# Can You Hear the Shape of a Market? Geometric Arbitrage and Spectral Theory

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Geometric Arbitrage Theory Background

#### 2.1. The Classical Market Model

- right continuity: ${\mathcal{A}}_{t}={\bigcap}_{s>t}{\mathcal{A}}_{s}$ for all $t\in [0,+\infty [$;
- ${\mathcal{A}}_{0}$ contains all null sets of ${\mathcal{A}}_{\infty}$.

**assets**indexed by $j=1,\dots ,N$, whose

**nominal prices**are given by the vector valued semimartingale $S:[0,+\infty [\times \Omega \to {\mathbb{R}}^{N}$, denoted by ${\left({S}_{t}\right)}_{t\in [0,+\infty [}$ and adapted to the filtration $\mathcal{A}$. The stochastic process ${\left({S}_{t}^{j}\right)}_{t\in [0,+\infty [}$ describes the price at time t of the jth asset in terms of unit of cash at time $t=0$. More precisely, we assume the existence of a 0th asset—the

**cash**, a strictly positive semimartingale that evolves according to ${S}_{t}^{0}=exp({\int}_{0}^{t}du\phantom{\rule{0.166667em}{0ex}}{r}_{u}^{0})$, where the integrable semimartingale ${\left({r}_{t}^{0}\right)}_{t\in [0,+\infty [}$ represents the continuous interest rate provided by the cash account. One always knows in advance what the interest rate on one’s own bank account is, but this can change from time to time. The cash account is therefore considered the locally riskless asset in contrast to the other assets, the risky ones. Subsequently, we will mainly utilize

**discounted prices**, defined as ${\widehat{S}}_{t}^{j}:={S}_{t}^{j}/{S}_{t}^{0}$, representing the asset prices in terms of a current unit of cash.

**numéraire**, then this asset must have a strictly positive price process. More precisely, a generic numéraire is a portfolio of the original assets $j=0,1,2,\dots ,N$, whose nominal price is represented by a strictly positive stochastic process ${\left({B}_{t}\right)}_{t\in [0,+\infty [}$. The discounted prices of the original assets are then represented in terms of the numéraire by the semimartingales ${\widehat{S}}_{t}^{j}:={S}_{t}^{j}/{B}_{t}$.

**strategy**$x={\left({x}_{t}\right)}_{t\in [0,+\infty [}$ is a predictable S-integrable process for which the Itô integral ${\int}_{0}^{t}x\xb7dS\ge -v$ a.s. for all $t\ge 0$ with ${x}_{0}=0$. A strategy is admissible if it is v-admissible for some $v\ge 0$.

**Definition**

**1**

**(Arbitrage).**Let the process ${\left({S}_{t}\right)}_{[0,+\infty [}$ be a semimartingale and ${\left({x}_{t}\right)}_{t\in [0,+\infty [}$ be an admissible self-financing strategy. Let us consider trading up to time $T\le \infty $. The portfolio wealth at time t is given by ${V}_{t}\left(x\right):={V}_{0}+{\int}_{0}^{t}{x}_{u}\xb7d{S}_{u}$, and we denote as ${K}_{0}$ the subset of ${L}^{0}(\Omega ,{\mathcal{A}}_{T},P)$ containing all such ${V}_{T}\left(x\right)$, where x is any admissible self-financing strategy. We define the following:

- ${C}_{0}:={K}_{0}-{L}_{+}^{0}(\Omega ,{\mathcal{A}}_{T},P)$;
- $C:={C}_{0}\cap {L}^{\infty}(\Omega ,{\mathcal{A}}_{T},P)$;
- $\overline{C}$: the closure of C in ${L}^{\infty}$ with respect to the norm topology;
- ${\mathcal{V}}^{{V}_{0}}:=\left\{{\left({V}_{t}\right)}_{t\in [0,+\infty [}\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}{V}_{t}={V}_{t}\left(x\right),\phantom{\rule{0.166667em}{0ex}}where\phantom{\rule{4.pt}{0ex}}x\phantom{\rule{4.pt}{0ex}}is\phantom{\rule{4.pt}{0ex}}{V}_{0}-\mathit{admissible}\right\}$;
- ${\mathcal{V}}_{T}^{{V}_{0}}:=\left\{{V}_{T}\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}{\left({V}_{t}\right)}_{t\in [0,+\infty [}\in {\mathcal{V}}^{{V}_{0}}\right\}$;

