Geometric Arbitrage Theory and Market Dynamics

# Geometric Arbitrage Theory and Market Dynamics

Simone Farinelli
Core Dynamics GmbH
Scheuchzerstrasse 43
CH-8006 Zurich
Email: simone@coredynamics.ch
###### Abstract

We have embedded the classical theory of stochastic finance into a differential geometric framework called Geometric Arbitrage Theory and show that it is possible to:

• Write arbitrage as curvature of a principal fibre bundle.

• Parameterize arbitrage strategies by its holonomy.

• Give the Fundamental Theorem of Asset Pricing a differential homotopic characterization.

• Characterize Geometric Arbitrage Theory by five principles and show they are consistent with the classical theory of stochastic finance.

• Derive for a closed market the equilibrium solution for market portfolio and dynamics in the cases where:

• Arbitrage is allowed but minimized.

• Arbitrage is not allowed.

• Prove that the no-free-lunch-with-vanishing-risk condition implies the zero curvature condition. The converse is in general not true and additionally requires the Novikov condition for the instantaneous Sharpe Ratio to be satisfied.

\setstretch

1.5

## 1 Introduction

This paper develops a conceptual structure - called Geometric Arbitrage Theory or GAT - embedding the classical stochastic finance into a stochastic differential geometric framework. The main contribution of this approach consists of modelling markets made of basic financial instruments together with their term structures as principal fibre bundles. Financial features of this market - like no arbitrage and equilibrium - are then characterized in terms of standard differential geometric constructions - like curvature - associated to a natural connection in this fibre bundle or to a stochastic Lagrangian structure that can be associated to it.
Several research areas can benefit from the GAT approach:

• Risk management, with the development of a consistent scenario generators reducing the complexity of the market, while maintaining the fundamental connections between financial instruments and allowing for a reconciliation of econometric forecasting with SDEs techniques. See Smith and Speed ([SmSp98]).

• Pricing, hedging and statistical arbitrage, with the development of generalized Black-Scholes equations accounting for arbitrage and the computation of positive arbitrage strategies in intraday markets. See Farinelli and Vazquez ([FaVa12]) for a practical application leading to an almost one probability growth portfolios with real assets.

Principal fibre bundle theory has been heavily exploited in theoretical physics as the language in which laws of nature can be best formulated by providing an invariant framework to describe physical systems and their dynamics. These ideas can be carried over to mathematical finance and economics. A market is a financial-economic system that can be described by an appropriate principle fibre bundle. A principle like the invariance of market laws under change of numéraire can be seen then as gauge invariance. 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]). Ilinski (see [Il00] and [Il01]) and Young ([Yo99]) proposed to view arbitrage as the curvature of a gauge connection, in analogy to some physical theories. Independently, Cliff and Speed ([SmSp98]) further developed Flesaker and Hughston seminal work ([FlHu96]) and utilized techniques from differential geometry (indirectly mentioned by allusive wording) to reduce the complexity of asset models before stochastic modelling. Perhaps due to its borderline nature lying at the intersection between stochastic finance and differential geometry, there was almost no further mathematical research, and the subject, unfairly considered as an exotic topic, remained confined to econophysics, (see [FeJi07], [Mo09] and [DuFiMu00]). We would like to demonstrate that Geometric Arbitrage Theory can be given a rigorous mathematical background and can bring new insights to mathematical finance by looking at the same concepts from a different perspective. That for we will utilize the formal background of stochastic differential geometry as in Schwartz ([Schw80]), Elworthy ([El82]), Eméry ([Em89]), Hackenbroch and Thalmaier ([HaTh94]), Stroock ([St00]) and Hsu ([Hs02]).

This paper is structured as follows. In section 2, after an introductory review of classical stochastic finance, the primitives of Geometric Arbitrage Theory are explained. Section 3 develops the foundations of GAT, allowing an interpretation of arbitrage as curvature of a principal fibre bundle representing the market and defining the quantity of arbitrage associated to a market or to a self-financing strategy. The no-free-lunch-with-vanishing-risk (or NFLVR for short) condition implies the vanishing of the curvature. The converse is in general not true and additionally requires the instantaneous Sharpe Ratio for the asset value dynamics to satisfy the Novikov condition. The NFLVR condition has the interpretation of a continuity equation satisfied by value density and current of the market, as fluid density and current in the hydrodynamics of an incompressible flow. If all market agents follow the principle of expected utility maximization, then the curvature vanishes and viceversa. Section 4 provides a guiding example for a market whose asset prices are Itô processes. In section 5 the connections between mathematical finance and differential topology are analyzed. Homotopic equivalent self-financing arbitrage strategies can be parameterized by the Lie algebra of the holonomy group of the principal fibre bundle. The no-free-lunch-with-vanishing-risk condition is seen to be equivalent to the triviality of the holonomy group or to the triviality of the homotopy group. This is a differential-homotopic formulation of the Fundamental Theorem of Asset Pricing. In section 6 we express the market model in terms of a stochastic Lagrangian system, whose dynamics is given by the stochastic Euler-Lagrange Equations. Symmetries of the Lagrange function can be utilized to derive first integrals of the dynamics by means of the stochastic version of Nöther’s Theorem. Equilibrium and non-equilibrium solutions are explicitly computed. Section 7 concludes.

