A brief prehistory of SPT
Robert Fernholz INTECH
SPT 2015 Columbia University May 8–9, 2015
1 / 24
Introduction
The prehistoric period of SPT is the span of time from about 1980 to the 2002 publication of the monograph Stochastic Portfolio Theory. This was a period during which a number of ideas were introduced, but before these ideas coalesced into a common framework. The main ideas that were introduced in this early period include: the logarithmic representation, the excess-growth rate, entropy and diversity, portfolio generating functions, relative arbitrage, leakage, portfolios based on rank, Atlas models, first-order models. I will discuss these ideas and try to explain the events and motivation that led to their development.
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High interest rates in the 1980s
In 1980, interest rates were high — at one point 3-month treasury bill rates were over 15%. The Black-Scholes option-pricing model was fashionable at that time, and it showed that with high interest rates, call options would have high prices.
3 / 24
High interest rates in the 1980s
In 1980, interest rates were high — at one point 3-month treasury bill rates were over 15%. The Black-Scholes option-pricing model was fashionable at that time, and it showed that with high interest rates, call options would have high prices. Because of the high-priced calls, option overwriting became a popular strategy. With an option overwriting strategy, a pension fund would write covered call options on its portfolio of stocks. It was claimed that this strategy would outperform a conventional stock portfolio over the long term. Could this possibly be true?
3 / 24
Option overwriting: 1980
The Black-Scholes model priced a call so that it will follow the value of a dynamic cash/stock portfolio. So, to understand option overwriting, we needed to understand cash/stock portfolios. With the high interest rates, it wasn’t unreasonable to let cash and stocks grow at the same exponential rate, so $(t) = $0 e gt , X (t) = X0 e gt+
W (t)
where $(·) is the cash value, X (·) is the stock price, g and constants, and W is a Brownian motion.
> 0 are
4 / 24
Option overwriting: 1980 Breiman (1961) had shown that logarithmic growth determines long-term behavior, so we needed to understand logarithmic growth. In our case, this was d log $(t) = g dt
d log X (t) = g dt + dW (t),
where these were Itˆo processes. However, the classical representation for stock prices did not use logarithms, but instead used the arithmetic representation given by d$(t) = d log $(t) $(t)
2 dX (t) = d log X (t) + dt, X (t) 2
by Itˆ o’s rule. This representation was used both for option pricing and for portfolio theory.
5 / 24
A cash/stock portfolio: 1980 In the arithmetic representation, the value Z⇡ of a cash/stock portfolio with ⇡ proportion of stock followed dZ⇡ (t) dX (t) =⇡ + 1 Z⇡ (t) X (t)
⇡
d$(t) . $(t)
By Itˆ o’s rule, the logarithmic representation of this was d log Z⇡ (t) + a.s., or
⇣ ⌘ 2 ⇡2 2 dt = ⇡ d log X (t) + dt + 1 2 2
d log Z⇡ (t) = g dt + ⇡
⇡2
⇡ d log $(t),
2
2
dt + ⇡ dW (t).
6 / 24
A cash/stock portfolio: 1980 In the arithmetic representation, the value Z⇡ of a cash/stock portfolio with ⇡ proportion of stock followed dZ⇡ (t) dX (t) =⇡ + 1 Z⇡ (t) X (t)
⇡
d$(t) . $(t)
By Itˆ o’s rule, the logarithmic representation of this was d log Z⇡ (t) + a.s., or
⇣ ⌘ 2 ⇡2 2 dt = ⇡ d log X (t) + dt + 1 2 2
d log Z⇡ (t) = g dt + ⇡
⇡2
⇡ d log $(t),
2
2
dt + ⇡ dW (t).
For 0 < ⇡ < 1 the middle term is positive, so the cash/stock portfolio will grow faster than either of its two components. Option overwriting might actually work!