**(NA) no-arbitrage**if and only if $C\cap {L}_{+}^{\infty}(\Omega ,{\mathcal{A}}_{T},P)=\left\{0\right\}$;**(NFLVR) no-free-lunch-with-vanishing-risk**if and only if $\overline{C}\cap {L}_{+}^{\infty}(\Omega ,{\mathcal{A}}_{T},P)=\left\{0\right\}$;**(NUPBR) no-unbounded-profit-with-bounded-risk**if and only if ${\mathcal{V}}_{T}^{{V}_{0}}$ isbounded in ${L}^{0}$ for some ${V}_{0}>0$.

**Theorem**

**1.**

**Theorem**

**2**

**(First fundamental theorem of asset pricing).**The market $(S,\mathcal{A})$ satisfies the NFLVR condition if and only if there exists an equivalent local martingale measure ${P}^{*}$.

**Remark**

**1.**

**Definition**

**2**

**(Complete market).**The market $(S,\mathcal{A})$ is

**complete**on $[0,T]$ if for all contingent claims $C\in {L}_{+}({P}^{*},{\mathcal{A}}_{T}):=\{C:\Omega \to [0,+\infty \left[\right|\phantom{\rule{0.166667em}{0ex}}C\phantom{\rule{4.pt}{0ex}}\mathrm{is}\phantom{\rule{4.pt}{0ex}}{\mathcal{A}}_{T}-\mathit{measurable},\phantom{\rule{4.pt}{0ex}}\mathit{and}\phantom{\rule{4.pt}{0ex}}{\mathbb{E}}_{0}^{{P}^{*}}\left[\right|C\left|\right]<+\infty \}$ there exists an admissible self-financing strategy x such that $C={V}_{T}\left(x\right)$.

**Theorem**

**3**

**(Second fundamental theorem of asset pricing).**Given that $(S,\mathcal{A})$ satisfies the NFLVR condition, the market is complete on $[0,T]$ if and only if the equivalent local martingale ${P}^{*}$ is unique.

**Definition**

**3**

**(Dominance).**The j-th security ${S}^{j}={\left({S}_{t}^{j}\right)}_{t\in [0.T]}$ is

**undominated**on $[0,T]$ if there is no admissible strategy ${\left({x}_{t}\right)}_{t\in [0,T]}$ such that:

**no dominance**condition (ND) on $[0,T]$ if and only if each ${S}^{j}$, $j=0,1,\dots ,N$ is undominated on $[0,T]$.

**Definition**

**4**

**(Economy).**An

**economy**consists of a market given by $(S,\mathcal{A})$ and a finite number of investors $k=1,\dots ,K$ characterized by their beliefs, information, preferences, and endowment. Moreover, there is a single consumption good that is perishable. The price of the consumption good in units of the cash account is denoted as $\Psi ={\left({\Psi}_{t}\right)}_{t\in [0,T]}$. We assume that Ψ is strictly positive. The k-th investor is characterized by the following quantities:

