scholarly journals Windings of planar processes, exponential functionals and Asian options

2018 ◽  
Vol 50 (3) ◽  
pp. 726-742 ◽  
Author(s):  
Wissem Jedidi ◽  
Stavros Vakeroudis
Keyword(s):  
Brownian Motion ◽  
Exit Time ◽  
Random Variable ◽  
Infinite Divisibility ◽  
Mathematical Finance ◽  
Asian Options ◽  
Product Representation ◽  
Exponential Functional ◽  
Planar Brownian Motion ◽  
Divisibility Properties

Abstract Motivated by a common mathematical finance topic, we discuss the reciprocal of the exit time from a cone of planar Brownian motion which also corresponds to the exponential functional of Brownian motion in the framework of planar Brownian motion. We prove a conjecture of Vakeroudis and Yor (2012) concerning infinite divisibility properties of this random variable and present a novel simple proof of the result of DeBlassie (1987), (1988) concerning the asymptotic behavior of the distribution of the Bessel clock appearing in the skew-product representation of planar Brownian motion, as t→∞. We use the results of the windings approach in order to obtain results for quantities associated to the pricing of Asian options.

scholarly journals An exponential functional of random walks

2003 ◽  
Vol 40 (2) ◽  
pp. 413-426 ◽  
Author(s):  
Tamás Szabados ◽  
Balázs Székely
Keyword(s):  
Brownian Motion ◽  
Random Walks ◽  
Lebesgue Measure ◽  
Elementary Proof ◽  
Random Variable ◽  
Financial Mathematics ◽  
Asian Options ◽  
Discrete Approximations ◽  
Absolutely Continuous ◽  
Exponential Functional

The aim of this paper is to investigate discrete approximations of the exponential functional of Brownian motion (which plays an important role in Asian options of financial mathematics) with the help of simple, symmetric random walks. In some applications the discrete model could be even more natural than the continuous one. The properties of the discrete exponential functional are rather different from the continuous one: typically its distribution is singular with respect to Lebesgue measure, all of its positive integer moments are finite and they characterize the distribution. On the other hand, using suitable random walk approximations to Brownian motion, the resulting discrete exponential functionals converge almost surely to the exponential functional of Brownian motion; hence their limit distribution is the same as in the continuous case, namely that of the reciprocal of a gamma random variable, and so is absolutely continuous with respect to Lebesgue measure. In this way, we also give a new and elementary proof of an earlier result by Dufresne and Yor.

scholarly journals An exponential functional of random walks

2003 ◽  
Vol 40 (02) ◽  
pp. 413-426 ◽  
Author(s):  
Tamás Szabados ◽  
Balázs Székely
Keyword(s):  
Brownian Motion ◽  
Random Walks ◽  
Lebesgue Measure ◽  
Elementary Proof ◽  
Random Variable ◽  
Financial Mathematics ◽  
Asian Options ◽  
Discrete Approximations ◽  
Absolutely Continuous ◽  
Exponential Functional

The aim of this paper is to investigate discrete approximations of the exponential functional of Brownian motion (which plays an important role in Asian options of financial mathematics) with the help of simple, symmetric random walks. In some applications the discrete model could be even more natural than the continuous one. The properties of the discrete exponential functional are rather different from the continuous one: typically its distribution is singular with respect to Lebesgue measure, all of its positive integer moments are finite and they characterize the distribution. On the other hand, using suitable random walk approximations to Brownian motion, the resulting discrete exponential functionals converge almost surely to the exponential functional of Brownian motion; hence their limit distribution is the same as in the continuous case, namely that of the reciprocal of a gamma random variable, and so is absolutely continuous with respect to Lebesgue measure. In this way, we also give a new and elementary proof of an earlier result by Dufresne and Yor.

scholarly journals Windings of Planar Processes and Applications to the Pricing of Asian Options

2016 ◽  
Author(s):  
Stavros Vakeroudis
Keyword(s):  
Brownian Motion ◽  
Mathematical Finance ◽  
Stable Processes ◽  
Asian Options ◽  
Planar Brownian Motion ◽  
Complex Valued

