Kelly Criterion 2007, Artykuły po angielsku

[ Pobierz całość w formacie PDF ]
1
Chapter 9
1
2
2
3
THE KELLY CRITERION IN BLACKJACK SPORTS BETTING,
AND THE STOCK MARKET
1
3
4
4
5
5
6
EDWARD O. THORP
Edward O. Thorp and Associates, Newport Beach, CA 92660, USA
6
7
7
8
8
9
Contents
9
10
10
11
Abstract
2
11
12
Keywords
2
12
13
1. Introduction
3
13
14
2. Coin tossing
4
14
15
3. Optimal growth: Kelly criterion formulas for practitioners
8
15
16
3.1. The probability of reaching a fixed goal on or before
n
trials
8
16
17
3.2. The probability of ever being reduced to a fraction
x
of this initial bankroll
10
17
18
3.3. The probability of being at or above a specified value at the end of a specified number of
trials
18
19
11
19
20
3.4. Continuous approximation of expected time to reach a goal
12
20
21
3.5. Comparing fixed fraction strategies: the probability that one strategy leads another after
n
trials
21
22
12
22
23
4. The long run: when will the Kelly strategy “dominate”?
14
23
24
5. Blackjack
15
24
25
6. Sports betting
17
25
26
7. Wall street: the biggest game
21
26
27
7.1. Continuous approximation
22
27
28
7.2. The (almost) real world
25
28
29
7.3. The case for “fractional Kelly”
27
29
30
7.4. A remarkable formula
30
30
31
8. A case study
31
31
32
8.1. The constraints
32
32
33
8.2. The analysis and results
32
33
34
8.3. The recommendation and the result
33
34
35
8.4. The theory for a portfolio of securities
34
35
36
36
37
37
38
1
Paper presented at: The 10th International Conference on Gambling and Risk Taking, Montreal, June 1997,
published in: Finding the Edge: Mathematical Analysis of Casino Games, edited by O. Vancura, J.A. Cor-
nelius, W.R. Eadington, 2000. Corrections added April 20, 2005.
Handbook of Asset and Liability Management, Volume 1
Edited by S.A. Zenios and W. Ziemba
Copyright
©
2006 Elsevier B.V. All rights reserved
DOI: 10.1016/S1872-0978(06)01009-X
38
39
39
40
40
41
41
42
42
43
43
2
E.O. Thorp
1
9. My experience with the Kelly approach
35
1
2
10. Conclusion
36
2
3
Acknowledgements
36
3
4
Appendix A. Integrals for deriving moments of
E
∞
36
4
5
Appendix B. Derivation of formula (3.1)
37
5
6
Appendix C. Expected time to reach goal
39
6
7
Uncited references
44
7
8
References
44
8
9
9
10
10
11
Abstract
11
12
12
The central problem for gamblers is to find positive expectation bets. But the gam-
bler also needs to know how to manage his money, i.e., how much to bet. In the stock
market (more inclusively, the securities markets) the problem is similar but more com-
plex. The gambler, who is now an “investor”, looks for “excess risk adjusted return”.
In both these settings, we explore the use of the Kelly criterion, which is to maximize
the expected value of the logarithm of wealth (“maximize expected logarithmic util-
ity”). The criterion is known to economists and financial theorists by names such as
the “geometric mean maximizing portfolio strategy”, maximizing logarithmic utility,
the growth-optimal strategy, the capital growth criterion, etc. The author initiated the
practical application of the Kelly criterion by using it for card counting in blackjack.
We will present some useful formulas and methods to answer various natural questions
about it that arise in blackjack and other gambling games. Then we illustrate its recent
use in a successful casino sports betting system. Finally, we discuss its application to
the securities markets where it has helped the author to make a thirty year total of 80
billion dollars worth of “bets”.
13
13
14
14
15
15
16
16
17
17
18
18
19
19
20
20
21
21
22
22
23
23
24
24
25
25
26
26
27
27
28
28
29
29
Keywords
30
30
31
Kelly criterion, Betting, Long run investing, Portfolio allocation, Logarithmic utility,
Capital growth
31
32
32
33
33
34
JEL classification
: C61, D81, G1
34
35
35
36
36
37
37
38
38
39
39
40
40
41
41
42
42
43
43
Ch. 9: The Kelly Criterion in Blackjack Sports Betting, and the Stock Market
3
1
1. Introduction
1
2
2
3
The fundamental problem in gambling is to find positive expectation betting op-
portunities. The analogous problem in investing is to find investments with excess
risk-adjusted expected rates of return. Once these favorable opportunities have been
identified, the gambler or investor must decide how much of his capital to bet. This
is the problem which we consider here. It has been of interest at least since the
eighteenth century discussion of the St. Petersburg Paradox (
Feller, 1966
) by Daniel
Bernoulli.
One approach is to choose a goal, such as to minimize the probability of total loss
within a specified number of trials,
N
. Another example would be to maximize the
probability of reaching a fixed goal on or before
N
trials (
Browne, 1996
).
