744 research outputs found
Critical transaction costs and 1-step asymptotic arbitrage in fractional binary markets
We study the arbitrage opportunities in the presence of transaction costs in
a sequence of binary markets approximating the fractional Black-Scholes model.
This approximating sequence was constructed by Sottinen and named fractional
binary markets. Since, in the frictionless case, these markets admit arbitrage,
we aim to determine the size of the transaction costs needed to eliminate the
arbitrage from these models. To gain more insight, we first consider only
1-step trading strategies and we prove that arbitrage opportunities appear when
the transaction costs are of order . Next, we characterize the
asymptotic behavior of the smallest transaction costs , called
"critical" transaction costs, starting from which the arbitrage disappears.
Since the fractional Black-Scholes model is arbitrage-free under arbitrarily
small transaction costs, one could expect that converges to
zero. However, the true behavior of is opposed to this
intuition. More precisely, we show, with the help of a new family of trading
strategies, that converges to one. We explain this apparent
contradiction and conclude that it is appropriate to see the fractional binary
markets as a large financial market and to study its asymptotic arbitrage
opportunities. Finally, we construct a -step asymptotic arbitrage in this
large market when the transaction costs are of order , whereas for
constant transaction costs, we prove that no such opportunity exists.Comment: 21 page
A probabilistic view on the deterministic mutation-selection equation: dynamics, equilibria, and ancestry via individual lines of descent
We reconsider the deterministic haploid mutation-selection equation with two
types. This is an ordinary differential equation that describes the type
distribution (forward in time) in a population of infinite size. This paper
establishes ancestral (random) structures inherent in this deterministic model.
In a first step, we obtain a representation of the deterministic equation's
solution (and, in particular, of its equilibrium) in terms of an ancestral
process called the killed ancestral selection graph. This representation allows
one to understand the bifurcations related to the error threshold phenomenon
from a genealogical point of view. Next, we characterise the ancestral type
distribution by means of the pruned lookdown ancestral selection graph and
study its properties at equilibrium. We also provide an alternative
characterisation in terms of a piecewise-deterministic Markov process.
Throughout, emphasis is on the underlying dualities as well as on explicit
results.Comment: J. Math. Biol., in pres
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