Wealth and price distribution by diffusive approximation in a repeated prediction market

Abstract

The approximate agents’ wealth and price invariant densities of a repeated prediction market model is derived using the Fokker–Planck equation of the associated continuous-time jump process. We show that the approximation obtained from the evolution of log-wealth difference can be reliably exploited to compute all the quantities of interest in all the acceptable parameter space. When the risk aversion of the trader is high enough, we are able to derive an explicit closed-form solution for the price distribution which is asymptotically correct

Betting, Selection, and Luck: A Long-Run Analysis of Repeated Betting Markets

“…In the third case, luck recovers the role of ultimate arbiter, traditionally attributed to it in games of chance. Notice that if one confines the analysis to specific families of strategies, like Kelly or fractional Kelly strategies, the third outcome becomes non-generic or disappears [ 8 , 9 ]. This explains why it was largely unobserved in several previous studies.…”

“…In a market populated by utility maximizers, the agent who trades knowing the correct probabilities always realizes a non-negative expected profit [ 7 ]. In the case of bettors using the fractional Kelly rule, a generalization of the Kelly rule that includes a risk-aversion parameter, sufficient and, apart from hairline cases, necessary conditions for strategy dominance or survival has been derived [ 8 , 9 ]. These conditions generalize and correct previous tentative results based on numerical simulations [ 10 ].…”

Far from the madding crowd: collective wisdom in prediction markets

“…This values does in general depend on all the parameters of the model. Bottazzi and Giachini (2016) show that (14) constitutes a good approximation of the actual distribution in all the points of the parameter space. Moreover, the approximation is better the lower the value of c. One can rewrite 14as…”

“…As a specific example, take π * = 0.2, π 1 = 0.1, π 2 = 0.32 and c = 0.96. Numerical simulations show that in this case it is E[p] = 0.3191 ± 0.00003.9 The derivation is sketched in Appendix D. For more details seeBottazzi and Giachini (2016) …”

Drift criteria for persistence of discrete stochastic processes on the line

“…An analysis similar to Section 5.1 is conducted inDindo and Massari (2020), where the market-dependent consensus model is derived from a proper normalization of state prices. In this case, an approximation of the price distribution can be derived through Poisson embedding in a continuous time process, as illustrated inBottazzi and Giachini (2017). A similar relative consumption dynamics arises also with Epstein-Zin preferences, seeDindo (2019) andBorovička (2020).…”