  In this paper, we derive and analyze a continuous of a binary option market
with exogenous information. The resulting non-linear system has a discontinuous
right hand side, which can be analyzed using zero-dimensional Filippov
surfaces. Under general assumptions on purchasing rules, we show that when
exogenous information is constant in the binary asset market, the price always
converges. We then investigate market prices in the case of changing
information, showing empirically that price sensitivity has a strong effect on
price lag vs. information. We conclude with open questions on general $n$-ary
option markets. As a by-product of the analysis, we show that these markets are
equivalent to a simple recurrent neural network, helping to explain some of the
predictive power associated with prediction markets, which are usually designed
as $n$-ary option markets.
