Mathai, Melrose, and Singer compute the index of projective elliptic operators.
problem Computing the index of projective elliptic operators on manifolds with Azumaya bundles.
method Equivariant index of transversally elliptic operators as pullbacks of projective elliptic operators.
result Comprehensive fractional index formula for projective elliptic operators.
We calculate the equivariant index formula for an infinite dimensional Clifford module canonically associated to any Riemannian manifold. It encompasses the fractional index formula of the projective Dirac operator by Mathai--Melrose--Singer.
The paper converts nonalternating forms of rational links into all-even forms and derives formulas for their braid index and HOMFLY polynomial.
problem Finding formulas for the braid index and HOMFLY polynomial of rational links.
method Algorithm to transform nonalternating forms into all-even forms and derivation of formulas.
result Formulas for the braid index and HOMFLY polynomial of rational links in terms of their reduced alternating form.
G-framework is presented by Peng [41] for measure risk under uncertainty. In this paper, we define fractional G-Brownian motion (fGBm). Fractional G-Brownian motion is a centered G-Gaussian process with zero mean and stationary increments in the sense of sub-linearity with Hurst index H∈(0,1). This process has sta…
A new option pricing model uses a time-varying Hurst exponent for more accurate financial predictions.
problem Inaccurate modeling of financial time series due to constant memory parameter limitations.
method Modeling price fluctuations with multifractional Brownian motion and deriving option pricing formula.
result Empirical performance shows the multifractional model fits market quotes better than standard models.
Paper defines multi-dimensional fractional Brownian motion under volatility uncertainty.
problem Volatility uncertainty in fractional Brownian motion.
method Definition and study of multi-dimensional fractional Brownian motion (G-fBm) with Hurst index.
result First results on stochastic calculus for G-fBm with Hurst index > 0.5.
In this note the fractional analytic index, for a projective elliptic operator associated to an Azumaya bundle, of DG/0402329 is related to the equivariant index of Atiyah and Singer for an associated transversally elliptic operator.
Derives an option-pricing formula for fractional markets with skew and smile.
problem Developing a pricing formula for financial options with skew and smile.
method Employed the Lévy-Khintchine theorem and fractional Gaussian noise to generalize the Black-Scholes-Merton formula.
result An exponentially convergent option-pricing formula for fractional markets.
Develops hedging formula in fractional model with costs.
problem Hedging in fractional Black-Scholes model with transaction costs.
method Explicit formula for hedging portfolio using fractional Brownian motion.
result Explicit formula for conditional-mean hedging portfolio.
Inspired by the LG/CY correspondence, we study the local index theory of the Schrödinger operator associated to a singularity defined on Cn by a quasi-homogeneous polynomial f. Under some mild assumption on f, we show that the small time heat kernel expansion of the corresponding Schrödinger operator e…
The paper extends Merton model to price equity warrants under subdiffusive fractional Brownian motion of the short rate.
problem Equity warrant pricing under subdiffusive fractional Brownian motion of the short rate.
method The paper applies subdiffusive mechanism to analyze equity warrant in a fractional Brownian motion environment, deriving a pricing formula for equity warrant.
result The paper provides a pricing formula for equity warrants under subdiffusive fractional Brownian motion model of the short rate.
The twisted Connes-Moscovici higher index theorem is generalized to the case of good orbifolds. The higher index is shown to be a rational number, and in fact non-integer in specific examples of 2-orbifolds. This results in a non-commutative geometry model that predicts the occurrence of fractional quantum numbers in t…
This paper develops a European option pricing formula for fractional market models. Although there exist option pricing results for a fractional Black-Scholes model, they are established without accounting for stochastic volatility. In this paper, a fractional version of the Constant Elasticity of Variance (CEV) model …
Braids with more twists than strands achieve their braid index.
problem Understanding the braid index of closures of braids with fractional Dehn twist coefficients.
method Characterizing fractional Dehn twist coefficients in terms of the Upsilon function and proving the braid index via homogenization of knot concordance homomorphisms.
result Proves that n-braids with fractional Dehn twist coefficient larger than n−1 realize the braid index of their closure. Analyzes American option pricing with fractional derivatives.
problem Characterizing American option prices under modified models.
method Uses fractional partial differential equations and approximation techniques.
result Proves convexity of American put prices and tail index impact.
