729 research outputs found
Uniform approximation of the Cox-Ingersoll-Ross process
The Doss-Sussmann (DS) approach is used for uniform simulation of the
Cox-Ingersoll-Ross (CIR) process. The DS formalism allows to express
trajectories of the CIR process through solutions of some ordinary differential
equation (ODE) depending on realizations of a Wiener process involved. By
simulating the first-passage times of the increments of the Wiener process to
the boundary of an interval and solving the ODE, we uniformly approximate the
trajectories of the CIR process. In this respect special attention is payed to
simulation of trajectories near zero. From a conceptual point of view the
proposed method gives a better quality of approximation (from a path-wise point
of view) than standard, or even exact simulation of the SDE at some discrete
time grid.Comment: 24 page
Path-wise approximation of the Cox-Ingersoll-Ross process
The Doss-Sussmann (DS) approach is used for simulating the
Cox-Ingersoll-Ross (CIR) process. The DS formalism allows for expressing
trajectories of the CIR process by solutions of some ordinary differential
equation (ODE) that depend on realizations of the Wiener process involved.
Via simulating the first-passage times of the increments of the Wiener
process to the boundary of an interval and solving an ODE, we approximately
construct the trajectories of the CIR process. From a conceptual point of
view the proposed method may be considered as an exact simulation approach
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Uniform approximation of the CIR process via exact simulation at random times
In this paper we uniformly approximate the trajectories of the
Cox-Ingersoll-Ross (CIR) process. At a sequence of random times the
approximate trajectories will be even exact. In between, the approximation
will be uniformly close to the exact trajectory. From a conceptual point of
view the proposed method gives a better quality of approximation in a
path-wise sense than standard, or even exact simulation of the CIR dynamics
at some deterministic time grid
Uniform approximation of the CIR process via exact simulation at random times
In this paper we uniformly approximate the trajectories of the Cox-Ingersoll-Ross (CIR) process. At a sequence of random times the approximate trajectories will be even exact. In between, the approximation will be uniformly close to the exact trajectory. From a conceptual point of view the proposed method gives a better quality of approximation in a path-wise sense than standard, or even exact simulation of the CIR dynamics at some deterministic time grid
Path-wise approximation of the Cox--Ingersoll--Ross process
The Doss-Sussmann (DS) approach is used for simulating the Cox-Ingersoll-Ross (CIR) process. The DS formalism allows for expressing trajectories of the CIR process by solutions of some ordinary differential equation (ODE) that depend on realizations of the Wiener process involved. Via simulating the first-passage times of the increments of the Wiener process to the boundary of an interval and solving an ODE, we approximately construct the trajectories of the CIR process. From a conceptual point of view the proposed method may be considered as an exact simulation approach
Numerical construction of a hedging strategy against the European claim
For evaluating a hedging strategy we have to know at every instant the solution of the Cauchy problem for a parabolic equation (the value of the hedging portfolio) and its derivatives (the deltas). We suggest to find these magnitudes by Monte Carlo simulation of the corresponding system of stochastic differential equations using weak solution schemes. It turns out that with one and the same control function a variance reduction can be achieved simultaneously for the claim value as well as for the deltas. We consider asset models with an instantaneous saving bond and the Jamshidian LIBOR rate model
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Forward and reverse representations for Markov chains
In this paper we carry over the concept of reverse probabilistic
representations developed in Milstein, Schoenmakers, Spokoiny (2004) for
diffusion processes, to discrete time Markov chains. We outline the
construction of reverse chains in several situations and apply this to
processes which are connected with jump-diffusion models and finite state
Markov chains. By combining forward and reverse representations we then
construct transition density estimators for chains which have root-N accuracy
in any dimension and consider some applications
Monte Carlo methods for pricing and hedging American options
We introduce a new Monte Carlo method for constructing the exercise boundary of an American option in a generalized Black-Scholes framework. Based on a known exercise boundary, it is shown how to price and hedge the American option by Monte Carlo simulation of suitable probabilistic representations in connection with the respective parabolic boundary value problem. The methods presented are supported by numerical simulation experiments
Transition density estimation for stochastic differential equations via forward-reverse representations
The general reverse diffusion equations are derived. They are applied to the problem of transition density estimation of diffusion processes between two fixed states. For this problem it is shown that density estimation based on forward-reverse representations allows for achieving essentially better results in comparison with usual kernel or projection estimation based on forward representations only
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