**Beliefs and information:**$({P}_{k},\mathcal{A})$. We assume that the investor’s beliefs ${P}_{k}$ are equivalent to P. All investors have the same information filtration $\mathcal{A}$.**Utility function:**${U}_{k}:[0,T]\times [0,+\infty [\to \mathbb{R}$ and μ is a probability measure on $[0,T]$ with $\mu \left(\right\{T\left\}\right)>0$ such that for every t in the support of μ, the function ${U}_{k}(t,\xb7)$ is strictly increasing. We also assume ${lim}_{v\to +\infty}{U}_{k}(T,v)=+\infty $. The utility that agent k derives from consuming ${c}_{t}\mu \left(dt\right)$ at each time $t\le T$ is as follows:$${\mathcal{U}}_{k}\left(c\right)={\mathbb{E}}_{0}^{k}\left[{\int}_{0}^{T}{U}_{k}(t,{c}_{t})\mu \left(dt\right)\right],$$where ${\mathbb{E}}^{k}$ is the expectation with respect to ${P}_{k}$. Since $\mu \left(\right\{T\left\}\right)>0$, the utility is strictly increasing in the final consumption ${c}_{T}$.**Initial wealth:**${v}_{k}$. Given a trading strategy $x=({x}^{1},\dots ,{x}^{N})$, the investor will be required to choose his initial holding ${x}_{0}^{0}$ in the cash account such that:$${v}_{k}={x}_{0}^{0}+\sum _{j=1}^{N}{x}_{0}^{j}{S}_{0}^{i}.$$**Stochastic endowment stream:**${\u03f5}_{t}^{k}$, $t<T$ of the commodity. This means that the investors receive ${\u03f5}_{t}^{k}\mu \left(dt\right)$ units of the commodity at time $t\le T$. The cumulative endowment of the k-th investor in units of the cash account is given by the following equation:$${\mathcal{E}}_{t}^{k}:={\int}_{0}^{t}{\Psi}_{s}{\u03f5}_{s}^{k}\mu \left(ds\right).$$

**Definition**

**5**

**(Consumption plan and strategy).**A pair ${({c}_{t}^{k},{x}_{t}^{k})}_{t\in [0,T]}$ is called

**admissible**if ${\left({c}_{t}^{k}\right)}_{t\in [0,T]}$ is progressively measurable with respect to the filtration $\mathcal{A}$, ${\left({x}_{t}^{k}\right)}_{t\in [0,T]}$ is admissible in the usual sense, and it generates a wealth process ${V}^{k}={\left({V}_{t}^{K}\right)}_{t\in [0,T]}$ with non-negative terminal wealth ${V}_{T}^{k}\ge 0$.

**Definition**

**6**

**(Equilibrium).**Given an economy $({\left\{{P}_{k}\right\}}_{k=1,\dots ,K},{\left({\mathcal{A}}_{t}\right)}_{t\in [0,T]},{\left\{{\u03f5}_{k}\right\}}_{k=1,\dots ,K},$${\left\{{U}_{k}\right\}}_{k=1,\dots ,K})$, a consumption good price index Ψ, financial assets $S={[{S}^{0},{S}^{1},\dots ,{S}^{N}]}^{\u2020}$, and investor consumption-investment plans $({\widehat{c}}^{k},{\widehat{x}}^{k})$ for $k=1,\dots K$, the pair $(\Psi ,S)$ is an

**equilibrium price process**if for all $t\le T$$P-$ a.s. the following conditions are satisfied:

**Securities markets clear:**$$\sum _{k=1}^{K}{\widehat{x}}_{t}^{k,j}={\alpha}^{j}\phantom{\rule{2.em}{0ex}}(j=0,1,\dots ,N),$$where ${\alpha}^{j}$ is the aggregate net supply of the j-th security. It is assumed that each ${\alpha}^{j}$ is non-random and constant over time, with ${\alpha}^{0}=0$ and ${\alpha}^{j}>0$ for $j=1,\dots ,N$.**Commodity markets clear:**$$\sum _{k=1}^{K}{\widehat{c}}_{t}^{k}=\sum _{k=1}^{K}{\u03f5}_{t}^{k}.$$**Investors’ choices are optimal:**$({\widehat{c}}^{k},{\widehat{x}}^{k})$ solves the k-th investor’s utility maximization problem$${u}_{k}\left(x\right):=sup\left\{{U}_{k}\left(c\right)\right|\phantom{\rule{0.166667em}{0ex}}c\phantom{\rule{4.pt}{0ex}}\mathit{admissible}\phantom{\rule{4.pt}{0ex}}\mathit{consumption}\phantom{\rule{4.pt}{0ex}}\mathit{plan},{x}^{k}=x\},$$and the optimal value is finite.