For a finite rank projective bundle over a compact manifold, so associated to a torsion, Dixmier-Douady, 3-class, w, on the manifold, we define the ring of differential operators `acting on sections of the projective bundle' in a formal sense. In particular, any oriented even-dimensional manifold carries a projective s…
Study prices compound and extendible options using mixed fractional Brownian motion with jumps.
problem Pricing compound and extendible options under mixed fractional Brownian motion with jumps.
method Analytic formula derived under risk-neutral measure, applied to extendible options, discussed special cases, provided numerical results.
result An analytic formula for pricing compound options derived.
The study values a new type of insurance-linked security called CocoCat bonds.
problem Valuing a new type of insurance-linked security called contingent convertible catastrophe bonds.
method Formalized design, derived analytical valuation formulae, used time-inhomogeneous compound Poisson process for natural catastrophe losses, and applied exponential change of measure and Girsanov-like transformation.
result CocoCat bond prices are most sensitive to interest rates, conversion fractions, and trigger levels.
The paper presents a series representation for European option pricing driven by fractional diffusion.
problem Pricing European options under space-time fractional diffusion.
method Uses Mellin-Barnes representation and residue summation in the complex plane.
result Derives a rapidly convergent double-series formula for option pricing.
Study examines pricing of target volatility options in fractional SABR model.
problem Pricing target volatility options in the lognormal fractional SABR model.
method Used Ito's calculus for a theoretical replicating strategy and derived approximations and closed-form expressions.
result Accuracy of approximations for target volatility option pricing in various parameter ranges.
The paper explores global index formulas for one-dimensional holomorphic foliations.
problem Global index formulas for one-dimensional holomorphic foliations.
method Microlocal point of view and short proofs for existing index formulas.
result Generalizations of existing index formulas.
The spectral eta-invariant of a self-adjoint elliptic differential operator on a closed manifold is rigid, provided that the parity of the order is opposite to the parity of dimension of the manifold. The paper deals with the calculation of the fractional part of the eta-invariant in this case. The method used to obtai…
In this paper we study BSE Index financial time series for fractal and multifractal behaviour. We show that Bombay stock Exchange (BSE)Index time series is mono-fractal and can be represented by a fractional Brownian motion.
New index formula for hypoelliptic operators on manifolds.
problem Index computation for hypoelliptic differential operators.
method Generalized index formula for *-maximally hypoelliptic operators.
result Explicit index computations for Hormander's sum of squares operators.
Formula for option pricing in a stochastic volatility model with jumps.
problem Developing a formula for European option pricing in a complex stochastic volatility model.
method Fractional integral of a diffusion process, martingale representation, and Itô calculus for processes with jumps.
result A first-order approximation formula for option prices.
A geometric model for twisted K-homology is introduced. It is modeled after the Mathai-Melrose-Singer fractional analytic index theorem in the same way as the Baum-Douglas model of K-homology was modeled after the Atiyah-Singer index theorem. A natural transformation from twisted geometric K-homology to the new g…
Socio-economic inequality is characterized from data using various indices. The Gini (g) index, giving the overall inequality is the most common one, while the recently introduced Kolkata (k) index gives a measure of 1−k fraction of population who possess top k fraction of wealth in the society. Here, we show t…
An index formula is proposed for contact transformations between contact manifolds equipped with CR structures or with fillings by symplectic manifolds. The formula generalizes the Atiyah-Singer formula and gives a conjectured formula for the index of Fourier integral operators, as well as Epstein's relative index for …
Proves a formula in Heegaard Floer homology using combinatorial methods.
problem Proving Lipshitz's Maslov index formula in Heegaard Floer homology.
method Combinatorial proof via Heegaard diagrams.
result Validated Lipshitz's Maslov index formula in Heegaard Floer homology.
New connection found between cluster algebras and knot theory.
problem Connecting cluster algebras to knot theory and Jones polynomials.
method Using continued fractions and snake graphs.
result Direct formula for the Jones polynomial of 2-bridge links.
Study finds curves with explicit formulas for curvature and torsion.
problem Finding curves with specific geometric properties.
method Developed a family of curves parametrized by arc length, dependent on angular and intrinsic fraction functions.
result Explicit formulas for curvature, torsion, and geodetic curvature found in terms of angular and intrinsic fraction functions.
In this paper, we establish an infinitesimal equivariant index formula in the noncommutative geometry framework using Greiner's approach to heat kernel asymptotics. An infinitesimal equivariant index formula for odd dimensional manifolds is also given. We define infinitesimal equivariant eta cochains, prove their regul…
Paper derives analytical formulas for NLD-CEV moments with regime switching.
problem Analytical tractability of NLD-CEV models under stochastic regimes.
method Hybrid system approach using Feynman-Kac formula for solving interconnected PDEs.
result Exact closed-form expressions for fractional-order conditional moments.
