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Dissertation: Self-Imposed Limits to Arbitrage
I document a multi-billion dollar discrepancy that lasted from 1992 to 1999 between two otherwise identical share classes of HSBC that did not suffer from the external limits to arbitrage that traditionally explain such other mispriced pairs as 3Com/Palm or Royal Dutch-Shell. Instead, I describe how self-imposed, internal limits to arbitrage such as restrictions on position size can result in persistent mispricings.I also show that relatively more trading volume coincides with relatively lower prices in HSBC, and the same effects holds for 3Com/Palm, Royal Dutch-Shell, and other large mispriced pairs. An increase of one standard deviation in their relative volume coincides with a decrease of about one quarter of a standard deviation in their relative price. Self-imposed limits to arbitrage explain this phenomenon as well, so long as the more expensive security also tends to have greater daily volume: a higher relative price, or a wider discrepancy, leads to more arbitrage activity, and arbitrageurs trade equal volume in both securities, bringing the relative volume down closer to one.
Finally, I distinguish self-imposed limits to arbitrage from limits imposed as a result of market transactions costs or risk measures by calculating the market implied overall maximum position size of arbitrageurs for mispriced pairs spanning different time periods and countries and having different volume characteristics. The results are roughly constant at about one hundred days of typical trading volume, consistent with self-imposed limits.
View the complete manuscript. Also available on SSRN. Also available are presentation slides.
Minimal Models of the Complexity of Security Prices
A representative investor who sells the market asset if it has two consecutive upticks or two consecutive downticks, and buys otherwise, generates realistic and complex security price paths if his order is delayed a few ticks. This simple, unique, and robust model is the smallest possible deterministic model of financial complexity, and its generalization leads to complex variety. Compared to a random walk, the minimal model generates time series with fatter tails and more frequent crashes, thus more closely matching the real world. It does all this without any parameter fitting.View the complete manuscript. Also available on SSRN. Also see this introductory poster.
The minimal models can best be understood through live demonstrations. Step-by-step trader dynamics |
 Prices Generated by Rule 54 |
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Lambda-Q Calculus (1996-1997)
Extending the Lambda Calculus to Express Quantumized Algorithms
I introduce a formal metalanguage called the lambda-q calculus for the specification of quantum programming languages. This metalanguage is an extension of the lambda calculus, which provides a formal setting for the specification of classical programming languages.As an intermediary step, I introduce a formal metalanguage called the lambda-p calculus for the specification of programming languages that allow true random number generation. I demonstrate how selected randomized algorithms can be programmed directly in the lambda-p calculus.
I also demonstrate how satisfiability can be solved in the lambda-q calculus.
View the complete manuscript. Available on arXiv.
The Lambda-Q Calculus Can Efficiently Simulate Quantum Computers
I show that the lambda-q calculus can efficiently simulate quantum Turing machines by showing how the lambda-q calculus can efficiently simulate a class of quantum cellular automaton that are equivalent to quantum Turing machines.I conclude by noting that the lambda-q calculus may be strictly stronger than quantum computers because NP-complete problems such as satisfiability are efficiently solvable in the lambda-q calculus but there is a widespread doubt that they are efficiently solvable by quantum computers.
View the complete manuscript. Available on arXiv.
Programming Complex Systems
Classical programming languages cannot model essential elements of complex systems such as true random number generation. I develop a formal programming language called the lambda-q calculus that addresses the fundamental properties of complex systems. This formal language allows the expression of quantumized algorithms, which are extensions of randomized algorithms in that probabilities can be negative, and events can cancel out.View the complete manuscript. Available on arXiv.