Noncommutative Geometry and Stochastic Calculus

category: papers

The present report contains an introduction to some elementary concepts in noncommutative differential geometry. The material extends upon ideas first presented by Dimakis and Mueller-Hoissen. In particular, stochastic calculus and the Ito formula are shown to arise naturally from introducing noncommutativity of functions (0-forms) and differentials (1-forms). The abstract construction allows for the straightforward generalization to lattice theories for the direct implementation of numerical models. As an elementary demonstration of the formalism, the standard Black-Scholes model for option pricing is reformulated.

There was also a short companion paper to this article:

Noncommutative geometry is a relatively new branch of mathematics pioneered by the Fields Medalist Alain Connes $[1]$ during the early ’80s. Since it’s inception, noncommutative geometry has established itself on the forefronts of modern research in mathematics and has been steadily etching a place for itself in theoretical physics. This is particularly true since the appearance of the influential paper by Seiberg and Witten $[2]$, which as of the time of this writing has been referenced nearly a thousand times in just three years according to the preprint archive (http://www.arxiv.org). This marks an amazing explosion of noncommutative geometry onto the scenes of theoretical physics.

The basic idea of noncommutative geometry stems from the observation that there is a deep relation between the collection (commutative algebra) of continuous functions and the (topological) space on which they reside. That is, given the collection of functions, the underlying space could be deduced. Conversely, given the underlying space, the collection of functions could be deduced. This can be summarized via the simple diagram

$COMMUTATIVE ALGEBRA\leftrightarrow SPACE.$

At the risk of over simplifying this absolutely intimidating branch of mathematics, the question that noncommutative geometry set out to answer was, “If the commutative algebra of functions gives rise to a concept of space, would a noncommutative algebra give rise to some kind of noncommutative space?” In other words,

$NONCOMMUTATIVE ALGEBRA\stackrel{?}{\leftrightarrow} NONCOMMUTATIVE SPACE.$

The answer is in the affirmative, and perhaps unsurprisingly, noncommutative spaces play a crucial role in quantum theory.

Since the initial flood of papers on noncommutative geometry *a la* Connes’ original framework, various other flavors have appeared. The noncommutative geometry of interest in the present paper retains the commutativity of functions, i.e.

$(f g)(x) = f(x)g(x) = g(x)f(x) = (g f)(x),$

but noncommutativity is introduced between functions and differentials. For instance, the function $f$ and the differential $d g$ need not commute in general, i.e.

$f(d g)\ne (d g)f.$

Although this is already a very specialized arena of noncommutative geometry, there is quite a vast array of applications that result from this simple extension of the standard calculus. A highly lucid introduction to the noncommutative geometry of commutative algebras with applications in physics is provided by Muller-Hoissen in $[3]$.

A somewhat surprising result presented in $[3]$ is that finite differences appear naturally within this framework even when considering continuum theories. As a consequence, the machinery is highly adapted to numerical work. Furthermore, an even more striking observation in $[3]$ is that noncommutative geometry naturally accommodates a slight generalization of stochastic calculus. Herein lies the applicability to mathematical finance.

For a moment, reflect on the basic class of objects required in order to build financial models such as the Black-Scholes equations. First, there are scalar functions representing the values of options $V$, tradeables $S$, and numeraires $B$, as well as functions for the number of units $\alpha$, $\Delta$, $\beta$, respectively, of each being held in a portfolio of total value $\Pi$. Next, there are the corresponding differentials $d V$, $d S$, $d B$, $d\alpha$, $d \Delta$, $d \beta$, and $d \Pi$. It is helpful to think of differentials as constituting a class of objects separate from that of scalar functions. To make the distinction between scalar functions and differentials as clear as possible, the former will be referred to as 0-forms, while the latter will be referred to as 1-forms. Hence, the Black-Scholes model begins with a collection of 0-forms and 1-forms.

