[MAS R-9914] Prices of tradables can only be expressed relative to eachother at any instant of time. This fundamental fact should thereforealso hold for contingent claims, i.e. tradable instruments, whoseprices depend on the prices of other tradables. We show that thisproperty induces local scale-invariance in the problem of pricingcontingent claims. Due to this symmetry we do {it not/} require anymartingale techniques to arrive at the price of a claim. If thetradables are driven by Brownian motion, we find, in a natural way,that this price satisfies a PDE. Both possess a manifestgauge-invariance. A unique solution can only be given when we imposerestrictions on the drifts and volatilities of the tradables, i.e.the underlying market structure. We give some examples of theapplication of this PDE to the pricing of claims. In the Black-Scholesworld we show the equivalence of our formulation with the standardapproach. It is stressed that the formulation in terms of tradablesleads to a significant conceptual simplification of thepricing-problem.#[MAS R-9919] This article is the second one in a series on the use of scaling invariance in finance. In the first paper, we introduced a new formalism for the pricing of derivative securities, which focusses on tradable objects only, and which completely avoids the use of martingale techniques. In this article we show the use of the formalism in the context of path-dependent options. We derive compact and intuitive formulae for the prices of a whole range of well known options such as arithmetic and geometric average options, barriers, rebates and lookback options. Some of these have not appeared in the literature before. For example, we find rather elegant formulae for double barrier options with moving barriers, continuous dividends and all possible configurations of the barriers. The strength of the formalism reveals itself in the ease with which these prices can be derived. This allowed us to pinpoint some mistakes regarding geometric mean options, which frequently appear in the literature. Furthermore, symmetries such as put-call transformations appear in a natural way within the framework.

Price theory and market structure (msc 91B24), Stochastic ordinary differential equations (msc 60H10), Heat and other parabolic equation methods (msc 58J35), Invariance and symmetry properties (msc 58J70)
Modelling, Analysis and Simulation [MAS]

Hoogland, J.K, & Neumann, C.D.D. (1999). Scaling invariance and contingent claim pricing. Modelling, Analysis and Simulation [MAS]. CWI.