overview of stochastic portfolio theorydgomes/austin/...synopsis the purpose of these lectures is to...
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OVERVIEW OF STOCHASTIC PORTFOLIOTHEORY
IOANNIS KARATZAS
INTECH Investment Technologies LLC, Princetonand
Department of Mathematics, Columbia University
(Builds on work of E.R. Fernholz, as well as A. Banner, C. Kardaras, S.
Pal, V. Papathanakos, T. Ichiba, D. Fernholz, J. Ruf, some of it joint.)
Talk at Summer School in Lisabon, Portugal July 2012
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SYNOPSIS
The purpose of these lectures is to offer an overview of StochasticPortfolio Theory, a rich and flexible framework introduced byE.R. Fernholz (2002) for analyzing portfolio behavior and equitymarket structure.
This theory is descriptive as opposed to normative, is consistentwith observable characteristics of actual markets and portfolios,and provides a theoretical tool which is useful for practicalapplications.
As a theoretical tool, this framework provides fresh insights intoquestions of market structure and arbitrage, and can be used toconstruct portfolios with controlled behavior.
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As a practical tool, Stochastic Portfolio Theory has been applied tothe analysis and optimization of portfolio performance and hasbeen the basis of successful investment strategies for close to 20years.
More importantly, SPT explains under what conditions it becomespossible to outperform a cap-weighted benchmark index – andthen, exactly how to do this by means of simple investment rules.
These typically take the form of adjusting the capitalizationweights of an index portfolio to more efficient combinations.
They do it by exploiting the natural volatilities of stock prices, andneed no forecasts of mean rates of return.
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SOME REFERENCES
I Fernholz, E.R. (2002). Stochastic Portfolio Theory.Springer-Verlag, New York.
I Fernholz, E.R., Karatzas, I. & Kardaras, C.(2005). Diversity and arbitrage in equity markets. Finance &Stochastics 9, 1-27.
I Fernholz, E.R. & Karatzas, I. (2005). Relativearbitrage in volatility-stabilized markets. Annals of Finance 1,149-177.
I Karatzas, I. & Kardaras, C. (2007). The numeraireportfolio and arbitrage in semimartingale markets. Finance &Stochastics 11, 447-493.
I Fernholz, E.R. & Karatzas, I. (2009) StochasticPortfolio Theory: An Overview. Handbook of NumericalAnalysis, volume “Mathematical Modeling and NumericalMethods in Finance” (A. Bensoussan, ed.) 89-168.
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1. THE FRAMEWORK
Equity market framework (Bachelier, Samuelson...) of the form
dB(t) = B(t)r(t) dt, B(0) = 1, (1)
dXi (t) = Xi (t)
(βi (t) dt +
N∑ν=1
σiν(t) dWν(t)
), i = 1, . . . , n.
Money-market B(·) and n stocks with strictly positivecapitalizations X1(·), · · · ,Xn(·).
Driven by the Brownian motion W (·) = (W1(·), · · · , WN(·))′ withN ≥ n. Probability space (Ω,F ,P) .
All processes are assumed to be progressively measurable withrespect to a filtration F = F(t)0≤t<∞ which represents the“flow of information” in the market.Not much needs to be assumed at this point about it... .
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We shall take r(·) ≡ 0 until further notice: investing in themoney-market will amount to hoarding, whereas borrowing fromthe money-market will incur no interest.
Arithmetic mean rates of return β(·) = (β1(·), . . . , βn(·))′
and Variance rates (αii (·))1≤i≤n satisfy for every T ∈ (0,∞)the integrability condition
n∑i=1
∫ T
0
(∣∣βi (t)∣∣+ αii (t)
)dt <∞ , a.s.
Here σ(·) = (σij(·))1≤i≤n, 1≤j≤N is the (n × N)−matrix of localvolatility rates, and α(·) = σ(·)σ′(·) is the (n × n)−matrix ofCovariance rates
αij(t) :=N∑ν=1
σiν(t)σjν(t) =1
Xi (t)Xj(t)· d
d t〈Xi ,Xj〉(t) .
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2. STRATEGIES and PORTFOLIOS
A small investor decides, at each time t and for every 1 ≤ i ≤ n,which proportion πi (t) of his current wealth V (t) to invest in thei th stock.
We require that each πi (t) be F(t)−measurable. Theproportion 1−
∑ni=1 πi (t) gets invested in the money market.
The wealth V (·) ≡ V v ,π(·) corresponding to an initial capitalv ∈ (0,∞) and an investment strategy π(·) =
(π1(·), · · · , πn(·)
)′satisfies V (0) = v and the linear Markowitz equation
dV (t)
V (t)=
n∑i=1
πi (t)dXi (t)
Xi (t)+
(1−
n∑i=1
πi (t)
)dB(t)
B(t)
= π′(t)[β(t)dt + σ(t)dW (t)
].
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Equivalently,
dV (t)
V (t)= βπ(t)dt +
N∑ν=1
σπν (t)dWν(t)
where
βπ(t) :=n∑
i=1
πi (t)βi (t) , σπν (t) :=n∑
i=1
πi (t)σiν(t) ,
are, respectively, the portfolio’s arithmetic rate-of-return, and theportfolio’s volatilities.
• We shall call investment strategy an F−progressivelymeasurable process π : [0,∞)× Ω→ Rn which satisfies for eachT ∈ (0,∞) the integrability condition∫ T
0
( ∣∣βπ(t)∣∣+(σπ(t)
)′σπ(t)
)dt < ∞ , a.s.
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The collection of all investment strategies will be denoted by Π .The wealth process corresponding to an investment strategyπ(·) ∈ Π and an initial capital v > 0 is strictly positive:
V v ,π(t) = v exp
∫ t
0
(βπ(s)− 1
2
(σπ(s)
)′σπ(s)
)ds
+
∫ t
0
(σπ(s)
)′dW (s)
> 0 .
• An investment strategy π(·) ∈ Π with
n∑i=1
πi (t) = 1 , ∀ 0 ≤ t <∞
almost surely, will be called portfolio.
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A portfolio never invests in the money market, and never borrowsfrom it. We shall say that a portfolio is bounded, if there exists areal constant K > 0 such that ‖π(t, ω)‖ ≤ K holds for all(t, ω) ∈ [0,∞)× Ω .
The collection of all portfolios will be denoted by by P .
We shall call π(·) a long-only portfolio, if it satisfies almost surely
π1(t) ≥ 0 , · · · , πn(t) ≥ 0 , ∀ 0 ≤ t <∞ .
Every long-only portfolio is also bounded.
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3. THE MARKET PORTFOLIO
Consider now the market portfolio µ(·) =(µ1(·), · · · , µn(·)
)′given by
µi (t) :=Xi (t)
X (t), i = 1, . . . , n ,
whereX (t) := X1(t) + . . .+ Xn(t) .
This invests in all stocks in proportion to their relativecapitalization weights.
Such an investment amounts to “owning the entire market”:the wealth process becomes
V v ,µ(·) = vX (·)/X (0) .
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4. RELATIVE ARBITRAGE
Given a real number T > 0 and any two investment strategiesπ(·) ∈ Π and ρ(·) ∈ Π , we shall say that π(·) is an arbitragerelative to ρ(·) over [0,T ], if we have
P(V 1,π(T ) ≥ V 1,ρ(T )
)= 1 and P
(V 1,π(T ) > V 1,ρ(T )
)> 0 .
• Strong relative arbitrage: P(V 1,π(T ) > V 1,ρ(T )
)= 1 .
• No arbitrage is possible relative to a strategy %∗(·) ∈ Π that hasthe so-called “numeraire property”:
V 1,π(·)/V 1,%∗(·) is a supermartingale, for every π(·) ∈ Π .
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4.a: Market Price of Risk (Optional)
Suppose there exists a market price of risk (or “relative risk”)ϑ : [0,∞)× Ω→ RN , an F−progressively measurable processthat satisfies for each T ∈ (0,∞) the requirements
σ(t)ϑ(t) = β(t) , ∀ 0 ≤ t ≤ T and
∫ T
0‖ϑ(t)‖2 dt <∞ .
• Whenever it exists, this ϑ(·) allows us to introduce a corres-ponding “delfator” Zϑ(·) ≡ Z (·) . This is an exponential localmartingale and supermartingale
Z (t) := exp
−∫ t
0ϑ′(s) dW (s)− 1
2
∫ t
0‖ϑ(s)‖2 ds
, 0 ≤ t <∞ .
A martingale, if and only if E(Z (T )) = 1 , ∀ T ∈ (0,∞).
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The existence of this deflator proscribes immediate (or egregious,or scalable) arbitrage opportunities, a.k.a. UP’s BR (UnboundedProfits with Bounded Risk).
