3
A Perspective on
Quantitative Finance:
Models for Beating
the Market
Ed Thorp
© 2003
Quantitative Finance Review 2003
T
his is a perspective on quantitative ﬁnance from my point of view, a 45year effort
to build mathematical models for “beating markets”, by which I mean achieving
riskadjusted excess returns.
I’d like to illustrate with models I’ve developed, starting with a relatively simple
example, the widely played casino game of blackjack or twentyone. What does
blackjack have to do with ﬁnance? A lot more than I ﬁrst thought, as we’ll see.
Blackjack
When I ﬁrst learned of the game in 1958, I was a new PhD in a part of mathematics known
as functional analysis. I had never gambled in a casino. I avoided negative expectation games.
I
knew
of
the
various
proofs
that
it
was
not
possible
to
gain
an
edge
in
virtually
all
the
standard casino gambling games. But an article by four young mathematicians presented a
“basic” strategy for blackjack that allegedly cut the House edge to a mere 0.6%. The authors
had developed their strategy for a complete randomly shufﬂed deck.
But in the game as played, successive rounds are dealt from a more and more depleted pack
of cards. To me, this “sampling without replacement”, or “dependence of trials”, meant that
the usual proofs that you couldn’t beat the game did not apply. The game might be beatable. I
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THE BEST OF WILMOTT
realized that the player’s expectations would ﬂuctuate, under best strategy, depending on which
depleted pack was being used. Would
the
ﬂuctuations be
enough to
give
favorable betting
opportunities? The domain of the expectation function for one deck had more than 33 million
points in a space with 10 independent variables, corresponding to the 10 different card values
in blackjack (for eight decks it goes up to 6
×
10
15
).
Voila! There “must be” whole continents of positive expectation. Now to ﬁnd them. The
paper I read had found the strategy and expectation for only one of these 33 million points.
That was only an approximation with a smallish but poorly known error term, and it took 12
years on desk calculators. And each such strategy had to address several mathematically distinct
decisions at the table, starting with 550 different combinations of the dealer’s up card and the
player’s initial two cards. Nonetheless, I had taken the ﬁrst step towards building a model: the
key idea or “inspiration” – the domain of the visionaries.
The
next
step
is
to
develop
and
reﬁne
the
idea
via
quantitative
and
technical
work
so
that it can be used in the real world. The brute force method would be to compute the basic
strategy and expectation for each of the 33 million mathematically distinct subsets of cards
and assemble a 33 millionpage “book”. Fortunately, using linear methods well known to me
from functional analysis, I was able to build a simpliﬁed approximate model for much of the
10dimensional expectation surface. My methods reduced the problem from 400 million years
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 Three '09
 EgonKalotay
 The Market, Statistical Arbitrage

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