Gauge Invariance, Geometry and Arbitrage
In this work, we identify the most general measure of arbitrage for any market model governed by Itô processes. We show that our arbitrage measure is invariant under changes of numéraire and equivalent probability. Moreover, such measure has a geometrical interpretation as a gauge connection. The connection has zero curvature if and only if there is no arbitrage. We prove an extension of the Martingale pricing theorem in the case of arbitrage. In our case, the present value of any traded asset is given by the expectation of future cash-flows discounted by a line integral of the gauge connection. We develop simple strategies to measure arbitrage using both simulated and real market data. We find that, within our limited data sample, the market is efficient at time horizons of one day or longer. However, we provide strong evidence for non-zero arbitrage in high frequency intraday data. Such events seem to have a decay time of the order of one minute.
The no-arbitrage principle is the cornerstone of modern financial mathematics. Put it simply, an arbitrage opportunity allows an agent to make a non-risky profit with zero or negative net investment (see [DeSc08]). Under the no-arbitrage assumption111In continuous time the no-arbitrage condition (NA) has to be sharpened to the no-free-lunch-with-vanishing-risk condition (NFLVR). Only under the (NFLVR) the fundamental theorem of asset pricing can be proved, see [DeSc08]. Therefore, in this paper, when referring to (NA) in continuous time, (NFLVR) is meant. one can assign in a complete market a unique price to the derivative of any traded assets using the replicating portfolio method (see [MuRu07] or [CvZa04]). The no-arbitrage principle can also be shown to be equivalent to a weaker form of economic equilibrium (cf. [CvZa04]) and can therefore be seen as a form of market efficiency (see [Fam98], [Ma03]). It is then not surprising that most financial and economic literature is based on the no-arbitrage assumption.
Nevertheless, the no-arbitrage principle represents a very strong assumption about market dynamics which must be tested empirically. Even when the market participants use no-arbitrage models, the ultimate price of any security which is traded in a centralized market is set by supply/demand and the complex dynamics of the order book. That is, “the market” sets the prices. It then makes sense to ask: how efficient are these final prices? In order to answer this question one needs a measure of arbitrage.
There is a large body of empirical studies on financial arbitrage. Most of these studies focus on measuring the “excess return” of particular trading strategies (see e.g. [JeTi93], [GaGoRo06]). Other studies try to find violations to general no-arbitrage relations between option prices (see e.g. [AcTi99]). However, there does not seem to be a consensus on whether the reported market “anomalies” are due to arbitrage, or simply to random fluctuations (see [Ma03], [Fam98]). Part of the problem is that there seems to be no general measure of arbitrage which can be applied to any traded asset. One of the main goals of this paper is to define such measure.
The second goal of this paper is to provide a geometrical interpretation of the arbitrage measure. In particular, it has been speculated long ago by Ilinski [Il97], [Il01] and Young [Y099] that arbitrage should be viewed as the “curvature” of a gauge connection, in analogy to some physical theories. The fact that gauge theories are the natural language to describe economics was first proposed by Malaney and Weinstein in the context of the economic index problem [Ma96], [We06]. The need for such mathematical language can easily be seen from the fact that prices are only relational. More precisely, let be the price vector of all goods in the economy at time , in some common unit (say USD). Since the measuring units are arbitrary, fundamental economic laws must be invariant under the transformations
where is a positive stochastic factor222For example, can be the EUR/USD exchange rate.. In physics, a (local) transformation such as Eq. (1) is known as a gauge transformation. These represents a redundancy of our description of the economic system. The laws of the economy should be gauge invariant. The need for a gauge theoretical approach to economics was highlighted recently in [Sm09]. The role of gauge invariance in option pricing has been studied in [HoNe99a], [HoNe99b], [HoNe00] and [HoNeVe01]. For an unrelated use of differential geometric methods in (no-arbitrage) option pricing see [La08]. A recent proposal for a gauge connection in finance, and its relation to arbitrage, was presented in [Far08], [Far09a] and [Far09b]. In fact, we will show that the curvature of the gauge connection proposed in [Far08], [Far09a] and [Far09b] is equivalent to our arbitrage measure.