6 / 24
The logarithmic representation: 1982–1995 The arithmetic representation of a stock price was dX (t) = ↵(t)X (t) dt + (t)X (t) dW (t), 2
with rate of return ↵ and variance rate representation of X is
. The equivalent logarithmic
d log X (t) = (t) dt + (t) dW (t), with
2
(t) = ↵(t)
(t) , 2
a.s.,
by Itˆ o’s rule.
7 / 24
The logarithmic representation: 1982–1995 The arithmetic representation of a stock price was dX (t) = ↵(t)X (t) dt + (t)X (t) dW (t), with rate of return ↵ and variance rate representation of X is
2
. The equivalent logarithmic
d log X (t) = (t) dt + (t) dW (t), with
2
(t) , a.s., 2 is called the growth rate process, and
(t) = ↵(t) by Itˆ o’s rule. The process
1⇣ lim log X (T ) T !1 T
Z
T 0
⌘ (t) dt = 0,
a.s.,
by the strong law of large numbers. 7 / 24
The logarithmic representation: 1982–1995 A market X1 , . . . , Xn is represented stock value processes d log Xi (t) =
i (t) dt
+
d X
⇠i⌫ (t) dW⌫ (t)
⌫=1
where d n, (W1 , . . . , Wd ) is a Brownian motion, and the processes i and ⇠i⌫ are adapted to the Brownian filtration. The covariance processes ij are defined by d
ij (t)
,
X dhlog Xi , log Xj it = ⇠i⌫ (t)⇠j⌫ (t), dt ⌫=1
for i, j = 1, . . . , n.
8 / 24
The logarithmic representation: 1982–1995 A market X1 , . . . , Xn is represented stock value processes d log Xi (t) =
i (t) dt
+
d X
⇠i⌫ (t) dW⌫ (t)
⌫=1
where d n, (W1 , . . . , Wd ) is a Brownian motion, and the processes i and ⇠i⌫ are adapted to the Brownian filtration. The covariance processes ij are defined by d
ij (t)
,
X dhlog Xi , log Xj it = ⇠i⌫ (t)⇠j⌫ (t), dt ⌫=1
for i, j = 1, . . . , n. Ironically, although cash played a key role in the origins of SPT, it was usually not included in these models.
8 / 24
The logarithmic representation: 1982–1995 With the logarithmic representation, a portfolio is defined by bounded weight processes ⇡1 , . . . , ⇡n that are adapted to the filtration and add up to one. In this case the value process would follow d log Z⇡ (t) =
n X
⇡i (t) d log Xi (t) +
⇤ ⇡ (t) dt,
a.s.,
i=1
where the excess-growth rate
⇤ ⇡
is given by
1⇣ X ⇡i (t) 2 n
⇤ ⇡ (t)
,
2 ii (t)
i=1
n X
⇡i (t)⇡j (t)
i,j=1
ij (t)
⌘
.
Since the excess growth rate a↵ects the log-return of the portfolio, it can be useful for portfolio optimization, and for a long-only portfolio, ⇤ ⇡ (t)
0,
a.s.
9 / 24
Relative return: 1982–1995 The market portfolio µ is the portfolio with market weights µi and value process Zµ , with µi (t) = Xi (t)/Zµ (t),
and
Zµ (t) = X1 (t) + · · · + Xn (t).
The relative (log-)return process for a portfolio ⇡ is d log(Z⇡ (t)/Zµ (t)) =
n X
⇡i (t) d log(Xi (t)/Zµ (t)) +
⇤ ⇡ (t) dt
i=1
=
n X
⇡i (t) d log µi (t) +
⇤ ⇡ (t) dt,
a.s.
i=1
10 / 24
Relative return: 1982–1995 The market portfolio µ is the portfolio with market weights µi and value process Zµ , with µi (t) = Xi (t)/Zµ (t),
and
Zµ (t) = X1 (t) + · · · + Xn (t).
