On the generation of arbitrage-free stock price models using Lie symmetry analysis

AbstractIn Bell and Stelljes (2009) a scheme for constructing explicitly solvable arbitrage-free models for stock prices is proposed. Under this scheme solutions of a second-order (1+1)-partial differential equation, containing a rational parameter p drawn from the interval [1/2,1], are used to generate arbitrage-free models of the stock price. In this paper Lie symmetry analysis is employed to propose candidate models for arbitrage-free stock prices. For all values of p, many solutions of the determining partial differential equation are constructed algorithmically using routines of Lie symmetry analysis. As such the present study significantly extends the work by Bell and Stelljes who found only two arbitrage-free models based on two simple solutions of the determining equation, corresponding to p=1/2 and p=1

Some Remarks on the Solution of Linearisable Second-Order Ordinary Differential Equations via Point Transformations

Transformations of differential equations to other equivalent equations play a central role in many routines for solving intricate equations. A class of differential equations that are particularly amenable to solution techniques based on such transformations is the class of linearisable second-order ordinary differential equations (ODEs). There are various characterisations of such ODEs. We exploit a particular characterisation and the expanded Lie group method to construct a generic solution for all linearisable second-order ODEs. The general solution of any given equation from this class is then easily obtainable from the generic solution through a point transformation constructed using only two suitably chosen symmetries of the equation. We illustrate the approach with three examples

On the Derivation of Nonclassical Symmetries of the Black–Scholes Equation via an Equivalence Transformation

The nonclassical symmetries method is a powerful extension of the classical symmetries method for finding exact solutions of differential equations. Through this method, one is able to arrive at new exact solutions of a given differential equation, i.e., solutions that are not obtainable directly as invariant solutions from classical symmetries of the equation. The challenge with the nonclassical symmetries method, however, is that governing equations for the admitted nonclassical symmetries are typically coupled and nonlinear and therefore difficult to solve. In instances where a given equation is related to a simpler one via an equivalent transformation, we propose that nonclassical symmetries of the given equation may be obtained by transforming nonclassical symmetries of the simpler equation using the equivalence transformation. This is what we illustrate in this paper. We construct four nontrivial nonclassical symmetries of the Black–Scholes equation by transforming nonclassical symmetries of the heat equation. For completeness, we also construct invariant solutions of the Black–Scholes equation associated with the determined nonclassical symmetries