## 2 Geometric Arbitrage Theory Fundamentals

### 2.1 The Classical Market Model

In this subsection we will summarize the classical set up, which will be rephrased in section (3) in differential geometric terms. We basically follow [HuKe04] and the ultimate reference [DeSc08].

We assume continuous time trading and that the set of trading dates is . This assumption is general enough to embed the cases of finite and infinite discrete times as well as the one with a finite horizon in continuous time. Note that while it is true that in the real world trading occurs at discrete times only, these are not known a priori and can be virtually any points in the time continuum. This motivates the technical effort of continuous time stochastic finance.

The uncertainty is modelled by a filtered probability space , where is the statistical (physical) probability measure, an increasing family of sub--algebras of and is a probability space. The filtration is assumed to satisfy the usual conditions, that is

• right continuity: for all .

• contains all null sets of .

The market consists of finitely many assets indexed by , whose nominal prices are given by the vector valued semimartingale denoted by adapted to the filtration . The stochastic process describes the price at time of the th asset in terms of unit of cash at time . More precisely, we assume the existence of a th asset, the cash, a strictly positive semimartingale, which evolves according to , where the predictable semimartingale represents the continuous interest rate provided by the cash account: one always knows in advance what the interest rate on the own bank account is, but this can change from time to time. The cash account is therefore considered the locally risk less asset in contrast to the other assets, the risky ones. In the following we will mainly utilize discounted prices, defined as , representing the asset prices in terms of current unit of cash.

We remark that there is no need to assume that asset prices are positive. But, there must be at least one strictly positive asset, in our case the cash. If we want to renormalize the prices by choosing another asset instead of the cash as reference, i.e. by making it to our numéraire, then this asset must have a strictly positive price process. More precisely, a generic numéraire is an asset, whose nominal price is represented by a strictly positive stochastic process , and which is a portfolio of the original assets . The discounted prices of the original assets are then represented in terms of the numéraire by the semimartingales .

We assume that there are no transaction costs and that short sales are allowed. Remark that the absence of transaction costs can be a serious limitation for a realistic model. The filtration is not necessarily generated by the price process : other sources of information than prices are allowed. All agents have access to the same information structure, that is to the filtration .

A strategy is a predictable stochastic process describing the portfolio holdings. The stochastic process represents the number of pieces of th asset portfolio held by the portfolio as time goes by. Remark that the Itô stochastic integral

 ∫t0x⋅dS=∫t0xu⋅dSu, (1)

and the Stratonovich stochastic integral

 ∫t0x∘dS:=∫t0x⋅dS+12∫t0d⟨x,S⟩=∫t0xu⋅dSu+12∫t0d⟨x,S⟩u (2)

are well defined for this choice of integrator () and integrand (), as long as the strategy is admissible. We mean by this that is a predictable semimartingale for which the Itô integral is a.s. -uniformly bounded from below. Thereby, the bracket denotes the quadratic covariation of two processes. In a general context strategies do not need to be semimartingales, but if we want the quadratic covariation in (2)and hence the Stratonovich integral to be well defined, we must require this additional assumption. For details about stochastic integration we refer to Appendix A in [Em89], which summarizes Chapter VII of the authoritative [DeMe80]. The portfolio value is the process defined by

 Vt:=Vxt:=xt⋅St. (3)

An admissible strategy is said to be self-financing if and only if the portfolio value at time is given by

 Vt=V0+∫t0xu⋅dSu. (4)

This means that the portfolio gain is the Itô integral of the strategy with the price process as integrator: the change of portfolio value is purely due to changes of the assets’ values. The self-financing condition can be rewritten in differential form as

 dVt=xt⋅dSt. (5)

As pointed out in [BjHu05], if we want to utilize the Stratonovich integral to rephrase the self-financing condition, while maintaining its economical interpretation (which is necessary for the subsequent constructions of mathematical finance), we write

 Vt=V0+∫t0xu∘dSu−12∫t0d⟨x,S⟩u (6)

or, equivalently

 dVt=xt∘dSt−12d⟨x,S⟩t. (7)

An arbitrage strategy (or arbitrage for short) for the market model is an admissible self-financing strategy , for which one of the following condition holds for some horizon :

• and ,

• and with .

