William T. Shaw

Differential equations and asymptotic solutions for arithmetic Asian options: 'Black-Scholes formulae' for Asian rate calls

In this article, we present a simplified means of pricing Asian options using partial differential equations (PDEs). We first provide a concise derivation of the well-known similarity reduction and exact Laplace transform solution. We then analyse the problem afresh as a power series in the volatility-scaled contract duration, with a view to obtaining an asymptotic solution for the low-volatility limit, a limit which presents difficulties in the context of the general Laplace transform solution. The problem is approached anew from the point of view of asymptotic expansions and the results are compared with direct, high precision, inversion of the Laplace transform and with numerical results obtained by V. Linetsky and J. Vecer. Our asymptotic formulae are little more complicated than the standard Black-Scholes formulae and, working to third order in the volatility-scaled expiry, are accurate to at least four significant figures for standard test problems. In the case of zero risk-neutral drift, we have the solution to fifth order and, for practical purposes, the results are effectively exact. We also provide comparisons with the hybrid analytic and finite-difference method of Zhang.

Sampling Student's T distribution - use of the inverse cumulative distribution function

With the current interest in copula methods, and fat-tailed or other non-normal distributions, it is appropriate to investigate technologies for managing marginal distributions of interest. We explore “Student’s” T distribution, survey its simulation, and present some new techniques for simulation. In particular, for a given real (not necessarily integer) value n of the number of degrees of freedom, we give a pair of power series approximations for the inverse, F−1 n , of the cumulative distribution function (CDF), Fn.We also give some simple and very fast exact and iterative techniques for defining this function when n is an even integer, based on the observation that for such cases the calculation of F−1 n amounts to the solution of a reduced-form polynomial equation of degree n − 1. We also explain the use of Cornish–Fisher expansions to define the inverse CDF as the composition of the inverse CDF for the normal case with a simple polynomial map. The methods presented are well adapted for use with copula and quasi-Monte-Carlo techniques.

CONFORMAL SUPERGRAVITY, TWISTORS AND THE SUPER BMS GROUP

The asymptotic symmetry group of General Relativity is the Bondi-Metzner-Sachs (BMS) group. We find the appropriate supergeneralization of this group for supergravity. The analysis is carried out using properties of the twistor equation and is valid for any space-time dimension.

A model of returns for the post-credit-crunch reality: hybrid Brownian motion with price feedback

Recent market events have reinvigorated the search for realistic return models that capture greater likelihoods of extreme movements. In this paper we model the medium-term log-return dynamics in a market with both fundamental and technical traders. This is based on a trade arrival model with variable size orders and a general arrival-time distribution. With simplifications we are led in the jump-free case to a local volatility model defined by a hybrid SDE mixing both arithmetic and geometric or CIR Brownian motions, whose solution in the geometric case is given by a class of integrals of exponentials of one Brownian motion against another, in forms considered by Yor and collaborators. The reduction of the hybrid SDE to a single Brownian motion leads to an SDE of the form considered by Nagahara, which is a type of ‘Pearson diffusion’, or, equivalently, a hyperbolic OU SDE. Various dynamics and equilibria are possible depending on the balance of trades. Under mean-reverting circumstances we arrive naturally at an equilibrium fat-tailed return distribution with a Student or Pearson Type~IV form. Under less-restrictive assumptions, richer dynamics are possible, including time-dependent Johnson-SU distributions and bimodal structures. The phenomenon of variance explosion is identified that gives rise to much larger price movements that might have a priori been expected, so that ‘25σ’ events are significantly more probable. We exhibit simple example solutions of the Fokker–Planck equation that shows how such variance explosion can hide beneath a standard Gaussian facade. These are elementary members of an extended class of distributions with a rich and varied structure, capable of describing a wide range of market behaviors. Several approaches to the density function are possible, and an example of the computation of a hyperbolic VaR is given. The model also suggests generalizations of the Bougerol identity. We touch briefly on the extent to which such a model is consistent with the dynamics of a ‘flash-crash’ event, and briefly explore the statistical evidence for our model.

Modelling bonds and credit default swaps using a structural model with contagion

This paper develops a two-dimensional structural framework for valuing credit default swaps and corporate bonds in the presence of default contagion. Modelling the values of related firms as correlated geometric Brownian motions with exponential default barriers, analytical formulae are obtained for both credit default swap spreads and corporate bond yields. The credit dependence structure is influenced by both a longer-term correlation structure as well as by the possibility of default contagion. In this way, the model is able to generate a diverse range of shapes for the term structure of credit spreads using realistic values for input parameters.

Green's function refinement as an approach to radar backscatter: general theory and applications to LGA scattering from the ocean

We present a new approach to the computation of radar returns from dielectric bodies whose boundary is the deformation of a plane surface. The method relies on combining a systematic improvement in the Greens function with a good approximate choice of local boundary condition. In this paper, the general theory is presented together with a simple application where the Greens function is that for a lossy dielectric half-space. We derive the root radar cross section (RCS) for a statistical surface and the mean backscatter RCS for a rough surface. We explore the applications to low-grazing-angle (LGA) scattering from statistical surfaces with an ocean-like spectrum and demonstrate that such a model explains some of the previously unexplained LGA phenomena, such as the absolute and relative levels of the vertical (VV) and horizontal (HH) channel RCS measurements.

Bounds for in-progress floating-strike Asian options using symmetry

This paper studies symmetries between fixed and floating-strike Asian options and exploits this symmetry to derive an upper bound for the price of a floating-strike Asian. This bound only involves fixed-strike Asians and vanillas, and can be computed simply given one of the many efficient methods for pricing fixed-strike Asian options. The bound coincides with the true price until after the averaging has begun and again at maturity. The bound is compared to benchmark prices obtained via Monte Carlo simulation in numerical examples.

Real classical strings

A general solution is constructed for the equations of motion of the classical relativistic string. All allowable string configurations are encompassed, apart from a special set of measure zero characterized by the vanishing of a certain invariant for which the motion of the string is restricted to a time-like hyperplane. The solution is given in terms of an essentially freely specifiable curve lying in the hypersurface PN, the space of ‘null twistors’.

Twistors, minimal surfaces and strings

It is shown that holomorphic curves on a projective twistor space T correspond to null holomorphic curves in complexified Minkowski spacetime. The real and imaginary parts of such a curve define a minimal 2-surface on real Minkowskian or real Euclidean 4-space. The minimal 2-surface equations can thereby be solved in terms of two free holomorphic functions of one variable. Pairs of holomorphic curves on twistor space describe complex minimal 2-surfaces. Such surfaces may also admit real string cross sections. Using this, solutions of the string equations in real Minkowskian 3-space are obtained in terms of two real analytic functions. The twistor transform of a general string in complex Minkowskian 4-space is obtained in terms of non-holomorphic curves in T.

Risk, VaR, CVaR and Their Associated Portfolio Optimizations When Asset Returns Have a Multivariate Student T Distribution

We show how to reduce the problem of computing VaR and CVaR with Student T return distributions to evaluation of analytical functions of the moments. This allows an analysis of the risk properties of systems to be carefully attributed between choices of risk function (e.g. VaR vs CVaR); choice of return distribution (power law tail vs Gaussian) and choice of event frequency, for risk assessment. We exploit this to provide a simple method for portfolio optimization when the asset returns follow a standard multivariate T distribution. This may be used as a semi-analytical verification tool for more general optimizers, and for practical assessment of the impact of fat tails on asset allocation for shorter time horizons.