Meta-CTA Trading Strategies based on the Kelly Criterion

Abstract

The influence of Commodity Trading Advisors (CTA) on the price process is explored with the help of a simple model. CTA managers are taken to be Kelly optimisers, which invest a fixed proportion of their assets in the risky asset and the remainder in a riskless asset. This requires regular adjustment of the portfolio weights as prices evolve. The CTA trading activity impacts the price change in the form of a power law. These two rules governing investment ratios and price impact are combined and lead through updating at fixed time intervals to a deterministic price dynamic. For different choices of the model parameters one gets qualitatively different dynamics. The result can be expressed as a phase diagram. Meta-CTA strategies can be devised to exploit the predictability inherent in the model dynamics by avoiding critical areas of the phase diagram or by taking a contrarian position at an opportune time.

Full text

arXiv:1610.10029v1 [q-fin.PM] 31 Oct 2016
Meta-CTA Trading Strategies based on the Kelly Criterion
Bernhard K. Meister
Department of Physics, Renmin University of China, Beijing, China 100872
(Dated: November 1, 2016)
The impact of Kelly-optimizing portfolio managers on the price process is explored
with the help of a simple model. Investments are restricted to one risky and one
riskless asset. The Kelly optimizers invest under the assumption that the risky asset
follows a geometric Brownian motion with fixed risk premium and volatility. They
keep a fixed proportion called the leverage ratio, defined as the ratio of the risk
premium and the volatility, in the risky asset. This requires regular adjustment of
the portfolio weights as prices evolve. The price impact relating trading activity and
price change is assumed to be given by a power law. These two rules are combined
and lead to a deterministic price dynamic. Price changes provide feedback to the
Kelly optimizers, who adjust their portfolios. This in turn leads to further price
changes and closes the loop. For different choices of the model parameters, most
importantly the leverage ratio and the power law coefficient of the price impact,
one gets qualitatively different dynamics. The results can be expressed as a phase
diagram. For different combinations of the two parameters one gets three types of
qualitatively different behaviour. In the first phase, the change of the price gradually
decreases to zero. In the second phase the sign of the price change switches at each
step. The third phase is most interesting, since the change of the price increases at
each step and leads to a run-away effect. This analysis is useful for understanding
the impact of Commodity Trading Advisors (CTA) on the underlying price process,
and is of increasing relevance, since the assets under management in this strategy
have ballooned. As a result, slippage becomes an issue and the profitabilities of
conventional CTA strategies are affected. In contrast, meta-CTA strategies can be
devised to exploit the predictability inherent in the toy model dynamics by avoiding
critical areas of the phase diagram or by taking a contrarian position at an opportune
time.
Keywords: Portfolio optimization, commodity trading advisors, Kelly criterion.
I. INTRODUCTION
The aim of this paper is to develop a simple model to explore some particular consequences
of Kelly optimizers on the price process. This is for example useful to understand some
of the side-effects of the explosive growth of so-called Commodity Trading Advisers (CTA)
over the last few decades. A range of different investment styles are subsumed under the
name Commodity Trading Advisers (Burghardt & Walls 2011, Lemprire et al. 2014). Such
strategies have existed for many years and invest in a variety of assets. For the purpose
of this paper we focus on funds that base trading decisions on computer models. Human
intervention, except when severe short-term market dislocations occur, is frowned upon. In
particular we are interested in funds that try to exploit trend-based strategies, implemented
through liquid financial instruments such as futures. The whole CTA space according to

