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26
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 205 (15 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 Distribution And Asymptotic Results For Exponential Functionals Of Lévy Processes
, 1997
"... The aim of this note is to study the distribution and the asymptotic behavior of the exponential functional A t := R t 0 e s ds, where ( s ; s 0) denotes a L'evy process. When A1 ! 1, we show that in most cases, the law of A1 is a solution of an integrodifferential equation ; moreover, ..."
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Cited by 121 (11 self)
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The aim of this note is to study the distribution and the asymptotic behavior of the exponential functional A t := R t 0 e s ds, where ( s ; s 0) denotes a L'evy process. When A1 ! 1, we show that in most cases, the law of A1 is a solution of an integrodifferential equation ; moreover, this law is characterized by its integral moments. When the process is asymptotically ffstable, we prove that t \Gamma1=ff log A t converges in law, as t !1, to the supremum of an ffstable L'evy process ; in particular, if E [ 1 ] ? 0, then ff = 1 and (1=t) log A t converges almost surely to E [ 1 ]. Eventually, we use Girsanov's transform to give the explicit behavior of E \Theta (a +A t ()) \Gamma1 as t ! 1, where a is a constant, and deduce from this the rate of decay of the tail of the distribution of the maximum of a diffusion process in a random L'evy environment.
Exponential functionals of Lévy processes
 Probabilty Surveys
, 2005
"... Abstract: This text surveys properties and applications of the exponential functional ∫ t exp(−ξs)ds of realvalued Lévy processes ξ = (ξt, t ≥ 0). 0 ..."
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Cited by 80 (6 self)
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Abstract: This text surveys properties and applications of the exponential functional ∫ t exp(−ξs)ds of realvalued Lévy processes ξ = (ξt, t ≥ 0). 0
Pricing equity derivatives subject to bankruptcy
 Mathematical Finance
, 2006
"... We solve in closed form a parsimonious extension of the Black–Scholes–Merton model with bankruptcy where the hazard rate of bankruptcy is a negative power of the stock price. Combining a scale change and a measure change, the model dynamics is reduced to a linear stochastic differential equation who ..."
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Cited by 54 (8 self)
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We solve in closed form a parsimonious extension of the Black–Scholes–Merton model with bankruptcy where the hazard rate of bankruptcy is a negative power of the stock price. Combining a scale change and a measure change, the model dynamics is reduced to a linear stochastic differential equation whose solution is a diffusion process that plays a central role in the pricing of Asian options. The solution is in the form of a spectral expansion associated with the diffusion infinitesimal generator. The latter is closely related to the Schrödinger operator with Morse potential. Pricing formulas for both corporate bonds and stock options are obtained in closed form. Term credit spreads on corporate bonds and implied volatility skews of stock options are closely linked in this model, with parameters of the hazard rate specification controlling both the shape of the term structure of credit spreads and the slope of the implied volatility skew. Our analytical formulas are easy to implement and should prove useful to researchers and practitioners in corporate debt and equity derivatives markets.
Infinitely Divisible Laws Associated With Hyperbolic Functions
, 2000
"... The infinitely divisible distributions on R + of random variables C t , S t and T t with Laplace transforms ` 1 cosh p 2 ' t ; / p 2 sinh p 2 ! t ; and / tanh p 2 p 2 ! t respectively are characterized for various t ? 0 in a number of different ways: by simple relation ..."
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Cited by 41 (8 self)
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The infinitely divisible distributions on R + of random variables C t , S t and T t with Laplace transforms ` 1 cosh p 2 ' t ; / p 2 sinh p 2 ! t ; and / tanh p 2 p 2 ! t respectively are characterized for various t ? 0 in a number of different ways: by simple relations between their moments and cumulants, by corresponding relations between the distributions and their L'evy measures, by recursions for their Mellin transforms, and by differential equations satisfied by their Laplace transforms. Some of these results are interpreted probabilistically via known appearances of these distributions for t = 1 or 2 in the description of the laws of various functionals of Brownian motion and Bessel processes, such as the heights and lengths of excursions of a onedimensional Brownian motion. The distributions of C¹ and S³ are also known to appear in the Mellin representations of two important functions in analytic number theory, the Riemann zeta function and ...
Spectral Expansions for Asian (Average Price) Options
, 2004
"... Arithmetic Asian or average price options deliver payoffs based on the average underlying price over a prespecified time period. Asian options are an important family of derivative contracts with a wide variety of applications in currency, equity, interest rate, commodity, energy, and insurance mark ..."
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Cited by 32 (4 self)
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Arithmetic Asian or average price options deliver payoffs based on the average underlying price over a prespecified time period. Asian options are an important family of derivative contracts with a wide variety of applications in currency, equity, interest rate, commodity, energy, and insurance markets. We derive two analytical formulas for the value of the continuously sampled arithmetic Asian option when the underlying asset price follows geometric Brownian motion. We use an identity in law between the integral of geometric Brownian motion over a finite time interval 0 t and the state at time t of a onedimensional diffusion process with affine drift and linear diffusion and express Asian option values in terms of spectral expansions associated with the diffusion infinitesimal generator. The first formula is an infinite series of terms involving Whittaker functions M and W. The second formula is a single real integral of an expression involving Whittaker function W plus (for some parameter values) a finite number of additional terms involving incomplete gamma functions and Laguerre polynomials. The two formulas allow accurate computation of continuously sampled arithmetic Asian option prices.