In physics, curvature is a gauge invariant measure of the path dependency of some physical process. For example, readers familiar with electrodynamics might recall the vector potential , where label the space-time directions. In differential geometry for theoretical physics, is known as a gauge connection. Now consider a charged particle which is traveling along some trajectory in space-time , . The interaction of this particle with the gauge potential is proportional to the line integral, . Now suppose that we make an infinitesimal change in the path of the particle , keeping the boundary conditions fixed: . The interaction changes by , where is known as the curvature of . Therefore, we see that for zero curvature , the line integral is independent of the path . Moreover, note that the curvature is invariant under a gauge transformation of the form , where is any function of space-time.
We will find a very similar construction in the case of mathematical finance. In particular, we will show that the arbitrage curvature defined in this paper measures the path dependency of the present value of a self-financing portfolio of traded assets with fixed final payoff. The no-arbitrage principle is then equivalent to a zero-curvature condition. In analogy with the electromagnetic curvature , we expect that any measure of arbitrage should be invariant under the gauge transformation in Eq. (1). Moreover, the fundamental theorem of asset pricing states that the no-arbitrage principle is equivalent to the existence of a probability measure with respect to which asset prices expressed in terms of a numéraire are Martingales [MuRu07]. Therefore, we expect that any measure of arbitrage should also be invariant under a change of probability. These are, in fact, two very important properties that will characterize our arbitrage measure.
This paper takes a “macroscopic” or phenomenological approach to arbitrage. More precisely, we will study arbitrage from the point of view of general stochastic models. We do not address the question of what is causing the arbitrage. Our main goal is to identify the gauge invariant financial observables that indicate an arbitrage opportunity. We believe that a proper understanding of such quantity, including its geometrical interpretation, is a first step towards a theory of non-equilibrium economics. Moreover, our methods can be applied to the construction of profitable trading strategies. Our main assumption is that the prices of all financial instruments can be described by Itô processes. Moreover, we ignore transaction costs333Note that, as pointed out in [ShVi97], many possible arbitrage opportunities disappear once one takes into account market frictions such as transaction costs. Therefore, it is important to keep in mind that even when we measure a non-trivial curvature in the market, it does not mean that it can always be exploited in a practical trading strategy..
The organization and main results of the paper are the following. In section 2 we present the class of models that we use in the rest of the paper. They are very general and include the case of stocks, bonds and commodities, and more complicated derivative products. We decompose the dynamics of these models in terms of their gauge transformation properties with respect to Eq. (1). We identify the gauge invariants and show that they represent an obstruction to the existence of a Martingale probability measure. We conclude section 2 with an example with three assets, and we derive a modified non-linear Black Scholes equation with arbitrage.
In section 3, we give a geometrical interpretation to the gauge invariant quantities defined in section 2. Our main goal is to identify the stochastic gauge invariants of section 2 with the curvature of a gauge connection. We begin with a review of the Malaney-Weinstein connection [Ma96], [We06], which is done in the context of differentiable economic paths. In section 3.1, we generalize the Malaney-Weinstein construction to stochastic processes and prove an asset pricing theorem. The main result of this section is that the present value of any self-financing portfolio of traded assets is given by the conditional expectation of future cash-flows, discounted by a line integral of the Malaney-Weinstein gauge connection. We show how the value functions of different portfolios replicating the same contingent claim are related to the arbitrage curvature. Finally, we show that the gauge connection recently proposed in [Far08], [Far09a] and [Far09b], is equivalent to the present construction. Readers interested in only the arbitrage measure and the detection techniques, can skip section 3. None of the results of section 4 require an understanding of the geometry of arbitrage. In section 4 we develop a simple algorithm to measure the arbitrage curvature using financial data. We explain the main sources of error in such measurement. The algorithm is applicable to any financial instrument. We provide examples with financial data of stock indexes and index futures. We find that, on long time scales, the market is very efficient. However, we provide strong evidence for non-zero curvature fluctuations at short time scales in the order of one minute. We conclude in section 5.