2
BarclaysHedge grew from $300 million in 1980 to more than $300 billion in 2014. Out of
these assets around $275 billion are estimated to be controlled by “systematic traders”. This
group matches more or less the subset we are considering, since systematic funds, in contrast
to discretionary ones, base trading decisions solely on computer models. These quantitative
models often rely on momentum or mean-reversion signals, i.e. are concerned with trend
following or trend reversion. Leverage is employed and funds can take long as well as short
positions. All the usual liquid financial assets are employed, with a preferences for assets
such as futures that allow naturally for long and short positions. Holding periods vary,
but are mostly longer than just intra-day, unlike high-frequency based statistical arbitrage
strategies. The toy model that we shall construct can be used to show how CTA trading can
impact the underlying instruments and effect their performance. In addition it will be shown
in a heuristic way how this insight can be used to modify and improve trading strategies.
Market micro-structure, except where it touches on general price impact, will be ignored
in what follows. Portfolio selection paired with risk control plays a prominent role for
CTA managers, and the hope is that the results we describe are of practical relevance to
the investment policies of portfolio managers. In particular, our approach may allow for
the construction of meta-strategies that are able to avoid the slippage associated with the
conventional crowded CTA strategies, and in the optimal case will lead to predatory trading
strategies that exploit the predictability of conventional CTA strategies. The individual
systematic CTA strategies are black boxes not known to outsiders. Nevertheless there seems
to be a significant overlap between different CTAs. In addition, their long-running and well-
documented success has led to some leakage of their ideas into the public domain.
The remainder of the paper is organised as follows. After a general description of the
toy model in the next subsection, the two main components called Kelly criterion and
price impact are briefly explored. In the subsequent section implications of the model are
developed and a phase diagram produced. It will be shown how different values of the main
parameters of price impact and the proportion of portfolio value in the risky asset will lead
to qualitatively different behaviour. Next, the key assumptions are introduced to gain a
qualitative understanding of how Kelly optimizers effect the underlying market.
II. MARKET IMPACT AND KELLY CRITERION
The toy model at the centre of our approach and laid out below is based on two elementary
assumptions. The model is constructed in such a way as to capture a set of stylized facts.
The assumptions are related to portfolio construction and adjustment as well as trading
impact. The result is a deterministic price dynamic, as the updating of positions, e.g. end-
of-day, leads through price impact to a modification of price, which in the next cycle leads
to an adjustment of the positions. This feedback loop continues indefinitely.
The two assumptions are as follows. (i) The Kelly criterion (i.e., the optimal growth
strategy) governs the portfolio selection. Past stock behaviour is used as a guide to es-
timate future performance parameters of drift, volatility and correlation for a standard
CTA-strategy, which ignores price impact. The use of information theoretic ideas in port-
folio optimization goes back to Kelly (1956) and was further developed both in theory and
practice by Thorp (1969). For a detailed review see MacLean et al. (2011). (ii) The relation
between investment amount and price impact is given by a power law relating price change

3
to investment amount, in other words, a relation of the form
(∆S)γ∼∆N, (1)
where the asset price is given by S, and the number of shares in the asset is given by N.
This is a generally accepted relationship, i.e. a stylized fact, and can be found the work of
econophysicists such as Stanley, Farmer, and others. The Kelly criterion and price impact
are combined to up-date in regular time intervals, e.g. daily, the portfolio allocation in a
self-contained way and produce a price dynamic for the underlying assets.
These key concepts are sufficient to develop a tractable deterministic model. First the
concepts are turned into equations, which can be analyzed and lead to a phase diagram
relating parameter choices to price behaviour. The stability properties of the resulting price
process can be deduced.
The market impact γis normally a number around 0.5 depending on asset class and
country with a vast phenomenological literature to support the estimates. It signifies how
price impact scales with amount of directed buy or sell activity. The necessary liquidity is
provided by dedicated market makers and nowadays to some extent also by High Frequency
Traders. In the next paragraphs the Kelly criterion is further explored.
For the portfolio construction we rely on the Kelly criterion introduced by Kelly (1956)
in the wake of earlier information-theoretic results by Shannon to find the optimal betting
amount in games with fixed known odds. It was later extended to the field of financial
investments by Thorp (1969) and others. The strategy is equivalent to myopic log-utility
maximization. In the process the entropy of the value process is maximized. Entropy up to
a multiplicative constant can be uniquely characterized by a small set of rules. For a modern
derivation from three axioms based on information loss see Baez et al. (2012). The question,
why investors should choose to maximize the log wealth, has a simple answer: according to
Breiman’s theorem (Breiman, 1960), the strategy gives the asymptotically optimal pay-out
and dominates any other strategy, i.e in the long run trounces any different strategy. The
Kelly criterion tells us for example that the optimal betting fraction is given by p−q, if a
gambler is faced with a bet, where the probability to double the money is pand to lose the
initial stake is q(p > q, p +q= 1).
The original idea has been extended to the general continuous time framework with N
arbitrarily correlated assets. The sensitivity to estimation errors in both expected drift
and correlation has been studied and sensitivities, like the greeks of option theory, can be
calculated. A general analysis of the Kelly criterion for the Ornstein-Uhlenbeck process can
be found in Lv & Meister (2009, 2010).
The Kelly criterion is not a panacea and has it critics: see, in particular, Samuelson
(1969, 1979). It is often regarded as too risky for practical investment, since (a) the chance
of losing ǫis 1−ǫfor ǫ∈(0,1), (b) parameters are unstable, and (c) the sensitivities are hard
to estimate in practice. One way of going beyond standard entropy is to consider the larger
class of Renyi entropies, which provide a range of portfolios associated with a continuous
risk aversion parameter.
In the continuous case with one risky asset with price process {St}t≥0assumed by the
Kelly-optimizers to evolve according to a geometric Brownian motion (GBM) with drift µt
and volatility σt,
dSt
St
=µtdt +σtdWt,(2)