**Definition**

**7**

**(Efficiency).**A market model given by S is called

**efficient**on $[0,T]$ with respect to ${\left({\mathcal{A}}_{t}\right)}_{t\in [0,T]}$, i.e., (E), if there exists a consumption good price index Ψ and an economy $({\left\{{P}_{k}\right\}}_{k=1,\dots ,K},{\left({\mathcal{A}}_{t}\right)}_{t\in [0,T]},{\left\{{\u03f5}_{k}\right\}}_{k=1,\dots ,K},{\left\{{U}_{k}\right\}}_{k=1,\dots ,K})$, for which $(\Psi ,S)$ is an equilibrium price process on $[0,T]$.

**Theorem**

**4**

**(Third fundamental theorem of asset pricing, characterization of efficiency).**Let $(S,\mathcal{A})$ be a market. The following statements are equivalent:

- (i)
- (E): $(S,\mathcal{A})$ is efficient in $[0,T]$;
- (ii)
- $(S,\mathcal{A})$ satisfies both (NFLVR) and (ND) on $[0,T]$;
- (iii)
- (EMM): There exists a probability ${P}^{*}$ equivalent to P such that S is a $({P}^{*},\mathcal{A})$ martingale on $[0,T]$.

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

**Definition**

**8.**

**gauge**is an ordered pair of two $\mathcal{A}$-adapted real valued semimartingales $(D,P)$, where $D={\left({D}_{t}\right)}_{t\ge 0}:[0,+\infty [\times \Omega \to \mathbb{R}$ is called a

**deflator**and $P={\left({P}_{t,s}\right)}_{t,s}:\mathcal{T}\times \Omega \to \mathbb{R}$, which is called a

**term structure**, is considered a stochastic process with respect to the time t, termed

**valuation date**, and $\mathcal{T}:=\{(t,s)\in [0,+\infty {[}^{2}\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}s\ge t\}$. The parameter $s\ge t$ is referred to as

**maturity date**. The following properties must be satisfied a.s. for all $t,s$ such that $s\ge t\ge 0$:

- (i)
- ${P}_{t,s}>0$;
- (ii)
- ${P}_{t,t}=1$.

**Remark**

**2.**

- Deflator: ${D}_{t}$ is the value of the financial instrument at time t expressed in terms of some numéraire. If we choose the cash account, the 0-th asset, as numéraire, then we can set ${D}_{t}^{j}:={\widehat{S}}_{t}^{j}=\frac{{S}_{t}^{j}}{{S}_{t}^{0}}\phantom{\rule{1.em}{0ex}}(j=1,\dots N)$.
- Term structure: ${P}_{t,s}$ is the value at time t (expressed in units of deflator at time t) of a synthetic zero coupon bond with maturity s delivering one unit of financial instrument at time s. It represents a term structure of forward prices with respect to the chosen numéraire.