In Chapter 9 of [DeSc08] the no arbitrage condition is given a topological characterization. In view of the fundamental Theorem of asset pricing, the no-arbitrage condition is substituted by a stronger condition, the so called no-free-lunch-with-vanishing-risk.

###### Definition 1.

Let be a semimartingale and and admissible strategy. We denote by , if such limit exists, and by the subset of containing all such . Then, we define

• .

• .

• : the closure of in with respect to the norm topology.

The market model satisfies

• the 1st order no-arbitrage condition or no arbitrage (NA) if and only if , and

• the 2nd order no-arbitrage condition or no-free-lunch-with-vanishing-risk (NFLVR) if and only if .

Delbaen and Schachermayer proved in 1994 (see [DeSc08] Chapter 9.4, in particular the main Theorem 9.1.1)

###### Theorem 2 (Fundamental Theorem of Asset Pricing in Continuous Time).

Let and be bounded semimartingales. There is an equivalent martingale measure for the discounted prices if and only if the market model satisfies the (NFLVR).

This is a generalization for continuous time of the Dalang-Morton-Willinger Theorem proved in 1990 (see [DeSc08], Chapter 6) for the discrete time case, where the (NFLVR) is relaxed to the (NA) condition. The Dalang-Morton-Willinger Theorem generalizes to arbitrary probability spaces the Harrison and Pliska Theorem (see [DeSc08], Chapter 2) which holds true in discrete time for finite probability spaces.

An equivalent alternative to the martingale measure approach for asset pricing purposes is given by the pricing kernel (state price deflator) method.

###### Definition 3.

Let be a semimartingale describing the price process for the assets of our market model. The positive semimartingale is called pricing kernel (or state price deflator) for if and only if is a -martingale.

As shown in [HuKe04] (Chapter 7, definitions 7.18, 7.47 and Theorem 7.48), the existence of a pricing kernel is equivalent to the existence of an equivalent martingale measure:

###### Theorem 4.

Let and be bounded semimartingales. The process admits an equivalent martingale measure if and only if there is a pricing kernel for (or for ).

In economic theory the value of an investment is given by the present value of its future cashflows. This idea can be mathematically formalized in terms of the market model presented so far by introducing the following

###### Definition 5 (Cashflows and Intensities).

Let be the valued semimartingale representing nominal prices, given a certain numéraire with value process . All process are adapted to the filtration . The asset stochastic cashflow intensities are given by the semimartingale defined as

 ct:=−limh→0+Et[St+h−Sth]+r0tSt, (8)

wherever the limit is defined. The components of a vector valued process satisfying the Itô integral equation

 Ct=∫t+t−dch (9)

are termed stochastic cashflows.

For example, a bond is identified with its future coupons and its nominal, and a stock is identified with all its future dividends. In the (straight) bond case the cashflow is deterministic, has discontinuities at the coupon payment dates and vanishes after maturity. In the stock case the cashflow is stochastic, has discontinuities at the dividend payment dates and has an unbounded support. In these two cases intensities exist as stochastic generalized functions.

###### Theorem 6.

Let and be bounded semimartingales, and the cash account be the numéraire. If the market model satisfies the NFLVR condition, then

 St=E∗t[∫+∞tdhchexp(−∫htdur0u)]=1βtEt[∫+∞tdhchβh], (10)

where denotes the risk neutral conditional expectation, and the state price deflator.

### 2.2 Geometric Reformulation of the Market Model: Primitives

We are going to introduce a more general representation of the market model introduced in section 2.1, which better suits to the arbitrage modelling task. In this subsection we extend the terminology introduced by [SmSp98] for the time discrete case to the generic one.

###### Definition 7.

A gauge is an ordered pair of two -adapted real valued semimartingales , where is called deflator and , which is called term structure, is considered as a stochastic process with respect to the time , termed valuation date and . The parameter is referred as maturity date. The following properties must be satisfied a.s. for all such that :

• ,

• .

###### Remark 8.

Deflators and term structures can be considered outside the context of fixed income. An arbitrary financial instrument is mapped to a gauge with the following economic interpretation:

• Deflator: is the value of the financial instrument at time expressed in terms of some numéraire. If we choose the cash account, the -th asset as numéraire, then we can set .

• Term structure: is the value at time (expressed in units of deflator at time ) of a synthetic zero coupon bond with maturity delivering one unit of financial instrument at time . It represents a term structure of forward prices with respect to the chosen numéraire.

We point out that there is no unique choice for deflators and term structures describing an asset model. For example, if a set of deflators qualifies, then we can multiply every deflator by the same positive semimartingale to obtain another suitable set of deflators. Of course term structures have to be modified accordingly. The term ”deflator” is clearly inspired by actuarial mathematics. In the present context it refers to a nominal asset value up division by a strictly positive semimartingale (which can be the state price deflator if this exists and it is made to the numéraire). There is no need to assume that a deflator is a positive process. However, if we want to make an asset to our numéraire, then we have to make sure that the corresponding deflator is a strictly positive stochastic process.