• For the purposes of these talks, as we shall see, it is importantto allow Z (·) to be a strict local martingale, that is, not to exclu-de the possibility E(Z (T )) < 1 for some (possibly all...) horizonsT ∈ (0,∞). This means, we still keep the door open to the exist-ence of arbitrage opportunities that cannot be scaled, that is tosay, to some (small) profits with bounded risk.
• Recall that the wealth process corresponding to an investmentstrategy π(·) ∈ Π and an initial capital v > 0 is strictly positive:
V v ,π(t) = v exp
∫ t
0
(βπ(s)− 1
2
(σπ(s)
)′σπ(s)
)ds
+
∫ t
0
(σπ(s)
)′dW (s)
> 0 .
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Whereas, in the presence of a market of risk process ϑ(·) we havein addition
dV v ,π(t)
V v ,π(t)= π′(t)σ(t)
[dW (t) + ϑ(t)dt
].
Let us pair this with the equation
dZ (t) = −Z (t)ϑ′(t)dW (t)
for the deflator Z (·)
Z (·) = exp
−∫ ·
0ϑ′(t) dW (t)− 1
2
∫ ·0‖ϑ(t)‖2 dt
we introduced a couple of slides ago. We see that the “deflatedwealth process” Z (·)V v ,π(·) is also a positive local martingale anda supermartingale for every π(·) ∈ Π , v > 0, namely
Z (t)V v ,π(t) = v +
∫ t
0Z (s)V v ,π(s)
(σ′(s)π(s)− ϑ(s)
)′dW (s) .
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• Suppose that the covariance process α(·) satisfies, for somereal number L > 0 , the a.s. boundedness condition
ξ′α(t)ξ = ξ′σ(t)σ′(t)ξ ≤ L‖ξ‖2, ∀ t ∈ [0,∞) , ξ ∈ Rn . (2)
If π(·) is arbitrage relative to ρ(·) and both are bounded portfo-lios, then Z (·) and Z (·)V v ,ρ(·) are strict local martingales:
E[Z (T )
]< 1 , E [ Z (T )V v ,ρ(T ) ] < v .
I In particular, if there exists a bounded portfolio π(·) which isarbitrage relative to µ(·) , we have
E[Z (T )
]< 1 , E [ Z (T )X (T ) ] < X (0) , E [ Z (T )Xi (T ) ] < Xi (0) .
Relative arbitrage becomes a “machine” for generating strict localmartingales.
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5. REMARKS
• Suppose there exists a real constant h > 0 for which we have
n∑i=1
µi (t)αii (t)−n∑
i=1
n∑i=1
µi (t)αij(t)µj(t) ≥ h , ∀ 0 ≤ t <∞ .
(3)Under this condition, we shall see that, for a sufficiently large realconstant c > 0 , the long-only modified entropic portfolio
E(c)i (t) =
µi (t)(c − logµi (t)
)∑nj=1 µj(t)
(c − logµj(t)
) , i = 1, · · · , n
(4)is a strong arbitrage relative to the market portfolio µ(·) over anytime-horizon [0,T ] with T > (2 log n)/h .
• It is still an open question, whether such relative arbitrage canbe constructed over arbitrary time-horizons under (3).
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• Another condition guaranteeing the existence of arbitragerelative to the market is, as we shall see, that there exist a realconstant h > 0 with
(µ1(t) · · ·µn(t)
)1/n
n∑i=1
αii (t)− 1
n
n∑i=1
n∑j=1
αij(t)
≥ h , ∀ t ≥ 0 .
(5)
Then with M(t) := (µ1(t) · · ·µn(t))1/n and for c > 0 largeenough, the long-only modified equally-weighted portfolio
ϕ(c)i (t) =
c
c + M(t)· 1
n+
M(t)
c + M(t)· µi (t) , i = 1, · · · , n ,
(6)(a convex combination of equal-weighting and the market),is strong arbitrage relative to µ(·) over any time horizon [0,T ]with T > (2 n1−(1/n))/h .
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• Consider now the a.s. strong non-degeneracy condition
ξ′α(t)ξ = ξ′σ(t)σ′(t)ξ ≥ ε‖ξ‖2, ∀ t ∈ [0,∞) , ξ ∈ Rn (7)
for some real number ε > 0 , on the covariance process α(·) .
Suppose the conditions (2), (3), namely
ξ′α(t)ξ = ξ′σ(t)σ′(t)ξ ≤ L‖ξ‖2, ∀ t ∈ [0,∞) , ξ ∈ Rn ,
n∑i=1
µi (t)αii (t)−n∑
i=1
n∑i=1
µi (t)αij(t)µj(t) ≥ h , ∀ 0 ≤ t <∞
and (7) all hold.
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Then, as we shall see, for any given constant p ∈ (0, 1) , thelong-only diversity-weighted portfolio
D(p)i (t) =
(µi (t))p∑nj=1(µj(t))p
, i = 1, · · · , n (8)
is again a strong arbitrage relative to the market portfolio oversufficiently long time-horizons.
• Appropriate modifications of this diversity-weightedportfolio do yield such arbitrage over any time-horizon [0,T ].This takes some work to prove. And the shorter the time-horizon,the bigger the amount of initial capital that is required to achievethe extra basis point’s worth of outperformance:
v ≥ v(T ) ≡ q(T )
(µ1(0))q(T )−1 , q(T ) := 1+
(2/ε δ T
)log(1/µ1(0)
).
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• Please note that these portfolios (entropic, equally-weighted,modified equally-weighted, diversity-weighted) are determinedentirely from the market weights µ1(t), · · · , µn(t) .
These market weights are perfectly easy to observe and tomeasure.
• Construction of these portfolios does not assume any knowledgeabout the exact structure of market parameters, such as the meanrates of return βi (·)’s, or the local covariance rates αij(·)’s.
To put it a bit more colloquially: does not require us to take theseparticular features of the model “too seriously”. Only as a general“framework”... .
To put it in the parlance of finance practice: these portfolios arecompletely “passive” (their construction requires neither estima-tion nor optimization).
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6. GROWTH RATES
• Equivalent way of representing the positive Ito process X (·) of(1), namely
dXi (t) = Xi (t)
(βi (t) dt +
N∑ν=1
σiν(t) dWν(t)
), i = 1, . . . , n ,
for the stock-price
d(
log Xi (t))
= γi (t) dt +N∑ν=1
σiν(t) dWν(t)︸ ︷︷ ︸ ,with the logarithmic mean rate of return for the i th stock
γi (t) := βi (t)− 1
2αii (t) .
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EXAMPLE
Stock XYZ doubles in good years (+100%) and halves in bad years(-50%). Years good and bad alternate independently and equallylikely (to wit, with probability 0.50), thus
β =1
2(+100%) +
1
2(−50%) =
1
2− 1
4= 0.25 ,
γ =1
2(log 2) +
1
2
(log
1
2
)= 0 .
On the other hand, log 2 ' 0.7 , so the variance is
α = σ2 =1
2(0.7)2 +
1
2(−0.7)2 ' 0.50 ,
and indeed
(0.25) = 0 + (1/2)(0.50) or β = γ + (1/2)α .
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• This logarithmic rate of return can be interpreted also as agrowth-rate, in the sense that
limT→∞
1
T
(log Xi (t)−
∫ T
0γi (t)dt
)= 0 a.s.
holds, under the assumption αii (·) ≤ L <∞ on the variance ofthe stock; recall
γi (t) := βi (t)− 1
2αii (t) .
More generally, under the condition
limT→∞
(log log T
T 2
∫ T
0αii (t)dt
)= 0 , a.s.
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• Similarly, the solution of the linear equation
dV (t)
V (t)=
n∑i=1
πi (t)dXi (t)
Xi (t)+
(1−
n∑i=1
πi (t)
)dB(t)
B(t)
= π′(t)[β(t)dt + σ(t) dW (t)
]for the wealth V (·) ≡ V v ,π(·) corresponding to an initial capitalv ∈ (0,∞) and portfolio π(·) = (π1(·), · · · , πn(·))′ , is given as
d(
log V v ,π(t))
= γπ(t) dt +N∑ν=1
σπν (t) dWν(t) (9)
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Portfolio growth-rate and volatilities
γπ(t) :=n∑
i=1
πi (t)γi (t) + γπ∗ (t) , σπν (t) :=n∑
i=1
πi (t)σiν(t).
Excess growth-rate
γπ∗ (t) :=1
2
n∑i=1
πi (t)αii (t)−n∑
i=1
n∑j=1
πi (t)αij(t)πj(t)
︸ ︷︷ ︸
.
