4,386 research outputs found
Hedging under arbitrage
It is shown that delta hedging provides the optimal trading strategy in terms
of minimal required initial capital to replicate a given terminal payoff in a
continuous-time Markovian context. This holds true in market models where no
equivalent local martingale measure exists but only a square-integrable market
price of risk. A new probability measure is constructed, which takes the place
of an equivalent local martingale measure. In order to ensure the existence of
the delta hedge, sufficient conditions are derived for the necessary
differentiability of expectations indexed over the initial market
configuration. The recently often discussed phenomenon of "bubbles" is a
special case of the setting in this paper. Several examples at the end
illustrate the techniques described in this work.Comment: Minor changes, accepted for publication in Journal of Mathematical
Financ
Markovian approximation in foreign exchange markets
In this paper we test the random walk hypothesis on the high frequency
dataset of the bid--ask Deutschemark/US dollar exchange rate quotes registered
by the inter-bank Reuters network over the period October 1, 1992 to September
30, 1993. Then we propose a stochastic model for price variation which is able
to describe some important features of the exchange market behavior. Besides
the usual correlation analysis we have verified the validity of this model by
means of other approaches inspired by information theory . These techniques are
not only severe tests of the approximation but also evidence some aspects of
the data series which have a clear financial relevance.Comment: 19 pages, LaTeX, uses elsart.cls and JournalOfFinance.sty, 7 eps
figures, submitted to J. of Int. Money and Financ
A Mathematical Approach to Order Book Modeling
Motivated by the desire to bridge the gap between the microscopic description
of price formation (agent-based modeling) and the stochastic differential
equations approach used classically to describe price evolution at macroscopic
time scales, we present a mathematical study of the order book as a
multidimensional continuous-time Markov chain and derive several mathematical
results in the case of independent Poissonian arrival times. In particular, we
show that the cancellation structure is an important factor ensuring the
existence of a stationary distribution and the exponential convergence towards
it. We also prove, by means of the functional central limit theorem (FCLT),
that the rescaled-centered price process converges to a Brownian motion. We
illustrate the analysis with numerical simulation and comparison against market
data
Price dynamics in a Markovian limit order market
We propose and study a simple stochastic model for the dynamics of a limit
order book, in which arrivals of market order, limit orders and order
cancellations are described in terms of a Markovian queueing system. Through
its analytical tractability, the model allows to obtain analytical expressions
for various quantities of interest such as the distribution of the duration
between price changes, the distribution and autocorrelation of price changes,
and the probability of an upward move in the price, {\it conditional} on the
state of the order book. We study the diffusion limit of the price process and
express the volatility of price changes in terms of parameters describing the
arrival rates of buy and sell orders and cancelations. These analytical results
provide some insight into the relation between order flow and price dynamics in
order-driven markets.Comment: 18 pages, 5 figure
Asymptotic Exponential Arbitrage and Utility-based Asymptotic Arbitrage in Markovian Models of Financial Markets
Consider a discrete-time infinite horizon financial market model in which the
logarithm of the stock price is a time discretization of a stochastic
differential equation. Under conditions different from those given in a
previous paper of ours, we prove the existence of investment opportunities
producing an exponentially growing profit with probability tending to
geometrically fast. This is achieved using ergodic results on Markov chains and
tools of large deviations theory.
Furthermore, we discuss asymptotic arbitrage in the expected utility sense
and its relationship to the first part of the paper.Comment: Forthcoming in Acta Applicandae Mathematica
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