In the standard Black-Scholes model, each of the 1-forms may be expressed in terms of a Wiener process $d W$ and time $d t$. For example, the spot price of a tradeable asset is often modeled via

$d S = S(\sigma d W + \mu d t).$

In this way, 1-forms may be thought of as constituting a two-dimensional vector space with bases $\{d W, d t\}$. The primary algebraic aspect of the stochastic calculus which differs from standard elementary calculus is in how two 1-forms are multiplied. Due to linearity, it suffices to consider the multiplication of basis elements. In stochastic calculus, the multiplication is given by

$d W d W = dt,\quad d W d t = d t d W = 0; d t d t = 0.$

As a result,

$d S d S = \sigma^2 S^2 d t.$

follows directly from the rules for multiplication. One may then derive the Ito formula

$\begin{aligned}
d V
&= \frac{\partial V}{\partial S} d S + \frac{\partial V}{\partial t} d t + \frac{1}{2} \frac{\partial^2 V}{\partial S^2} dS dS \\
&= \frac{\partial V}{\partial S} d S + \left(\frac{\partial V}{\partial t} + \frac{\sigma^2 S^2}{2} \frac{\partial^2 V}{\partial S^2}\right) d t.\end{aligned}$

from which standard self-financing and no-arbitrage arguments lead to the Black-Scholes equations. Note that the only difference between the stochastic calculus and the standard elementary calculus as far as formal algebra manipulations are concerned lies in the way 1-forms are multiplied. Hence, one could argue that the Black-Scholes equation follows from a formal algebraic argument.

The formulation via noncommutative geometry follows a similar formal algebraic approach. First, the spaces of 0-forms and 1-forms are defined as usual. Next, a derivation $d$ is defined that takes a 0-form $f$ and returns the 1-form $d f$ satisfying the product rule

$d(f g) = (d f)g + f(d g).$

Note that this is already different from the stochastic calculus, which does not satisfy the product rule. Furthermore, the order in which the terms are written is important due to the fact that 0-forms and 1-forms no longer commute. Once again, as with the stochastic calculus, the space of 1-forms is spanned by the basis $\{d W, d t\}$. However, instead of defining the product of 1-forms as is done in the stochastic calculus, the commutative relations between 0-forms and 1-forms is specified according to the rules

$[d W,W] = dt,\quad [d W,t] = [d t,W] = 0,\quad [d t,t] = 0,$

where

$[d f,g] \coloneq (d f)g - g(d f)$

is the commutator. It is quite remarkable that the Ito formula may be derived via noncommutative geometry by simply specifying the commutative relations. Once the Ito formula has been derived, the derivation of the Black-Scholes equations follow.

A new feature of noncommutative geometry that is not present in stochastic calculus is the concept of left and right components of 1-forms. A general 1-form may be written in terms of left components

$\alpha = \stackrel{\leftarrow}{\alpha_1} d W + \stackrel{\leftarrow}{\alpha_2} d t$

or in terms of right components

$\alpha = d W \stackrel{\rightarrow}{\alpha_1} + d t\stackrel{\rightarrow}{\alpha_2}.$

In general, the left and right components do not coincide due to the noncommutativity. This fact leads to a novel concept of left and right martingales. As discussed in the report, a right martingale is a process that satisfies the heat equation, while a left martingale is a process that satisfies the time-reversed heat equation. This follows from the fact that there are left and right versions of the Ito formula.

It should be pointed out that the report does not represent the culmination of an intense research effort. Rather, this report represents a humble beginning. The author expects that the framework of noncommutative geometry in mathematical finance will ultimately become just as standard as stochastic calculus. Due to its natural adaptability to numerical modeling, it may even become more prominent. The intention of the report is to expose researchers at this early stage to some of the elementary concepts in noncommutative geometry in the hopes that they will help pick up the reigns and develop new models that further push the envelope in financial modeling.

- A. Connes, Noncommutative Geometry. San Diego: Academic Press, 1994.
- N. Seiberg and E. Witten, “String theory and noncommutative geometry,” JHEP 9909, vol. 032, 1999. (http://www.arxiv.org/abs/hep-th/9908142).
- F. Muller-Hoissen, “Introduction to noncommutative geometry of commutative algebras and applications in physics,” in Proceedings of the 2nd Mexican School on Gravitation and Mathematical Physics, (Konstanz), Science Network Publishing, 1998. (Available online at http://kaluza.physik.uni-konstanz.de/2MS/mh/mh.html) mexico.

Revised on September 29, 2011 at 10:42:53
by
Eric Forgy