Portfolio variance
aππ(t) :=N∑ν=1
(σπν (t))2 =n∑
i=1
n∑j=1
πi (t)αij(t)πj(t) .
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7. RELATIVE COVARIANCE STRUCTURE
• Variance/Covariance Process, not in absolute terms, but relativeto the portfolio π(·):
Aπij(t) :=d∑ν=1
(σiν(t)− σπν (t)
)(σjν(t)− σπν (t)) , 1 ≤ i , j ≤ n .
If the covariance matrix α(t) is positive-definite, than the relativecovariance matrix
Aπ(t) = Aπij(t) 1≤i , j≤n
has rank n − 1 and its null space is spanned by the vector π(t) .
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• The excess growth-rate
γπ∗ (t) :=1
2
n∑i=1
πi (t)αii (t)−n∑
i=1
n∑j=1
πi (t)αij(t)πj(t)
has, for any two portfolios π(·) ∈ P , ρ(·) ∈ P , the invarianceproperty
γπ∗ (t) = 12
n∑i=1
πi (t)Aρii (t)−n∑
i=1
n∑j=1
πi (t)Aρij(t)πj(t)
,
and consequently (read the above with ρ(·) ≡ π(·) , and recall thatthe null space of the relative covariance matrix Aπij(t)1≤i ,j≤n isspanned by π(t)):
γπ∗ (t) =1
2
n∑i=1
πi (t)Aπii (t) .
In particular, we have γπ∗ (·) ≥ 0 for a long-only portfolio.28 / 111
• Now let us consider the market portfolio π ≡ µ . The excessgrowth rate of the market portfolio
γµ∗ (t) = 12
( n∑i=1
µi (t)αii (t)−n∑
i=1
n∑i=1
µi (t)αij(t)µj(t))
can then be interpreted as a measure of intrinsic volatilityavailable in the market:
γµ∗ (t) = 12
n∑i=1
µi (t)Aµii (t) ,
where
µi (t) :=Xi (t)
X (t), σµν (t) :=
n∑i=1
µi (t)σiν(t) ,
Aµij (t) :=d∑ν=1
(σiν(t)− σµν (t)
)(σjν(t)− σµν (t)) =
d〈µi , µj〉(t)
µi (t)µj(t)dt.
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Thus the excess growth rate of the market portfolio
γµ∗ (t) = 12
n∑i=1
µi (t)Aµii (t)
is also a weighted average, according to capitalization, of thevariances of individual stocks – not in absolute terms, but relativeto the market. This quantity will be very important in what follows.
• Related to the dynamics of the log-market-weights
d(logµi (t)) =(γi (t)− γµ(t)
)dt +
N∑ν=1
(σiν(t)− σµν (t)
)dWν(t)
for all stocks i = 1, . . . , n .
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Equivalently, in arithmetic terms
dµi (t)
µi (t)=(γi (t)− γµ(t) +
1
2Aµij (t)
)dt
+N∑ν=1
(σiν(t)− σµν (t)
)dWν(t) . (10)
It is now clear from this, that
d〈µi , µj〉(t)
µi (t)µj(t)dt=
d∑ν=1
(σiν(t)− σµν (t)
)(σjν(t)− σµν (t))
=d
dt〈logµi , logµj〉(t) = Aµij (t) .
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TWO-STOCK EXAMPLE (or “PARABLE”)
Suppose there are only two, perfectly negatively correlated, stocksA and B. We toss a fair coin, independently from day to day; whenthe toss comes up heads, stock A doubles and stock B halves inprice (and vice-versa, if the toss comes up tails).
Clearly, each stock has a growth rate of zero: holding any one ofthem produces nothing in the long term.
• What happens if we hold both stocks? Suppose we invest $100in each; one of them will rise to $200 and the other fall to $50, fora guaranteed total of $250, representing a net gain of 25%; thewinner has gained more than the loser has lost.
If we rebalance to $125 in each stock (so as to maintain the equalproportions we started with), and keep doing this day after day, welock in a long-term growth rate of 25%.
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Indeed, taking n = 2 and
γ1 = γ2 = 0 , α11 = α22 = −α12 = −α21 = 0.50
andπ1 = π2 = 0.50
in
γπ =n∑
i=1
πi γi +1
2
n∑i=1
πi αii −n∑
i=1
n∑j=1
πi αij πj
=
1
2
(π1
(1− π1
)α11 + π2
(1− π2
)α22
)− π1π2α12
we get the same growth rate that we computed a moment ago:
γπ = γπ∗ = 0.25 .
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MORAL OF THIS TALE
• In the presence of “sufficient intrinsic volatility”, setting targetweights and rebalancing to them can capture this volatility andturn it into growth.
(And this can occur even without precise estimates of modelparameters, and without refined optimization.)
We have encountered several variations on this parable already,and will encounter a few more below. In particular, we shallquantify what “sufficient intrinsic volatility” means.
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8. PORTFOLIO DIVERSIFICATION AND MARKETVOLATILITY AS DRIVERS OF GROWTH
Now let us suppose that, for some real number ε > 0 , condition(7) holds:
ξ′α(t)ξ = ξ′σ(t)σ′(t)ξ ≥ ε‖ξ‖2, ∀ t ∈ [0,∞) , ξ ∈ Rn .
That is, we have a strictly nondegenerate covariance structure.Then an elementary computation shows
γπ(t)−n∑
i=1
πi (t)γi (t) = γπ∗ (t) ≥(ε/2)(
1− max1≤i≤n
πi (t))≥(ε/2)η > 0 ,
as long as for some η ∈ (0, 1) we have
max1≤i≤n
πi (t) ≤ 1− η .
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To wit, such a portfolio’s growth rate γπ(t) will dominate, andstrictly, the average growth rate of the constituent assets
n∑i=1
πi (t) γi (t)
(Fernholz & Shay, Annals of Finance (1982)):
γπ(t) ≥n∑
i=1
πi (t)γi (t) +(ε/2)η .
In words: Under the above condition of “sufficient volatility”, eventhe slightest bit of portfolio diversification can not only decreasethe portfolio’s variance, as is well known, but also enhance itsgrowth.
We shall see below additional incarnations of this principle.
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¶ To see just how significant such an enhancement can be,consider any fixed-proportion, long-only portfolio π(·) ≡ p , forsome vector p ∈ ∆n with 1−max1≤i≤n pi =: η > 0 , and with
∆n :=
(p1, · · · , pn) : p1 ≥ 0, · · · , pn ≥ 0 , p1 + · · ·+ pn = 1.
For any portfolio π(·) and T ∈ (0,∞) , we have the identity
log
(V 1,π(t)
V 1,µ(t)
)=
∫ T
0γπ∗ (t)dt +
n∑i=1
∫ T
0πi (t)d logµi (t) .
(11)(This is a rather simple consequence of
d(
log V v ,π(t))
= γπ(t)dt +N∑ν=1
σπν (t) dWν(t)
in (9), slide 25.)
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From
log
(V 1,π(t)
V 1,µ(t)
)=
∫ T
0γπ∗ (t) dt +
n∑i=1
∫ T
0πi (t) d logµi (t) ,
of the previous slide, we get the a.s. comparisons
1
Tlog
(V 1,p(T )
V 1,µ(T )
)−
n∑i=1
piT
log
(µi (T )
µi (0)
)=
=1
T
∫ T
0γp∗(t)dt ≥ εη
2> 0 .
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Suppose now that the market is coherent, meaning that noindividual stock crashes relative to the rest of the market:
limT→∞
1
Tlogµi (T ) = 0 , ∀ i = 1, · · · , n .
Then passing to the limit as T →∞ in
1
Tlog
(V 1,p(T )
V 1,µ(T )
)−
n∑i=1
piT
log
(µi (T )
µi (0)
)≥ εη
2> 0
we see that the wealth corresponding to any such fixed-proportion,long-only portfolio, grows exponentially at a rate strictly higherthan that of the overall market:
lim infT→∞
1
Tlog
(V 1,p(T )
V 1,µ(T )
)≥ εη
2> 0 , a.s.
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Remark:
Tom Cover’s (1991) “universal portfolio”
Πi (t) :=
∫∆n pi V 1,p(t) dp∫
∆n V 1,p(t) dp, i = 1, · · · , n
has value
V 1,Π(t) =
∫∆n V 1,p(t) dp∫
∆n dp∼ max
p∈∆nV 1,p(t) .
Note the “total agnosticism” of this portfolio regarding the detailsof the particular, underlying model.
z Up to now we have not even tried to select portfolios in an“optimal” fashion. Here then are some Portfolio Optimizationproblems; some of them are classical, while for others very little isknown.