###### Example 9.

Stock Index
Let us consider a total return stock index, where the dividends are reinvested.

• stock index value at time expressed in terms of the cash asset (risk free discounting).

• price of a forward on the stock index issued at time maturing at time expressed in terms of .

###### Example 10.

Zero Bonds
Let us consider a family of maturing zero bonds.

• value of a zero bond maturing at time = value of one unit of cash at time expressed in terms of the cash asset itself.

• price of a zero bond issued at time and delivering one unit of cash at time expressed in terms of .

Deflators typically represent for a currency the time evolution of inflation or deflation. Quotients of deflators are exchange rates.

###### Example 11.

Exchange Rates

 DUSDtDCHFt=FXCHF→USDt. (11)

### 2.3 Geometric Reformulation of the Market Model: Portfolios

We want now to introduce transforms of deflators and term structures in order to group gauges containing the same (or less) stochastic information. That for, we will consider deterministic linear combinations of assets modelled by the same gauge (e. g. zero bonds of the same credit quality with different maturities).

###### Definition 12.

Let be a deterministic cashflow intensity (possibly generalized) function. It induces a gauge transform by the formulae

 Dπt:=Dt∫+∞0dhπhPt,t+hPπt,s:=∫+∞0dhπhPt,s+h∫+∞0dhπhPt,t+h. (12)
###### Remark 13.

The cashflow intensity specifies the bond cashflow structure. The bond value at time expressed in terms of the market model numéraire is given by . The term structure of forward prices for the bond future expressed in terms of the bond current value is given by .

A gauge transform is well defined if and only if the integrals are convergent, which is the case if . A gauge transform with positive cashflows always maps a gauge to another gauge. A generic gauge transform does not, since the positivity of term structures is not a priori preserved. Therefore, when referring to a generic gauge transform, it is necessary to specify its domain of definition, that is the set of gauges which are mapped to other gauges.

We can use gauge transforms to construct portfolios of instruments already modelled by a known gauge.

###### Example 14.

Coupon Bonds
Let the gauge describing the family of zero bonds. To model a family of straight coupon bonds with coupon rate and term to maturity let us choose the (generalized) cashflow intensity function

 πt:=T−1∑s=1gδt−s+(1+g)δt−T. (13)

Thereby denotes the Dirac-delta generalized function.

• value at time of a coupon bond issued at time .

• price of a synthetic zero bond issued at time and delivering at time a coupon bond (issued at time ), expressed in terms of .

###### Proposition 15.

Gauge transforms induced by cashflow vectors have the following property:

 ((D,P)π)ν=((D,P)ν)π=(D,P)π∗ν, (14)

where denotes the convolution product of two cashflow vectors or intensities respectively:

 (π∗ν)t:=∫t0dhπhνt−h. (15)

The convolution of two non-invertible gauge transform is non-invertible. The convolution of a non-invertible with an invertible gauge transform is non-invertible.

###### Definition 16.

An invertible gauge transform is called non-singular. Two gauges are said to be in same orbit if and only if there is a non-singular gauge transform mapping one onto the other. A singular gauge transform defines a partial ordering in the set of gauges. is said to be in a higher orbit than .

It is therefore possible to construct gauges in a lower orbit from higher orbits, but not the other way around. Orbits represent assets containing equivalent information. For every orbit it suffices therefore to specify only one gauge.

###### Definition 17.

A gauge with term structure satisfies the positive interest condition if and only if for all the function is strictly monotone decreasing. Such a gauge is said to be positive. A gauge not satisfying this property is termed principal gauge. The term structure can be written as a functional of the instantaneous forward rate f defined as

 ft,s:=−∂∂slogPt,s,Pt,s=exp(−∫stdhft,h). (16)

and

 rt:=lims→t+ft,s (17)

is termed short rate.

###### Remark 18.

Since is a -stochastic process (semimartingale) depending on a parameter , the -derivative can be defined deterministically, and the expressions above make sense pathwise in a both classical and generalized sense. In a generalized sense we will always have a derivative for any ; this corresponds to a classic -continuous derivative if is a -function of for any fixed and .

We see that the positive interest condition is satisfied if and only if for all , . The positive interest condition is associated with the storage requirement. Whenever it is always more valuable to get a piece of a financial object today than in the future, then it should be modelled with a gauge satisfying the positive interest condition. Examples are: non perishable goods, currencies, price indices for equities and real estates, total return indices. Examples of financial quantities not satisfying the positive interest condition and thus reflecting items which are not storable, are: inflation indices, short rates, dividend indices for equities, rental indices for real estates.

