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On some exponential functionals of Brownian motion
 Adv. Appl. Prob
, 1992
"... Abstract: This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, expl ..."
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Cited by 97 (9 self)
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Abstract: This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, explicit expressions for the heat kernels on hyperbolic spaces, diffusion processes in random environments and extensions of Lévy’s and Pitman’s theorems are discussed.
On the valuation of arithmetic–average Asian options: explicit formulas
, 1999
"... In a recent significant advance, using Laguerre series, the valuation of Asian options has been reduced in [D] to computing the negative moments of Yor’s accumulation processes for which functional recursion rules are given. Stressing the role of Theta functions, this paper now solves these recursio ..."
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Cited by 8 (3 self)
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In a recent significant advance, using Laguerre series, the valuation of Asian options has been reduced in [D] to computing the negative moments of Yor’s accumulation processes for which functional recursion rules are given. Stressing the role of Theta functions, this paper now solves these recursion rules and expresses these negative moments as linear combinations of certain Theta integrals. Using the Jacobi transformation formula, very rapidly and very stably convergent series for them are derived. In this way a computable series for Black–Scholes price of the Asian option results which is numerically illustrated. Moreover, the Laguerre series approach of [D] is made rigorous, and extensions and modifications are discussed. The key for this is the analysis of the integrability and growth properties of the Asia density in [Y], basic problems which seem to be addressed here for the first time. 1. Introduction: Asian
Increment sizes of the principal value of Brownian local time
, 1999
"... this paper, we mention an interesting property: stopped at some suitably chosen random times, the principal values give all the possible symmetric stable processes (cf. Biane and Yor [3], Fitzsimmons and Getoor [9], Bertoin [2]). It is easily seen that Y (\Delta) inherits a scaling property from Bro ..."
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Cited by 2 (2 self)
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this paper, we mention an interesting property: stopped at some suitably chosen random times, the principal values give all the possible symmetric stable processes (cf. Biane and Yor [3], Fitzsimmons and Getoor [9], Bertoin [2]). It is easily seen that Y (\Delta) inherits a scaling property from Brownian motion, namely, for any fixed a ? 0, t 7! a
reality: Hybrid Brownian motion with price
, 2009
"... A model of returns for the postcreditcrunch ..."
A Model of Returns for the PostCreditCrunch environment
, 2009
"... The market events of 20072009 have reinvigorated the search for realistic return models that capture greater likelihoods of extreme movements. In this paper we model the mediumterm logreturn dynamics in a market with both fundamental and technical traders. This is based on a Poisson trade arrival ..."
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The market events of 20072009 have reinvigorated the search for realistic return models that capture greater likelihoods of extreme movements. In this paper we model the mediumterm logreturn dynamics in a market with both fundamental and technical traders. This is based on a Poisson trade arrival model with variable size orders. With simplifications we are led to an SDE mixing both arithmetic and geometric Brownian motions, whose solution is an given by a class of integrals of exponentials of Brownian motions, in forms considered by Yor and collaborators. The reduction of the hybrid SDE to a single Brownian motion leads to an SDE of the form considered by Nagahara, which is a type of “Pearson diffusion”, or equivalently a hyperbolic OU SDE. Various dynamics and equilibria are possible depending on the balance of trades. Under meanreverting circumstances we arrive naturally at an equilibrium fattailed return distribution with a Student or Pearson Type IV form. Under less restrictive assumptions richer dynamics are possible. The phenomenon of variance explosion is identified that gives rise to much larger price movements that might have a priori been expected, so that “25σ ” events are realistic. We exhibit simple example solutions of the FokkerPlanck equation that shows how such variance explosion can hide beneath a standard Gaussian facade. These are elementary members of an extended class of distributions with a rich and varied structure, capable of describing a wide range of market behaviours. Several approaches to the density function are possible, and an example of the computation of a hyperbolic VaR is given. The model also suggests generalizations of the Bougerol identity.