The calculation of expectations for classes of diffusion processes by Lie symmetry methods

This paper uses Lie symmetry methods to calculate certain expectations for a large class of It\^{o} diffusions. We show that if the problem has sufficient symmetry, then the problem of computing functionals of the form $E_x(e^{-\lambda X_t-\int_0^tg(X_s) ds})$ can be reduced to evaluating a single integral of known functions. Given a drift $f$ we determine the functions $g$ for which the corresponding functional can be calculated by symmetry. Conversely, given $g$, we can determine precisely those drifts $f$ for which the transition density and the functional may be computed by symmetry. Many examples are presented to illustrate the method.Comment: Published in at http://dx.doi.org/10.1214/08-AAP534 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Exact Pricing and Hedging Formulas of Long Dated Variance Swaps under a 3/23/2 Volatility Model

This paper investigates the pricing and hedging of variance swaps under a $3/2$ volatility model. Explicit pricing and hedging formulas of variance swaps are obtained under the benchmark approach, which only requires the existence of the num\'{e}raire portfolio. The growth optimal portfolio is the num\'{e}raire portfolio and used as num\'{e}raire together with the real world probability measure as pricing measure. This pricing concept provides minimal prices for variance swaps even when an equivalent risk neutral probability measure does not exist.Comment: 23 pages, 5 figure

LIE SYMMETRY APPROACH TO THE CEV MODEL

Abstract: Using a Lie algebraic approach we explicitly provide both the probabilitydensity function of the constant elasticity of variance (CEV) process andthe fundamental solution for the associated pricing equation. In particular wereduce the CEV stochastic differential equation (SDE) to the SDE characterizingthe Cox, Ingersoll and Ross (CIR) model, being the latter easier to treat.The fundamental solution for the CEV pricing equation is then obtained followingtwo methods. We first recover a fundamental solution via the invariantsolution method, while in the second approach we exploit Lie classical result onclassification of linear partial differential equations (PDEs). In particular wefind a map which transforms the pricing equation for the CIR model into anequation of the form v\u3c4 = vyy 12 Ay2 v whose fundamental solution is known. Then,by inversion, we obtain a fundamental solution for the CEV pricing equation

Solving stochastic differential equations with Cartan's exterior differential systems

The aim of this work is to use systematically the symmetries of the (one dimensional) bacward heat equation with potentiel in order to solve certain one dimensional It\^o's stochastic differential equations. The special form of the drift (suggested by quantum mechanical considerations) gives, indeed, access to an algebrico-geometric method due, in essence, to E.Cartan, and called the Method of Isovectors. A V singular at the origin, as well as a one-factor affine model relevant to stochastic finance, are considered as illustrations of the method

Symmetries of stochastic differential equations using Girsanov transformations

Aiming at enlarging the class of symmetries of an SDE, we introduce a family of stochastic transformations able to change also the underlying probability measure exploiting Girsanov Theorem and we provide new determining equations for the infinitesimal symmetries of the SDE. The well-defined subset of the previous class of measure transformations given by Doob transformations allows us to recover all the Lie point symmetries of the Kolmogorov equation associated with the SDE. This gives the first stochastic interpretation of all the deterministic symmetries of the Kolmogorov equation. The general theory is applied to some relevant stochastic models