The relative (log-)return process for a portfolio ⇡ is d log(Z⇡ (t)/Zµ (t)) =
n X
⇡i (t) d log(Xi (t)/Zµ (t)) +
⇤ ⇡ (t) dt
i=1
=
n X
⇡i (t) d log µi (t) +
⇤ ⇡ (t) dt,
a.s.
i=1
An immediate consequence of this representation is that for ⇡ = µ, n X
µi (t) d log µi (t) =
⇤ µ (t) dt
< 0,
a.s.
i=1
The larger stocks have negative relative growth rates, on average. 10 / 24
Capital distribution and market entropy: 1995 In the mid 1990s, there was a great concentration of capital into the largest stocks. This was causing anxiety for equity managers, who usually were overweighted in smaller stocks. We needed a method to measure what was happening. Entropy (Shannon) seemed like a possibility, S(x) ,
n X
xi log xi ,
i=1
where {x1 , . . . , xn } is a discrete probability distribution. The idea was to take n X S(µ(t)) = µi (t) log µi (t), i=1
where µ = (µ1 , . . . , µn ), and see how it behaved. We used Itˆo’s rule to generate
11 / 24
Capital distribution and market entropy: 1995 In the mid 1990s, there was a great concentration of capital into the largest stocks. This was causing anxiety for equity managers, who usually were overweighted in smaller stocks. We needed a method to measure what was happening. Entropy (Shannon) seemed like a possibility, S(x) ,
n X
xi log xi ,
i=1
where {x1 , . . . , xn } is a discrete probability distribution. The idea was to take n X S(µ(t)) = µi (t) log µi (t), i=1
where µ = (µ1 , . . . , µn ), and see how it behaved. We used Itˆo’s rule to generate dS(µ(t)) =??????????????? This was a failure.
11 / 24
The entropy-weighted portfolio: 1995 However, if we take log-entropy, then traces of a portfolio appear among the scattered pieces. In fact, d log S(µ(t)) = d log Z⇡ (t)/Zµ (t)
⇤ µ (t)
S(µ(t))
dt,
a.s.,
where ⇡ is the portfolio with weights ⇡i (t) =
µi (t) log µi (t) , S(µ(t))
for i = 1, . . . , n. This was called the entropy-weighted portfolio, and its relative return followed Z t ⇤ µ (s) log Z⇡ (t)/Zµ (t) = log S(µ(t)) + ds, a.s. 0 S(µ(s))
12 / 24
Functionally generated portfolios: 1995 It eventually became clear that any positive C 2 generating function S defined in a neighborhood of the unit simplex in Rn would generate a portfolio ⇡ such that log Z⇡ (t)/Zµ (t) = log S(µ(t)) + ⇥(t),
a.s.,
where ⇥ is a function of bounded variation. The portfolio weights are ⇣ ⇡i (t) = Di log S(µ(t)) + 1
n X j=1
⌘ µj (t)Dj log S(µ(t)) µi (t),
and d⇥(t) =
n X 1 Dij S(µ(t))µi (t)µj (t)dhlog µi , log µj it . 2S(µ(t)) i,j=1
13 / 24
Diversity-weighted indexing: 1995 Of particular interest was the function Dp (x) =
n ⇣X i=1
xip
⌘1/p
for 0 < p < 1. It was called a measure of diversity, and it generates the portfolio ⇡ with weights ⇡i (t) / µpi (t), with log Z⇡ (t)/Zµ (t) = log Dp (µ(t)) + (1
p)
Z
t 0
⇤ ⇡ (s) ds,
a.s.
14 / 24
Diversity-weighted indexing: 1995 Of particular interest was the function Dp (x) =
n ⇣X i=1
xip
⌘1/p
for 0 < p < 1. It was called a measure of diversity, and it generates the portfolio ⇡ with weights ⇡i (t) / µpi (t), with log Z⇡ (t)/Zµ (t) = log Dp (µ(t)) + (1
p)
Z
t 0
⇤ ⇡ (s) ds,
a.s.