###### Definition 19.

The cash flow intensity , first derivative of the Dirac delta generalized function, defines the short rate transform,

 D[−1]t=Dt∫+∞0dh−δ′hPt,t+h=−DtrtP[−1]t,s=∫+∞0dh−δ′hPt,s+h∫+∞0dh−δ′hPt,t+h=ft,srtPt,s (18)

while the cash flow intensity , Heavyside function, defines the perpetuity transform

 D[+1]t=Dt∫+∞0dhPt,t+hP[+1]t,s=∫+∞0dhPt,s+h∫+∞0dhPt,t+h. (19)
###### Notation 20.

Repeated application of perpetuity and short rate transforms are given by:

 [0]:=δ: Dirac delta % generalized function[+1]:=Θ: Heavyside function[+k]t:=tk−1(k−1)!(k≥2)[−1]t:=−δ′: first derivative % of Dirac delta[−k]t:=(−1)kδ(k): k-th % derivative of Dirac delta,(k≥2) (20)

Thereby, for any integers one has (cf. [Ho03] Chapter IV).

The short rate and the perpetuity transform are inverse to another, as one can see from Proposition 15 and . The short rate transform can be applied only to a positive gauge producing a gauge which possibly does not satisfy the positive interest rate condition. The perpetuity transform is a gauge transform that can be applied to any gauge producing always a positive gauge.

###### Proposition 21.

A gauge satisfies the positive interest condition if and only if it can be obtained as the perpetuity transform of some other gauge.

The positive interest condition is difficult to satisfy for a stochastic model of a gauge.

###### Example 22.

Fixed Income, Equity and Real Estate Gauges

###### Remark 23.

The special choice of vanishing interest rate or flat term structure for all assets corresponds to the classical model, where only asset prices and their dynamics are relevant. We will analyze this case in detail in the guiding example presented in section 4.

## 3 Arbitrage Theory in a Differential Geometric Framework

Now we are in the position to rephrase the asset model presented in subsection 2.1 in terms of a natural geometric language. That for, we will unify Smith’s and Ilinski’s ideas to model a simple market of base assets. In Smith and Speed ([SmSp98]) there is no explicit differential geometric modelling but the use of an allusive terminology (e.g. gauges, gauge transforms). In Ilinski ([Il01]) there is a construction of a principal fibre bundle allowing to express arbitrage in terms of curvature. Our construction of the principal fibre bundle will differ from Ilinski’s one in the choice of the group action and the bundle covering the base space. Our choice encodes Smith’s intuition in differential geometric language.

In this paper we explicitly model no derivatives of the base assets, that is, if derivative products have to be considered, then they have to be added to the set of base assets. The treatment of derivatives of base assets is tackled in ([FaVa12]). Given base assets we want to construct a portfolio theory and study arbitrage. Since arbitrage is explicitly allowed, we cannot a priori assume the existence of a risk neutral measure or of a state price deflator. In terms of differential geometry, we will adopt the mathematician’s and not the physicist’s approach. The market model is seen as a principal fibre bundle of the (deflator, term structure) pairs, discounting and foreign exchange as a parallel transport, numéraire as global section of the gauge bundle, arbitrage as curvature. The Ambrose-Singer Theorem allows to parameterize arbitrage strategies as element of the Lie algebra of the holonomy group. The no-free-lunch-with-vanishing-risk condition is proved to be equivalent to a zero curvature condition or to a continuity equation allowing for an hydrodynamics study of arbitrage flows.

### 3.1 Market Model as Principal Fibre Bundle

As a concise general reference for principle fibre bundles we refer to Bleecker’s book ([Bl81]). More extensive treatments can be found in Dubrovin, Fomenko and Novikov ([DuFoNo84]), and in the classical Kobayashi and Nomizu ([KoNo96]). Let us consider -in continuous time- a market with assets and a numéraire. A general portfolio at time is described by the vector of nominals , for an open set . Following Definition 7, the asset model induces for the gauge

 (Dj,Pj)=((Djt)t∈[0,+∞[,(Pjt,s)s≥t), (21)

where denotes the deflator and the term structure. This can be written as

 Pjt,s=exp(−∫stfjt,udu), (22)

where is the instantaneous forward rate process for the -th asset and the corresponding short rate is given by . For a portfolio with nominals we define

 Dxt:=N∑j=1xjDjtfxt,u:=N∑j=1xjDjt∑Nj=1xjDjtfjt,uPxt,s:=exp(−∫stfxt,udu). (23)

The short rate writes

 rxt:=limu→0+fxt,u=N∑j=1xjDjt∑Nj=1xjDjtrjt. (24)

The image space of all possible strategies reads

 M:={(x,t)∈X×[0,+∞[}. (25)

In subsection 2.3 cashflow intensities and the corresponding gauge transforms were introduced. They have the structure of an Abelian semigroup

 G:=E′([0,+∞[,R)={F∈D′([0,+∞[)∣supp(F)⊂[0,+∞[ is compact}, (26)

where the semigroup operation on distributions with compact support is the convolution (see [Ho03], Chapter IV), which extends the convolution of regular functions as defined by formula (15).