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

**Definition**

**9.**

**gauge transform**$(D,P)\mapsto \pi (D,P):={(D,P)}^{\pi}:=({D}^{\pi},{P}^{\pi})$ by the following formula:

**Proposition**

**1.**

**Definition**

**10.**

**instantaneous forward rate**f defined as follows:

**short rate**.

**Remark**

**3.**

#### 2.4. Arbitrage Theory in a Differential Geometric Framework

#### 2.4.1. Market Model as Principal Fibre Bundle

**Definition**

**11.**

**Market Fibre Bundle**is defined as the fibre bundle of the following gauges:

**Theorem**

**5.**

#### 2.4.2. Nelson $\mathcal{D}$ Weak Differentiable Market Model

**Definition**

**12.**

**Nelson $\mathcal{D}$ weak differentiable market model**for N assets is described by N gauges, which are Nelson $\mathcal{D}$ weak differentiable with respect to the time variable. More exactly, for all $t\in [0,+\infty [$ and $s\ge t$, there is an open time interval $I\ni t$ such that for the deflators ${D}_{t}:={[{D}_{t}^{1},\dots ,{D}_{t}^{N}]}^{\u2020}$ and the term structures ${P}_{t,s}:={[{P}_{t,s}^{1},\dots ,{P}_{t,s}^{N}]}^{\u2020}$, the latter seen as processes in t and parameter s, there exists a $\mathcal{D}$ weak t-derivative (see Appendix A). The short rates are defined by ${r}_{t}:={lim}_{s\to {t}^{-}}\frac{\partial}{\partial s}log{P}_{ts}$.

**closed**if it represented by a closed curve. A

**weak $\mathcal{D}$-admissible strategy**is predictable and $\mathcal{D}$- weak differentiable.

**Remark**

**4.**

**Proposition**

**2.**

#### 2.4.3. Arbitrage as Curvature

**Theorem**

**6.**

- Parallel transport along the nominal directions (x-lines) corresponds to a multiplication by an exchange rate;
- Parallel transport along the time direction (t-line) corresponds to a division by a stochastic discount factor.

**Proof.**

**Proposition**

**3**

**(Curvature Formula).**Let R be the curvature. Then, the following quality holds:

**Theorem**

**7**

**(No-Arbitrage).**The following assertions are equivalent:

- (i)
- The market model (with base assets and futures with discounted prices D and P) satisfies the no-free-lunch-with-vanishing-risk condition;
- (ii)
- There exists a positive martingale $\beta ={\left({\beta}_{t}\right)}_{t\ge 0}$ such that deflators and short rates satisfy, for all portfolio nominals and all times, the condition$${r}_{t}^{x}=-\mathcal{D}log\left({\beta}_{t}{D}_{t}^{x}\right);$$
- (iii)
- There exists a positive martingale $\beta ={\left({\beta}_{t}\right)}_{t\ge 0}$ such that deflators and term structures satisfy, for all portfolio nominals and all times, the condition$${P}_{t,s}^{x}=\frac{{\mathbb{E}}_{t}\left[{\beta}_{s}{D}_{s}^{x}\right]}{{\beta}_{t}{D}_{t}^{x}}.$$

**Definition**

**13.**

**zero curvature (ZC)**if and only if the curvature vanishes a.s.

**Corollary**

**1.**

**Theorem**

**8.**

**Remark**

**5.**

- 1.
**The components of r are equal:**For example, in the classical model, where there are no term structures (i.e., $r\equiv 0$),- (a)
**D and r are constant over time:**NFLVR is satisfied;- (b)
**D and r are deterministic and not constant over time:**NFLVR is never satisfied.

- 2.
**The components of r are not equal:**- (a)
**D and r are constant over time:**NFLVR is never satisfied;- (b)
**D and r are deterministic and not constant over time:**NFLVR can be satisfied if (ii) or (iii) hold true.

#### 2.4.4. Expected Utility Maximization and the CAPM Formula

**Definition**

**14**

**(EUM).**The

**expected utility maximization**problem for the final wealth over the period $[0,t]$ for a given utility function u reads as follows:

**Theorem**

**9.**

**Theorem**

**10**

**(CAPM).**Let

**Proof.**

**Remark**

**6.**

**Remark**

**7.**

## 3. Spectral Theory

#### 3.1. Cash Flows as Sections of the Associated Vector Bundle

**Definition**

**15**

**(Cash Flow Bundle).**By choosing the fiber $V:={\mathbb{R}}^{[0,+\infty [}$ and the representation $\rho :G\to \mathrm{GL}\left(V\right)$ induced by the gauge transform definition, and therefore satisfying the homomorphism relation $\rho ({g}_{1}*{g}_{2})=\rho \left({g}_{1}\right)\rho \left({g}_{2}\right)$, we obtain the associated vector bundle $\mathcal{V}$. Its sections represent cash flow streams—expressed in terms of the deflators—generated by portfolios of the base assets. If $v={\left({v}_{t}^{x}\right)}_{(t,x)\in M}$ is the deterministic cash flow stream, then its value at time t is equal to:

- the deterministic quantity ${v}_{t}^{x}$, if the value is measured in terms of the deflator ${D}_{t}^{x}$;
- the stochastic quantity ${v}_{t}^{x}{D}_{t}^{x}$, if the value is measured in terms of the numéraire (e.g., the cash account for the choice ${D}_{t}^{j}:={\widehat{S}}_{t}^{j}$ for all $j=1,\dots ,N$).