A different approach, much studied by economists and others, is to value money
using a utility function. These are typically defined for all non-negative real numbers,
have extended real number values, and are non-decreasing (more money is at least as
good as less money). Some examples are
U(x)
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
16
=
x
a
,0
a<
∞
, and
U(x)
=
log
x
,
16
17
17
. Once a utility function is specified, the object
is to maximize the expected value of the utility of wealth.
Daniel Bernoulli used the utility function log
x
to “solve” the St. Petersburg Para-
dox. (But his solution does not eliminate the paradox because every utility function
which is unbounded above, including log, has a modified version of the St. Petersburg
Paradox.) The utility function log
x
was revisited by
Kelly (1956)
where he showed
that it had some remarkable properties. These were elaborated and generalized in an
important paper by
Breiman (1961)
.
Markowitz (1959)
illustrates the application to se-
curities. For a discussion of the Kelly criterion (the “geometric mean criterion”) from
a finance point of view, see
McEnally (1986)
. He also includes additional history and
references.
I was introduced to the Kelly paper by Claude Shannon at M.I.T. in 1960, shortly
after I had created the mathematical theory of card counting at casino blackjack. Kelly’s
criterion was a bet on each trial so as to maximize
E
log
X
, the expected value of
the logarithm of the (random variable) capital
X
. I used it in actual play and intro-
duced it to the gambling community in the first edition of Beat the Dealer (
Thorp,
. If all blackjack bets paid even money, had positive expectation and were in-
dependent, the resulting Kelly betting recipe when playing one hand at a time would
be extremely simple: bet a fraction of your current capital equal to your expectation.
This is modified somewhat in practice (generally down) to allow for having to make
some negative expectation “waiting bets”, for the higher variance due to the occur-
rence of payoffs greater than one to one, and when more than one hand is played at a
time.
Here are the properties that made the Kelly criterion so appealing. For ease of un-
derstanding, we illustrate using the simplest case, coin tossing, but the concepts and
conclusions generalize greatly.
=−∞
18
18
19
19
20
20
21
21
22
22
23
23
24
24
25
25
26
26
27
27
28
28
29
29
30
30
31
31
32
32
33
33
34
34
35
35
36
36
37
37
38
38
39
39
40
40
41
41
42
42
43
43
where log means log
e
, and log 0
4
E.O. Thorp
1
2. Coin tossing
1
2
2
3
Imagine that we are faced with an infinitely wealthy opponent who will wager even
money bets made on repeated independent trials of a biased coin. Further, suppose that
on each trial our win probability is
p>
1
/
2 and the probability of losing is
q
3
4
4
5
p
.
Our initial capital is
X
0
. Suppose we choose the goal of maximizing the expected value
E(X
n
)
after
n
trials. How much should we bet,
B
k
,onthe
k
th trial? Letting
T
k
=
=
1
−
5
6
6
7
1ifthe
7
8
k
th trial is a win and
T
k
=−
1ifitisaloss,then
X
k
=
X
k
−
1
+
T
k
B
k
for
k
=
1
,
2
,
3
,...
,
8
X
0
+
k
=
1
T
k
B
k
. Then
9
and
X
n
=
9
10
10
11
n
n
11
12
E(X
n
)
=
X
0
+
E(B
k
T
k
)
=
X
0
+
(p
−
q)E(B
k
).
12
13
k
=
1
k
=
1
13
14
q>
0 in this even payoff situation,
then in order to maximize
E(X
n
)
we would want to maximize
E(B
k
)
at each trial.
Thus, to maximize expected gain we should bet
all of our resources
at each trial. Thus
B
1
=
−
14
15
15
16
16
17
X
0
and if we win the first bet,
B
2
=
2
X
0
, etc. However, the probability of ruin is
17
18
18
1 so ruin is almost sure. Thus the
“bold” criterion of betting to maximize expected gain is usually undesirable.
Likewise, if we play to minimize the probability of eventual ruin (i.e., “ruin” occurs
if
X
k
=
−
p
n
and with
p<
1, lim
n
→∞
[
1
−
p
n
]=
19
19
20
20
21
0onthe
k
th outcome) the well-known gambler’s ruin formula in
Feller (1966)
shows that we minimize ruin by making a
minimum
bet on each trial, but this unfortu-
nately also minimizes the expected gain. Thus “timid” betting is also unattractive.
This suggests an intermediate strategy which is somewhere between maximizing
E(X
n
)
(and assuring ruin) and minimizing the probability of ruin (and minimizing
E(X
n
)
). An asymptotically optimal strategy was first proposed by
Kelly (1956)
.
In the coin-tossing game just described, since the probabilities and payoffs for each
bet are the same, it seems plausible that an “optimal” strategy will involve always wa-
gering the same fraction
f
of your bankroll. To make this possible we shall assume
from here on that capital is infinitely divisible. This assumption usually does not matter
much in the interesting practical applications.