###### Definition 24.

The Market Fibre Bundle is defined as the fibre bundle of gauges

 B:={(Dπtx,Pπt,⋅x)|(x,t)∈M,π∈G∗}. (27)

The cashflow intensities defining invertible transforms constitute an Abelian group

 G∗:={π∈G| it exists ν∈G such that π∗ν=[0]}⊂E′([0,+∞[,R). (28)

From Proposition 15 we obtain

###### Theorem 25.

The market fibre bundle has the structure of a -principal fibre bundle given by the action

 (29)

The group acts freely and differentiably on to the right.

### 3.2 Numéraire as Global Section of the Bundle of Gauges

If we want to make an arbitrary portfolio of the given assets specified by the nominal vector to our numéraire, we have to renormalize all deflators by an appropriate gauge transform so that:

• The portfolio value is constantly over time normalized to one:

 DxNum,πNumt≡1. (30)
• All other assets’ and portfolios’ are expressed in terms of the numéraire:

 Dx,πNumt=FXx→xNumt:=DxtDxNumt. (31)

It is easily seen that the appropriate choice for the gauge transform making the portfolio to the numéraire is given by the global section of the bundle of gauges defined by

 πNum,xt:=FXx→xNumt. (32)

Of course such a gauge transform is well defined if and only if the numéraire deflator is a positive semimartingale.

### 3.3 Cashflows as Sections of the Associated Vector Bundle

By choosing the fiber and the representation induced by the gauge transform definition, and therefore satisfying the homomorphism relation , we obtain the associated vector bundle . Its sections represents cashflow streams - expressed in terms of the deflators - generated by portfolios of the base assets. If is the deterministic cashflow stream, then its value at time is equal to

• the deterministic quantity , if the value is measured in terms of the deflator ,

• the stochastic quantity , if the value is measured in terms of the numéraire (e.g. the cash account for the choice for all ).

In the general theory of principal fibre bundles, gauge transforms are bundle automorphisms preserving the group action and equal to the identity on the base space. Gauge transforms of are naturally isomorphic to the sections of the bundle (See Theorem 3.2.2 in [Bl81]). Since is Abelian, right multiplications are gauge transforms. Hence, there is a bijective correspondence between gauge transforms and cashflow intensities admitting an inverse. This justifies the terminology introduced in Definition 12.

### 3.4 Derivatives of Stochastic Processes

One of the main contribution of this paper is to reformulate stochastic finance in a natural geometric language. In stochastic differential geometry one would like to lift the constructions of stochastic analysis from open subsets of to dimensional differentiable manifolds. To that aim, chart invariant definitions are needed and hence a stochastic calculus satisfying the usual chain rule and not Itô’s Lemma is required. (cf. [HaTh94], Chapter 7, and the remark in Chapter 4 at the beginning of page 200). That is why we will be mainly concerned in the following by stochastic integrals and derivatives meant in the sense of Stratonovich and not of Itô. Following [Gl11] and [CrDa07] we introduce the following

###### Definition 26.

Let be a real interval and be a vector valued stochastic process on the probability space . The process determines three families of -subalgebras of the -algebra :

• ”Past” , generated by the preimages of Borel sets in by all mappings for .

• ”Future” , generated by the preimages of Borel sets in by all mappings for .

• ”Present” , generated by the preimages of Borel sets in by the mapping .

Let be continuous. Assuming that the following limits exist, Nelson’s stochastic derivatives are defined as

 DQt:=limh→0+E[Qt+h−Qth∣∣∣Pt]: % forward derivative,D∗Qt:=limh→0+E[Qt−Qt−hh∣∣∣Ft]: backward % derivative,DQt:=DQt+D∗Qt2: mean % derivative. (33)

Let the set of all processes such that , and are continuous mappings from to . Let the completion of with respect to the norm

 ∥Q∥:=supt∈I(∥Qt∥L2(Ω,A)+∥DQt∥L2(Ω,A)+∥D∗Qt∥L2(Ω,A)). (34)
###### Remark 27.

The stochastic derivatives , and correspond to Itô’s, to the anticipative and, respectively, to Stratonovich’s integral (cf. [Gl11]). The process space contains all Itô processes. If is a Markov process, then the sigma algebras (”past”) and (”future”) in the definitions of forward and backward derivatives can be substituted by the sigma algebra (”present”), see Chapter 6.1 and 8.1 in ([Gl11]).

