Brownian Type Boundary Crossing Probabilities for Piecewise Linear Boundary Functions

2004 ◽  
Vol 33 (7) ◽  
pp. 1445-1464 ◽  
Author(s):  
Arnold Janssen ◽  
Michael Kunz
Keyword(s):  
Piecewise Linear ◽  
Boundary Crossing ◽  
Linear Boundary ◽  
Boundary Functions ◽  
Type Boundary ◽  
Crossing Probabilities

Boundary crossing probability for Brownian motion and general boundaries

1997 ◽  
Vol 34 (1) ◽  
pp. 54-65 ◽  
Author(s):  
Liqun Wang ◽  
Klaus Pötzelberger
Keyword(s):  
Brownian Motion ◽  
Piecewise Linear ◽  
Linear Functions ◽  
Boundary Crossing ◽  
Time Interval ◽  
Linear Boundary ◽  
Piecewise Linear Functions ◽  
Boundary Crossing Probability ◽  
Crossing Probability ◽  
Crossing Probabilities

An explicit formula for the probability that a Brownian motion crosses a piecewise linear boundary in a finite time interval is derived. This formula is used to obtain approximations to the crossing probabilities for general boundaries which are the uniform limits of piecewise linear functions. The rules for assessing the accuracies of the approximations are given. The calculations of the crossing probabilities are easily carried out through Monte Carlo methods. Some numerical examples are provided.

Boundary crossing probability for Brownian motion and general boundaries

1997 ◽  
Vol 34 (01) ◽  
pp. 54-65 ◽  
Author(s):  
Liqun Wang ◽  
Klaus Pötzelberger
Keyword(s):  
Brownian Motion ◽  
Piecewise Linear ◽  
Linear Functions ◽  
Boundary Crossing ◽  
Time Interval ◽  
Linear Boundary ◽  
Piecewise Linear Functions ◽  
Boundary Crossing Probability ◽  
Crossing Probability ◽  
Crossing Probabilities

An explicit formula for the probability that a Brownian motion crosses a piecewise linear boundary in a finite time interval is derived. This formula is used to obtain approximations to the crossing probabilities for general boundaries which are the uniform limits of piecewise linear functions. The rules for assessing the accuracies of the approximations are given. The calculations of the crossing probabilities are easily carried out through Monte Carlo methods. Some numerical examples are provided.

Time-Dependent Barrier Options and Boundary Crossing Probabilities

2003 ◽  
Vol 10 (2) ◽  
pp. 325-334
Author(s):  
A. Novikov ◽  
V. Frishling ◽  
N. Kordzakhia
Keyword(s):  
Piecewise Linear ◽  
Time Dependent ◽  
Piecewise Linear Approximation ◽  
Boundary Crossing ◽  
Barrier Options ◽  
Standard Brownian Motion ◽  
Linear Boundary ◽  
Fair Price ◽  
Black Scholes ◽  
Crossing Probabilities

Abstract The problem of pricing of time-dependent barrier options is considered in the case when interest rate and volatility are given functions in Black–Scholes framework. The calculation of the fair price reduces to the calculation of non-linear boundary crossing probabilities for a standard Brownian motion. The proposed method is based on a piecewise-linear approximation for the boundary and repeated integration. The numerical example provided draws attention to the performance of suggested method in comparison to some alternatives.

Approximations of boundary crossing probabilities for a Brownian motion

1999 ◽  
Vol 36 (4) ◽  
pp. 1019-1030 ◽  
Author(s):  
Alex Novikov ◽  
Volf Frishling ◽  
Nino Kordzakhia
Keyword(s):  
Brownian Motion ◽  
Power Series ◽  
Numerical Integration ◽  
Piecewise Linear ◽  
Boundary Crossing ◽  
Square Root ◽  
Girsanov Transformation ◽  
Piecewise Linear Approximations ◽  
Infinite Power ◽  
Crossing Probabilities

Using the Girsanov transformation we derive estimates for the accuracy of piecewise approximations for one-sided and two-sided boundary crossing probabilities. We demonstrate that piecewise linear approximations can be calculated using repeated numerical integration. As an illustrative example we consider the case of one-sided and two-sided square-root boundaries for which we also present analytical representations in a form of infinite power series.

scholarly journals Explicit Bounds for Approximation Rates of Boundary Crossing Probabilities for the Wiener Process

2005 ◽  
Vol 42 (1) ◽  
pp. 82-92 ◽  
Author(s):  
K. Borovkov ◽  
A. Novikov
Keyword(s):  
Wiener Process ◽  
Convergence Rates ◽  
Piecewise Linear ◽  
Boundary Crossing ◽  
Barrier Option ◽  
Curvilinear Boundary ◽  
Approximation Rates ◽  
Explicit Bounds ◽  
Barrier Option Pricing ◽  
Crossing Probabilities

We give explicit upper bounds for convergence rates when approximating both one- and two-sided general curvilinear boundary crossing probabilities for the Wiener process by similar probabilities for close boundaries of simpler form, for which computation of the boundary crossing probabilities is feasible. In particular, we partially generalize and improve results obtained by Pötzelberger and Wang in the case when the approximating boundaries are piecewise linear. Applications to barrier option pricing are also discussed.