2 Stochastic Models and Gauge Invariance
Let be the set of all traded securities at any point in time in the market. We will use greek indices to label members of the set . We denote the price of security by . Our main assumption is that the dynamics of all such securities is described by Itô processes of the form [So06] and [Sh00]:
where are standard Brownian motion such that are independently and normally distributed random variables with,
We make no assumptions about the number of Brownian terms, and hence the completeness of the market. The set of Brownian motions represent all the randomness in the market. Therefore, they induce a natural filtration for our probability space. The coefficients and can also be stochastic processes adapted444 In simple terms, a process adapted to means that it does not depend on future values of the Brownian motion. In other words, can only depend on up to time . to the filtration . However, they are assumed to satisfy suitable conditions to ensure the existence of the price processes (see [LaLa06]). This class of models is very general and includes stocks, bonds, options, etc. Moreover, the case of fat tails in the distribution of returns is also included, since this is known to be generated by stochastic volatilities .
Looking back at Eq. (2), we can see that the tangent space has a natural decomposition into the directions which contain all the randomness () and those orthogonal to it. Therefore, we will make the following decomposition of the drift term in Eq. (2):
where is the space spanned by basis vectors such that
We will refer to as the null space of the market. Note that this space is orthogonal to all the randomness in the tangent space . However, we need to remember that might be trivial. The definition of in Eq. (4) is not unique if the vectors are linearly dependent. This is the case of, for example, an incomplete market with more Brownian motions than traded securities. Moreover, is unique up to rotations in the null space. As we will see, the quantities are the unique gauge invariant measures of arbitrage. The two main goals of this paper are to give a geometric interpretation to the parameters , and to set up a procedure to measure them using financial data.
Since prices are relative and only reflect an exchange rate between two products, the units used to measure are arbitrary. Therefore, the dynamics of the market must be invariant under a change of measuring units. In mathematical finance, this is known as a change of numéraire [MuRu07], and it can be interpreted as a gauge transformation,
where is a positive stochastic process which is adapted to the filtration . Another symmetry, which is special to the particular models of Eq. (2), is a transformation of the probability measure. This is not really a gauge symmetry, but corresponds rather to a change of variables of the form .
Proposition 2.1: Consider a change of numéraire of the form , where is a positive stochastic process adapted to the filtration , and . Then, the coefficients of the Itô processes, Eqs. (2) and (4), transform as
Finally, under a transformation of the probability measure given by the Radon-Nykodým derivative , we have the mapping of standard Brownian motions
In particular, it follows that , and are invariant under such transformations.
Proof: The result in Eq. (8) above follow from a simple application of Itô rule to the product :
where we defined . Note that both and are unchanged by these gauge transformations. In particular, suppose that . Then, it follows from Eq. (5) that .
So far we have taken the existence of the basis vectors for granted. A constructive procedure to find such basis, if non-trivial, is given by the following proposition.
Proposition 2.2: Let be the symmetric and real matrix with component , where . Moreover, define as the matrix of all ones, e.g. , . Then, the matrix defined by
is gauge invariant. Let be the null space of matrix such that for any non-trivial . Then . In particular, the space is spanned by the orthonormal zero-modes of which are orthogonal to the vector .
Proof: First we need to prove that the space of vectors such that , , is in one-to-one correspondence with the zero modes of : . Obviously, if , it follows that is also a zero-mode of . To prove the converse, suppose that , but , where for at least one value of . Then, , which can only be true if .
Now we turn our attention to the matrix , defined in Eq. (13). Using the gauge transformation , we can see that transforms as
It is then straightforward to verify the gauge invariance of the matrix . Next we recall that the space is spanned by (non-trivial) orthonormal zero-modes of such that they also satisfy . One can define a similar space for . It is easy to verify that any vector is also a vector in . On the other hand, for any vector , it follows from Eq. (13) that . Thus, we have proven that .
It is easy to verify that . Therefore, the vector is a particular zero-mode of G. Now take any other zero-mode of , call it , which is orthogonal to . It follows that, . Therefore, . This completes the proof.
So far we have talked about the full set of securities of the market. However, it is clear that the decomposition in Eq. (4) can be done for any subset of the market. That is, suppose we observe a subset of the prices , . Moreover, suppose we find that within this subset one still can find some zero-modes obeying, , and . One can then easily lift these vectors to the full set by taking . This represents a particular choice of basis in the null space . By observing a sub-sector of the market, we will only have access to some of the components of . For notational convenience, we will not distinguish between the full market and a subset of it in what follows.
