Kelly Betting Can Be Too Conservative

Kelly betting is a prescription for optimal resource allocation among a set of gambles which are typically repeated in an independent and identically distributed manner. In this setting, there is a large body of literature which includes arguments that the theory often leads to bets which are "too aggressive" with respect to various risk metrics. To remedy this problem, many papers include prescriptions for scaling down the bet size. Such schemes are referred to as Fractional Kelly Betting. In this paper, we take the opposite tack. That is, we show that in many cases, the theoretical Kelly-based results may lead to bets which are "too conservative" rather than too aggressive. To make this argument, we consider a random vector X with its assumed probability distribution and draw m samples to obtain an empirically-derived counterpart Xhat. Subsequently, we derive and compare the resulting Kelly bets for both X and Xhat with consideration of sample size m as part of the analysis. This leads to identification of many cases which have the following salient feature: The resulting bet size using the true theoretical distribution for X is much smaller than that for Xhat. If instead the bet is based on empirical data, "golden" opportunities are identified which are essentially rejected when the purely theoretical model is used. To formalize these ideas, we provide a result which we call the Restricted Betting Theorem. An extreme case of the theorem is obtained when X has unbounded support. In this situation, using X, the Kelly theory can lead to no betting at all.Comment: Accepted in 2016 IEEE 55th Conference on Decision and Control (CDC

On Solving Robust Log-Optimal Portfolio: A Supporting Hyperplane Approximation Approach

A {log-optimal} portfolio is any portfolio that maximizes the expected logarithmic growth (ELG) of an investor's wealth. This maximization problem typically assumes that the information of the true distribution of returns is known to the trader in advance. However, in practice, the return distributions are indeed {ambiguous}; i.e., the true distribution is unknown to the trader or it is partially known at best. To this end, a {distributional robust log-optimal portfolio problem} formulation arises naturally. While the problem formulation takes into account the ambiguity on return distributions, the problem needs not to be tractable in general. To address this, in this paper, we propose a {supporting hyperplane approximation} approach that allows us to reformulate a class of distributional robust log-optimal portfolio problems into a linear program, which can be solved very efficiently. Our framework is flexible enough to allow {transaction costs}, {leverage and shorting}, {survival trades}, and {diversification considerations}. In addition, given an acceptable approximation error, an efficient algorithm for rapidly calculating the optimal number of hyperplanes is provided. Some empirical studies using historical stock price data are also provided to support our theory.Comment: submitted for possible publicatio

Partition-dependent framing effects in lab and field prediction markets

Many psychology experiments show that individually judged probabilities of the same event can vary depending on the partition of the state space (a framing effect called "partition-dependence"). We show that these biases transfer to competitive prediction markets in which multiple informed traders are provided economic incentives to bet on their beliefs about events. We report results of a short controlled lab study, a longer field experiment (betting on the NBA playoffs and the FIFA World Cup), and naturally-occurring trading in macro-economic derivatives. The combined evidence suggests that partition-dependence can exist and persist in lab and field prediction markets

The Impact of Execution Delay on Kelly-Based Stock Trading: High-Frequency Versus Buy and Hold

Stock trading based on Kelly's celebrated Expected Logarithmic Growth (ELG) criterion, a well-known prescription for optimal resource allocation, has received considerable attention in the literature. Using ELG as the performance metric, we compare the impact of trade execution delay on the relative performance of high-frequency trading versus buy and hold. While it is intuitively obvious and straightforward to prove that in the presence of sufficiently high transaction costs, buy and hold is the better strategy, is it possible that with no transaction costs, buy and hold can still be the better strategy? When there is no delay in trade execution, we prove a theorem saying that the answer is ``no.'' However, when there is delay in trade execution, we present simulation results using a binary lattice stock model to show that the answer can be ``yes.'' This is seen to be true whether self-financing is imposed or not.Comment: Has been accepted to the IEEE Conference on Decision and Control, 201

Adaptive Bet-Hedging Revisited: Considerations of Risk and Time Horizon

Models of adaptive bet-hedging commonly adopt insights from Kelly's famous work on optimal gambling strategies and the financial value of information. In particular, such models seek evolutionary solutions that maximize long term average growth rate of lineages, even in the face of highly stochastic growth trajectories. Here, we argue for extensive departures from the standard approach to better account for evolutionary contingencies. Crucially, we incorporate considerations of volatility minimization, motivated by interim extinction risk in finite populations, within a finite time horizon approach to growth maximization. We find that a game-theoretic competitive-optimality approach best captures these additional constraints, and derive the equilibria solutions under straightforward fitness payoff functions and extinction risks. We show that for both maximal growth and minimal time relative payoffs the log-optimal strategy is a unique pure-strategy symmetric equilibrium, invariant with evolutionary time horizon and robust to low extinction risks.Comment: Accepted for publication in Bulletin of Mathematical Biolog

SP Betting as a Self-Enforcing Implicit Cartel

A large share of the UK off-course horse racing betting market involves winning payouts determined at Starting Prices (SP). This implies that gamblers can bet with off-course bookies on any horse before a race at the final pre-race odds as set by on-course bookies for that horse. Given the oligopolistic structure of the off-course gambling market in the UK, a market that is dominated by a small number of large bookmaking firms, we study the phenomenon of SP as a type of self-enforcing implicit collusion. We show that given the uncertainty about a race outcome, and their ability to influence the prices set by on-course bookies, agreeing to lay bets at SP is superior for off-course bookies as compared with offering fixed odds. We thus extend the results of Rotemberg and Saloner (1990) to markets with uncertainty about both demand and outcomes, We test our model by studying the predicted effects of SP betting on the behavior of on-course bookies. Using data drawn from both the UK and Australian on-course betting markets, we show that the differences between these markets are consistent with the predicted effects of SP betting in the UK off-course market and its absence from the Australian market.

Sports betting: a new asset class to bet on

This dissertation has the aim to present a complete overview of the current features and activities related to the sports betting industry and to explain the reasons why it can be considered a new asset class to invest on. The first chapter explains the main features of both fixed-odds and exchange betting market, the second describes the activity of sport trading, while the third presents a deep investigation concerning the market efficiency. Chapter 4 shows the arbitrage opportunities implementable in this market, that come from the efficiency study of the previous chapter. Before the conclusion, a personal study about the value betting arbitrage opportunity is presented, confirming that abnormal returns are achievable

Applications of Chance Constrained Optimization in Operations Management

In this thesis we explore three applications of chance constrained optimization in operations management. We first investigate the effect of consumer demand estimation error on new product production planning. An inventory model is proposed, whereby demand is influenced by price and advertising. The effect of parameter misspecification of the demand model is empirically examined in relation to profit and service level feasibility, and conservative approaches to estimating their effect on consumer demand is determined. We next consider optimization in Internet advertising by introducing a chance constrained model for the fulfillment of guaranteed display Internet advertising campaigns. Lower and upper bounds using Monte Carlo sampling and convex approximations are presented, as well as a branching heuristic for sample approximation lower bounds and an iterative algorithm for improved convex approximation upper bounds. The final application is in risk management for parimutuel horse racing wagering. We develop a methodology to limit potential losing streaks with high probability to the given time horizon of a gambler. A proof of concept was conducted using one season of historical race data, where losing streaks were effectively contained within different time periods for superfecta betting