Efficient evaluation of expectations of functions of a L\'evy process and its extremum

We prove simple general formulas for expectations of functions of a L\'evy process and its running extremum. Under additional conditions, we derive analytical formulas using the Fourier/Laplace inversion and Wiener-Hopf factorization, and discuss efficient numerical methods for realization of these formulas. As applications, the cumulative probability distribution function of the process and its running maximum and the price of the option to exchange the power of a stock for its maximum are calculated. The most efficient numerical methods use the sinh-acceleration technique and simplified trapezoid rule. The program in Matlab running on a Mac with moderate characteristics achieves the precision E-7 and better in several milliseconds, and E-14 - in a fraction of a second

Eigenvalue bounds in the gaps of Schrodinger operators and Jacobi matrices

We consider $C=A+B$ where $A$ is selfadjoint with a gap $(a,b)$ in its spectrum and $B$ is (relatively) compact. We prove a general result allowing $B$ of indefinite sign and apply it to obtain a $(\delta V)^{d/2}$ bound for perturbations of suitable periodic Schrodinger operators and a (not quite)Lieb-Thirring bound for perturbations of algebro-geometric almost periodic Jacobi matrices

Lookback option pricing using the Fourier transform B-spline method

We derive a new, efficient closed-form formula approximating the price of discrete lookback options, whose underlying asset price is driven by an exponential semimartingale process, which includes (jump) diffusions, Lévy models, affine processes and other models. The derivation of our pricing formula is based on inverting the Fourier transform using B-spline approximation theory. We give an error bound for our formula and establish its fast rate of convergence to the true price. Our method provides lookback option prices across the quantum of strike prices with greater efficiency than for a single strike price under existing methods. We provide an alternative proof to the Spitzer formula for the characteristic function of the maximum of a discretely observed stochastic process, which yields a numerically efficient algorithm based on convolutions. This is an important result which could have a wide range of applications in which the Spitzer formula is utilized. We illustrate the numerical efficiency of our algorithm by applying it in pricing fixed and floating discrete lookback options under Brownian motion, jump diffusion models, and the variance gamma process

Valuation Of Continuously Monitored Double Barrier Options And Related Securities

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/92059/1/j.1467-9965.2010.00469.x.pd

Graded quiver varieties, quantum cluster algebras and dual canonical basis

Inspired by a previous work of Nakajima, we consider perverse sheaves over acyclic graded quiver varieties and study the Fourier-Sato-Deligne transform from a representation theoretic point of view. We obtain deformed monoidal categorifications of acyclic quantum cluster algebras with specific coefficients. In particular, the (quantum) positivity conjecture is verified whenever there is an acyclic seed in the (quantum) cluster algebra. In the second part of the paper, we introduce new quantizations and show that all quantum cluster monomials in our setting belong to the dual canonical basis of the corresponding quantum unipotent subgroup. This result generalizes previous work by Lampe and by Hernandez-Leclerc from the Kronecker and Dynkin quiver case to the acyclic case. The Fourier transform part of this paper provides crucial input for the second author's paper where he constructs bases of acyclic quantum cluster algebras with arbitrary coefficients and quantization.Comment: 42 pages, minor corrections, references update

Spectral Theory for Perturbed Krein Laplacians in Nonsmooth Domains

We study spectral properties for $H_{K,\Omega}$, the Krein--von Neumann extension of the perturbed Laplacian $-\Delta+V$ defined on $C^\infty_0(\Omega)$, where $V$ is measurable, bounded and nonnegative, in a bounded open set $\Omega\subset\mathbb{R}^n$ belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class $C^{1,r}$, $r>1/2$. In particular, in the aforementioned context we establish the Weyl asymptotic formula #\{j\in\mathbb{N} | \lambda_{K,\Omega,j}\leq\lambda\} = (2\pi)^{-n} v_n |\Omega| \lambda^{n/2}+O\big(\lambda^{(n-(1/2))/2}\big) {as} \lambda\to\infty, where $v_n=\pi^{n/2}/ \Gamma((n/2)+1)$ denotes the volume of the unit ball in $\mathbb{R}^n$, and $\lambda_{K,\Omega,j}$, $j\in\mathbb{N}$, are the non-zero eigenvalues of $H_{K,\Omega}$, listed in increasing order according to their multiplicities. We prove this formula by showing that the perturbed Krein Laplacian (i.e., the Krein--von Neumann extension of $-\Delta+V$ defined on $C^\infty_0(\Omega)$) is spectrally equivalent to the buckling of a clamped plate problem, and using an abstract result of Kozlov from the mid 1980's. Our work builds on that of Grubb in the early 1980's, who has considered similar issues for elliptic operators in smooth domains, and shows that the question posed by Alonso and Simon in 1980 pertaining to the validity of the above Weyl asymptotic formula continues to have an affirmative answer in this nonsmooth setting.Comment: 60 page

Operators and Boundary Problems in Finance, Economics and Insurance: Peculiarities, Efficient Methods and Outstanding Problems

The price V of a contingent claim in finance, insurance and economics is defined as an expectation of a stochastic expression. If the underlying uncertainty is modeled as a strong Markov process X, the Feynman–Kac theorem suggests that V is the unique solution of a boundary problem for a parabolic equation. In the case of PDO with constant symbols, simple probabilistic tools explained in this paper can be used to explicitly calculate expectations under very weak conditions on the process and study the regularity of the solution. Assuming that the Feynman–Kac theorem holds, and a more general boundary problem can be localized, the local results can be used to study the existence and regularity of solutions, and derive efficient numerical methods. In the paper, difficulties for the realization of this program are analyzed, several outstanding problems are listed, and several closely efficient methods are outlined