Under the no-arbitrage assumption (see [MuRu07], [CvZa04]), it is always possible to find a common positive discount factor and an equivalent probability measure such that the discounted prices are martingales: , where . This is known as the Martingale representation theorem (see [So06] and [Sh00]). In our language, this means that there is a gauge transformation mapping to -Martingales. In other words, if there is no arbitrage, price processes are gauge-equivalent to -Martingales for some probability measure . The result of the Martingale representation theorem can only be obtained if one is able to write for some adapted process . The reason is that the stochastic integral is a Martingale: . By proposition 2.2 there is neither a change of probability nor a choice of a positive discount factor for which the vector is mapped to (in contrast to and all s which can indeed be made vanish). Therefore, it is easy to see that the term parametrizes the obstruction to the existence of a Martingale probability measure for any discounted price process .
As s are gauge invariant quantities, one expects that they should be observables. In the next section we will relate this quantity to a gauge connection and its curvature. In section 4 we will show that such quantity can indeed be observed, and we explain simple strategies to measure it. Before concluding this section, it is instructive to a study particular example with three assets.
2.1 An Example
Consider the case of three assets , , where is a savings account and are some other risky assets. All prices are measured in the same common units. We will assume only one Brownian motion. Therefore, the dynamics of the prices is described by
For later convenience, we assume that the interest rate is deterministic. In order to do the decomposition in Eq. (4) we need to find a basis for the null space . In this case, since there is only one Brownian motion and two risky assets, there will be only one null direction. To calculate it, we start by identifying the matrix:
One can now construct the matrix using Eq. (13). The explicit form of is not very illuminating. The unormalized eigenvectors of are found to be
In order to find a basis for the null space defined in Eq. (5), we need to project or into the space orthogonal to the vector . To do this, we define the projection matrix
where is the all-ones matrix. Note that projects into the space orthogonal to . Our choice for the normalized null vector is then
It is easy to verify that obeys the properties given in Eq. (5).
For , Eqs. (22) - (24) reduce to the familiar no-arbitrage Black-Scholes dynamics. As usual, is interpreted as the market price of risk. Note that in this example, both risky assets are exposed to the same market risk factor . The volatility measures the coupling to such risk. Under the no-arbitrage assumption, both assets should give the same expected return per unit of risk. This is . However, we see that if , and have different expected returns, even when they are exposed to the same risk. This discloses an arbitrage opportunity.
There is a very interesting consequence of Eqs. (22) - (24) when is any function of , e.g. an option. For simplicity, consider the case where the only time dependence in is of the form , where is a differentiable function of and . Moreover, the interest rate is assumed to be deterministic. In this case one can derive a non-linear version of the Black Scholes equation with arbitrage. For ease of notation let . Under our assumptions we will have that . Then, using Itô rule we find
where we identify
where from Eq. (23)
Therefore, after some algebra Eq. (27) becomes a modified non-linear Black Scholes partial differential equation:
Note that for this reduces to the familiar Black Scholes equation. The non-linear Black Scholes equation is a special case of the more general pricing theorem presented in section 3.
It is important to remember that the arbitrage parameter, in Eq. (29), can in general depend on time and the stock price. Therefore, in principle almost any deformation of the option price is possible. It follows that the Eq. (29) can be solved only if the arbitrage dynamics is known. For example, consider the case where we set
for some constant . Then, the option price obeys the usual Black-Scholes equation but with the “wrong” volatility:
This is a simple example of the well-known volatility arbitrage.
3 The Gauge Connection
The application of differential geometric ideas in economics can be traced to the work of Malaney and Weinstein ([Ma96], [We06]). It was found that the solution to the apparent discrepancy among different economic growth indices could be solved by the appropriate choice of a covariant derivative. Such derivative has the property that a self-financing basket of goods is seen as “constant”. More technically, a self-financing basket is interpreted as being “parallel transported” along a one dimensional curve in the a base manifold spanned by prices and portfolio nominals. Then, there is a natural geometric index to measure the growth of such basket, which was shown to be identical to the so-called Divisa Index. It is very illuminating to review this construction to gain intuition about the relation between arbitrage and curvature. In what follows, all quantities are assumed to be deterministic and differentiable. We will return to the stochastic case in the next subsection.
where are the portofolio nominals, , and the base space is parametrized by the coordinates . Note that under a change of numéraire , the connection transforms as
This is the analog of the transformation rule of the vector potential in electrodynamics.
