Optimal stopping for Brownian motion with applications to sequential analysis and option pricing

2005 ◽  
Vol 130 (1-2) ◽  
pp. 21-47 ◽  
Author(s):  
Tze Leung Lai ◽  
Tiong Wee Lim
Keyword(s):  
Brownian Motion ◽  
Option Pricing ◽  
Optimal Stopping ◽  
Sequential Analysis

scholarly journals Corrected random walk approximations to free boundary problems in optimal stopping

2007 ◽  
Vol 39 (03) ◽  
pp. 753-775
Author(s):  
Tze Leung Lai ◽  
Yi-Ching Yao ◽  
Farid Aitsahlia
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Optimal Stopping ◽  
Continuous Time ◽  
Sequential Analysis ◽  
Free Boundary Problems ◽  
Boundary Problems ◽  
Computational Tools ◽  
Time Optimal ◽  
New Applications

Corrected random walk approximations to continuous-time optimal stopping boundaries for Brownian motion, first introduced by Chernoff and Petkau, have provided powerful computational tools in option pricing and sequential analysis. This paper develops the theory of these second-order approximations and describes some new applications.

scholarly journals Corrected random walk approximations to free boundary problems in optimal stopping

2007 ◽  
Vol 39 (3) ◽  
pp. 753-775 ◽  
Author(s):  
Tze Leung Lai ◽  
Yi-Ching Yao ◽  
Farid Aitsahlia
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Optimal Stopping ◽  
Continuous Time ◽  
Sequential Analysis ◽  
Free Boundary Problems ◽  
Boundary Problems ◽  
Computational Tools ◽  
Time Optimal ◽  
New Applications

Corrected random walk approximations to continuous-time optimal stopping boundaries for Brownian motion, first introduced by Chernoff and Petkau, have provided powerful computational tools in option pricing and sequential analysis. This paper develops the theory of these second-order approximations and describes some new applications.

On wald-type optimal stopping for Brownian motion

1997 ◽  
Vol 34 (1) ◽  
pp. 66-73 ◽  
Author(s):  
S. E. Graversen ◽  
G. Peškir
Keyword(s):  
Brownian Motion ◽  
Optimal Stopping ◽  
Stopping Time ◽  
Maximal Inequality ◽  
Stopping Times ◽  
Standard Brownian Motion ◽  
Square Root ◽  
Real Analysis ◽  
Optimal Stopping Problems ◽  
Wald's Identity

The solution is presented to all optimal stopping problems of the form supτE(G(|Β τ |) – cτ), where is standard Brownian motion and the supremum is taken over all stopping times τ for B with finite expectation, while the map G : ℝ+ → ℝ satisfies for some being given and fixed. The optimal stopping time is shown to be the hitting time by the reflecting Brownian motion of the set of all (approximate) maximum points of the map . The method of proof relies upon Wald's identity for Brownian motion and simple real analysis arguments. A simple proof of the Dubins–Jacka–Schwarz–Shepp–Shiryaev (square root of two) maximal inequality for randomly stopped Brownian motion is given as an application.

Optimal Stopping of the Maximum Process

2014 ◽  
Vol 51 (03) ◽  
pp. 818-836 ◽  
Author(s):  
Luis H. R. Alvarez ◽  
Pekka Matomäki
Keyword(s):  
Brownian Motion ◽  
Free Boundary ◽  
Optimal Stopping ◽  
Original Problem ◽  
Alternative Route ◽  
Linear Diffusion ◽  
Maximum Process ◽  
Running Maximum ◽  
Parameterized Family ◽  
Optimal Stopping Problems

We consider a class of optimal stopping problems involving both the running maximum as well as the prevailing state of a linear diffusion. Instead of tackling the problem directly via the standard free boundary approach, we take an alternative route and present a parameterized family of standard stopping problems of the underlying diffusion. We apply this family to delineate circumstances under which the original problem admits a unique, well-defined solution. We then develop a discretized approach resulting in a numerical algorithm for solving the considered class of stopping problems. We illustrate the use of the algorithm in both a geometric Brownian motion and a mean reverting diffusion setting.

Option pricing under the fractional stochastic volatility model

2021 ◽  
Vol 63 ◽  
pp. 123-142
Author(s):  
Yuecai Han ◽  
Zheng Li ◽  
Chunyang Liu
Keyword(s):  
Brownian Motion ◽  
Option Pricing ◽  
Fractional Brownian Motion ◽  
Stochastic Volatility ◽  
Option Price ◽  
Stochastic Volatility Model ◽  
Option Prices ◽  
European Call Option ◽  
Volatility Model ◽  
The Impact

We investigate the European call option pricing problem under the fractional stochastic volatility model. The stochastic volatility model is driven by both fractional Brownian motion and standard Brownian motion. We obtain an analytical solution of the European option price via the Itô’s formula for fractional Brownian motion, Malliavin calculus, derivative replication and the fundamental solution method. Some numerical simulations are given to illustrate the impact of parameters on option prices, and the results of comparison with other models are presented. doi:10.1017/S1446181121000225

Bounds for the American perpetual put on a stock index

2001 ◽  
Vol 38 (01) ◽  
pp. 55-66 ◽  
Author(s):  
V. Paulsen
Keyword(s):  
Brownian Motion ◽  
Optimal Stopping ◽  
Geometric Brownian Motion ◽  
Stock Index ◽  
Decomposition Technique ◽  
Early Exercise

Let us consider n stocks with dependent price processes each following a geometric Brownian motion. We want to investigate the American perpetual put on an index of those stocks. We will provide inner and outer boundaries for its early exercise region by using a decomposition technique for optimal stopping.

scholarly journals Option pricing of geometric Asian options in a subdiffusive Brownian motion regime

2020 ◽  
Vol 5 (5) ◽  
pp. 5332-5343
Author(s):  
Zhidong Guo ◽  
◽  
Xianhong Wang ◽  
Yunliang Zhang ◽  
Keyword(s):  
Brownian Motion ◽  
Option Pricing ◽  
Asian Options
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