Quadratic Variation of Functionals of Brownian Motion

The modern theory of option pricing rests on Itô calculus, which is a second-order calculus based on the quadratic variation of a stochastic process. One can instead develop a first-order stochastic calculus, which is based on the running minimum of a stochastic process, rather than its quadratic variation. We focus here on the analog of geometric Brownian motion (GBM) in this alternative stochastic calculus. The resulting stochastic process is a positive continuous martingale whose laws are easy to calculate. We show that this analog behaves locally like a GBM whenever its running minimum decreases, but behaves locally like an arithmetic Brownian motion otherwise. We provide closed form valuation formulas for vanilla and barrier options written on this process. We also develop a reflection principle for the process and use it to show how a barrier option on this process can be hedged by a static postion in vanilla options.

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DIFFERENTIAL FORMULAS OF STOCHASTIC FUNCTIONS

Science and Technology Development Journal ◽

10.32508/stdj.v12i7.2263 ◽

2009 ◽

Vol 12 (7) ◽

pp. 29-34

Author(s):

Dam Ton Duong

Keyword(s):

Stochastic Process ◽

Brownian Motion ◽

Stochastic Processes ◽

Quadratic Variation ◽

Stochastic Functions

Based on the quadratic variation theorem of the Brownian motion, we have established the basic rules of stochastic differetial calculus operations. Theorem 1. If X,Y, are positive-valued stochastic processes satisfying respectively the following stochastic differenntial equations Then a, b R: Where Theorem 2 Suppose is the Hermite type stochastic process of then

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Strong approximation of continuous local martingales by simple random walks

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/012.2004.41.1.6 ◽

2004 ◽

Vol 41 (1) ◽

pp. 101-126

Author(s):

B. Székely ◽

T. Szabados

Keyword(s):

Brownian Motion ◽

Random Walks ◽

Rates Of Convergence ◽

Quadratic Variation ◽

Local Martingale ◽

Motion Time ◽

Time Changes ◽

Necessary And Sufficient ◽

Nested Sequence ◽

The Given

The aim of this paper is to represent any continuous local martingale as an almost sure limit of a nested sequence of simple, symmetric random walk, time changed by a discrete quadratic variation process. One basis of this is a similar construction of Brownian motion. The other major tool is a representation of continuous local martingales given by Dambis, Dubins and Schwarz (DDS) in terms of Brownian motion time-changed by the quadratic variation. Rates of convergence (which are conjectured to be nearly optimal in the given setting) are also supplied. A necessary and sufficient condition for the independence of the random walks and the discrete time changes or equivalently, for the independence of the DDS Brownian motion and the quadratic variation is proved to be the symmetry of increments of the martingale given the past, which is a reformulation of an earlier result by Ocone [8].

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