### 3.5 Stochastic Parallel Transport and Holonomy

Let us consider the projection of onto

 p:B≅M×G∗⟶M(x,t,g)↦(x,t) (35)

and its tangential map

 T(x,t,g)p:T(x,t,g)B≅RN×R×R[0,+∞[⟶T(x,t)M≅RN×R. (36)

The vertical directions are

 V(x,t,g)B:=ker(T(x,t,g)p)≅R[0,+∞[, (37)

and the horizontal ones are

 H(x,t,g)B≅RN+1. (38)

A connection on is a projection . More precisely, the vertical projection must have the form

 Πv(x,t,g):T(x,t,g)B⟶V(x,t,g)B(δx,δt,δg)↦(0,0,δg+Γ(x,t,g).(δx,δt)), (39)

and the horizontal one must read

 Πh(x,t,g):T(x,t,g)B⟶H(x,t,g)B(δx,δt,δg)↦(δx,δt,−Γ(x,t,g).(δx,δt)), (40)

such that

 Πv+Πh=1B. (41)

Stochastic parallel transport on a principal fibre bundle along a semimartingale is a well defined construction (cf. [HaTh94], Chapter 7.4 and [Hs02] Chapter 2.3 for the frame bundle case) in terms of Stratonovich integral. Existence and uniqueness can be proved analogously to the deterministic case by formally substituting the deterministic time derivative with the stochastic one corresponding to the Stratonovich integral.

Following Ilinski’s idea ([Il01]), we motivate the choice of a particular connection by the fact that it allows to encode foreign exchange and discounting as parallel transport.

###### Theorem 28.

With the choice of connection

 Γ(x,t,g).(δx,δt):=g(DδxtDxt−rxtδt), (42)

the parallel transport in has the following financial interpretations:

• Parallel transport along the nominal directions (-lines) corresponds to a multiplication by an exchange rate.

• Parallel transport along the time direction (-line) corresponds to a division by a stochastic discount factor.

Recall that time derivatives needed to define the parallel transport along the time lines have to be understood in Stratonovich’s sense. We see that the bundle is trivial, because it has a global trivialization, but the connection is not trivial.

###### Proof.

Let us consider a curve in for and an element of the fiber over the starting point . The parallel transport of along is the solution of the first order differential equation

 {Πv(x(τ),t(τ),g(τ))(Dx(τ),Dt(τ),Dg(τ))=0g(τ1)=g1, (43)

which in our case writes

 ⎧⎪ ⎪⎨⎪ ⎪⎩Dg(τ)=−g(τ)(DDx(τ)t(τ)Dx(τ)t(τ)−rx(τ)t(τ)Dt(τ))g(τ1)=g1, (44)

Recall that the time derivative is Nelson’s derivative corresponding to the Stratonovich integral, see subsection 3.4. Now, if is a nominal direction, then and . Thus

 ⎧⎪⎨⎪⎩Dg(τ)=−g(τ)∑Nj=1Dxj(τ)Djt∑Nj=1xj(τ)Djtg(τ1)=g1, (45)

which means

 g(τ)=g1∑Nj=1xj(τ1)Djt∑Nj=1xj(τ)Djt (46)

corresponding to a multiplication by an exchange rate at time from portfolio to portfolio .
If is the time direction, then , and , . Thus

 {Dg(τ)=g(τ)rxτg(τ1)=g1, (47)

which means

 g(τ)=g1exp(∫ττ1rxudu) (48)

corresponding to a division by the stochastic discount rate for portfolio from time to time . ∎

###### Remark 29.

Malaney and Weinstein ([Ma96]) already introduced a connection in the deterministic case in the context of self-financing basket of goods for divisa indices. Recently, [FaVa12] have elaborated a stochastic version of the Malaney-Weinstein connection proving its equivalence with the connection defined in (42).

Holonomy is the group generated by the parallel transport along closed curves. We distinguish the local from the global case.

###### Definition 30.

The holonomy group based at is defined as

 Holb(χ):={g∈G∣b and b.g can be joined by an horizontal curve in B}. (49)

The local holonomy group based at is defined as

 Hol0b(χ):={g∈G∣b and b.g can be joined by a % contractible horizontal curve in B}. (50)

If and are connected, then holonomy and local holonomy depend on the base point only up to conjugation. In this paper we will always assume connectivity for both and and therefore drop the reference to the basis point , with the understanding that the definition is good up to conjugation.

### 3.6 Nelson D Differentiable Market Model

We continue to reformulate the classic asset model introduced in subsection 2.1 in terms of stochastic differential geometry.