**Cash Flow Bundle**.

#### 3.2. The Connection Laplacian associated with the Market Model

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

**Definition**

**16.**

**Dirichlet boundary condition:**$${B}_{D}{\left(f\right):=f|}_{\partial M}.$$**Neumann boundary condition:**$${B}_{N}\left(f\right):=\left({\nabla}_{\nu}^{\mathcal{V}}f\right){|}_{\partial M},$$

**Proposition**

**6.**

**Remark**

**8.**

**Remark**

**9.**

**Theorem**

**11.**

**Proof.**

**Remark**

**10.**

**Remark**

**11.**

**Corollary**

**2.**

**Remark**

**12.**

#### 3.3. Arbitrage Bubbles

**Definition**

**17**

**(Spectral Lower Bound).**The highest spectral lower bound of the connection Laplacian on the cash flow bundle $\mathcal{V}$ is given by the following:

**Proposition**

**7.**

**Definition**

**18**

**(Basic Assets’ Arbitrage Fundamental Prices and Bubbles).**Let ${\left({C}_{t}\right)}_{t\in [0,T]}$ be the ${\mathbb{R}}^{N}$ cash flow stream stochastic process associated with the N assets of the market model with a given spectral lower bound ${\lambda}_{0}$ and the Radon–Nikodym subspace ${\mathcal{K}}_{{\lambda}_{0}}$. For a given choice of $\phi \in {\mathcal{K}}_{{\lambda}_{0}}$, the approximated fundamental value of the assets with a stochastic ${\mathbf{R}}^{N}$-valued price process ${\left({S}_{t}\right)}_{t\in [0,T]}$ is defined as follows:

**minimal arbitrage measure**.

**Proposition**

**8.**

**Definition**

**19**

**(Scalar curvature).**The market

**integral scalar curvature**at time t for the portfolio x is defined as follows:

**free lunch/no-arbitrage/rip-off strategy**if and only if

**vector valued integral curvature**of the portfolio is defined as the vector of integral scalar curvatures for the portfolio single asset components:

**Remark**

**13.**

**Theorem**

**12**

**(Bubble decomposition).**Let τ denote the maturity time of all risky assets in the market model. ${S}_{t}$ admits a unique (up to the P-evanescent set) decomposition into a sum of fundamental and bubble values:

**Proof.**

**Theorem**

**13**

**(Bubble types).**Let $T=+\infty $, and denote as τ the maturity time of all risky assets in the market model. If there exists a non-trivial bubble ${B}_{t}^{j}$ in an asset’s price for $j=1,\dots ,N$, then there exists at least one probability measure ${P}^{*}$ equivalent to P, for which we have three and only three possibilities:

**Type1:**${B}_{t}^{j}$ is local super- or submartingale with respect to both P and ${P}^{*}$, if $P[\tau =+\infty ]>0$;**Type2:**${B}_{t}^{j}$ is local super- or submartingale with respect to both P and ${P}^{*}$, but not uniformly integrable super- or submartingale, if ${B}_{t}^{j}$ is unbounded but with $P[\tau <+\infty ]=1$;**Type3:**${B}_{t}^{j}$ is a strict local super- or sub- P- and ${P}^{*}$-martingale, if τ is a bounded stopping time.