4
and the riskless asset increasing with the short rate rtgiven by
dBt
Bt
=rtdt, (3)
with B0= 1, the optimal portfolio, with value process {Vt}t≥0, maximizes the exponential
growth. The result is given in terms of Λt, which is the proportion of the available assets
invested in the risky asset at time t. We further set the drift µtequal to rt+σtλt, where λt
is the market price of risk. The value process of the portfolio has to satisfy
θtSt+φtBt=Vt,(4)
where θtis the number of shares in the portfolio, and φtis the number of units in the money
market account. In addition, the self-financing equation is
θtdSt+φtdBt=dVt.(5)
These relations allow us to calculate the optimal amount θt. First we evaluate the differential
of the growth of the value process
dlog(Vt) = dVt
Vt−1
2
(dVt)2
V2
t
=θtdSt+φtdBt
Vt−1
2
(dSt)2
V2
t
=rtdt +λtσtθtSt
Vt−1
2
σ2
tθ2
tS2
t
V2
tdt +θtStσt
Vt
dWt.(6)
The drift of the value process {Vt}t≥0is maximized by taking the derivative with respect to
the freely chosen number of shares θt, to obtain
θ∗
t:= λt
σt
Vt
St
,
and as a consequence
φ∗
t:= 1−λt
σtVt
Bt
,
It can be rewritten, and provides the definition of the leverage ratio
Λ∗
t:= λt
σt
=θ∗
t
St
Vt
.(7)
This derivation holds for a general class of adapted processes for the drift, volatility, interest
rate and market price of risk. The proportion of assets in shares is equal to the ratio of
the market price of risk, which is equal to the Sharpe ratio, and the volatility, and can be
rewritten as (µt−rt)/σ2
t. In subsequent sections we assume the leverage ratio, the risky
asset price volatility, the interest rate and the market price of risk to be slowly changing
variables. As a consequence, we assume the CTA investors to take these quantities to be
constant and not to be readjusted with the same speed as the portfolio weights. In other