A self-financing portfolio can be seen as being parallel transported with the connection as
where is the covariant derivative along the trajectory . The solution to this equation is simply,
where is a particular self-financing trajectory , , and is known as the Divisa Index.
The dependence of on the choice of curve is parametrized by the curvature of the gauge connection, which is given by
Note that the curvature is invariant under a gauge transformation, as . In the approximation where economic agents are price takers, the price trajectory is given exogenously, and we are only allowed to make changes in the portfolio nominals . In other words, we can write in Eq. (36). One can then restrict the curvature to the submanifold corresponding to the coordinates. The induced curvature in this submanifold is given by
In this case, the path dependency of the Divisa Index, Eq. (35), can be written as,
where represents a variation to the trajectory of the portfolio nominals. Therefore, we see that Eq. (35) is independent on the path only if the price trajectories obey the zero-curvature condition , . The zero-curvature condition implies that the prices of all securities evolve by the same common inflation factor.
The relation between curvature and arbitrage goes as follow. Suppose that the prices obeyed the zero-curvature condition given above. It follows that, for any self-financing portfolio, we have
for . In particular, if it follows that . Therefore, it is not possible to make wealth without a positive initial investment. On the other hand, suppose that the curvature is not zero. Consider two portfolio trajectories and such that, say at some time , for the same initial wealth . Now construct the difference portfolio with nominals and wealth function
Then, at time we have
while . In other words, we have made wealth out of nothing. In the next section we show how this construction carries over to the stochastic case.
3.1 The Stochastic Gauge Connection
In the previous section we illustrated the relation between curvature, path dependency and arbitrage, using the Malaney-Weinstein connection. However, this construction only works for differentiable economic trajectories in the base space . Nevertheless, we have found a direct analog of the Malaney-Weinstein connection for Itô processes, which we summarize in the following theorem. In order to avoid technical complications, we restrict our attention to an economy on a finite interval of time .
Theorem 3.1: Consider any self-financing portfolio , so that . Then, there exist a (non-unique) equivalent probability measure under which the price processes obey
Moreover, the present value of given some final payoff , , is given by
where is some self-financing trajectory, and is given by the expectation of the Malaney-Weinstein connection,
Finally, the path dependency of the present value of the portfolio, with fixed final payoff, is parametrized by
where are the components of the curvature two-form defined in the reduced base space ,
Proof: We start by writing the portfolio return as
Now consider the combination , where we take (c.f. Eq. (2))
A simple application of Itô rule gives
A further application of Itô rule gives
where is defined in Eq. (44), with .
Now consider making a change of probability measure such that . It is easy to see that, under , the price processes will obey Eq. (42) of the theorem. Moreover, the Radon-Nykodým derivative is given by
In order to prove that can be written as an expectation of the Malaney-Weinstein connection, we recall that
The last result of the theorem, Eq. (45), follows simply by making a small change in the portfolio nominals, and keeping the boundary conditions on fixed.
Note that the curvature of is zero if and only if , which is equivalent to the no-arbitrage condition. Moreover, the probability measure might not be unique, as the choice of in general is not. This also implies that is not unique in general.
A special case of a self-financing portfolio is a
portfolio containing just one base asset.
Corollary 3.2: For all assets in the market model
In particular, under the no-arbitrage assumption , we recover the classic Martingale pricing theorem:
In section 2.1 we derived a modified Black-Scholes equation for the case of three assets. Now we can use the result of Corollary 3.2 to prove a generalization of such equation. Consider the following vector of assets
We will label the components of this vector by , . Moreover, we assume that are smooth functions of the vector of underlying prices, , and describes a savings account with deterministic interest rate . The functions describe a set of European-style contingent claims with final payoff , for some fixed . Finally, we need to assume that are either deterministic or some function of the underlying prices . These assumptions ensure that the expectation values in the RHS of Eq. (56) are functions of and only, and so our assumption, , is self-consistent. Under these assumptions we can prove the following corollary.
Corollary 3.3 (Modified Black-Scholes Equation): Under the assumptions given above, Corollary 3.2 implies that the European-style contingent claims , , obey the non-linear Black-Scholes equations