scholarly journals Explicit Bounds for Approximation Rates of Boundary Crossing Probabilities for the Wiener Process

2005 ◽  
Vol 42 (01) ◽  
pp. 82-92 ◽  
Author(s):  
K. Borovkov ◽  
A. Novikov
Keyword(s):  
Wiener Process ◽  
Convergence Rates ◽  
Piecewise Linear ◽  
Boundary Crossing ◽  
Barrier Option ◽  
Curvilinear Boundary ◽  
Approximation Rates ◽  
Explicit Bounds ◽  
Barrier Option Pricing ◽  
Crossing Probabilities

We give explicit upper bounds for convergence rates when approximating both one- and two-sided general curvilinear boundary crossing probabilities for the Wiener process by similar probabilities for close boundaries of simpler form, for which computation of the boundary crossing probabilities is feasible. In particular, we partially generalize and improve results obtained by Pötzelberger and Wang in the case when the approximating boundaries are piecewise linear. Applications to barrier option pricing are also discussed.

Approximations of boundary crossing probabilities for a Brownian motion

1999 ◽  
Vol 36 (04) ◽  
pp. 1019-1030 ◽  
Author(s):  
Alex Novikov ◽  
Volf Frishling ◽  
Nino Kordzakhia
Keyword(s):  
Brownian Motion ◽  
Power Series ◽  
Numerical Integration ◽  
Piecewise Linear ◽  
Boundary Crossing ◽  
Square Root ◽  
Girsanov Transformation ◽  
Piecewise Linear Approximations ◽  
Infinite Power ◽  
Crossing Probabilities

Using the Girsanov transformation we derive estimates for the accuracy of piecewise approximations for one-sided and two-sided boundary crossing probabilities. We demonstrate that piecewise linear approximations can be calculated using repeated numerical integration. As an illustrative example we consider the case of one-sided and two-sided square-root boundaries for which we also present analytical representations in a form of infinite power series.

Boundary crossing probability for Brownian motion

2001 ◽  
Vol 38 (1) ◽  
pp. 152-164 ◽  
Author(s):  
Klaus Pötzelberger ◽  
Liqun Wang
Keyword(s):  
Brownian Motion ◽  
Piecewise Linear ◽  
Approximation Error ◽  
Simulation Method ◽  
Boundary Crossing ◽  
Boundary Crossing Probability ◽  
Crossing Probability ◽  
Approximation Errors ◽  
Crossing Probabilities ◽  
Asymptotic Upper Bound

Wang and Pötzelberger (1997) derived an explicit formula for the probability that a Brownian motion crosses a one-sided piecewise linear boundary and used this formula to approximate the boundary crossing probability for general nonlinear boundaries. The present paper gives a sharper asymptotic upper bound of the approximation error for the formula, and generalizes the results to two-sided boundaries. Numerical computations are easily carried out using the Monte Carlo simulation method. A rule is proposed for choosing optimal nodes for the approximating piecewise linear boundaries, so that the corresponding approximation errors of boundary crossing probabilities converge to zero at a rate of O(1/n2).

Boundary crossing probability for Brownian motion

2001 ◽  
Vol 38 (01) ◽  
pp. 152-164 ◽  
Author(s):  
Klaus Pötzelberger ◽  
Liqun Wang
Keyword(s):  
Brownian Motion ◽  
Piecewise Linear ◽  
Approximation Error ◽  
Simulation Method ◽  
Boundary Crossing ◽  
Boundary Crossing Probability ◽  
Crossing Probability ◽  
Approximation Errors ◽  
Crossing Probabilities ◽  
Asymptotic Upper Bound

Wang and Pötzelberger (1997) derived an explicit formula for the probability that a Brownian motion crosses a one-sided piecewise linear boundary and used this formula to approximate the boundary crossing probability for general nonlinear boundaries. The present paper gives a sharper asymptotic upper bound of the approximation error for the formula, and generalizes the results to two-sided boundaries. Numerical computations are easily carried out using the Monte Carlo simulation method. A rule is proposed for choosing optimal nodes for the approximating piecewise linear boundaries, so that the corresponding approximation errors of boundary crossing probabilities converge to zero at a rate of O(1/n 2).

Minimal distributions in a stochastic partial ordering and bounds for Gaussian processes

1983 ◽  
Vol 20 (03) ◽  
pp. 529-536
Author(s):  
W. J. R. Eplett
Keyword(s):  
Gaussian Process ◽  
Gaussian Processes ◽  
Partial Ordering ◽  
Boundary Crossing ◽  
Conditional Probabilities ◽  
Life Distribution ◽  
Life Distributions ◽  
Bounded Below ◽  
Crossing Probabilities

A natural requirement to impose upon the life distribution of a component is that after inspection at some randomly chosen time to check whether it is still functioning, its life distribution from the time of checking should be bounded below by some specified distribution which may be defined by external considerations. Furthermore, the life distribution should ideally be minimal in the partial ordering obtained from the conditional probabilities. We prove that these specifications provide an apparently new characterization of the DFRA class of life distributions with a corresponding result for IFRA distributions. These results may be transferred, using Slepian's lemma, to obtain bounds for the boundary crossing probabilities of a stationary Gaussian process.

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