5
words, for the time-scale we are interested in the price changes of the underlying asset as
well as the connected portfolio weights are significant, but the parameters are held constant.
One could ask, if there is an elegant way of implementing this optimal portfolio using
derivatives. Indeed, such a derivative exist. Let us look at the European call option in
the Black-Scholes model with constant volatility and interest rate. The value of a call with
strike Kis given by
C(St, K, σ, r, T −t) = StN(d1)−Ke−r(T−t)N(d2),(8)
where N(·) denotes the normal cumulative distribution function, and d1=d2+σ√T−t=
1
σ√T−tlog(St/Kt) + (r+σ2/2)(T−t). The value process of the call can be replicated by
a combination of stock and bond. The stock weight is given by, what is commonly called ∆,
N(d1) and the bond weight is given by −KN(d2) exp(−rT ). If these weights coincide with
the optimal leverage in stocks and bonds as calculated before, then the portfolio with value
C(St, K, σ, r, T −t) achieves the stated purpose. This can be further elaborated by noting
that for
N(d1) = λ
σ
C(St, K, σ, r, T −t)
St
N(d2) = −1−λ
σC(St, K, σ, r, T −t)
Ker(T−t)(9)
equivalence is achieved. There is not always a single call available to reproduce the optimal
portfolio. If for example λ≤σ, then there is no solution. On the other hand, due to market
completeness derivatives can never outperform the optimal stock plus bond portfolio. In
addition, since N(d1)≥N(d2), we have the further constraint
λ
σ
K
Ste−r(T−t)−1≥ −1.
One can determine numerically combinations of Kand T−tfor fixed St,σ,rand λtthat
satisfy the constraints. As time progresses the position would shift in strike or maturity. A
special cases is treated in the Appendix.
The risky asset does not have to follow a geometric Brownian motion, but can instead
belong to the class of geometric Levy models (GLM) of the form
ˆ
St=ˆ
S0ert+R(λ,σ)t+σXt−ψ(σ)t(10)
with the expectation value heσXti=eψ(σ)t; for background and details see Brody et al.
(2012). where σ, λ &rare chosen to be constants, ψis called the Levy exponent, and
R(λ, σ) has the role of a risk premium. In this case the self-financing condition containing
differentials is not directly applicable and instead we use the expansion of the logarithm
log(1 + x) = x−x2/2 + O(x3). The value process in the GLM case is
ˆ
θˆ
St+ˆ
φBt=ˆ
Vt.(11)
and for computational convenience we set ˆ
V0= 1 and B0= 1. We maximize again the
logarithmic utility
hlog(ˆ
θˆ
St+ (1 −ˆ
θˆ
S0)Bt)i(12)

6
over all ˆ
θ. The logarithmic utility function in the short time limit, relevant for the optimi-
sation, since the portfolio can be adjusted at any time, becomes
hlog(ˆ
θˆ
St+ (1 −ˆ
θˆ
S0)Bt)i − log( ˆ
V0) = Dlog 1 + ˆ
θˆ
St+ (1 −ˆ
θˆ
S0)Bt−ˆ
V0
ˆ
V0E
=D(ˆ
Vt−1) −(ˆ
Vt−1)2/2 + ∞
X
k=2
(−1)k(Vt−1)k
kE.
The first term in the expansion is
ˆ
Vt−1 = ˆ
θˆ
S0(1 + rt +R(λ, σ)t)e(σXt−ψ(σ)t)−(1 + rt)+rt +O(t2),
(13)
and the expectation value of this term is
hˆ
Vt−1i=ˆ
θˆ
S0Dexp(σXt−tψ(σ))E(1 + rt +R(λ, σ)t)−ˆ
θˆ
S0(1 + rt) + rt +O(t2)
=ˆ
θˆ
S0R(λ, σ)t+rt +O(t2).(14)
The expectation value of the second term is
h(ˆ
Vt−1)2i=ˆ
θ2ˆ
S2
0[tψ(2σ)−2tψ(σ)] + O(t2),(15)
while higher order terms can be ignored, since they are of order O(t2). The combination of
terms gives
hlog( ˆ
Vt)i=ˆ
θˆ
S0R(λ, σ)t+rt −1
2ˆ
θˆ
S02tψ(2σ)−2tψ(σ)+O(t2).(16)
The optimal leverage ratio for GLM in the limit of t→0 is
ˆ
Λ = R(λ, σ)
ψ(2σ)−2ψ(σ).(17)
As an example, for geometric Brownian motion with R(λ, σ) = λ σ,ψ(σ) = σ2/2, and the
process {Xt}t≥0set equal to {Wt}t≥0one reproduces the result derived above. The optimal
ratio can also be calculated for jump diffusion and other popular models. Sensitivities of
the optimal ratio to the various quantities appearing in the formula like σand λ, as done in
Lv et al. (2009 & 2010), can be determined.
In the next section for a specific price process and under some additional suitable sim-
plifications the dynamics of the optimal value as well as the underlying asset process are
described. This leads to a deterministic dynamic with an associated phase diagram.
III. THE RESULTING PORTFOLIO DYNAMIC IN THE ONE RISKY ASSET
CASE
The portfolio dynamics can be studied in a variety of ways. Here we look at a particular
simple case, where both the market price of risk and volatility estimate, λand σ, for the