Motivated by a common Mathematical Finance topic, this paper surveys several results concerning windings of 2-dimensional processes, including planar Brownian motion, complex-valued Ornstein-Uhlenbeck processes and planar stable processes. In particular, we present Spitzer's asymptotic Theorem for each case. We also relate this study to the pricing of Asian options.

scholarly journals On some exponential functionals of Brownian motion

1992 ◽  
Vol 24 (03) ◽  
pp. 509-531 ◽  
Author(s):  
Marc Yor
Keyword(s):  
Brownian Motion ◽  
Bessel Functions ◽  
Exit Time ◽  
Fixed Time ◽  
Mathematical Finance ◽  
Time Interval ◽  
Bessel Processes ◽  
Fixed Time Interval ◽  
Last Exit Time ◽  
Subordination Result

In this paper, distributional questions which arise in certain mathematical finance models are studied: the distribution of the integral over a fixed time interval [0, T] of the exponential of Brownian motion with drift is computed explicitly, with the help of computations previously made by the author for Bessel processes. The moments of this integral are obtained independently and take a particularly simple form. A subordination result involving this integral and previously obtained by Bougerol is recovered and related to an important identity for Bessel functions. When the fixed time T is replaced by an independent exponential time, the distribution of the integral is shown to be related to last-exit-time distributions and the fixed time case is recovered by inverting Laplace transforms.

scholarly journals On some exponential functionals of Brownian motion

1992 ◽  
Vol 24 (3) ◽  
pp. 509-531 ◽  
Author(s):  
Marc Yor
Keyword(s):  
Brownian Motion ◽  
Bessel Functions ◽  
Exit Time ◽  
Fixed Time ◽  
Mathematical Finance ◽  
Time Interval ◽  
Bessel Processes ◽  
Fixed Time Interval ◽  
Last Exit Time ◽  
Subordination Result

In this paper, distributional questions which arise in certain mathematical finance models are studied: the distribution of the integral over a fixed time interval [0, T] of the exponential of Brownian motion with drift is computed explicitly, with the help of computations previously made by the author for Bessel processes. The moments of this integral are obtained independently and take a particularly simple form. A subordination result involving this integral and previously obtained by Bougerol is recovered and related to an important identity for Bessel functions. When the fixed time T is replaced by an independent exponential time, the distribution of the integral is shown to be related to last-exit-time distributions and the fixed time case is recovered by inverting Laplace transforms.

scholarly journals The First Exit Time of Planar Brownian Motion from The Interior Of a Parabola

2001 ◽  
Vol 29 (2) ◽  
pp. 882-901 ◽  
Author(s):  
Rodrigo Bañuelos ◽  
R. Dante DeBlassie ◽  
Robert Smits
Keyword(s):  
Brownian Motion ◽  
Exit Time ◽  
First Exit Time ◽  
Planar Brownian Motion

scholarly journals Asymptotics for the discrete-time average of the geometric Brownian motion and Asian options

2017 ◽  
Vol 49 (2) ◽  
pp. 446-480 ◽  
Author(s):  
Dan Pirjol ◽  
Lingjiong Zhu
Keyword(s):  
Brownian Motion ◽  
Discrete Time ◽  
Moment Generating Function ◽  
Mathematical Finance ◽  
Geometric Brownian Motion ◽  
Time Averaging ◽  
Asian Options ◽  
Time Average ◽  
Averaging Time ◽  
The Moment

Abstract The time average of geometric Brownian motion plays a crucial role in the pricing of Asian options in mathematical finance. In this paper we consider the asymptotics of the discrete-time average of a geometric Brownian motion sampled on uniformly spaced times in the limit of a very large number of averaging time steps. We derive almost sure limit, fluctuations, large deviations, and also the asymptotics of the moment generating function of the average. Based on these results, we derive the asymptotics for the price of Asian options with discrete-time averaging in the Black–Scholes model, with both fixed and floating strike.

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