Formula calculates index for CR operators on surfaces with boundary punctures.
problem Computing the index for Cauchy-Riemann operators on surfaces with boundary punctures.
method Large antilinear deformations method, generalized to punctured surfaces.
result Involves a non-standard weighted count of boundary zeros in the Euler characteristic term.
Based on empirical market data, a stochastic volatility model is proposed with volatility driven by fractional noise. The model is used to obtain a risk-neutrality option pricing formula and an option pricing equation.
Derives an index formula for families of end-periodic Dirac operators.
problem Calculating the index of families of end-periodic Dirac operators.
method Using the renormalized Chern character and Fourier-Laplace transform of the Bismut superconnection.
result Establishes an index formula involving a new end-periodic eta form.
The extended Heisenberg algebra for a contact manifold has a symbolic calculus that accommodates both Heisenberg pseudodifferential operators as well as classical pseudodifferential operators. We derive here a formula for the index of Fredholm operators in this extended calculus. This formula incorporates in a single e…
Empirical studies show that the volatility may exhibit correlations that decay as a fractional power of the time offset. The paper presents a rigorous analysis for the case when the stationary stochastic volatility model is constructed in terms of a fractional Ornstein Uhlenbeck process to have such correlations. It is…
A new method for analyzing multifractal cross correlations in complex systems.
problem Characterizing long-range cross-correlations in complex systems.
method Multifractal Cross Wavelet Analysis (MFXWT)
result MFXWT accurately captures joint multifractality in binomial multifractal measures but may produce spurious results for bivariate fractional Brownian motions.
The article recovers tensor fields from partial data using weighted divergent ray transforms.
problem Recovering tensor fields from partial data.
method Weighted divergent ray transforms, unique continuation property of fractional Laplacian, explicit reconstruction formulas.
result Recovery of symmetric m-tensor fields and unique continuation for vector fields and symmetric 2-tensor fields. Formula for index of Dirac-type operators on stratified spaces.
problem Calculating the index of Dirac-type operators on complex geometric structures.
method Defined a closed domain, proved self-adjoint and Fredholm properties, established index formula.
result Proved a formula for the Chern character of the index of Dirac-type operators.
Formula for index in Lorentzian spacetimes.
problem Developing an index formula for spacetimes with boundary.
method Reduction from equivariant to non-equivariant, Lorentzian spectral flow.
result Equivalence of equivariant index and spectral flow in Lorentzian spacetimes.
Topological degrees of continuous mappings between manifolds of even dimension are studied in terms of index theory of pseudo-differential operators. The index formalism of non-commutative geometry is used to derive analytic integral formulas for the index of a 0:th order pseudo-differential operator twisted by a Hölde…
The study examines European option pricing using a generalized tempered stable distribution.
problem Investigating the pricing of European options under a generalized tempered stable distribution.
method Fitting the Generalized Tempered Stable (GTS) distribution to S\&P 500 Index returns, applying the Esscher transform, and using the Extended Black-Scholes and Generalized Black-Scholes formulas.
result The GTS distribution yields consistent European option prices for deep OTM and ITM options, but underprices near-the-money and in-the-money options compared to the Black-Scholes model.
Study prices currency options using fractional delta hedging with transaction costs.
problem Pricing European currency options with transaction costs in fractional Black Scholes model.
method Applied delta hedging strategy to derive pricing formula and PDE.
result Fractional Black Scholes model with transaction costs is a satisfactory model.
First variation of fractional k-dimensional measure for submanifolds
problem Computing the first variation of a fractional k-dimensional measure for submanifolds method First variation computation
result Definition of a nonlocal mean-curvature vector for embedded submanifolds
The paper analyzes option pricing under subdiffusive fractional Brownian motion.
problem Option pricing with a short rate following subdiffusive fractional Merton model.
method Incorporates stochastic short rate into fractional Black-Scholes equation and derives explicit formulas.
result Explicit formulas for call and put options derived under subdiffusive fractional Merton model.
Given a proper, cocompact action of a Lie groupoid, we define a higher index pairing between invariant elliptic differential operators and smooth groupoid cohomology classes. We prove a cohomological index formula for this pairing by applying the van Est map and algebraic index theory. Finally we discuss in examples th…