In 1998, INTECH was granted a patent for investment strategies based on functionally generated portfolios. After that, INTECH managed a strategy based on Dp called diversity-weighted indexing for a number of years. 14 / 24
Diversity and arbitrage: 1996–1998 A market X1 , . . . , Xn is diverse if there exists a and t, µi (t) < 1 , a.s.
> 0 such that for all i
In a diverse market with a strongly nondegenerate covariance structure, a form of arbitrage can be shown to exist. A (strong) relative arbitrage exists if there is a portfolio ⇡ and a constant T > 0 such that P Z⇡ (T )/Z⇡ (0) > Zµ (T )/Zµ (0) = 1.
15 / 24
Diversity and arbitrage: 1996–1998 A market X1 , . . . , Xn is diverse if there exists a and t, µi (t) < 1 , a.s.
> 0 such that for all i
In a diverse market with a strongly nondegenerate covariance structure, a form of arbitrage can be shown to exist. A (strong) relative arbitrage exists if there is a portfolio ⇡ and a constant T > 0 such that P Z⇡ (T )/Z⇡ (0) > Zµ (T )/Zµ (0) = 1. Dp will work, but historically the generating function that was first used for this purpose was n
S(µ(t)) = 1
1X 2 µi (t), 2 i=1
because the portfolio it generates has a number of benign properties. 15 / 24
The problem of “leakage”: 1996–1998
INTECH managed a diversity-weighted version of the S&P 500 for a few years in the late 1990s. However, there was a problem with diversity-weighted indexing. Small stocks dropped out of the S&P 500 from time to time, and since diversity weighting overweights small stocks, this a↵ected the diversity-weighted index more than the cap-weighted index. We called this phenomenon leakage.
16 / 24
The problem of “leakage”: 1996–1998
INTECH managed a diversity-weighted version of the S&P 500 for a few years in the late 1990s. However, there was a problem with diversity-weighted indexing. Small stocks dropped out of the S&P 500 from time to time, and since diversity weighting overweights small stocks, this a↵ected the diversity-weighted index more than the cap-weighted index. We called this phenomenon leakage. At first we measured leakage statistically, but statistical treatment alone was unsatisfactory. We needed a theoretical model. To understand leakage for portfolios of the top 500 stocks in the market, we needed to understand portfolios based on rank.
16 / 24
Semimartingale local time: 1996–1998 Rank can be expressed in terms of maxima and minima, and maxima and minima can be expressed in terms of absolute values. Absolute values of continuous semimartingales involve semimartingale local time. By the Tanaka-Meyer formula, the local time at the origin ⇤X for a continuous semimartingale X satisfies Z t ⌘ 1⇣ ⇤X (t) = X (t) X (0) sgn X (s) dX (s) , a.s. 2 0
17 / 24
Semimartingale local time: 1996–1998 Rank can be expressed in terms of maxima and minima, and maxima and minima can be expressed in terms of absolute values. Absolute values of continuous semimartingales involve semimartingale local time. By the Tanaka-Meyer formula, the local time at the origin ⇤X for a continuous semimartingale X satisfies Z t ⌘ 1⇣ ⇤X (t) = X (t) X (0) sgn X (s) dX (s) , a.s. 2 0 Portfolio generating functions based on rank could be defined. E.g., for m < n, the function SL (x) = x(1) + · · · + x(m) generates the cap-weighted index ⌫ of the m largest stocks, with ⌫i (t) / µi (t) for i = 1, . . . , m, and d log Z⌫ (t)/Zµ (t) = d log SL (µ(t))
1 ⌫(m) (t) d⇤m,m+1 (t), 2
where ⇤m,m+1 is the semimartingale local time of (log µ(m) at the origin.
log µ(m+1) )
17 / 24
Leakage in a diversity-weighted portfolio: 1998 For 1 < m < n, let 0 < p < 1 be a constant and define S(x) =
m ⇣X i=k
p x(k)
⌘1/p
.
