Joanna Goard

Stochastic Volatility Models and the Pricing of VIX Options

In this paper, we examine and compare the performance of a variety of continuous‐time volatility models in their ability to capture the behavior of the VIX. The “3/2‐ model” with a diffusion structure which allows the volatility of volatility changes to be highly sensitive to the actual level of volatility is found to outperform all other popular models tested. Analytic solutions for option prices on the VIX under the 3/2‐model are developed and then used to calibrate at‐the‐money market option prices.

New solutions to the bond-pricing equation via Lie's classical method

Using Lies classical method of group invariants, we provide new and simple solutions to the bond-pricing partial differential equation. All of our solutions satisfy the final condition for the bond price. In finding these solutions, we used one-factor interest-rate modelling, in which our short-term interest rate follows a random walk which allows the volatility of interest rate changes to be highly sensitive to the level of riskless rate, and the market price of risk is arbitrary. As well, in one of our simple solutions, the nonlinear drift of our risk-neutral interest rate contains an arbitrary function of time, which may be freely chosen.

The method of generalised conditional symmetries and its various implementations

We relate Kaptsovs method of B-determining equations for finding invariant solutions of PDEs to the nonclassical method and to the method of generalised conditional symmetries. An extension of Kaptsovs method is then used to find new solutions of degenerate diffusion equations.

Noninvariant Boundary Conditions

It is generally believed that in order to solve initial and boundary value problems using Lie symmetry methods, the boundary and initial conditions need to be left invariant by the infinitesimal symmetry generator which admits the invariant solution. In this article we give less restrictive conditions on the imposed initial and boundary values in order that they be recoverable with a particular symmetry generator.

Symmetry Reductions of Equations for Solute Transport in Soil

Solute transport in saturated soil is represented by anonlinear system consisting of a Fokker–Planck equation coupled toLaplaces equation. Symmetries, reductions and exact solutions are foundfor two dimensional transient solute transport through some nontrivialwedge and spiral steady water flow fields. In particular, the mostgeneral complex velocity potential is determined, such that the soluteequation admits a stretching group of transformations that wouldnormally be possessed by a point source solution.

Generalised conditional symmetries and Nucci's method of iterating the nonclassical symmetries method

Abstract It is shown that Nuccis method of constructing heir equations by iterating the non-classical symmetries method is equivalent to the generalised conditional symmetries method.

Finding Symmetries by incorporating initial conditions as side conditions

It is generally believed that in order to solve initial value problems using Lie symmetry methods, the initial condition needs to be left invariant by the infinitesimal symmetry generator that admits the invariant solution. This is not so. In this paper we incorporate the imposed initial value as a side condition to find ‘infinitesimals’ from which solutions satisfying the initial value can be recovered, along with the corresponding symmetry generator.

Tractable forms of the bond pricing equation

So far, a small number of analytically tractable single-factor models have been devised for the well-known bond pricing equation (BPE). In this paper, new tractable models are formulated in a systematic manner. First, the BPE is transformed to a standard canonical form in which only one coefficient function appears. In some interesting cases, this single coefficient function is identically zero, leaving nothing more to solve than the classical heat equation. In many cases, the canonical form allows a general solution by standard mathematical techniques such as separation of variables and Laplace transforms. In other cases, the general solution of the BPE is reduced to a single inverse Laplace transform.

Comparison of the performance of a time-dependent short-interest rate model with time-independent models

The coefficients in the stochastic differential equation that the short interest rate follows are of vital importance in the subsequent modelling of bond prices and other interest rate products. Empirical tests have previously been performed by various authors who compare a variety of popular short‐rate models. Most recently, Ahn and Gao compared their model with affine‐drift models and showed that their model with a non‐linear drift function outperforms the others. This paper compares the model developed by Goard, which is a time‐dependent generalization of the Ahn–Gao model, with the Ahn–Gao model itself. It is found that the time‐dependent model using a second‐order Fourier series in time, outperforms the Ahn–Gao model for all data sets considered.

Exact Solutions for a strike reset put option and a shout call option

Abstract A strike reset put option is a European put option with the added feature that it gives the holder the right to ‘shout’ during the life of the option and thus reset the strike to the then prevailing spot price. The related shout call option is similar to the European call option but has the added feature that the holder can ‘shout’ during the life of the contract to both lock in a profit and reset the strike. This paper derives exact solutions to both of these exotic options.