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9. PORTFOLIO OPTIMIZATION
Problem #1: Quadratic criterion, linear constraint(Markowitz, 1952). Minimize the portfolio variance
aππ(t) =n∑
i=1
n∑j=1
πi (t)αij(t)πj(t) (12)
among all portfolios π(·) ∈ P that keep the rate-of-return at leastequal to a given constant: βπ(t) =
∑ni=1 πi (t)βi (t) ≥ β0 .
Problem #2: Quadratic criterion, quadratic constraint.Minimize the portfolio variance aππ(t) of (12) among all portfoliosπ(·) ∈ P with growth-rate at least equal to a given constant γ0 :
n∑i=1
πi (t)βi (t) ≥ γ0 + 12
n∑i=1
n∑j=1
πi (t)αij(t)πj(t) .
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Problem #3: Maximize the probability of reaching a given“ceiling” c before reaching a given “floor” f , with0 < f < 1 < c <∞ .
More specifically, maximize over π(·) ∈ P the probability
P [Tπc < Tπf ] , with Tπc := inf t ≥ 0 : X π(t) = c .
In the case of constant coefficients γi and αij , and with
Γn the collection of vectors p ∈ Rn with p1 + · · ·+ pn = 1 ,
the solution to this problem is given by the vector π ∈ Γn thatmaximizes the mean-variance, or signal-to-noise, ratio (Pestien &Sudderth, Mathematics of Operations Research 1985):
γπ
aππ=
∑ni=1 πi (γi + 1
2αii )∑ni=1
∑nj=1 πiαijπj
− 12 .
Open Question: How about (more) general coefficients?
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Problem #4: Maximize the probability P [Tπc < T ∧ Tπf ]of reaching a given “ceiling” c before reaching a given “floor” fwith 0 < f < 1 < c <∞ , by a given “deadline” T ∈ (0,∞).
Always with constant coefficients, suppose there is a vectorπ = (π1, . . . , πn)′ that maximizes both the signal-to-noise ratioand the variance,
γπ
aππ=
∑ni=1 πi (γi + 1
2αii )∑ni=1
∑nj=1 πi αij πj
− 1
2and aππ =
n∑i=1
n∑j=1
πi αij πj ,
over all π1 ≥ 0, . . . , πn ≥ 0 with∑n
i=1 πi = 1 .
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Then the portfolio π is optimal for the above criterion (Sudderth& Weerasinghe, Mathematics of Operations Research, 1989).
This is a huge assumption; it is satisfied, for instance, under the(very stringent) condition that, for some β ≤ 0 , we have
βi = γi + 12 αii = β , for all i = 1, . . . , n .
Open Question: As far as I can tell, nobody seems to know thesolution to this problem, if such “simultaneous maximization” isnot possible.
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Problem #5: Minimize over portfolios π(·) the expected timeE [Tπc ] until a given “ceiling” c ∈ (1,∞) is reached.
Again with constant coefficients, it turns out that it is enough tomaximize, over vectors π ∈ Rn with
∑ni=1 πi = 1 , the drift in the
equation for log X π(·), namely the portfolio growth-rate
γπ =n∑
i=1
πi(γi + 1
2αii
)− 1
2
n∑i=1
n∑j=1
πiαijπj .
(See Heath, Orey, Pestien & Sudderth, SIAM Journal on Control& Optimization, 1987.)
Again, how about (more) general coefficients?
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Problem #6: Suppose we can find a portfolio %∗(·) ∈ P thatsatisfies(p − %∗(t)
)′(β(t)− α(t) %∗(t)
)≤ 0 , ∀ p ∈ Γn , 0 ≤ t <∞
Then for every π(·) ∈ P we have the numeraire-like property
V 1,π(·) /V 1,%∗(·) is a supermartingale,
as well as
lim supT→∞
1
Tlog
(V 1,π(T )
V 1,%∗(T )
)≤ 0 , a.s.,
E[
log V 1,π(T )]≤ E
[log V 1,%∗(T )
], ∀ T ∈ (0,∞) .
Unlike Cover’s universal (and “agnostic”) portfolio, this one needsconsiderable γνωσις (knowledge) of both the covariance structureof the market and its growth rates.
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Discussion:
In none of these problems did we need to assume the existence ofan equivalent martingale measure – or even of a deflator Z (·) .(As Constantinos Kardaras showed in his dissertation, thesolvability of quite general utility maximization problems onlyneeds the existence of a numeraire portfolio %∗(·).)
In most of them, we needed to “take our model quite seriously”, tothe extent that the solution assumed knowledge (γνωσις) of boththe covariance structure of the market and of the assets’ growthrates.
And in some (rather special) such problems, the solution onlyneeds estimates of the covariance structure of the market – not atrivial task, but much easier than estimating growth rates ofindividual assets.
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FUNCTIONALLY GENERATED PORTFOLIOS
Let us recall the expression
log
(V 1,π(t)
V 1,µ(t)
)=
∫ T
0γπ∗ (t)dt +
n∑i=1
∫ T
0πi (t)d logµi (t)
of (11) for the relative performance of an arbitrary portfolio π(·)with respect to the market.
There is a class of very special portfolios π(·) , described solely interms of the market weights µ1(·), . . . , µ1(·) and nothing else, forwhich the stochastic integrals disappear completely from theright-hand side and the remaining (Lebesgue) also depend solelyon market weights and have a particular sign. This allows forpathwise comparisons of relative performance; or, to put it a bitdifferently, for the construction of arbitrage relative to the market,under appropriate conditions.
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We start with a concave, smooth function S : ∆n+ → R+ , and
consider the portfolio πS(·) generated by it:
πSi (t) := µi (t)
Di log S(µ(t)) + 1−n∑
j=1
µj(t) · Dj log S(µ(t))
.
(13)Then an application of Ito’s rule gives the “master equation”
log
(V πS
(T )
V µ(T )
)= log
(S(µ(T ))
S(µ(0))
)+
∫ T
0g(t) dt . (14)
Here, thanks to our assumptions, the quantity g(·) is nonnegative:
g(t) :=−1
S(µ(t))
∑i
∑j
D2ijS(µ(t)) · µi (t)µj(t)Aµij (t) . (15)
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πSi (t) := µi (t)
Di log S(µ(t)) + 1−n∑
j=1
µj(t) · Dj log S(µ(t))
g(t) :=
−1
S(µ(t))
∑i
∑j
D2ijS(µ(t)) · µi (t)µj(t)Aµij (t)
Recall the entries of the relative covariance matrix:
Aµij (t) =d∑ν=1
(σiν(t)− σµν (t)
)(σjν(t)− σµν (t)) =
d〈µi , µj〉(t)
µi (t)µj(t)dt.
Significance: Stochastic integrals have been excised in (14) , i.e.,
log
(V πS
(T )
V µ(T )
)= log
(S(µ(T ))
S(µ(0))
)+
∫ T
0g(t) dt ,
and we can begin to make comparisons that are valid withprobability one (a.s.)...
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Proof of the “Master Equation” (14): To ease notation we set
hi (t) := Di log S(µ(t)) and N(t) :=n∑
j=1
µj(t) hj(t) ,
so (13), that is
πi (t) = µi (t)
Di log S(µ(t)) + 1−n∑
j=1
µj(t) · Dj log S(µ(t))
,
reads:
πi (t) =(hi (t) + N(t)
)µi (t) , i = 1, · · · n .
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Then the terms on the right-hand side of
d log
(V π(t)
V µ(t)
)=
n∑i=1
πi (t)
µi (t)dµi (t)−1
2
n∑i=1
n∑j=1
πi (t)πj(t)Aµij (t)
dt ,
an equivalent version of
log
(V 1,π(t)
V 1,µ(t)
)=
∫ T
0γπ∗ (t) dt +
n∑i=1
∫ T
0πi (t) d logµi (t)
in (11), become
n∑i=1
πi (t)
µi (t)dµi (t) =
n∑i=1
hi (t) dµi (t) + N(t) · d( n∑
i=1
µi (t))
=n∑
i=1
hi (t) dµi (t) ,
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whereas∑n
i=1
∑nj=1 πi (t)πj(t)Aµij (t) becomes
=n∑
i=1
n∑j=1
(hi (t)+N(t)
)(hj(t)+N(t)
)µi (t)µj(t)Aµij (t)
=n∑
i=1
n∑j=1
hi (t)hj(t)µi (t)µj(t)Aµij (t) .
