we

can have DijS(·) = 0, for i 6= j. If DijS(·) 6= 0 for some i 6= j, then we have to deal with

the term τ_ijµ(t), for i 6= j — which may be both positive and negative. It will make the whole estimation much harder.

Moreover, even though log S(µ(T )) − log S(µ(0)) may be negative, but by Remark 4 τ_iiµ(t) > 0, for all t > 0, and we know f00 < 0 from (B.d); these facts ensure that the last term of (47) is always positive, the question is that does this positive term large enough to offset the negative part.

From (B.d), we also know that f is a convex function, with this observation, we can use the convexity to give estimations to f (x1) + · · · + f (xn), or S(x). The upper bound is

well-known, as for the lower bound is a important step to this proof. Combine (47) with upper bound (45) and lower bound (46), we have

log Zπ(T ) Zµ(T )  ≥ log S(µ(T )) − log S(µ(0)) + Z T 0 1 2S(µ(s)) −f 00 (ys)
τ_m(s)m(s)µ (s)ys2ds

≥ log [(n − 1)f (y_T) + f (1 − (n − 1)yT)] − log

 nf 1 n  + Z T 0 −f00(ys)τ_m(s)m(s)µ (s)ys2 2hf (ys) + (n − 1)f  1−ys n−1 i ds, 12

where ys= µ(n)(s).

From (A), we know that there exist a positive function l defined on [0,1_n], such that τ_m(t)m(t)µ (t) ≥ l(µ_m(t)(t)) = l(yt), for all t ≥ 0

almost surely. It follows that log Zπ(T )

Zµ(T )



≥ log [(n − 1)f (yT) + f (1 − (n − 1)yT)]−log

 nf 1 n  + Z T 0 −f00(ys)l(ys)ys2 2 h f (ys) + (n − 1)f  1−ys n−1 i ds = S1(yT) − log  nf 1 n  + Z T 0 Θ1(ys)ds, (48) where S1(x) = log [(n − 1)f (x) + f (1 − (n − 1)x)] , Θ1(x) = −f00(x)l(x)x2 2hf (x) + (n − 1)f1−x_n−1i .

Next, consider the question : given T > t0 > 0, and f as above, does there exist a

deterministic function h(·) such that, for all t ∈ [t0, T ],

         S1(h(t)) + Z t t0 Θ1(s)ds = log  nf 1 n  , h(t0) = 1 n . (49)

If so, what is the connection between h(t) and yt?

To answer these questions, we assume first that such a function h(·) exist, then, differ-entiate the first equation of (49) with respect to t, we find

0 = d dt  S1(h(t)) + Z t t0 Θ1(h(s))ds  = S₁0(h(t))h0(t) + Θ1(h(t)) ,

Moreover, if the function h(t) is monotone, and let g(t) be the inverse function of h(t), then by Inverse Function Theorem, we have

g0(h(t)) = 1 h0(t) = − S₁0(h(t)) Θ1(h(t)) ⇒ g0(x) = −S 0 1(x) Θ1(x) , (50)

Combine with the initial condition h(t0) = 1_n, or g(1_n) = t0, we can derive

g(x) = t0+ Z x 1 n −S 0 1(r) Θ1(r) dr . (51)

Conversely, if g(·) is defined as above, and g is a continuous monotone function, then define h = g−1, h will be a solution of (49). Therefore, here comes another question — when is g monotone ? Since f00(x) < 0, f (x) > 0 and l(x) > 0, for x ∈ (0, 1/n], it is not hard to see

that Θ1(x) > 0, and S10(x) > 0, for x ∈ (0, 1/n], it follows that g(x) is decreasing on [0,1/n].

It remains to check g(0) < ∞.

To see g(0) is well defined, note that by choosing f properly, (B.c), (B.f ) and Remark 7 implies that g(0) := t0+ Z 0 1 n −S 0 1(r) Θ1(r) dr = t0+ Z 1 n 0 (n − 1)f0(r) − (n − 1)f0(1 − (n − 1)r) (n − 1)f (r) + f (1 − (n − 1)r) 2  f (r) + (n − 1)f  1−r n−1  −f00_(r)l(r)r2 dr ≤ t0+ Z _n1 0 (n − 1)f0(r) f (1) 2nf _n1
−f00_(r)l(r)r2dr ≤ t0+ 2n(n − 1) Z 1 n 0 f0(r) −f00_(r)l(r)r2dr < ∞.

Hence, h is a continuous decreasing function defined on [t0, g(0)] with h(t0) = _n1 and

h(g(0)) = 0.

Now, given t0 = T /2, by (B.6), we may assume this f satisfies

Z 1 n 0 f0(r) −f00_(r)l(r)r2dr ≤ T 4n(n − 1) . This implies g(0) ≤ T . Then, define a stopping time η as

η = inf{t ≥ t0|yt> h(t)} , (52)

Note that t0 ≤ η ≤ g(0) ≤ T a.s., the fact is easy to observe from the path behavior, and

by the path continuity, we have yτ = h(τ ).

