If BB and f(B)f(B) are Brownian motions, then ff is affine

It is shown that if the processes $B$ and $f(B)$ are both Brownian motions (without a random time change) then $f$ must be an affine function. As a by-product of the proof, it is shown that the only functions which are solutions to both the Laplace equation and the eikonal equation are affine.Comment: 4 page

A Black–Scholes inequality: applications and generalisations

Abstract: The space of call price curves has a natural noncommutative semigroup structure with an involution. A basic example is the Black–Scholes call price surface, from which an interesting inequality for Black–Scholes implied volatility is derived. The binary operation is compatible with the convex order, and therefore a one-parameter sub-semigroup gives rise to an arbitrage-free market model. It is shown that each such one-parameter semigroup corresponds to a unique log-concave probability density, providing a family of tractable call price surface parametrisations in the spirit of the Gatheral–Jacquier SVI surface. An explicit example is given to illustrate the idea. The key observation is an isomorphism linking an initial call price curve to the lift zonoid of the terminal price of the underlying asset

Magnetic anisotropies and its optical manipulation in epitaxial bismuth ferrite/ferromagnet heterostructure

© 2018 Elsevier B.V. Exchange-coupled BiFeO3/ferromagnet thin film heterostructures are promising for a fast and power efficient control of magnetization of the ferromagnetic layer. In the paper, the results of the study of magnetic properties of the epitaxial Fe2B/BiFeO3heterostructure on (0 0 1)-SrTiO3substrate with ferromagnetic resonance (FMR) spectroscopy are presented. The hierarchy of magnetic anisotropies that determine the angular variation of the resonance field in-plane and in-/out-of the plane of the system include the tetragonal four-fold and uniaxial terms for the Fe2B layer, uniaxial term for antiferromagnetic BiFeO3layer and the exchange coupling at the interface. We find out that the exchange bias direction can be switched by a strong enough applied magnetic field. Both the exchange bias and an in-plane FMR resonance field of the heterostructure are strongly affected by the illumination with λ = 405 nm light. An optical tunability of the exchange bias and ferromagnetic resonance fields of the heterostructure has been demonstrated

Implied volatility of basket options at extreme strikes

In the paper, we characterize the asymptotic behavior of the implied volatility of a basket call option at large and small strikes in a variety of settings with increasing generality. First, we obtain an asymptotic formula with an error bound for the left wing of the implied volatility, under the assumption that the dynamics of asset prices are described by the multidimensional Black-Scholes model. Next, we find the leading term of asymptotics of the implied volatility in the case where the asset prices follow the multidimensional Black-Scholes model with time change by an independent increasing stochastic process. Finally, we deal with a general situation in which the dependence between the assets is described by a given copula function. In this setting, we obtain a model-free tail-wing formula that links the implied volatility to a special characteristic of the copula called the weak lower tail dependence function