Financial derivatives and Lie symmetries

Abstract

Derivatives in finance have become pervasive in recent decades. Tremendous impetus was given to this aspect of finance by the pioneering papers of Black and Scholes (1973) and Merton (1973) and has attracted considerable interest since. In general the evolution partial differential equations were solved by means of the traditional methods of partial differential equations and by ansatz previously useful in similar contexts. Here we illustrate the benefits of an algorithmic approach using the method of symmetry analysis introduced by Sophus Lie in the nineteenth century. We demonstrate the utility of this analytic approach with several examples chosen from the field of financial mathematics.

Keywords:

• lie symmetries
• financial mathematics
• evolution partial differential equations

Notes

¹ So named after the observation under a microscope by Robert Brown roughly a century and a half before of the apparently random motion of particles of pollen suspended in water (Brown, 1828).

² One would hope that this initial condition would not apply in financial matters! Unfortunately there are some instances of financial instability in which such an initial condition is far too accurate a model. Note that Ibragimov and Wafo's (1997) paper, with more realistic conditions, appeared earlier, but Gazizov and Ibragimov's (1998) paper had already been presented at a seminar in the Department of Physics at the University of the Witwatersrand, Johannesburg in 1996.

³ The solution symmetries, Γ_∞, can play no role in this because their action on u(T, x) = U produces a linear combination of linearly independent solutions.

⁴ Observe that equation (25) is merely equation (6), but where the parameters are functions of time. This model with time-dependent parameters has been considered with a nonalgebraic approach by Hull and White (1990).

⁵ The actual value is immaterial since (25) is linear and so admits a constant rescaling.