23,141 research outputs found
Optimal discretization of hedging strategies with directional views
We consider the hedging error of a derivative due to discrete trading in the
presence of a drift in the dynamics of the underlying asset. We suppose that
the trader wishes to find rebalancing times for the hedging portfolio which
enable him to keep the discretization error small while taking advantage of
market trends. Assuming that the portfolio is readjusted at high frequency, we
introduce an asymptotic framework in order to derive optimal discretization
strategies. More precisely, we formulate the optimization problem in terms of
an asymptotic expectation-error criterion. In this setting, the optimal
rebalancing times are given by the hitting times of two barriers whose values
can be obtained by solving a linear-quadratic optimal control problem. In
specific contexts such as in the Black-Scholes model, explicit expressions for
the optimal rebalancing times can be derived
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
Fast randomized iteration: diffusion Monte Carlo through the lens of numerical linear algebra
We review the basic outline of the highly successful diffusion Monte Carlo
technique commonly used in contexts ranging from electronic structure
calculations to rare event simulation and data assimilation, and propose a new
class of randomized iterative algorithms based on similar principles to address
a variety of common tasks in numerical linear algebra. From the point of view
of numerical linear algebra, the main novelty of the Fast Randomized Iteration
schemes described in this article is that they work in either linear or
constant cost per iteration (and in total, under appropriate conditions) and
are rather versatile: we will show how they apply to solution of linear
systems, eigenvalue problems, and matrix exponentiation, in dimensions far
beyond the present limits of numerical linear algebra. While traditional
iterative methods in numerical linear algebra were created in part to deal with
instances where a matrix (of size ) is too big to store, the
algorithms that we propose are effective even in instances where the solution
vector itself (of size ) may be too big to store or manipulate.
In fact, our work is motivated by recent DMC based quantum Monte Carlo schemes
that have been applied to matrices as large as . We
provide basic convergence results, discuss the dependence of these results on
the dimension of the system, and demonstrate dramatic cost savings on a range
of test problems.Comment: 44 pages, 7 figure
A modified particle method for semilinear hyperbolic systems with oscillatory solutions
We introduce a modified particle method for semi-linear hyperbolic systems with highly oscillatory solutions. The main feature of this modified particle method is that we do not require different families of characteristics to meet at one point. In the modified particle method, we update the ith component of the solution along its own characteristics, and interpolate the other components of the solution from their own characteristic points to the ith characteristic point. We prove the convergence of the modified particle method essentially independent of the small scale for the variable coefficient Carleman model. The same result also applies to the non-resonant Broadwell model. Numerical evidence suggests that the modified particle method also converges essentially independent of the small scale for the original Broadwell model if a cubic spline interpolation is used
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