7
asset {St}t≥0are assumed to remain unchanged and as a consequence the leverage ratio
is also a fixed quantity. In addition, the interest rate ris taken to be constant. These
three quantities are considered in this section as externally fixed parameters, i.e estimated
by the Kelly investors either solely from a fixed set of historical data or also based on
forward looking data only occasionally updated incorporating future earnings estimates or
the company growth rate as the case could be for equities. One could claim that this is not
entirely consistent, since the leverage ratio is only true in a probabilistic setting, but the
model we consider here is deterministic. This apparent paradox can be resolved by assuming
that in this toy model the portfolio investors are not aware of the deterministic nature of
the price process and believe the risky asset to be governed by a geometric Brownian motion
with fixed drift and volatility.
The portfolio value of a Kelly investor is then given by
Vt=λ
σVt+1−λ
σVt,(18)
where λ/σ is the optimal leverage ratio Λ derived above.The change of the portfolio value
due to the self-financing condition is given by
dVt=λ
σVt
dSt
St
+1−λ
σVt
dBt
Bt
(19)
and
dVt
Vt
= ΛdSt
St
+1−ΛdBt
Bt
.(20)
The change in the number of shares in the portfolio is given by
∆θt= Λ Vt+dVt
St+dSt−ΛVt
St
,
= ΛVt
StdVt
Vt−dSt
St−σ2dt +O(dS3
t).(21)
After some simplifications the combination of the last two equations leads to
∆θt∝hdVt
Vt−dSt
Sti∼h(Λ −1)dSt
St
+ (1 −Λ)rdti.(22)
We can further assume rdt ∼0, since to neglect the change of the value of the money market
account seems reasonable for short time intervals, in particular in the current interest rate
environment. The last equation can then be rewritten as
∆θt∼h(Λ −1)dSt
Sti.(23)
For example, we therefore have
∆θt∼dVt
Vt−dSt
St≥0,if dSt>0 & Λ >1.(24)

8
As a result, there are two conditions in this dynamical system that govern the changes
in the number of shares held
∆θt∼h(Λ −1)dSt
Sti,∗(25)
∆θt∼dSt
Stγ,∗∗ (26)
which are applied alternately.
FIG. 1: Price-change versus position-size-change diagram. The x-axis is dSt/Stand the y-axis
corresponds to the change of the number of shares in the portfolio at each adjustment step. The
straight line with the slope dependent on (Λ −1) relates price change to portfolio change and the
square-root function relates change in number of shares to price impact. As a consequence the
market drifts either to the stable fixed point at the origin or, if it starts to the right of the unstable
fixed point, away to infinity..
These two relations are sufficient to already develop some intuition about the dynamics
of the price process and the related portfolio changes as shown in Figure 1. The size of Λ
governs the slope of the straight line from the origin. If Λ is larger one then the number
of shares added at each time stage, assuming the right sign for the slow changing variables
like the drift and volatility, is positive. The number is either ever increasing as is the case
to the right of the unstable fixed point, where the two curves intersect, or is ever decreasing
and approaching zero as the process moves towards the left fixed point. The analysis above
holds true, if we assume all the ‘slow’ parameters like r, λ&σare fixed in time. A realistic
dynamic adjustment of these parameters, i.e. considering them as time dependent variables,
plus unavoidable noise leads to more complex behaviour more closely aligned with what one

9
can see in the financial market. This is not surprising, since the added layer of complexity
allows a fine tuning of the system.
The resulting dynamics can be most easily described graphically, and some additional
insights can be garnered from Figure 1, where the dynamics of the price process vis-a-vis
the portfolio changes are given by arrows. One see that the portfolio as well as the asset
price are adjusted to bounce between the two curves. The upper curve relates the position
adjustment to price changes by the price impact equation governed by γwhich is normally
around 0.5. The lower curve relates the portfolio adjustment based on the leverage ratio Λ
to the asset price change. The leverage ratio is in each step modified to maximize the growth
of the portfolio. As a result of the dynamics, the market drifts either to the stable fixed
point at the origin or, if it starts to the right of the unstable fixed point, away to infinity.
γ
λ
1/2
1
I
III
II
I
FIG. 2: Phase diagram. The phase diagram relates for a fixed initial change of position size the
price impact coefficient γaround the value of 0.5 to the leverage ratio Λ on the x-axis. The diagram
falls into three regions. In the region I the price process oscillates around zero and moves in the
deterministic case towards zero. In the region II the price process monotonically decreases and
tends towards zero. In region III the price process monotonically increases and the situation is
unstable. .
In Figure 2 a rough phase diagram is displayed around the value of γof 0.5 representing
the y-axis and the leverage ratio Λ the x-axis. It shows the existence of three phases. Phase
I represent the region, where the price change oscillates between positive and negative at
each step and price changes get progressively smaller. Not displayed is the region, for larger
values of γ, where the oscillations between positive and negative values get bigger with time
and an instability arises. The Phase II represents the region, where the changes of the price
in each time step always have the same sign, but decrease with time. In Phase III the
price process explodes as price changes always have the same sign and get bigger with each