In this case, Dp (⌫) = S(µ)/SL (µ) generates the diversity-weighted portfolio ⇡ of the m largest stocks. The value processes Z⇡ and Z⌫ follow d log Z⇡ (t)/Z⌫ (t) = d log Dp (⌫(t)) + (1 p) ⇡⇤ (t) dt 1 + ⌫(m) (t) ⇡(m) (t) d⇤m,m+1 (t), 2
a.s.
The last term measures leakage, and it is negative since local time is nondecreasing and ⇡(m) (t) > ⌫(m) (t), a.s.
18 / 24
1e−07
WEIGHT
1e−05
1e−03
1e−01
Stability of the distribution of capital: 2000
1
5
10
50
100
500
1000
5000
RANK
U.S. capital distribution, 1929 to 1999 19 / 24
Asymptotic stability: 2001 When a stock market is observed by rank, there appeared to be a long-term stability to the distribution of capital. Accordingly, a market X1 , . . . , Xn was defined to be asymptotically stable if: 1. lim t
1
t!1
log µ(n) (t) = 0 (coherence);
2. for k = 1, . . . , n
1,
3. for k = 1, . . . , n
1,
where
k,k+1
and
k:k+1
lim t
1
⇤k,k+1 (t) =
lim t
1
hlog µ(k)
t!1 t!1
k,k+1
> 0;
log µ(k+1) it =
2 k:k+1 ;
are positive constants.
20 / 24
Asymptotic stability: 2001 When a stock market is observed by rank, there appeared to be a long-term stability to the distribution of capital. Accordingly, a market X1 , . . . , Xn was defined to be asymptotically stable if: 1. lim t
1
t!1
log µ(n) (t) = 0 (coherence);
2. for k = 1, . . . , n
1,
3. for k = 1, . . . , n
1,
where
k,k+1
and
k:k+1
lim t
1
⇤k,k+1 (t) =
lim t
1
hlog µ(k)
t!1 t!1
1 T !1 T lim
T 0
> 0;
log µ(k+1) it =
2 k:k+1 ;
are positive constants.
In this case, for k = 1, . . . , n Z
k,k+1
1,
log µ(k) (t) log µ(k+1) (t) dt ⇠ = log(k) log(k + 1)
k 2
2 k:k+1
,
a.s.,
k,k+1
which is the approximate slope of the tangent to the distribution curve. 20 / 24
The “Atlas” model: 2001 Consider the market X1 , . . . , Xn defined by d log Xi (t) = ng 1{Xi (t)=X(n) (t)} dt + dWi (t) where g and are positive constants and W1 , . . . , Wn is a Brownian motion. In this case, it can be shown that k,k+1
and since
2 k:k+1
1 lim T !1 T
= 2kg ,
a.s.,
=2
2
Z
log µ(k) (t) log µ(k+1) (t) dt ⇠ = log(k) log(k + 1)
T 0
, we have, a.s., 2 k:k+1
k 2
k,k+1 2
=
2g
.
This model was called the Atlas model, and its capital distribution is a Pareto distribution. 21 / 24
First-order models: 2001 Consider the market X1 , . . . , Xn defined by d log Xi (t) = grt (i) dt + where rt (i) is the rank of Xi (t); the gk and
rt (i) dWi (t), k
are constants such that
g1 + · · · + gn = 0, and g1 + · · · + gk < 0 for k < n; and W1 , . . . , Wn is a Brownian motion. In this model, k,k+1
=
2 g1 + · · · + gk ,
a.s.,
so the distribution curve satisfies, Z 2 k k2 + k+1 1 T log µ(k) (t) log µ(k+1) (t) lim dt ⇠ , = T !1 T 0 log(k) log(k + 1) 4 g1 + · · · + gk
a.s.
This was called a first-order model, and it allowed us to create a rank-wise approximation of a real stock market. 22 / 24
Various publications (and unpublications) I
(1982) With B. Shay. Stochastic portfolio theory and stock market equilibrium. Journal of Finance 37, 615–624.
