w4reading_3 - 3 A Perspective on Quantitative Finance...

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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 45-year effort to build mathematical models for “beating markets”, by which I mean achieving risk-adjusted 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 twenty-one. 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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34 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 million-page “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 10-dimensional expectation surface. My methods reduced the problem from 400 million years
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w4reading_3 - 3 A Perspective on Quantitative Finance...

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