10
time step. The portfolio adjustment increases also with time and the free float is eventually
unable to accommodate the liquidity requirements. Explosive growth is unsustainable and
leads to a market break-down and eventually to a dramatic price correction. In the next
section strategies are briefly described to exploit this predictability.
IV. META-CTA STRATEGIES
A natural extension of the analysis of the earlier sections is to consider, how other mar-
ket participants can exploit rigid CTA strategies and how CTA managers can make their
strategies more flexible. This leads to meta-CTA strategies.
The first step in the construction of a meta-CTA strategy is to determine, in which of the
three phase the market is situated. If the process is placed squarely in Phase III, then caution
is required, since the simplified price process without noise will reach an instability due to
larger and larger movements, and a major reversion is to be expected. A meta-CTA investor
should as a consequence deviate from the rigid CTA strategy and reduce exposure, whereas a
predatory market participant should take a position possibly through derivatives like vanilla
options to exploit the eventual price reversion accompanied by a spike in volatility. A fast
moving market participant might also try to exploit first the run-up of the asset and then
the eventual correction using some well-defined rules, whose suitability might be profitably
checked against historical data.
The other two Phases provide less direct opportunities. If the market is in Phase II, a
move towards an equilibrium is expected. This suggests the absence of a large reversion as
well as a decline of volatility, which could be exploited through the selling of gamma. If
the market is in Phase I, volatility is likely to decline and the progressively decreasing price
swings lead to a lack of directional opportunities.
As just described, out of the three phases, the most substantial opportunities exist in
Phase III, where a run-away effect creates potentially an exploitable instability. These phe-
nomenological ideas can be further buttressed by the study of market data and by including
noise in the theoretical model to enable sensitivity analysis.
V. CONCLUSION
The toy model studied in the paper shows what CTA managers are confronted with, if they
rigidly adhere to a fixed view of the underlying price process.
The weakest assumption of the paper, namely that the Kelly based investors misjudge
the market by assuming it to be described by a known geometric Brownian motion, is
maybe less debilitating than it might initially appear, since the amount of data required to
reasonably estimate the asset drift is not available in markets with regular paradigm shifts.
As a consequence, a geometric Brownian motion model with fixed parameters for the risky
asset is often taken as a starting point in finance.
What might seem surprising is the ability to create interesting price behaviour and port-
folio dynamics without the addition of extraneous noise, i.e. in a deterministic setting. Of
course, noise as well as a sensitivity analysis with regards to the various variables employed
would make the model more realistic and useful. This will be studied separately.
It is comforting that even such a simple model already provides some food for thought
and raises some practical questions. Is it for example possible to determine in some heuristic

