Peter Laurence

Asymptotics of Implied Volatility in Local Volatility Models

Using an expansion of the transition density function of a 1-dimensional time inhomogeneous diffusion, we obtain the first and second order terms in the short time asymptotics of European call option prices. The method described can be generalized to any order. We then use these option prices approximations to calculate the first order and second order deviation of the implied volatility from its leading value and obtain approximations which we numerically demonstrate to be highly accurate. The analysis is extended to degenerate diffusions using probabilistic methods, i.e. the so called principle of not feeling the boundary.

Static-arbitrage upper bounds for the prices of basket options

In this paper we investigate the possible values of basket options. Instead of postulating a model and pricing the basket option using that model, we consider the set of all models which are consistent with the observed prices of vanilla options, and, within this class, find the model for which the price of the basket option is largest. This price is an upper bound on the prices of the basket option which are consistent with no-arbitrage. In the absence of additional assumptions it is the lowest upper bound on the price of the basket option. Associated with the bound is a simple super-replicating strategy involving trading in the individual calls.

The ballooning spectrum of rotating plasmas

Ballooning modes are shown to be part of the spectrum by using a ‘‘singular sequence’’ of localized modes. We show that the modes arise from Alfven and slow magnetosonic waves propagating along rays confined inside the plasma. Different ballooning modes are seen, depending on the particular rotating frame of observation, indicating that there are accumulation points of eigenvalues. The effect of rigidly rotating flow is seen to be destabilizing due to an analog of the Rayleigh–Taylor instability associated with density gradients in the presence of a centrifugal force. Flow shear also modifies the stability criterion. A certain component of the flow shear will eliminate the ballooning modes.

Multi-Asset Stochastic Local Variance Contracts

Variance swaps now trade actively over-the-counter (OTC) on both stocks and stock indices. Also trading OTC are variations on variance swaps which localize the payoff in time, in the underlying asset price, or both. Given that the price of the underlying asset evolves continuously over time, it is well known that there exists a semirobust hedge for these localized variance contracts. Remarkably, the hedge succeeds even though the stochastic process describing the instantaneous variance is never specified. In this paper, we present a generalization of these results to the case of two or more underlying assets.

Asymptotic Massey products, induced currents and Borromean torus links

We introduce a class of currents which allows a new and very explicit form for the Massey product of a third order link as a line integral. The explicit form permits the introduction of an asymptotic Massey product analogous to that introduced previously for Gauss’s integral by V. Arnold. The average third order asymptotic Massey product is shown to be equal to Berger’s third order helicity for divergence-free vector fields in linked tori.

Second Order Expansion for Implied Volatility in Two Factor Local Stochastic Volatility Models and Applications to the Dynamic \lambda -Sabr Model

Using an expansion of the transition density function of a two dimensional time inhomogeneous diffusion, we obtain the first and second order terms in the short time asymptotics of the local volatility function in a family of time inhomogeneous local-stochastic volatility models. With the local volatility function at our disposal, we show how recent results (Gatheral et al., Math. Financ. 22:591–620, 2012, [28]) for one dimensional diffusions can be applied to also determine expansions for call prices as well as for the implied volatility. The results are worked out in detail in the case of the dynamic Sabr model, thus generalizing earlier work by Hagan et al. (Wilmott Mag. 84–108, 2003, [31]), Hagan and Lesniewski (Springer Proceedings in Mathematics and Statistics, vol. 110, 2015, [32]) and by Henry-Labordere (Springer Proceedings in Mathematics and Statistics, vol. 110, 2015, Geometry, and Modeling in Finance. Chapman & Hall/CRC Financial Mathematics Series, 2008, [39, 40]).