With this stopping time η, define a corresponding portfolio ˜π(·) by setting ˜ π(t) =    π(t), t < η, µ(t), t ≥ η. (53) (48) ⇒ log Zπ˜(T ) Zµ(T )  = log Zπ(η) Zµ(η)  ≥ S₁(yη) − log  nf 1 n  + Z η 0 Θ1(ys)ds = S1(h(η)) − log  nf 1 n  + Z t0 0 Θ1(ys)ds + Z η t0 Θ1(ys)ds

It remains to connect h(t) with yt, the key step is to choose f such that Θ1(x) is

decreasing on [0, 1/n], if we can do this, then combine with (49), we have log Z˜π(T ) Zµ(T )  = S1(h(η)) − log  nf 1 n  + Z t0 0 Θ1(ys)ds + Z η t0 Θ1(ys)ds ≥ S₁(h(η)) − log  nf 1 n  + Z t0 0 Θ1  1 n  ds + Z η t0 Θ1(h(s))ds = t0Θ1  1 n  = T 2Θ1  1 n  > 0. 14

Hence, for any T > 0, a strong relative arbitrage opportunity exists over the time horizon [0, T ].

To check that Θ1(·) is decreasing, observe that the denominator of Θ1(·) is increasing,

and by (B.e), the numerator of Θ1(·) is decreasing, then Θ1(·) is indeed decreasing.

Remark 8. η defined in (52) is a stopping time since yt= µ(n)(t) is a continuous adapted

process and h(t) is a deterministic continuous function, moreover, F is a right-continuous filtration. With these facts, for each t ≥ 0, we have

{η < t} = [

t0≤s<t,s∈Q

{y_s> h(s)} ∈ F (t).

This implies that η is a stopping time. Furthermore, for the reason that η is a stopping time, ˜π defined in (53) is indeed a portfolio.

Theorem 1. For any T > 0, a strong relative arbitrage opportunity exists over the time horizon [0, T ] in any market of the form (1) satisfying the condition

τ_m(t)m(t)µ (t) ≥ C

µp_m(t)(t), for all t > 0 (54) almost surely, where C > 0 is a constant and p is a constant with 0 < p ≤ 1.

Proof. Consider l(x) = Cx−p and the family of functions {fα(·)}α≥1 defined by the formula

fα(y) =        1 p Z ∞ −p log y e−rrαdr, if 0 < y ≤ 1, 0, if y = 0. Then we have τ_m(t)m(t)µ (t) ≥ l(µ_m(t)(t)), for all t ≥ 0 almost surely. Also,

(a) fα(x) > 0, for all x ∈ (0, 1] and fα(0) = 0, for any α ≥ 1.

(b) xf_α0(x) = xp(−p log x)α for all x ∈ (0, 1) and α ≥ 1 By L’Hˆopital’s rule lim x→0x p_{(− log x)}α _{= lim} x→0 (− log x)α x−p = limx→0 α(− log x)α−1 px−p = · · · = 0 Hence xf_α0(x) is bounded on (0, 1).

(c) f_α0(x) = xp−1(−p log x)α > 0, for all x ∈ (0, 1) and α ≥ 1. (d) For any α ≥ 1, f_α00(x) = d

dxx

p−1_{(−p log x)}α

= (p − 1)xp−2(−p log x)α+ xp−1α(−p log x)α−1(−px−1)

= (p − 1)xp−2(−p log x)α− pxp−2_{α(−p log x)}α−1 _{< 0, for all x ∈ (0, 1).}

(e) For any α ≥ 1, d dx[−f 00 α(x)x2l(x)] = d dx[−f 00 α(x)Cx2−p] = d dxC(1 − p)(−p log x) α_{+ Cpα(−p log x)}α−1 = −C(1 − p)pαx−1(−p log x)α−1− Cp2α(α − 1)x−1(−p log x)α−2 ≤ 0, 15

for all x ∈ (0, 1). (f) Observe that

−f_α00(x)x2l(x) = C(1 − p)(−p log x)α+ Cpα(−p log x)α−1 ≥ Cpα(−p log x)α−1. Also, f_α0(x) > 0 for all x ∈ (0, 1) and α ≥ 1, these imply

f_α0(x) −f00 α(x)x2l(x) ≤ f 0 α(x) Cpα(−p log x)α−1 = xp−1(−p log x)α Cpα(−p log x)α−1 = 1 Cpαx p−1_{(−p log x),} for all x ∈ (0, 1). Moreover, Z 1 n 0 f_α0(x) −f00 α(x)x2l(x) dx ≤ Z 1 n 0 1 Cpαx p−1_{(−p log x)dx} let xp = y = Z n−p 0 1 Cp2_α(− log y)dy = 1

Cp2_α(−y log y + y)| n−p 0 = 1 Cp2_α p log n np .