I
(1998) With R. Garvy and J. Hannon. Diversity-weighted indexing. Journal of Portfolio Management 24(2), 74–82.
I
(1999) On the diversity of equity markets. Journal of Mathematical Economics 31(3), 393–417.
I
(1999) Portfolio generating functions. In M. Avellaneda (Ed.), Quantitative Analysis in Financial Markets, World Scientific, River Edge, NJ.
I
(1999) Arbitrage in Equity Markets. INTECH technical report, Princeton, NJ.
I
(2001) Equity portfolios generated by functions of ranked market weights. Finance and Stochastics 5, 469–486.
I
(2001) Stable Models for the Distribution of Equity Capital. SSRN id261150.
I
(2002) Stochastic Portfolio Theory. Springer, New York. 23 / 24
Various publications (and unpublications) I
(1982) With B. Shay. Stochastic portfolio theory and stock market equilibrium. Journal of Finance 37, 615–624.
I
(1998) With R. Garvy and J. Hannon. Diversity-weighted indexing. Journal of Portfolio Management 24(2), 74–82.
I
(1999) On the diversity of equity markets. Journal of Mathematical Economics 31(3), 393–417.
I
(1999) Portfolio generating functions. In M. Avellaneda (Ed.), Quantitative Analysis in Financial Markets, World Scientific, River Edge, NJ.
I
(1999) Arbitrage in Equity Markets. INTECH technical report, Princeton, NJ.
I
(2001) Equity portfolios generated by functions of ranked market weights. Finance and Stochastics 5, 469–486.
I
(2001) Stable Models for the Distribution of Equity Capital. SSRN id261150.
I
(2002) Stochastic Portfolio Theory. Springer, New York. 23 / 24
Various publications (and unpublications) I
(1982) With B. Shay. Stochastic portfolio theory and stock market equilibrium. Journal of Finance 37, 615–624.
I
(1998) With R. Garvy and J. Hannon. Diversity-weighted indexing. Journal of Portfolio Management 24(2), 74–82.
I
(1999) On the diversity of equity markets. Journal of Mathematical Economics 31(3), 393–417.
I
(1999) Portfolio generating functions. In M. Avellaneda (Ed.), Quantitative Analysis in Financial Markets, World Scientific, River Edge, NJ.
I
(1999) Arbitrage in Equity Markets. INTECH technical report, Princeton, NJ.
I
(2001) Equity portfolios generated by functions of ranked market weights. Finance and Stochastics 5, 469–486.
I
(2001) Stable Models for the Distribution of Equity Capital. SSRN id261150.
I
(2002) Stochastic Portfolio Theory. Springer, New York. 23 / 24
Various publications (and unpublications) I
(1982) With B. Shay. Stochastic portfolio theory and stock market equilibrium. Journal of Finance 37, 615–624.
I
(1998) With R. Garvy and J. Hannon. Diversity-weighted indexing. Journal of Portfolio Management 24(2), 74–82.
I
(1999) On the diversity of equity markets. Journal of Mathematical Economics 31(3), 393–417.
I
(1999) Portfolio generating functions. In M. Avellaneda (Ed.), Quantitative Analysis in Financial Markets, World Scientific, River Edge, NJ.
I
(1999) Arbitrage in Equity Markets. INTECH technical report, Princeton, NJ.
I
(2001) Equity portfolios generated by functions of ranked market weights. Finance and Stochastics 5, 469–486.
I
(2001) Stable Models for the Distribution of Equity Capital. SSRN id261150.
I
(2002) Stochastic Portfolio Theory. Springer, New York. 23 / 24
Various publications (and unpublications) I
(1982) With B. Shay. Stochastic portfolio theory and stock market equilibrium. Journal of Finance 37, 615–624.
I
(1998) With R. Garvy and J. Hannon. Diversity-weighted indexing. Journal of Portfolio Management 24(2), 74–82.
I
(1999) On the diversity of equity markets. Journal of Mathematical Economics 31(3), 393–417.