(Again, because µ(t) spans the null subspace of Aµij (t)1≤i ,j≤n .)Thus, using the dynamics of market weights in (10), the aboveequation gives
d log
(V π(t)
V µ(t)
)=
n∑i=1
hi (t) dµi (t)
− 1
2
n∑i=1
n∑j=1
hi (t)hj(t)µi (t)µj(t)Aµij (t) dt . (16)
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On the other hand, we have
D2ij log S(x) =
(D2ijS(x)/S(x)
)− Di log S(x) · Dj log S(x) ,
so we get
d log S(µ(t)) =n∑
i=1
hi (t) dµi (t)+1
2
n∑i=1
n∑j=1
D2ij log S(µ(t))d〈µi , µj〉(t)
=n∑
i=1
hi (t)dµi (t)
+1
2
n∑i=1
n∑j=1
(D2ijS(µ(t))
S(µ(t))−hi (t)hj(t)
)µi (t)µj(t)Aµij (t)dt
by Ito’s rule. Comparing this last expression with (16) and recallingthe notation of (15), we deduce (14), namely:
d log S(µ(t) = d log (V π(t)/V µ(t))− g(t)dt .
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For instance:
I S(·) ≡ w , a positive constant, generates the market portfolio.
I S(x) = w1x1 + · · ·+ wnxn generates the passive portfolio thatbuys at time t = 0, and holds up until time t = T , a fixednumber of shares wi in each asset i = 1, · · · , n.
(The market portfolio corresponds to the special case
w1 = · · · = wn = w
of equal numbers of shares across assets.)
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I The geometric mean
S(x) ≡ G(x) := (x1 · · · xn)1/n
generates the equal weighted portfolio
ϕi (·) ≡ 1/n , i = 1, · · · , n ,
with drift equal to the excess growth rate:
gϕ(·) ≡ γ∗ϕ(·) =1
2n
n∑i=1
αii (·)−1
n
n∑i=1
n∑j=1
αij(·)
.
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Discussion on Equal Weighting:
The equal weighted portfolio ϕ(·) maintains the same weights inall stocks at all times; it accomplishes this by selling those stockswhose price rises relative to the rest, and by buying stocks whoseprice falls relative to the others. Because of this built-in aspect of“buying-low-and-selling-high”, the equally weighted portfolio canbe used as a simple prototype for studying systematically theperformance of statistical arbitrage strategies in equity markets;see Fernholz & Maguire (2006) for details.
It has been observed empirically, that such a portfolio canoutperform the market (we shall see a rigorous result along theselines in a short while). Of course, implementing such a strategynecessitates very frequent trading and can incur substantialtransaction costs for an investor who is not a broker/dealer.
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It can also involve considerable risk: whereas the second term onthe right-hand side of
log V 1,ϕ(T ) =1
nlog
(X1(T ) · · ·Xn(T )
X1(0) · · ·Xn(0)
)+
∫ T
0γ∗ϕ(t) dt ,
or of
log
(V 1,ϕ(T )
V 1,µ(T )
)=
1
nlog
(µ1(T ) · · ·µn(T )
µ1(0) · · ·µn(0)
)+
∫ T
0γ∗ϕ(t) dt ,
is increasing it T , the first terms on the right-hand sides of theseexpressions can fluctuate quite a bit.
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• The diversity-weighted portfolio D(p)(·) of
D(p)i (t) =
(µi (t))p∑nj=1(µj(t))p
, i = 1, · · · , n
with 0 < p < 1 , stands between these two extremes, of capitaliza-tion weighting (as in S&P 500) and of equal weighting (as in theValue-Line Index). It is generated by the concave function
S(p)(x) :=(xp
1 + · · ·+ xpn
)1/p,
and has drift proportional to the excess growth rate:
g(·) ≡ (1− p) γD(p)
∗ (·) .
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D(p)i (t) =
(µi (t))p∑nj=1(µj(t))p
, i = 1, · · · , n
With p = 0 this becomes equal weighting ϕi (·) ≡ 1/n, 1 ≤ i ≤ n.With p = 1 we get the market portfolio µ(·) .
This portfolio over-weighs the small-cap stocks and under-weighsthe large-cap stocks, relative to the market weights (the valleys areexalted, at least somewhat, and the mountains and hills madelow (er).). It tries to capture some of the “buy-low/sell-high”characteristics of equal weighting, but without deviating too muchfrom market capitalizations and without incurring a lot of tradingcosts or excessive risk.
It can be viewed as an “enhanced market portfolio” or “enhancedcapitalization index”, in this sense.
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I Another way to “bridge” the extremes of equal-weighting andcapitalization-weighting goes as follows. Consider thegeometric mean
G(x) := (x1 · · · xn)1/n
and, for any given c ∈ (0,∞), its modification
Gc(x) := c+G(x), which satisfies: c < Gc(x) ≤ c+(1/n) .
This modified geometric mean function generates themodified equally-weighted portfolio
ϕ(c)i (t) =
c
c + G(µ(t))· 1
n+
G(µ(t))
c + G(µ(t))· µi (t) ,
for i = 1, · · · , n that we saw already in (6). These weightsare convex combination of the equal-weighting and marketportfolios; and we have
gϕ(c)
(t) =G(µ(t))
c + G(µ(t))γ∗ϕ(t) .
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I In a similar spirit, consider the entropy function
H(x) := −n∑
i=1
xi log xi , x ∈ ∆n+ .
This entropy function generates the entropic portfolio E(·),with weights
Ei (t) =−µi (t) log µi (t)
H(µ(t)), i = 1, · · · , n
and drift-process
gE(t) =γ∗µ(t)
H(µ(t)).
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I Now take again the entropy function
H(x) := −n∑
i=1
xi log xi , x ∈ ∆n+
and, for any given c ∈ (0,∞), look at its modification
Sc(x) := c + H(x), which satisfies: c < Sc(x) ≤ c + log n .
This modified entropy function generates the modifiedentropic portfolio E(c)(·) of (4), with weights
E(c)i (t) =
µi (t)(c − logµi (t)
)c + H(µ(t))
, i = 1, · · · , n
and drift-process given by
gE(c)
(t) =γ∗µ(t)
c + H(µ(t)).
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11. SUFFICIENT INTRINSIC VOLATILITY LEADS TOARBITRAGE RELATIVE TO THE MARKET
Broadly accepted practitioner wisdom upholds that sufficientvolatility creates growth opportunities in a financial market.
We have already encountered an instance of this phenomenon insection 8: we saw there that, in the presence of a strongnon-degeneracy condition on the market’s covariance structure,“reasonably diversified” long-only portfolios with constant weightsrepresent superior long-term growth opportunities relative to theoverall market.
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We shall examine in Proposition 1 below another instance of thisphenomenon.
More precisely, we shall try again to put the above intuition on aprecise quantitative basis, by identifying now the excess growthrate of the market portfolio – which also measures the market’sintrinsic volatility – as a driver of growth; to wit, as a quantitywhose “availability” or “sufficiency” (boundedness away from zero)can lead to opportunities for strong arbitrage and for superiorlong-term growth, relative to the market.
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Proposition 1: Assume that over [0,T ] there is “sufficientintrinsic volatility” (excess growth):∫ T
0γµ∗ (t)dt ≥ hT , or γµ∗ (t) ≥ h , 0 ≤ t ≤ T
holds a.s., for some constant h > 0 . Take
T > T∗ :=H(µ(0))
h, and H(x) := −
n∑i=1
xi log xi
the entropy function. Then the modified entropic portfolio (from acouple of slides ago)
E(c)i (t) :=
µi (t) (c − logµi (t))∑nj=1 µj(t) (c − logµj(t))
, i = 1, · · · , n
is generated by the function
Sc(x) := c + H(x)
on ∆n+ ; and for c > 0 sufficiently large, it represents a strong
arbitrage relative to the market.66 / 111
• Sketch of Argument for Proposition 1: Note that the functionSc(·) := c + H(·) is bounded both from above and below:
0 < c < Sc(x) ≤ c + log n , x ∈ ∆n+ .
The master equation now shows that
log
(V 1,E(c)
(T )
V 1,µ(T )
)= log
(c + H(µ(T ))
c + H(µ(0))
)+
∫ T
0gE
(c)(t)dt
is strictly positive, provided
T >1
h
(c + log n
)log(
1 +log n
c
)−→ log n
h
as c →∞ .
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This is because the first term on the right-hand side of
log
(V 1,E(c)
(T )
V 1,µ(T )
)= log
(c + H(µ(T ))
c + H(µ(0))
)+
∫ T
0gE
(c)(t)dt
dominates
− log( c + log n
c
)and, under the conditions of the proposition, the second term∫ T
0gE
(c)(t)dt = · · · =
∫ T
0
γµ∗ (·)c + H(·)
dt ≥∫ T
0
γµ∗ (·)c + log n
dt
dominates hT / (c + log n) .