###### Definition 31.

A Nelson differentiable market model for assets is described by gauges which are Nelson differentiable with respect to the time variable. More exactly, for all and there is an open time interval such that for the deflators and the term structures , the latter seen as processes in and parameter , there exist a -derivative. The short rates are defined by .

A strategy is a curve in the portfolio space parameterized by the time. This means that the allocation at time is given by the vector of nominals . We denote by the lift of to , that is . A strategy is said to be closed if it represented by a closed curve. A -admissible strategy is predictable and -differentiable.

In general the allocation can depend on the state of the nature i.e. for . Unless otherwise specified strategies will always be -admissible for an appropriate time interval.

###### Proposition 32.

A -admissible strategy is self-financing if and only if

 D(xt⋅Dt)=xt⋅DDt−12⟨x,D⟩torDxt⋅Dt=−12⟨x,D⟩t, (51)

almost surely.

###### Proof.

The strategy is self-financing if and only if

 xt⋅Dt=x0⋅D0+∫t0xu⋅dDu, (52)

which is, symbolizing Itô’s ”differential”, equivalent to

 D(xt⋅Dt)=xt⋅Dt. (53)

The selfinancing condition can be expressed by means of the anticipative ”differential” as

 xt⋅Dt=x0⋅D0+∫t0xu⋅d∗Du−∫t0d⟨x,D⟩u, (54)

which is equivalent to

 D∗(xt⋅Dt)=xt⋅D∗Dt−⟨x,D⟩t. (55)

By summing equations (53) and (55) we obtain

 D(xt⋅Dt)=12(D+D∗)(xt⋅Dt)=xt⋅DDt−12⟨x,D⟩t. (56)

To prove the second statement in expression 51 we consider the integration by part formula for Itô’s integral

 ∫t0xu⋅dDu+∫t0Du⋅dxu=xt⋅Dt−x0⋅D0−⟨x,D⟩t, (57)

which, expressed in terms of Stratonovich’s integral, leads to

 ∫t0xu∘dDu−12⟨x,D⟩t+∫t0Du∘dxu−12⟨x,D⟩t=xt⋅Dt−x0⋅D0−⟨x,D⟩t. (58)

By taking Stratonovich’s derivative on both side we get

 D(xt⋅Dt)=Dxt⋅Dt+xt⋅DDt, (59)

which, together with the first statement in expression (51) proves the second one. ∎

For the reminder of this paper unless otherwise stated we will deal only with differentiable market models, differentiable strategies, and, when necessary, with differentiable state price deflators. All Itô processes are differentiable, so that the class of considered admissible strategies is very large.

### 3.7 Arbitrage as Curvature

The Lie algebra of is

 g=R[0,+∞[ (60)

and therefore commutative. The -valued connection -form writes as

 χ(x,t,g)(δx,δt)=(DδxtDxt−rxtδt)g, (61)

or as a linear combination of basis differential forms as

 χ(x,t,g)=(1DxtN∑j=1Djtdxj−rxtdt)g. (62)

The -valued curvature -form is defined as

 R:=dχ+[χ,χ], (63)

meaning by this, that for all and for all

 R(x,t,g)(ξ,η):=dχ(x,t,g)(ξ,η)+[χ(x,t,g)(ξ),χ(x,t,g)(η)]. (64)

Remark that, being the Lie algebra commutative, the Lie bracket vanishes. After some calculations we obtain

 R(x,t,g)=gDxtN∑j=1Djt(rxt+Dlog(Dxt)−rjt−Dlog(Djt))dxj∧dt, (65)

and can prove following results which characterizes arbitrage as curvature.

###### Theorem 33 (No Arbitrage).

The following assertions are equivalent:

• The market model satisfies the no-free-lunch-with-vanishing-risk condition.

• There exists a positive semimartingale such that deflators and short rates satisfy for all portfolio nominals and all times the condition

 rxt=−Dlog(βtDxt). (66)
• There exists a positive semimartingale such that deflators and term structures satisfy for all portfolio nominals and all times the condition

 Pxt,s=Et[βsDxs]βtDxt. (67)

The following assertions are equivalent and follow from the above ones:

• The local holonomy group of the principal fibre bundle is trivial.

• The curvature form vanishes everywhere on .

###### Proof.

• (i)(iii): By Theorems 2 and 4 the no-free-lunch-with-vanishing-risk property is equivalent to the existence of a positive state price deflator, that is of a positive semimartingale such that the market value at time of the any contingent claim at time of the form is

 1βtEt[βsDxs], (68)

where denotes conditional expectation. Since prices are expressed in units of the deflator to which they relate the formula writes

 DxtPxt,s=1βtEt[βsDxs]. (69)
• (ii)(iii): (iii) is the integral version of (ii), which is the differential version of (iii).

• (ii)(v):

 Dlog(Dxt)+rxt