Generalization of Brownian Motion with Autoregressive Increments

Details Generalization of Brownian Motion with Autoregressive Increments Generalization of Brownian Motion with Autoregressive Increments

2011-04-25

arxiv.org/abs/1104.4741v1

Mathematics

Probability

26 pages

Scientific paper

This paper introduces a generalization of Brownian motion with continuous sample paths and stationary, autoregressive increments. This process, which we call a Brownian ray with drift, is characterized by three parameters quantifying distinct effects of drift, volatility, and autoregressiveness. A Brownian ray with drift, conditioned on its state at the beginning of an interval, is another Brownian ray with drift over the interval, and its expected path over the interval is a ray with a slope that depends on the conditioned state. This paper shows how Brownian rays can be applied in finance for the analysis of queues or inventories and the valuation of options. We model a queue's net input process as a superposition of Brownian rays with drift and derive the transient distribution of the queue length conditional on past queue lengths and on past states of the individual Brownian rays comprising the superposition. The transient distributions of Regulated Brownian Motion and of the Regulated Brownian Bridge are obtained as limiting cases. For the valuation of options, we model a security price on which the option is written as a Geometric Brownian Ray (GBR), thus generalizing the familiar model of a security price as a Geometric Brownian Motion (GBM). We show that the rational price of an option under GBR assumptions is given by the same Black-Scholes-Merton formula that applies under GBM assumptions, but that the formula's parameter that characterizes the volatility of the security price under GBM assumptions must be generalized under GBR assumptions to reflect autoregressiveness.

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