To put it a bit differently: if you have a constant wind on yourback, sooner all later you’ll overtake any obstacle – e.g., theconstant log
((c + log n)/c
).
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This leads to strong arbitrage relative to the market, for sufficientlylarge T > 0 ; indeed to
P(
V 1,E(c)(T ) > V 1,µ(T )
)= 1 .
(Intuition, as before: you can generate such relative arbitrage ifthere is “enough intrinsic volatility” in the market... .)
Major Open Question: Is such relative arbitrage possibleover arbitrary time-horizons, under the conditions ofProposition 1 ?
We shall discuss below two special cases, where the answer to thisquestion is known – and is affirmative.
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0.0
0.5
1.0
1.5
2.0
2.5
1927 1932 1937 1942 1947 1952 1957 1962 1967 1972 1977 1982 1987 1992 1997 2002
YEAR
CUM
ULAT
IVE
EXCE
SS G
ROW
TH
Figure 1: Cumulative Excess Growth∫ ·
0γµ∗ (t)dt for the U.S. Stock
Market during the period 1926-1999.
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The previous figure plots the cumulative excess growth∫ ·
0 γ∗µ(t) dt
for the U.S. equities market over most of the twentieth century.Note the conspicuous bumps in the curve, first in the GreatDepression period in the early 1930s, then again in the “irrationalexuberance” period at the end of the century. The data used forthis chart come from the monthly stock database of the Center forResearch in Securities Prices (CRSP) at the University of Chicago.
The market we construct consists of the stocks traded on the NewYork Stock Exchange (NYSE), the American Stock Exchange(AMEX) and the NASDAQ Stock Market, after the removal of allREITs, all closed-end funds, and those ADRs not included in theS&P 500 Index. Until 1962, the CRSP data included only NYSEstocks. The AMEX stocks were included after July 1962, and theNASDAQ stocks were included at the beginning of 1973. Thenumber of stocks in this market varies from a few hundred in 1927to about 7500 in 2005.
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Proposition 2:Introduce the “modified intrinsic volatility”
ζ∗(t) :=(µ1(t) · · ·µn(t)
)1/n
[n∑
i=1
αii (t)−1
n
n∑i=1
n∑j=1
αij(t)
]
and assume that over the given horizon [0,T ] we have a.s.:∫ T
0
ζ∗(t)dt ≥ h T , or ζ∗(t) ≥ h , 0 ≤ t ≤ T
for some constant h > 0 . Then, with Q(t) := (µ1(t) · · ·µn(t))1/n and for sufficiently
large c > 0 , the modified equally-weighted portfolio of (6)
ϕ(c)i (t) =
c
c + Q(t)· 1
n+
Q(t)
c + Q(t)· µi (t) , i = 1, · · · , n ,
is an arbitrage relative to the market over [0,T ], provided T > (2n1−(1/n))/h .
The proof is rather similar to that of Proposition 1. The modifiedequally-weighted portfolio is generated by the function c + Q(·) ,and we use the “master formula” as before.
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12. NOTIONS OF MARKET DIVERSITY
Major Open Question: Is such relative arbitrage possibleover arbitrary time-horizons, under the conditions ofProposition 1 ?∫ T
0γµ∗ (t)dt ≥ hT , or γµ∗ (t) ≥ h , 0 ≤ t ≤ T .
Partial Answer #1: YES, if the variance/covariance matrixα(·) = σ(·)σ′(·) has all its eigenvalues bounded away from zeroand infinity: to wit, if we have (a.s.)
κ|| ξ||2 ≤ ξ′α(t)ξ ≤ K|| ξ||2 , ∀ t ≥ 0 , ξ ∈ Rd (17)
for suitable constants 0 < κ < K <∞ .
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In this case one can show (Bob Fernholz, Kostas Kardaras)
κ
2
(1− π(1)(t)
)≤ γπ∗ (t) ≤ 2K
(1− π(1)(t)
)(18)
for the maximal weight of any long-only portfolio π(·) , namely
π(1)(t) := max1≤i≤n
πi (t) .
Thus, under the structural assumption
κ|| ξ||2 ≤ ξ′α(t)ξ ≤ K|| ξ||2 , ∀ t ≥ 0 , ξ ∈ Rd
of (17), the “sufficient intrinsic volatility” (a.s.) condition ofProposition 1, namely
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∫ T
0γµ∗ (t)dt ≥ hT , or γµ∗ (t) ≥ h , 0 ≤ t ≤ T ,
is equivalent to the (a.s.) requirement of Market Diversity∫ T
0µ(1)(t)dt ≤ (1− δ)T , or max
0≤t≤Tµ(1)(t) ≤ 1− δ
for some δ ∈ (0, 1) (weak diversity and strong diversity,respectively).
(The maximal relative capitalization never gets above a certainpercentage. In the S&P 500 universe, no company has ever beenmore than 15% of the total market capitalization; in the last 40years, this has been more like 6%.)
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0 100 200 300 400 500
01
23
45
WE
IGH
T (
%)
RANK
Figure 2: Capital Distribution for the S&P 500 Index. December 30,1997 (solid line), and December 29, 1999 (broken line).
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Proposition 3:Suppose (weak) diversity prevails, and the lowest eigenvalue of the covariance matrix isbounded away from zero. For fixed p ∈ (0, 1) , consider the simple “diversity-weighted”portfolio
D(p)i (t) ≡ Di (t) :=
(µi (t))p∑n
j=1(µj(t))p, ∀ i = 1, . . . , n ,
generated by the concave function
S(p)(x) ≡ S(x) = (xp1 + · · ·+ xp
n )1/p .
Then this portfolio leads to arbitrage relative to the market, over sufficiently long timehorizons.
With p = 0 this becomes equal weighting ϕi (·) ≡ 1/n, 1 ≤ i ≤ n.With p = 1 we get the market portfolio µ(·) .
(Recall in this vein the modified equal-weighted portfolio of (6),which “interpolates” between equal-weighting and cap-weighting ina rather different manner.)
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With respect to the market portfolio, this “diversity-weighted”portfolio
D(p)i (t) ≡ Di (t) :=
(µi (t))p∑nj=1(µj(t))p
, ∀ i = 1, . . . , n ,
de-emphasizes the “upper (big cap) end” of the market, andover-emphasizes the “lower (small cap) end” – but observes allrelative rankings. It does all this in a completely passive way,without estimating or optimizing anything.
. Appropriate modifications of this rule generate such arbitrageover arbitrary time-horizons; for detais, see FKK (2005).
For extensive discussion of the actual performance of this“diversity-weighted portfolio” as well as of the “pure entropicportfolio” (with c = 0) we saw before, see Fernholz (2002).
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Proof of Proposition 3: For this “diversity-weighted” portfolioD(p)(·) we have from the “master equation” (14) the formula
log
(V 1,D(p)
(T )
V 1,µ(T )
)= log
(S(p)(µ(T ))
S(p)(µ(0))
)+ (1− p)
∫ T
0γ D(p)
∗ (t)dt .
• First term on RHS tends to be mean-reverting, and is certainlybounded:
1 =n∑
j=1
xj ≤n∑
j=1
(xj)p =
(S(p)(x)
)p≤ n1−p .
Measure of Diversity: minimum occurs when one company is theentire market, maximum when all companies have equal relativeweights.
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• We remarked already, that the biggest weight of D(p)(·) doesnot exceed the largest market weight:
D(p)(1)(t) := max
1≤i≤nD
(p)i (t) =
(µ(1)(t)
)p∑nk=1
(µ(k)(t)
)p ≤ µ(1)(t) .
By weak diversity over [0,T ], there is a number δ ∈ (0, 1) forwhich ∫ T
0
(1− µ(1)(t)
)dt > δ T
holds.
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Thus, from the strict non-degeneracy of the covariance matrix wehave a.s.
κ
2
(1− π(1)(t)
)≤ γπ∗ (t)
as in (18), and thus:
2
κ
∫ T
0γ D(p)
∗ (t) dt ≥∫ T
0
(1−D(p)
(1)(t))dt ≥
∫ T
0
(1−µ(1)(t)
)dt > δT .
• From these two observations we get
log
(V 1,D(p)
(T )
V 1,µ(T )
)> (1− p)
[κT
2· δ − 1
p· log n
],
so for a time-horizon
T > T∗ := (2 log n)/(pκδ)
sufficiently large, the RHS is strictly positive.
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−20
−10
010
2030
40
%
1956 1959 1962 1965 1968 1971 1974 1977 1980 1983 1986 1989 1992 1995 1998 2001 2004
1
2
3
Figure 3: Simulation of a diversity-weighted portfolio, 1956–2005.(1: generating function; 2: drift process; 3: relative return.)
log
(V 1,D(p)
(T )
V 1,µ(T )
)= log
(S(p)(µ(T ))
S(p)(µ(0))
)+ (1− p)
∫ T
0γ D(p)
∗ (t)dt .