THE SPECTRAL DECOMPOSITION OF THE OPTION VALUE
, 2004
"... This paper develops a spectral expansion approach to the valuation of contingent claims when the underlying state variable follows a onedimensional diffusion with the infinitesimal variance a 2 (x), drift b(x) and instantaneous discount (killing) rate r(x). The Spectral Theorem for selfadjoint ope ..."
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Cited by 23 (10 self)
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This paper develops a spectral expansion approach to the valuation of contingent claims when the underlying state variable follows a onedimensional diffusion with the infinitesimal variance a 2 (x), drift b(x) and instantaneous discount (killing) rate r(x). The Spectral Theorem for selfadjoint operators in Hilbert space yields the spectral decomposition of the contingent claim value function. Based on the Sturm–Liouville (SL) theory, we classify Feller’s natural boundaries into two further subcategories: nonoscillatory and oscillatory/nonoscillatory with cutoff Λ ≥ 0 (this classification is based on the oscillation of solutions of the associated SL equation) and establish additional assumptions (satisfied in nearly all financial applications) that allow us to completely characterize the qualitative nature of the spectrum from the behavior of a, b and r near the boundaries, classify all diffusions satisfying these assumptions into the three spectral categories, and present simplified forms of the spectral expansion for each category. To obtain explicit expressions, we observe that the Liouville transformation reduces the SL equation to the onedimensional Schrödinger equation with a potential function constructed from a, b and r. If analytical solutions are available for the Schrödinger equation, inverting the Liouville transformation yields analytical solutions for the original SL equation, and the spectral representation for the diffusion process can be constructed explicitly. This produces an explicit spectral decomposition of the contingent claim value function.
Onedimensional disordered supersymmetric quantum mechanics: a brief survey, in Supersymmetry and Integrable Models, edited by
 Lecture Notes in Physics
, 1998
"... ‡ Unité de recherche des Universités Paris 11 et Paris 6 associée au CNRS. Abstract. We consider a onedimensional model of localization based on the Witten Hamiltonian of supersymmetric quantum mechanics. The low energy spectral properties are reviewed and compared with those of other models with o ..."
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Cited by 13 (8 self)
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‡ Unité de recherche des Universités Paris 11 et Paris 6 associée au CNRS. Abstract. We consider a onedimensional model of localization based on the Witten Hamiltonian of supersymmetric quantum mechanics. The low energy spectral properties are reviewed and compared with those of other models with offdiagonal disorder. Using recent results on exponential functionals of a Brownian motion we discuss the statistical properties of the ground state wave function and their multifractal behaviour. IPNO/TH 9721 1
Limiting laws for long Brownian bridges perturbed by their onesided maximum
 III, in "Period. Math. Hungar
"... In homage to Professors E. Csaki and P. Revesz. Abstract. Results of penalization of a onedimensional Brownian motion (Xt), by its onesided maximum (St = Xu), which were recently obtained by the authors are improved with the sup 0≤u≤t considerationin the present paper of the asymptotic behaviour ..."
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Cited by 10 (3 self)
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In homage to Professors E. Csaki and P. Revesz. Abstract. Results of penalization of a onedimensional Brownian motion (Xt), by its onesided maximum (St = Xu), which were recently obtained by the authors are improved with the sup 0≤u≤t considerationin the present paper of the asymptotic behaviour of the likewise penalized Brownian bridges of length t, as t → ∞, or penalizations by functions of (St, Xt), and also the study of the speed of convergence, as t → ∞, of the penalized distributions at time t.
Exact Pricing of Asian Options: An Application of Spectral Theory (Nov. 2001 working paper submitted for publication
, 2001
"... Arithmetic Asian or average price (rate) options deliver payoffs based on the average underlying price over a prespeciÞed time period. Asian options are an important family of derivative contracts with a wide variety of applications in currency, equity, interest rate, commodity, energy, and insura ..."
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Cited by 8 (0 self)
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Arithmetic Asian or average price (rate) options deliver payoffs based on the average underlying price over a prespeciÞed time period. Asian options are an important family of derivative contracts with a wide variety of applications in currency, equity, interest rate, commodity, energy, and insurance markets. We derive two analytical formulae for the price of the arithmetic Asian option when the underlying asset price follows geometric Brownian motion. The mathematics of the Asian option turns out to be related to the Schrödinger equation with Morse (1929) potential. Our derivation relies on the spectral theory of singular SturmLiouville (Schrödinger) operators and associated eigenfunction expansions. The Þrst formula is an inÞnite series of terms involving Whittaker functions M and W. The second formula is a single real integral of an expression involving Whittaker function W plus (for some parameter values) a Þnite number of additional terms involving incomplete Gamma functions and Laguerre polynomials. The two formulae allow exact computation of Asian option prices.