CLOSED FORM SOLUTIONS FOR QUADRATIC AND INVERSE QUADRATIC TERM STRUCTURE MODELS

We find fundamental solutions in closed form for a family of parabolic equations with two spatial variables, whose symmetry groups had been determined in an earlier paper by Finkel [12]. We show how these results can be applied in finance to yield closed form solutions for special affine and quadratic two factor term structure models as well as a new class of models with inverse square behavior. The latter can be considered a partial extension to two factors of pricing models related to the Bessel process devised by Albanese and Campolieti [3] and Albanese et al. [2]. A by-product of our results is that Lies reduction method in this setting leads only to fundamental solutions that can be factorized as products of functions that depend jointly on time and on one spatial coordinate. Thus all the results in this paper extend immediately to n factor models.

Two-dimensional magnetohydrodynamic equilibria with prescribed topology

We establish existence of weak solutions to the equilibrium equations of magnetohydrodynamics with prescribed topology. This is carried out in two settings. In the first we consider the variational problem of minimizing total energy in a torus under the assumption of axisymmetry, and prescription of mass and flux profiles. Existence of weak solutions implies that the prescription of topology is a natural constraint. Both the compressible and incompressible cases are considered. In the second setting we adapt examples of B. C. Low and R. Wolfson [13] and J. J. Aly and T. Amari [1, 2, 3] associated with Parkers explanation of the extreme heating of the solar corona and other solar phenomena. Existence of solutions with fixed topology is a first crucial step in rigorously examining the relationship between topology and the existence of current sheets. We use a decomposition introduced in [8, 9, 11] that captures much of the topology of level sets for certain classes of Sobolev functions. This decomposition is preserved under weak limits and so is useful for prescribing topological constraints. The approach is especially suited to the use of domain deformations.

On the representation of inhomogeneous linear force-free fields

It is shown that there is a false assumption hidden in the description of a relaxed state with inhomogeneous boundary conditions as the vector sum of a potential field, satisfying the boundary conditions, and a sum of eigenfunctions of the associated eigenvalue problem expanded by certain coefficients. In particular, although the Jensen and Chu formula (1984) can provide the correct expansion coefficients, it contains an implicit paradox in its derivation according to a general vector theorem. The same paradox led Chu et al. (1999) to be concerned about a contradiction obtained by taking the curl of their magnetic field expansion which, if permitted, becomes inconsistent with a current normal to the surface. The assumption that the curl can be commuted across an infinite sum of terms is the mechanism leading to these, apparently paradoxical, conclusions. Two mechanisms for resolving this apparent paradox are possible, one of which will be described in some detail below and the other discussed further in a forthcoming, more theoretical paper (Laurence et al., 2000). The decomposition of the magnetic field above is valid with convergence in the mean squared sense, but a decomposition of the current needs to be reinterpreted in terms of negative Sobolev spaces. To avoid this, and remain in a more easily managable and familiar setting, we derive the expansion coefficients in a way that involves the commuting of the inverse curl (as opposed to the curl) and the series. The resulting series converges in a mean square sense. When this is done the calculation can conform to the general vector theorem and a new gauge-invariant expression for the coefficients is obtained. However the consequence of the non-commutability is nullified in the Jensen and Chu formula, in both simply and multiply connected domains, by the important extra requirement of a boundary condition on the vector potential eigenfunctions; this excludes magnetic field eigenfunctions that carry flux, but there remains a complete set for the expansion and all flux is carried by the potential field. The two formulas are then identical. On a different issue, it is shown that if the general expansion is taken over a half-space, by combining positive and negative eigenvalue terms, then the coefficients are anisotropic, that is they are tensors except when evaluated at the first eigenvalue. A specific example is presented to illustrate the situation and to validate the new method of deriving the coefficients.

A lower bound for the energy of magnetic fields supported in linked tori

Abstract Given a divergence free field B in R 3 , the problem arises in fluids and magnetofluids, of finding lower bounds on the E 3/2 = ∫ | B | 3/2 d V norm of the field which depend only on the topology of the trajectories of the field and fluxes. In this Note, using the notion of third order helicity, we present such a lower bound in the case when it is known that a higher order linking is present, i.e., that the magnetic field has support in (or larger than) three solid tori linked like the Borromean rings, and is tangent to their boundaries.