Hence, for each T > 0, choose β =  1 Cp2_T p log n np  ≥ 1, then Z _n1 0 f_β0(x) −f_β00(x)x2_l(x)dx ≤ T.

Remark 9. Propositions 3.1 and 3.8 of Fernholz and Karatzas (2005) state that strong relative arbitrage opportunities exist over long enough time horizons in any market satisfying the condition

Γ(t) ≤ Z t

0

γ_µ,p∗ (s)ds < ∞, a.s. (55) for some p > 0 and continuous, strictly increasing function Γ : [0, ∞) → [0, ∞) with Γ(0) = 0 and Γ(∞) = ∞, where γ_µ,p∗ (·) is the generalized excess growth rate of the market and defined as γ_µ,p∗ (t) = 1 2 n X i=1 (µi(t))pτ_iiµ(t).

For markets that satisfying (54) for some p ∈ (0, 1], we have γ_µ,p∗ (t) = 1 2 n X i=1 (µi(t))pτiiµ(t) ≥ 1 2(µm(t)(t)) p_τµ m(t)m(t)(t) ≥ C 2, it follows that Z t 0 γ_µ,p∗ (s)ds ≥ Ct 2 , a.s. 16

Hence, we have a long-term strong relative arbitrage. Conversely, if we have the stronger condition

γ_µ,p∗ (t) ≥ C ∀ t, a.s., (56) for some p ∈ (0, 1], then for n = 2, by (21) we have µ2₁(t)τ₁₁µ(t) = µ2₂(t)τ₂₂µ(t). Without loss of generality, we may assume µ2(t) ≥ µ1(t), it follows that

µp₁(t)τ₁₁µ(t) ≥ 1 2 " µp₁(t) + µ1(t) µ2(t) 2−p µp₁(t) # τ₁₁µ(t) = 1 2(µ p 1(t)τ µ 11(t) + µ p 2(t)τ µ 22(t)) ≥ C.

Therefore, the stronger condition (56) leads to short-term relative arbitrage for n = 2, but we are still unable to show that short-term relative arbitrage exist for n ≥ 3 under condition (56).

Remark 10. Our sufficient condition (54) is weaker than that in Banner and Fernholz (2008). In fact in the next section, we provide a market model which may not satisfy the sufficient condition in Banner and Fernholz (2008). However, it satisfies our sufficient condition (54).

5

Generalized Volatility-Stabilized Market Model

We consider d log Xi(t) = δ 2µp_i(t)dt + 1 µp/2_i (t) dWi(t), i = 1, · · · , n, (57)

where δ ≥ 0 and p ∈ (0, 1] are both constants .

When p = 1, the theory developed by Bass and Perkins (2002) shows that the resulting system of stochastic differential equations determines the distribution of the ∆n-valued diffusion process (X1(t), · · · , Xn(t)) uniquely.

For this model, we have αij(t) = δijµ−p_i (t), hence the relative variances τ_iiµ(·) are given

by τ_iiµ(t) = αii(t) − n X j=1 αij(t)µj(t) − n X j=1 αij(t)µj(t) + n X i=1 n X j=1 αij(t)µi(t)µj(t) = 1 µp_i(t)− 1 µp_i(t)µi(t) − 1 µp_i(t)µi(t) + n X j=1 1 µp_j(t)µ 2 j(t) ≥ 1 µp_i(t) 1 − 2µi(t) + µ 2 i(t)
= 1 µp_i(t)(1 − µi(t)) 2_,

for all i = 1, · · · , n and for all t > 0. Hence, for n ≥ 2, we obtain τ_m(t)m(t)µ (t) ≥ 1 µp_m(t)(t) 1 − µm(t)(t)
2 ≥ n − 1 n 2 1 µp_m(t)(t), for all t > 0. By Theorem 1, we have the following :

Theorem 2. Assume that n ≥ 2, then the generalized volatility-stabilized market as (57) has a strong relative arbitrage opportunity over any time horizon [0, T ] for any T > 0.

Remark 11. For generalized volatility-stabilized market with p ∈ (0, 1), the following may not be true for any C > 0 :

τ_m(t)m(t)µ (t) ≥ C µm(t)(t)

, for all t ≥ 0 almost surely. (58) Indeed, we have µ_m(t)(t)τ_m(t)m(t)µ (t) = µ1−p_m(t)(t) 1 − 2µ_m(t)(t)
+ µ_m(t)(t) n X j=1 1 µp_j(t)µ 2 j(t) ≤ µ1−p_m(t)(t) + nµm(t)(t) ≤ (n + 1)µ1−p_m(t)(t),

and if for any C > 0, P  µ1−p_m(t)(t) < C n + 1 for some t ≥ 0  > 0, (59) then P  τ_m(t)m(t)µ (t) ≥ C µm(t)(t) for all t ≥ 0  < 1, for any C > 0.

