Abstract
This paper provides new explicit results for some boundary crossing distributions in a multidimensional geometric Brownian motion framework when the boundary is a piecewise constant function of time. Among their various possible applications, they enable accurate and efficient analytical valuation of a large number of option contracts traded in the financial markets belonging to the classes of barrier and look-back options.
Highlights
This paper provides new explicit results for some boundary crossing distributions in a multidimensional geometric Brownian motion framework when the boundary is a piecewise constant function of time
The joint law of the maximum or the minimum of a real-valued Brownian motion and its endpoint over a finite time interval is a central result in the study of Brownian motion, with regard to the many applications of the theory in finance, medical imaging, robotics, and biology
In the Proposition, we introduce a third correlated geometric Brownian motion that will serve as the endpoint of the joint distribution, and we show that this can still be analytically valued
Summary
The joint law of the maximum or the minimum of a real-valued Brownian motion and its endpoint over a finite time interval is a central result in the study of Brownian motion, with regard to the many applications of the theory in finance, medical imaging, robotics, and biology. It can be obtained as a consequence of the “reflection principle,” which derives from the strong Markov property of Brownian motion Freedman 1.
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