An Introduction to Theta-Calculus

An Introduction to Theta-Calculus

Theta-calculus is a mathematical calculus for the description of sequential processes, financial contracts and multiperiod strategies in game theory. This calculus allows the explicit notation of all trading strategies and financial products that can currently not be written in an explicit mathematical form. All kinds of contracts, strategies and multiperiod games can then be captured in terms of their quantitative implications by a vocabulary of three basic effects: waiting, transacting and deciding. Each elementary activity is represented by an operator that can be interpreted in an operator sequence as a chronologically ordered list of events, and the operator term also represents an explicit formula for the evaluation of the respective strategy's final result. Theta-calculus is especially useful for the notation of financial products, many of which can currently not be represented explicitly.

Quantitative finance is one of the most actively researched scientific fields dealing with processes. Many aspects of stochastic and deterministic processes as well as decision and game theories are found in finance. Despite a vast background of mathematical theory and concepts, prevailing financial calculus is unable to formalize one of the most fundamental aspects of trading and market analysis: financial calculus lacks an explicit mathematical notation for human trading activities and many financial products. Starting with the American option, everything that requires non-trivial intertemporal decisions or optimizations cannot be represented in a reasonable form from which the product evaluation can be derived algebraically.

Elaborate contract types and investment objectives are typically specified in prose form only and typically use specific terminology that is hard to interpret by an uninvolved reader. Such representations are difficult to evaluate mathematically and have to be translated into formulas and computer code individually. Those who feel inclined to pursue greater generality are mostly struck by an inflation of parameters and mathematical concepts.

To meet the industry demand for a technical portfolio representation, there has been development in the extension of existing programming dialects. The most notable results are MLFi, based on the functional programming language Caml, and the XML standard FpML. However, neither provides a mathematical framework for the derivation of theoretical properties: their vocabulary is huge and still being extended, and there is a large distance from product representation to the evaluation procedure, which raises the fear of model ambiguity and inconsistency.