If we bet according to
B
i
=
21
22
22
23
23
24
24
25
25
26
26
27
27
28
28
29
29
30
30
31
31
32
32
33
1, this is sometimes called “fixed
fraction” betting. Where
S
and
F
are the number of successes and failures, respectively,
in
n
trials, then our capital after
n
trials is
X
n
=
fX
i
−
1
, where 0
f
33
34
34
35
X
0
(
1
+
f)
S
(
1
−
f)
F
, where
S
+
F
=
n
.
35
0. Thus “ruin” in the technical sense
of the gambler’s ruin problem cannot occur. “Ruin” shall henceforth be reinterpreted to
mean that for arbitrarily small positive
ε
, lim
n
→∞
[
0
)
=
36
36
37
37
38
Pr
(X
n
ε)
]=
1. Even in this sense,
38
as we shall see, ruin
can
occur under certain circumstances.
We note that since
e
n
log
X
n
X
0
39
39
40
40
41
1
/n
X
n
X
0
,
41
42
=
42
43
43
Since the game has a positive expectation, i.e.,
p
given by 1
With
f
in the interval 0
<f <
1, Pr
(X
n
=
 Ch. 9: The Kelly Criterion in Blackjack Sports Betting, and the Stock Market
5
1
the quantity
1
2
log
X
n
X
0
1
/n
S
n
log
(
1
F
n
2
3
G
n
(f )
=
=
+
f)
+
log
(
1
−
f)
3
4
4
5
measures the exponential rate of increase per trial. Kelly chose to maximize the expected
value of the growth rate coefficient,
g(f )
, where
5
6
6
7
E
log
X
n
X
0
1
/n
E
S
f)
7
F
n
8
g(f )
=
=
n
log
(
1
+
f)
+
log
(
1
−
8
9
9
10
=
p
log
(
1
+
f)
+
q
log
(
1
−
f).
10
11
(
1
/n)
log
X
0
so for
n
fixed, maximizing
g(f )
is
the same as maximizing
E
log
X
n
. We usually will talk about maximizing
g(f )
in the
discussion below. Note that
=
(
1
/n)E(
log
X
n
)
−
11
12
12
13
13
14
14
p
q
p
−
q
−
f
15
g
(f )
=
f
−
f
=
f)
=
0
15
1
+
1
−
(
1
+
f)(
1
−
16
16
17
when
f
=
f
∗
=
p
−
q
.
17
18
Now
g
(f )
18
19
f)
2
<
0
so that
g
(f )
is monotone strictly decreasing on
f)
2
19
=−
p/(
1
+
−
q/(
1
−
20
20
21
[
0
,
1
)
.Also
g
(
0
)
=
p
−
q>
0 and
21
22
lim
f
→
1
−
g
(f )
=−∞
. Therefore by the continuity of
g
(f )
,
g(f )
has a unique maxi-
22
23
mum at
f
=
f
∗
, where
g(f
∗
)
=
p
log
p
+
q
log
q
+
log 2
>
0. Moreover,
g(
0
)
=
0 and
23
24
lim
f
→
q
−
g(f )
=−∞
so there is a unique number
f
c
>
0, where 0
<f
∗
<f
c
<
1,
24
25
0. The nature of the function
g(f )
is now apparent and a graph of
g(f )
versus
f
appears as shown in
Figure 1
.
The following theorem recounts the important advantages of maximizing
g(f )
.The
details are omitted here but proofs of (i)–(iii), and (vi) for the simple binomial case
can be found in
Thorp (1969)
; more general proofs of these and of (iv) and (v) are in
Breiman (1961)
.
=
25
26
26
27
27
28
28
29
29
30
30
31
31
32
Theorem 1.
(i)
If g(f ) >
0
, then
lim
n
→∞
X
n
=∞
almost surely, i.e., for each M,
32
33
Pr
[
1;
(ii)
If g(f ) <
0
, then
lim
n
→∞
X
n
]=
33
34
=
0
almost surely
;
i.e., for each ε>
0
,
34
35
Pr
[
lim sup
n
→∞
X
n
<ε
]=
1;
35
36
0
a.s.
(iv)
Given a strategy Φ
∗
which maximizes E
log
X
n
and any other “essentially differ-
ent” strategy Φ
(
not necessarily a fixed fractional betting strategy
)
, then
lim
n
→∞
(iii)
If g(f )
=
0
, then
lim sup
n
→∞
X
n
=∞
a.s. and
lim inf
n
→∞
X
n
=
36
37
37
38
38
39
a.s.
(v)
The expected time for the current capital X
n
to reach any fixed preassigned goal
C is, asymptotically, least with a strategy which maximizes E
log
X
n
.
(vi)
Suppose the return on one unit bet on the i
th
trial is the binomial random variable
U
i
;
further, suppose that the probability of success is p
i
, where
1
/
2
<p
i
<
1
. Then
X
n
(Φ
∗
)/X
n
(Φ)
=∞
39
40
40
41
41
42
42
43
43
Note that
g(f )
such that
g(f
c
)
lim inf
n
→∞
X
n
>M
[ Pobierz całość w formacie PDF ]
  • zanotowane.pl
  • doc.pisz.pl
  • pdf.pisz.pl
  • klobuckfatima.xlx.pl