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.

Problèmes de premier passage résolubles par la méthode de séparation des variables

1995 ◽  
Vol 32 (02) ◽  
pp. 337-348
Author(s):  
Mario Lefebvre
Keyword(s):  
Brownian Motion ◽  
Stochastic Processes ◽  
Generating Function ◽  
Moment Generating Function ◽  
First Passage Times ◽  
Standard Brownian Motion ◽  
First Passage ◽  
The Moment ◽  
Passage Times

In this paper, bidimensional stochastic processes defined by ax(t) = y(t)dt and dy(t) = m(y)dt + [2v(y)]1/2 dW(t), where W(t) is a standard Brownian motion, are considered. In the first section, results are obtained that allow us to characterize the moment-generating function of first-passage times for processes of this type. In Sections 2 and 5, functions are computed, first by fixing the values of the infinitesimal parameters m(y) and v(y) then by the boundary of the stopping region.

scholarly journals First Hitting Problems for Markov Chains That Converge to a Geometric Brownian Motion

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Mario Lefebvre ◽  
Moussa Kounta
Keyword(s):  
Brownian Motion ◽  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
Discrete Time ◽  
Geometric Brownian Motion ◽  
Expected Number ◽  
Discrete Time Markov Chain

We consider a discrete-time Markov chain with state space {1,1+Δx,…,1+kΔx=N}. We compute explicitly the probability pj that the chain, starting from 1+jΔx, will hit N before 1, as well as the expected number dj of transitions needed to end the game. In the limit when Δx and the time Δt between the transitions decrease to zero appropriately, the Markov chain tends to a geometric Brownian motion. We show that pj and djΔt tend to the corresponding quantities for the geometric Brownian motion.

Continuous-time methods in the study of discretely sampled functionals of Lévy processes. I. The positive process case

2007 ◽  
Vol 39 (1) ◽  
pp. 245-270 ◽  
Author(s):  
Michael Schröder
Keyword(s):  
Lévy Processes ◽  
Generalized Inverse ◽  
Moment Generating Function ◽  
Levy Processes ◽  
Asian Options ◽  
Inverse Gaussian ◽  
Contingent Claim Valuation ◽  
Reduction Series ◽  
Generalized Inverse Gaussian ◽  
The Moment

In this paper we develop a constructive approach to studying continuously and discretely sampled functionals of Lévy processes. Estimates for the rate of convergence of the discretely sampled functionals to the continuously sampled functionals are derived, reducing the study of the latter to that of the former. Laguerre reduction series for the discretely sampled functionals are developed, reducing their study to that of the moment generating function of the pertinent Lévy processes and to that of the moments of these processes in particular. The results are applied to questions of contingent claim valuation, such as the explicit valuation of Asian options, and illustrated in the case of generalized inverse Gaussian Lévy processes.

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.

Problèmes de premier passage résolubles par la méthode de séparation des variables

1995 ◽  
Vol 32 (2) ◽  
pp. 337-348 ◽  
Author(s):  
Mario Lefebvre
Keyword(s):  
Brownian Motion ◽  
Stochastic Processes ◽  
Generating Function ◽  
Moment Generating Function ◽  
First Passage Times ◽  
Standard Brownian Motion ◽  
First Passage ◽  
The Moment ◽  
Passage Times

In this paper, bidimensional stochastic processes defined by ax(t) = y(t)dt and dy(t) = m(y)dt + [2v(y)]1/2dW(t), where W(t) is a standard Brownian motion, are considered. In the first section, results are obtained that allow us to characterize the moment-generating function of first-passage times for processes of this type. In Sections 2 and 5, functions are computed, first by fixing the values of the infinitesimal parameters m(y) and v(y) then by the boundary of the stopping region.

scholarly journals The moment problem for some Wiener functionals: corrections to previous proofs (with an appendix by H. L. Pedersen)

2005 ◽  
Vol 42 (3) ◽  
pp. 851-860 ◽  
Author(s):  
Per Hörfelt
Keyword(s):  
Brownian Motion ◽  
Moment Problem ◽  
Geometric Brownian Motion ◽  
Geometric Inequalities ◽  
Gauss Space ◽  
The Moment

In this paper, we describe a class of Wiener functionals that are ‘indeterminate by their moments’, that is, whose distributions are not uniquely determined by their moments. In particular, it is proved that the integral of a geometric Brownian motion is indeterminate by its moments and, moreover, shown that previous proofs of this result are incorrect. The main result of this paper is based on geometric inequalities in Gauss space and on a generalization of the Krein criterion due to H. L. Pedersen.

scholarly journals Continuous-time methods in the study of discretely sampled functionals of Lévy processes. I. The positive process case

2007 ◽  
Vol 39 (01) ◽  
pp. 245-270
Author(s):  
Michael Schröder
Keyword(s):  
Lévy Processes ◽  
Generalized Inverse ◽  
Moment Generating Function ◽  
Levy Processes ◽  
Asian Options ◽  
Inverse Gaussian ◽  
Contingent Claim Valuation ◽  
Reduction Series ◽  
Generalized Inverse Gaussian ◽  
The Moment

In this paper we develop a constructive approach to studying continuously and discretely sampled functionals of Lévy processes. Estimates for the rate of convergence of the discretely sampled functionals to the continuously sampled functionals are derived, reducing the study of the latter to that of the former. Laguerre reduction series for the discretely sampled functionals are developed, reducing their study to that of the moment generating function of the pertinent Lévy processes and to that of the moments of these processes in particular. The results are applied to questions of contingent claim valuation, such as the explicit valuation of Asian options, and illustrated in the case of generalized inverse Gaussian Lévy processes.

scholarly journals The moment problem for some Wiener functionals: corrections to previous proofs (with an appendix by H. L. Pedersen)

2005 ◽  
Vol 42 (03) ◽  
pp. 851-860 ◽  
Author(s):  
Per Hörfelt
Keyword(s):  
Brownian Motion ◽  
Moment Problem ◽  
Geometric Brownian Motion ◽  
Geometric Inequalities ◽  
Gauss Space ◽  
The Moment

In this paper, we describe a class of Wiener functionals that are ‘indeterminate by their moments’, that is, whose distributions are not uniquely determined by their moments. In particular, it is proved that the integral of a geometric Brownian motion is indeterminate by its moments and, moreover, shown that previous proofs of this result are incorrect. The main result of this paper is based on geometric inequalities in Gauss space and on a generalization of the Krein criterion due to H. L. Pedersen.

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