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16
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 119 (12 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.
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 35 (5 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.
A.N.: Path Integral Approach for Superintegrable Potentials on the ThreeDimensional Hyperboloid
 Phys. Part.Nucl
, 1997
"... In this paper the Feynman path integral technique is applied for superintegrable potentials on twodimensional spaces of nonconstant curvature: these spaces are Darboux spaces DI and DII, respectively. On DI there are three and on DII four such potentials, respectively. We are able to evaluate the ..."
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Cited by 12 (7 self)
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In this paper the Feynman path integral technique is applied for superintegrable potentials on twodimensional spaces of nonconstant curvature: these spaces are Darboux spaces DI and DII, respectively. On DI there are three and on DII four such potentials, respectively. We are able to evaluate the path integral in most of the separating coordinate systems, leading to expressions for the Green functions, the discrete and continuous wavefunctions, and the discrete energyspectra. In some cases, however, the discrete spectrum cannot be stated explicitly, because it is either determined by a transcendental equation involving parabolic cylinder functions (Darboux space I), or by a higher order polynomial equation. The solutions on DI in particular show that superintegrable systems are not necessarily degenerate. We can also show how the limiting cases of flat space (constant curvature zero) and the twodimensional hyperboloid (constant negative curvature) emerge. CONTENTS 1 Contents 1
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 7 (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.
Magnetic bottles on the Poincaré halfplane: spectral asymptotics
"... wwwfourier.ujfgrenoble.fr/prepublications.html We consider a magnetic Laplacian −∆A = (id + A) ⋆ (id + A) on the Poincaré upperhalf plane H, when the magnetic field dA is infinite at infinity and such that −∆A has pure discret spectrum. We obtain the asymptotic behavior of the counting function ..."
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Cited by 4 (4 self)
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wwwfourier.ujfgrenoble.fr/prepublications.html We consider a magnetic Laplacian −∆A = (id + A) ⋆ (id + A) on the Poincaré upperhalf plane H, when the magnetic field dA is infinite at infinity and such that −∆A has pure discret spectrum. We obtain the asymptotic behavior of the counting function of the eigenvalues.
The Schrödinger Operator with Morse Potential on the Right Half Line
, 2009
"... This paper studies the Schrödinger operator with Morse potential Vk(u) = 1 4 e2u + ke u on a right halfline [u0, ∞), and determines the Weyl asymptotics of eigenvalues for constant boundary conditions at the endpoint u0. In consequence it obtains information on the location of zeros of the Whittak ..."
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Cited by 2 (0 self)
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This paper studies the Schrödinger operator with Morse potential Vk(u) = 1 4 e2u + ke u on a right halfline [u0, ∞), and determines the Weyl asymptotics of eigenvalues for constant boundary conditions at the endpoint u0. In consequence it obtains information on the location of zeros of the Whittaker function Wκ,µ(x), for fixed real parameters κ,x with x> 0, viewed as an entire function in the complex variable µ. In this case all zeros lie on the imaginary axis, with the exception, if k < 0 of a finite number of real zeros which lie in the interval κ  < k. We obtain an asymptotic formula for the number of zeros N(T) = {ρ  Wκ,ρ(x) = 0, Im(ρ)  < T} of the form N(T) = 2 2 πT log T + π (2log 2−1−log x)T +O(1). Parallels are observed with zeros of the Riemann zeta function.
quantph/9808060 ON THE PATH INTEGRAL TREATMENT FOR AN AHARONOV–BOHM FIELD ON THE HYPERBOLIC PLANE
, 1998
"... In this paper I discuss by means of path integrals the quantum dynamics of a charged particle on the hyperbolic plane under the influence of an Aharonov–Bohm gauge field. The path integral can be solved in terms of an expansion of the homotopy classes of paths. I discuss the interference pattern of ..."
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Cited by 1 (0 self)
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In this paper I discuss by means of path integrals the quantum dynamics of a charged particle on the hyperbolic plane under the influence of an Aharonov–Bohm gauge field. The path integral can be solved in terms of an expansion of the homotopy classes of paths. I discuss the interference pattern of scattering by an Aharonov–Bohm gauge field in the flat space limit, yielding a characteristic oscillating behavior in terms of the field strength. In addition, the cases of the isotropic Higgsoscillator and the Kepler–Coulomb The Aharonov–Bohm gauge field has a long history, beginning in 1959 by a classical paper by Aharonov and Bohm [Aharonov and Bohm (1959)]. The effect has been well studied and well confirmed [Anandan and Safko (1994)], but not necessarily well understood. It describes the motion of charged particles, i.e. electrons, which are scattered by an infinitesimal thin solenoid.
Weak Coherent State Path Integrals
, 2003
"... Weak coherent states share many properties of the usual coherent states, but do not admit a resolution of unity expressed in terms of a local integral. They arise e.g. in the case that a group acts on an inadmissible fiducial vector. Motivated by the recent Affine Quantum Gravity Program, the presen ..."
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Weak coherent states share many properties of the usual coherent states, but do not admit a resolution of unity expressed in terms of a local integral. They arise e.g. in the case that a group acts on an inadmissible fiducial vector. Motivated by the recent Affine Quantum Gravity Program, the present article studies the path integral representation of the affine weak coherent state matrix elements of the unitary timeevolution operator. Since weak coherent states do not admit a resolution of unity, it is clear that the standard way of constructing a path integral, by time slicing, is predestined to fail. Instead a welldefined path integral with Wiener measure, based on a continuoustime regularization, is used to approach this problem. The dynamics is rigorously established for linear Hamiltonians, and the difficulties presented by more general Hamiltonians are addressed. I.
unknown title
, 712
"... Magnetic bottles on the Poincaré halfplane: spectral asymptotics ..."
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