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−30
−20
−10
010
2030
YEAR
%
1927 1932 1937 1942 1947 1952 1957 1962 1967 1972 1977 1982 1987 1992 1997 2002
Figure 4: Cumulative Change in Market Diversity, 1927-2004.
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• Remark: Consider a market that satisfies the strongnon-degeneracy condition as in (7), for some κ ∈ (0,∞) :
ξ′α(t)ξ = ξ′σ(t)σ′(t)ξ ≥ κ‖ξ‖2, ∀ t ∈ [0,∞) , ξ ∈ Rn .
If all its stocks i = 1, . . . , n have the same growth-rateγi (·) ≡ γ(·), then
limT→∞
1
T
∫ T
0γµ∗ (t) dt = 0, a.s.
In particular, such a market cannot be diverse on long timehorizons: once in a while a single stock dominates such a market,then recedes; sooner or later another stock takes its place asabsolutely dominant leader; and so on.
The same can be seen to be true for a market that satisfies theabove strong nondegeneracy condition as in (7) and its assets haveconstant, though not necessarily equal, growth rates.
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• Here is a quick argument: from γi (·) ≡ γ(·) andX (·) = X1(·) + · · ·Xn(·) we have
limT→∞
1
T
(log X (T )−
∫ T
0γµ(t)dt
)= 0 ,
limT→∞
1
T
(log Xi (T )−
∫ T
0γ(t)dt
)= 0
for all 1 ≤ i ≤ n . But then
limT→∞
1
T
(log X(1)(T )−
∫ T
0γ(t)dt
)= 0 , holds a.s.
for the biggest stock X(1)(·) := max1≤i≤n Xi (·) , and we noteX(1)(·) ≤ X (·) ≤ n X(1)(·) .
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Therefore, from X(1)(·) ≤ X (·) ≤ n X(1)(·) we deduce
limT→∞
1
T
(log X (T )− log X(1)(T )
)= 0 , thus
limT→∞
1
T
∫ T
0
(γµ(t)− γ(t)
)dt = 0 .
But
γµ(t) =n∑
i=1
µi (t)γ(t) + γµ∗ (t) = γ(t) + γµ∗ (t) ,
because all growth rates are equal.
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13. STABILIZATION BY VOLATILITY
Major Open Question: Is such relative arbitrage possibleover arbitrary time-horizons, under the conditions ofProposition 1 ?∫ T
0γµ∗ (t)dt ≥ hT , or γµ∗ (t) ≥ h , 0 ≤ t ≤ T .
Partial Answer #2: YES, for the (non-diverse) so-calledVOLATILITY-STABILIZED model that we broach now.
Consider the abstract market model
d(
log Xi (t))
=α dt
2µi (t)+
1õi (t)
dWi (t)
for i = 1, · · · , n with d = n ≥ 2 and α ≥ 0 .
In other words, we assign the largest volatilities and the largestlog-drifts to the smallest stocks.
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This model amounts to solving in the positive orthant of Rn thesystem of degenerate stochastic differential equations, fori = 1, · · · , n :
dXi (t) =1 + α
2
(X1(t) + · · · + Xn(t)
)dt
+
√Xi (t)
(X1(t) + · · ·+ Xn(t)
)· dWi (t) .
General theory: Bass & Perkins (TAMS 2002). Shows this systemhas a weak solution, unique in distribution, so the model iswell-posed.
Better still: It is possible to describe this solution fairlyexplicitly, in terms of Bessel processes.
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• Since we have
αij(t) =δijµi (t)
in this model, an elementary computation gives the quantities
γ µ∗ (t) =1
2
n∑i=1
µi (t)(1− µi (t)
)αij(t) =
n − 1
2=: h > 0 ,
aµµ(·) ≡ 1
for the market portfolio µ(·) , and
γ µ(·) ≡ (1 + α)n − 1
2=: γ > 0 .
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Despite the erratic, widely fluctuating behavior of individualstocks, the overall market performance is remarkably stable. Inparticular, the total market capitalization is
X (t) = X1(t) + . . .+ Xn(t) = x · e γt +B(t) ,
for the scalar Brownian motion
B(t) :=n∑ν=1
∫ t
0
√µν(s) dWν(s) , 0 ≤ t <∞ .
• We call this phenomenon stabilization by volatility: bigvolatility swings for the smallest stocks, and smaller volatilityswings for the largest stocks, end up stabilizing the overall marketby producing constant, positive overall growth and variance.
(Note κ = 1 but K =∞, so
κ|| ξ||2 ≤ ξ′α(t)ξ ≤ K|| ξ||2 , ∀ t ≥ 0, ξ ∈ Rd
in (17), slide 73, fails.)90 / 111
• We saw that the condition γµ∗ (·) ≥ h > 0 of Proposition 1 issatisfied here, with h = (n − 1)/2 . Thus the model admitsarbitrage relative to the market, at least on time-horizons [0,T ]with
T > T∗ , where T∗ :=2 H(µ(0))
n − 1<
2 log n
n − 1.
The upper estimate (2 log n)/(n − 1) is a rather small number ifn = 5000 as in Wilshire 5000.
• This makes plausible the earlier claim, proved by A. Banner andD. Fernholz (Annals of Finance 2008) not just for the volatility-stabililized model but for quite general growth rates in
d(
log Xi (t))
= γi (t)dt +1√µi (t)
dWi (t) , i = 1, · · · , n
that such arbitrage is now possible on any time-horizon.
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• On the other hand, the condition
ζ∗(·) ≥ h > 0
of Proposition 2 (slide 66) is also satisfied here, with h = n − 1 .This follows from the geometric mean / harmonic mean inequality
ζ∗(t) =(µ1(t) · · ·µn(t)
)1/n
n∑i=1
αii (t)− 1
n
n∑i=1
n∑j=1
αij(t)
=(µ1(t) · · ·µn(t)
)1/n ·n∑
i=1
(1− 1
n
)αii (t)
≥ n1
µ1(t) + · · ·+ 1µn(t)
· n − 1
n
n∑i=1
1
µi (t)= n − 1 .
• What is the long-term-growth behavior of an individual stock?
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A little bit of Stochastic Analysis provides the Representations
Xi (t) =(Ri (Λ(t))
)2, 0 ≤ t <∞ , i = 1, · · · , n
and
X (t) = X1(t) + · · ·+ Xn(t) = x e γt+B(t) =(R(Λ(t))
)2.
Here
4 Λ(t) :=
∫ t
0X (s) ds = x
∫ t
0e γs+B(s) ds ,
whereas R1(·), · · · ,Rn(·) are independent Bessel processes indimension m := 2(1 + α) , and
R(u) :=
√(R1(u)
)2+ · · · +
(Rn(u)
)2.
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That is, with W1(·), · · · , Wn(·) independent scalar Brownianmotions, we have
dRi (u) =m − 1
2Ri (u)du + dWi (u) , i = 1, · · · , n .
Finally,
R(u) :=
√(R1(u)
)2+ · · · +
(Rn(u)
)2
is Bessel process in dimension mn.
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We are led to the skew representation (Irina Goia, Soumik Pal)
R2i (u) = R2(u) · µi
(4
∫ u
0
dv
R2(v)
), 0 ≤ u <∞ .
Here the vector µ(·) =(µ1(·), . . . , µn(·)
)of market-weights
µi (·) =(R2
i /R2)(
Λ(·))
is independent of the Bessel process R(·) , thus also of thechange-of-clock Λ(·) which is defined in terms of this Besselprocess R(·) via the integral equation
4 Λ(·) =
∫ ·0
R2 (Λ(t))dt , equivalently Λ−1(·) = 4
∫ ·0
dv
R2(v).
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This vector µ(·) = (µi (·))ni=1 of market-weights is a so-calledvector Jacobi process with values in ∆n
+ and the dynamics
dµi (t) = (1 + α)(1− nµi (t)
)dt +
(1− µi (t)
)√µi (t)dβi (t)
−µi (t)∑j 6=i
√µj(t) dβj(t) ,
for i = 1, · · · , n .
Here, β1(·), · · · , βn(·) are independent, standard Brownianmotions.
In particular, the processes µ1(·), · · · , µn(·) have local variancesµi (t)(1− µi (t)) and covariances −µi (t)µj(t) .