11
way in which part of the phase diagram a stock or other asset currently hovers? Once the
phase is determined one can adjust ones investment behaviour, e.g. Phase II has lower
volatility and allows higher leverage, whereas Phase III requires a more cautious approach.
This has to be compared with the volatility and drift assumptions for the original CTA
model to show how a Meta-CTA strategy differs from the un-reflected version. It would be
also of interest to see how the dynamics change, if one links the slow changing variables like
asset drift and volatility to recent changes of the asset and adds noise. Another questions
of interest is how the model can be used to discover points where trends are reversed, i.e.
how trend reversal is a natural aspect of the unstable Phase III.
Another point of interest would be to link the approach described to the predictable
rebalancing of leveraged exchanged traded funds (ETF), which in some Asian markets have
a significant market share and are responsible for the majority of ETF trading activity. The
link between leveraged ETF rebalancing and equity volatility was already explored in Shum
et al. (2015) and it would be interesting to see, how the methodology introduced in this
paper could distinguish between different phases.
Helpful comments by D.C. Brody and L.P. Hughston, including the derivation in Section
II of the leverage ratio Λ∗
tbased on the self-financing criterion, which is an improvement to
the calculations one normally finds in the literature, are gratefully acknowledged.
[Baez et al. 2011] J. Baez, J. Fritz and T. Leinster (2011) A characterization of entropy in terms
of information loss. Entropy 13.11, 1945-1957.
[Breiman 1960] L. Breiman (1960) Optimal gambling systems for favorable games. Proc. Fourth
Berkeley Symp. Math. Statist. Probab. 1, 60-77, Univ. California Press.
[Brody et al. 2012] D. C. Brody, L. P. Hughstan and E. Mackie (2012) General Theory of Geo-
metric Levy Models for Dynamic Asset Pricing. Proc. R. Soc. A June 8, 468 1778-1798.
[Bughardt & Walls 2011] G. Burghardt and B. Walls (2011) Managed Futures for Institutional
Investors: Analysis and Portfolio Construction. New York: Wiley.
[Kelly 1956] J. L. Kelly (1956) A new interpretation of information rate. Bell System Technical
Journal 35 (4), 917-926.
[Lemperiere et al. 2014] Y. Lemprire, C. Deremble, P. Seager, M. Potters and J. P. Bouchaud
(2014) Two centuries of trend following. arXiv:1404.3274.
[Lv & Meister 2009] Y. D. Lv and B. K. Meister (2009) Application of the Kelly criterion to
Ornstein-Uhlenbeck processes. LNICST, Vol. 4, 1051-1062.
[Lv & Meister 2010] Y. D. Lv and B. K. Meister (2010) Applications of the Kelly Crite-
rion to multi-dimensional diffusion processes. International Journal of Theoretical and Ap-
plied Finance 13, 93-112. Reprinted (2011) in The Kelly Criterion: Theory and Practice
(L. C. MacLean, E. O. Thorp and W. T. Ziemba, eds). Singapore: World Scientific Publish-
ing Company. 285-300.
[MacLean et al. 2011] L. C. MacLean, E. O. Thorp & W. T. Ziemba, editors (2011) The Kelly
Criterion: Theory and Practice. Singapore: World Scientific Publishing Company.
[Pierou et al. 2002] V. Plerou, P. Gopikrishnan, X. Gabaix and H. E. Stanley (2002) Quantifying
stock price response to demand fluctuations. Physical Review E, 66, 027104.
[Samuelson 1969] P. A. Samuelson (1969) Lifetime portfolio selection by dynamic stochastic pro-
gramming. Review of Economics and Statistics 51, 239-246.

12
[Samuelson 1979] P. A. Samuelson (1979) Why we should not make mean log of wealth big though
years to act are long. Journal of Banking and Finance 3, 305-307.
[Shum et al. 2015] P. Shum, W. Hejazi, E. Haryanto, A. Rodier (2015) Intraday share price volatil-
ity and leveraged ETF rebalancing. Review of Finance (online).
[Thorp 1969] E. O. Thorp (1969) Optimal gambling systems for favorable games. Revue de
l’Institut International de Statistique, 273-293.
APPENDIX: AN EXAMPLE OF A CALL OPTION AS A KELLY OPTIMAL
PORTFOLIO
Let us find the at-the-money call option Cwith r= 0 and positive market price of risk λ≥0,
such that the value of the Call option matches the optimal portfolio. Two constraints have
to be matched. The stock and bond weights of the replicating portfolio of the option have
to coincide with the weights calculated with the help of the Kelly criterion. By fixing the
various quantities, the equations containing the normal cumulative distribution functions
N(d1) and N(d2) can be combined and simplify to
N(d1) = 1
2−σ
λ
(27)
using N(d1) = 1 −N(−d1) = N(−d2). There are always solutions for N(d1) to satisfy the
equation above, if λ/σ > 1. The function N(d1) lies in this case only in the interval [1/2,1],
since both λand σare positive. If the combination σ∗and T∗−t∗is a solution, then so is
σ∗/α and (T∗−t∗)α2for any positive α.