Therefore, these model may not satisfy the sufficient condition given in (58), however, we give a weaker sufficient condition here to make sure short term arbitrage for any p ∈ (0, 1).

6

Some Related Results

6.1 Diffusion Models

1. In Fernholz and Karatzas(2010), the authors considered a diffusion model as the follow-ing: dXi(t) = Xi(t) bi(X(t))dt + n X k=1 sik(X(t))dWk(t) ! , Xi(0) = xi> 0, i = 1, · · · , n, (60) where X(t) := (X1(t), · · · , Xn(t)). Let aij(x) := n X k=1 sik(x)sjk(x) , ∀ 1 ≤ i, j ≤ n , and define U (T, x) := inf  w > 0|∃π(·) ∈ H s.t Zwx,π(T ) ≥ X(T ) X(0)x, a.s.  .

the smallest relative amount of initial capital x, starting with which one can match or exceed the market portfolio at time T . (Note that if U (T, x) < 1, then it means that we can have strong relative arbitrage (investment strategy) with respect to market portfolio µ at time T .) Then, under some appropriate conditions, they showed U satisfies

           ∂U ∂τ (τ, x) = 1 2 n X i=1 n X j=1 xixjaij(x)D2ijU (τ, x) + n X i=1 xi   n X j=1 xjaij(x) x1+ · · · + xn  DiU (τ, x) , U (0+, x) = 1 , x ∈ (0, ∞)n . (61) 18

Moreover, under some further conditions, they proved that if

U (T, x) < 1 , for some (T, x) ∈ (0, ∞) × (0, ∞)n, then

U (T, x) < 1 , ∀ (T, x) ∈ (0, ∞) × (0, ∞)n .

This shows the existence of long-term relative arbitrage implies the existence of short-term arbitrage. It is worth noting that such an arbitrage opportunity may be made by an investment strategy rather than a bounded portfolio.

6.2 Arbitrage and Diversity

In Fernholz and Karatzas (2009), they claimed that in weakly diverse markets, i.e. 1

T Z T

0

max

1≤i≤nµi(t)dt ≤ 1 − δ a.s. for some δ ∈ (0, 1),

which satisfy the strict non-degeneracy condition, that is

x0α(t)x ≥ kxk2 for all t ∈ [0, ∞) and x ∈ Rn for some  > 0,

one can construct simple long-only portfolios µ(p)(·), for some fixed p ∈ (0, 1), which lead to strong arbitrage relative to the market portfolio over [0, T ], where

µ(p)_i (t) := (µi(t)) p Pn j=1(µj(t))p , ∀ i = 1, · · · , n and T ≥ 2 pδlog n.

Furthermore, under these same conditions, they can even construct long-only portfolios η(·), which achieve strong relative arbitrage to the market over arbitrarily short time horizon, where ηi(t) := 1 qVµ_(t) (µ1(0))q − V b π_(t)  qVµ_(t) (µ1(0))q −_bπ(t)Vbπ(t)  and b π(t) := qe1+ (1 − q)µ(t), 0 ≤ t ≤ ∞ with q > 1 + 2 δ2_T log  1 µ1(0)  .

Note that these portfolios may be unbounded, which makes such portfolios more difficult for implementation in reality.

References

[1] Banner, A. and Fernholz, D. (2005) On Atlas models of equity markets. Annals of Ap-plied Probability 15, 2296-2330.

[2] Banner, A. and Fernholz, D. (2008) Short-term relative arbitrage in volatility-stabilized markets, Annals of Finance.

[3] Bass. R. and Perkins, E. (2002) Degenerate stochastic differential equations with H¨ older-continuous co¨eficients and super-Markov chains. Transactions of the American Mathemat-ical Society 355, 373-405.

[4] Fernholz, E.R. (1999) Portfolio generating functions. In M. Avellaneda (ed.), Quantita-tive Analysis in Financial Markets, River Edge, NJ. World Scientific.

[5] Fernholz, E.R. (2002) Stochastic Portfolio Theory. Springer-Verlag, New York.

[6] Fernholz, E.R. and Karatzas, I. (2005) Relative arbitrage in volatility-stabilized markets. Annals of Finance 1, 149-177.

[7] Fernholz, E.R. and Karatzas, I. (2009) Stochastic Portfolio Theory: A Survey. In Hand-book of Numerical Analysis and Numerical Methods in Finance (A. Bensoussan, ed.) 89-168. Elsevier, Amsterdam.

[8] Fernholz, E.R. and Karatzas, I. (2009) On optimal arbitrage. Annals of Applied Proba-bility 20, 1179-1204.