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This structure suggests that the invariant measure for the∆n
+−valued diffusion µ(·) = (µi (·))ni=1 of market weights, is thedistribution of the vector(
Q1
Q1 + · · ·+ Qn, · · · , Qn
Q1 + · · ·+ Qn
),
where Q1, · · · , Qn are independent random variables withcommon distribution
2−(1+α)
Γ(1 + α)qα e −q/2 dq , 0 < q <∞ ,
(chi-square with “2(1 + α)-degrees-of-freedom”).
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• From these representations, one obtains the (a.s.) long-termgrowth rates of the entire market and of the largest stock
limT→∞
1
Tlog X (T ) = lim
T→∞
1
Tlog
(max
1≤i≤nXi (T )
)= γ ;
the long-term growth rates for individual stocks
limT→∞
1
Tlog Xi (T ) = γ , i = 1, · · · , n (19)
for α > 0 .
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The long-term volatilities
limT→∞
1
T
∫ T
0
dt
µi (t)=
2 γ
α= n +
n − 1
α
(for α > 0 , using the Birkhoff ergodic theorem); that this model isnot diverse; and much more...
NOTE: When α = 0 , the equation (19) holds only in probability;the (a.s.) limit-superior is γ , whereas the (a.s.) limit-inferior is−∞ . (Spitzer’s 0-1 law for planar Brownian motion).
Crashes.... Failure of diversity... .
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13.a: Some Concluding Remarks
We have exhibited simple conditions, such as “sufficient level ofintrinsic volatility” and “diversity”, which lead to arbitragesrelative to the market.
These conditions are descriptive as opposed to normative, andcan be tested from the predictable characteristics of the modelposited for the market. In contrast, familiar assumptions, such asthe existence of an equivalent martingale measure (EMM), arenormative in nature, and cannot be decided on the basis ofpredictable characteristics in the model; see example in [Karatzas& Kardaras] (2007).
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14. RANK, LEAKAGE, AND THE SIZE EFFECT
An important generalization of the ideas and methods in thissection concerns generating functions that record market weightsnot according to their name (or index) i , but according to theirrank. To present this generalization, let us start by recalling theorder statistics notation
max1≤i≤n
πi (t) =: π(1)(t) ≥ π(2)(t) ≥ . . . ≥ π(n)(t) := min1≤i≤n
πi (t)
and consider for each 0 ≤ t <∞ the random permutation(pt(1), · · · , pt(n)
)of (1, · · · , n) with
µpt(k)(t) = µ(k)(t), and pt(k) < pt(k+1) if µ(k)(t) = µ(k+1)(t)
for k = 1, . . . , n. In words: pt(k) is the name (index) of the stockthat occupies the kth rank in terms of relative capitalization attime t; ties are resolved by resorting to the lowest index.
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Using Ito’s rule for convex functions of semimartingales one canobtain the following analogue of
dµi (t)
µi (t)=(γi (t)−γµ(t)+
1
2Aµij (t)
)dt +
N∑ν=1
(σiν(t)−σµν (t)
)dWν(t)
in (10) for the dynamics of the ranked market-weights
dµ(k)(t)
µ(k)(t)=(γpt(k)(t)− γµ(t) +
1
2Aµ(kk)(t)
)dt
+1
2
(dLk,k+1(t)− dLk−1,k(t)
)(20)
+N∑ν=1
(σpt(k)ν(t)− σµν (t)
)dWν(t)
for each k = 1, . . . , n − 1.
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Here the quantity Lk,k+1(t) ≡ ΛΞk (t) is the semimartingale localtime at the origin, accumulated by the nonnegative process
Ξk(t) := logµ(k)(t)− logµ(k+1)(t) , 0 ≤ t <∞ (21)
up to the calendar time t ; it measures the cumulative effect of thechanges, or turnover, that have occurred during the time-interval[0, t] between ranks k , k + 1 . We are also setting
L 0,1(·) ≡ 0 , Lm,m+1(·) ≡ 0
andAµ(k`)(·) := Aµpt(k)pt(`)
(·) .
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A derivation of this result, under appropriate conditions that we donot broach here, can be found on pp. 76-79 of Fernholz (2002);see also Banner & Ghomrasni (2007) for generalizations.
With this setup, we have then the following generalization of the“master equation” : consider a function G : ∆n
+ → (0,∞) exactlyas assumed there, written in the form
G(x1, · · · , xn) = G(x(1), · · · , x(n)
), ∀ x ∈ U
for some G ∈ C2(U) and U an open neighborhood of ∆n+ . The
effective domain of definition of this function is U ∩Wn , where
Wn :=
(x(1), · · · , x(n)) ∈ Rn : x(1) ≥ · · · ≥ x(n)
is the Weyl chamber of descending order statistics in Rn .
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Then with the shorthand x(·) := (x(1), · · · , x(n))′ and
µ(·)(t) :=(µ(1)(t), · · · , µ(n)(t)
)′, Aµ(k`)(t) := Aµpt(k)pt(`)
(t) ,
let us consider the rank-based portfolio $(·) defined for1 ≤ k ≤ n , by analogy with (13), as
$ pt(k)(t)
µ(k)(t)= Dk logG
(µ(·)(t)
)+ 1−
n∑`=1
µ(`)(t)D` logG(µ(·)(t)
).
(22)
It can be can shown that the performance of this portfolio, relativeto the market, is given for any 0 ≤ T <∞ and by analogy with(14), as
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log
(V $(T )
V µ(T )
)= log
(G(µ(·)(T )
)G(µ(·)(0)
) )+ Γ(T ) , (23)
where we have set
Γ(T ) := −∫ T
0
n∑k=1
n∑`=1
D2k`G
(µ(·)(t)
)2G(µ(·)(t)
) µ(k)(t)µ(`)(t)Aµ(k`)(t) dt
(24)
+1
2
n−1∑k=1
($ pt(k+1)(t)− $ pt(k)(t)
)dLk,k+1(t) .
We say that $(·) is the rank-based portfolio generated by thefunction G(·). (Functional generation by rank, as opposed to byname.) For details see Fernholz (2002), pp. 79-83.
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• For instance,G(x(·)) = x(1)
generates in (22) the portfolio
$pt(k)(t) = δ1k , k = 1, · · · , n , 0 ≤ t <∞
that invests only in the largest stock at all times.The relative performance
log
(V 1,$(T )
V 1,µ(T )
)= log
(µ(1)(T )
µ(1)(0)
)−1
2L1,2(T ) , 0 ≤ T <∞
of this portfolio will suffer (leakage) in the long run, if there aremany changes in leadership: in order for the biggest stock to dowell relative to the market, it must crush all competition!
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14.a: The Size Effect
This is the tendency of small stocks to have higher long-termreturns relative to their larger brethren. The formula of (23) offersa simple, structural explanation of this observed phenomenon, thus:
• Fix an integer m ∈ 2, · · · , n − 1 and consider the functions
GL(x) = x(1) + · · ·+ x(m) and GS(x) = x(m+1) + · · ·+ x(n) .
These generate, respectively, a large-stock portfolio
ζpt(k)(t) =µ(k)(t)
GL(µ(t)), 1 ≤ k ≤ m and ζpt(k)(t) = 0, m+1 ≤ k ≤ n
and a small-stock portfolio
ηpt(k)(t) =µ(k)(t)
GS(µ(t)), m+1 ≤ k ≤ n and ηpt(k)(t) = 0, 1 ≤ k ≤ m.
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According to (23) and (22), the performances of these portfolios,relative to the market, are given respectively by
log
(V 1,ζ(T )
V 1,µ(T )
)= log
(GL(µ(T ))
GL(µ(0))
)−1
2
∫ T
0ζ(m)(t) dLm,m+1(t) ,
log
(V 1,η(T )
V 1,µ(T )
)= log
(GS(µ(T ))
GS(µ(0))
)+
1
2
∫ T
0η(m)(t) dLm,m+1(t) .
Therefore,
log
(V 1,η(T )
V 1,ζ(T )
)= log
(GS(µ(T ))GL(µ(0))
GL(µ(T ))GS(µ(0))
)(25)
+1
2
∫ T
0
(ζ(m)(t) + η(m)(t)
)dLm,m+1(t) .
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If there is “stability” in the market, in the sense that the ratio ofthe relative capitalization of small to large stocks does not changemuch over time, then the first term on the right-hand side of (25)does not change much, whereas the second term keeps increasingand accounts for the better relative performance of the smallstocks.
Note that this argument does not need to invoke any assumptionabout the putative greater riskiness of the smaller stocks at all.
Note also that the expression of (25) can be used to estimate thelocal times Lm,m+1(·) .
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LOCA
L TI
ME
YEAR
010
0020
0030
0040
00
90 91 92 93 94 95 96 97 98 99
Figure 5: The estimated local time processes Lm,m+1(·) for m = 10, 20,40, ... , 5120.
111 / 111