I
(1999) Portfolio generating functions. In M. Avellaneda (Ed.), Quantitative Analysis in Financial Markets, World Scientific, River Edge, NJ.
I
(1999) Arbitrage in Equity Markets. INTECH technical report, Princeton, NJ.
I
(2001) Equity portfolios generated by functions of ranked market weights. Finance and Stochastics 5, 469–486.
I
(2001) Stable Models for the Distribution of Equity Capital. SSRN id261150.
I
(2002) Stochastic Portfolio Theory. Springer, New York. 23 / 24
Various publications (and unpublications) I
(1982) With B. Shay. Stochastic portfolio theory and stock market equilibrium. Journal of Finance 37, 615–624.
I
(1998) With R. Garvy and J. Hannon. Diversity-weighted indexing. Journal of Portfolio Management 24(2), 74–82.
I
(1999) On the diversity of equity markets. Journal of Mathematical Economics 31(3), 393–417.
I
(1999) Portfolio generating functions. In M. Avellaneda (Ed.), Quantitative Analysis in Financial Markets, World Scientific, River Edge, NJ.
I
(1999) Arbitrage in Equity Markets. INTECH technical report, Princeton, NJ.
I
(2001) Equity portfolios generated by functions of ranked market weights. Finance and Stochastics 5, 469–486.
I
(2001) Stable Models for the Distribution of Equity Capital. SSRN id261150.
I
(2002) Stochastic Portfolio Theory. Springer, New York. 23 / 24
Various publications (and unpublications) I
(1982) With B. Shay. Stochastic portfolio theory and stock market equilibrium. Journal of Finance 37, 615–624.
I
(1998) With R. Garvy and J. Hannon. Diversity-weighted indexing. Journal of Portfolio Management 24(2), 74–82.
I
(1999) On the diversity of equity markets. Journal of Mathematical Economics 31(3), 393–417.
I
(1999) Portfolio generating functions. In M. Avellaneda (Ed.), Quantitative Analysis in Financial Markets, World Scientific, River Edge, NJ.
I
(1999) Arbitrage in Equity Markets. INTECH technical report, Princeton, NJ.
I
(2001) Equity portfolios generated by functions of ranked market weights. Finance and Stochastics 5, 469–486.
I
(2001) Stable Models for the Distribution of Equity Capital. SSRN id261150.
I
(2002) Stochastic Portfolio Theory. Springer, New York. 23 / 24
Various publications (and unpublications) I
(1982) With B. Shay. Stochastic portfolio theory and stock market equilibrium. Journal of Finance 37, 615–624.
I
(1998) With R. Garvy and J. Hannon. Diversity-weighted indexing. Journal of Portfolio Management 24(2), 74–82.
I
(1999) On the diversity of equity markets. Journal of Mathematical Economics 31(3), 393–417.
I
(1999) Portfolio generating functions. In M. Avellaneda (Ed.), Quantitative Analysis in Financial Markets, World Scientific, River Edge, NJ.
I
(1999) Arbitrage in Equity Markets. INTECH technical report, Princeton, NJ.
I
(2001) Equity portfolios generated by functions of ranked market weights. Finance and Stochastics 5, 469–486.
I
(2001) Stable Models for the Distribution of Equity Capital. SSRN id261150.
I
(2002) Stochastic Portfolio Theory. Springer, New York. 23 / 24
Various publications (and unpublications) I
(1982) With B. Shay. Stochastic portfolio theory and stock market equilibrium. Journal of Finance 37, 615–624.
I
(1998) With R. Garvy and J. Hannon. Diversity-weighted indexing. Journal of Portfolio Management 24(2), 74–82.
I
(1999) On the diversity of equity markets. Journal of Mathematical Economics 31(3), 393–417.
I
(1999) Portfolio generating functions. In M. Avellaneda (Ed.), Quantitative Analysis in Financial Markets, World Scientific, River Edge, NJ.
I
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Thank you!
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