**Proof.**

**Remark**

**14.**

**Definition**

**20**

**(Contingent Claim’s Arbitrage Fundamental Price and Bubble).**Let us consider in the context of Definition (18) a European option given by the contingent claim with a unique payoff $H\left({S}_{T}\right)$ at time T for an appropriate real-valued function H of N real variables. The fundamental price of the contingent claim and its corresponding arbitrage bubble is defined in the case of base assets paying no dividends:

**Remark**

**15.**

**Remark**

**16.**

**Proposition**

**9**

**(Put-Call Parity for Fundamental Prices).**Let us consider the market model with $N=1$ for the base assets (i.e., cash and one risky asset). Then, the fundamental price processes:

- ${\left({C}_{t}^{*}\right)}_{t\in [0,T]}$ of a call option on ${\left({S}_{t}\right)}_{t\in [0,T]}$ with strike price $K>0$ at time T;
- ${\left({P}_{t}^{*}\right)}_{t\in [0,T]}$ of a put option on ${\left({S}_{t}\right)}_{t\in [0,T]}$ with strike price $K>0$ at time T;
- ${\left({F}_{t}^{*}\right)}_{t\in [0,T]}$ of a forward on ${\left({S}_{t}\right)}_{t\in [0,T]}$ with forward price $K>0$ at time T;

**Proof.**

**Remark**

**17.**

**Corollary**

**3.**

## 4. Topological Obstructions to Arbitrage

#### 4.1. Topological Obstruction to Arbitrage Induced by the Gauss-Bonnet-Chern Theorem

**Definition**

**21.**

- (i)
- W is a complex (real) vector bundle over the oriented Riemannian manifold $(M,g)$ with a Hermitian (Riemannian) structure $\langle \xb7,\xb7\rangle $;
- (ii)
- $\nabla :{C}^{\infty}(M,W)\to {C}^{\infty}(M,{T}^{*}M\otimes W)$ is a connection on M;
- (iii)
- $\gamma :\mathrm{Cl}(M,g)\to \mathrm{Hom}\left(W\right)$ is a real algebra bundle homomorphism from the Clifford bundle over M to the real bundle of complex (real) endomorphisms of W, i.e., W is a bundle of Clifford modules;

**Dirac bundle**if the following conditions are satisfied:

- (iv)
- $\gamma {\left(v\right)}^{*}=-\gamma \left(v\right)$, $\forall v\in TM$, i.e., the Clifford multiplication by tangent vectors is fiberwise skew-adjoint with respect to the Hermitian (Riemannian) structure $\langle \xb7,\xb7\rangle $;
- (v)
- $\nabla \langle \xb7,\xb7\rangle =0$, i.e., the connection is Leibnizian (Riemannian). In other words, it satisfies the product rule:$$d\langle \phi ,\psi \rangle =\langle \nabla \phi ,\psi \rangle +\langle \phi ,\nabla \psi \rangle ,\phantom{\rule{1.em}{0ex}}\forall \phi ,\psi \in {C}^{\infty}(M,W);$$
- (vi)
- $\nabla \gamma =0$, i.e., the connection is a module derivation. In other words, it satisfies the product rule:$$\begin{array}{cc}\hfill \nabla \left(\gamma \left(w\right)\phi \right)=\gamma \left({\nabla}^{g}w\right)\phi +\gamma \left(w\right)\nabla \phi ,& \phantom{\rule{1.em}{0ex}}\forall \phi ,\psi \in {C}^{\infty}(M,W),\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{1.em}{0ex}}\forall w\in {C}^{\infty}(M,\mathrm{Cl}(M,g)).\hfill \end{array}$$

**Example**

**1.**

**(Exterior algebra bundle as a Dirac Bundle).**Let $(M,g)$ be a ${C}^{\infty}$ Riemannian manifold of dimension m. The tangent and the cotangent bundles are identified by the ♭-map defined by ${v}^{\u266d}\left(w\right):=g(v,w)$. Its inverse is denoted as ♯. The exterior algebra can be seen as a Dirac bundle after the following choices:

- $W:=\Lambda \left({T}^{*}M\right)={\u2a01}_{j=0}^{m}{\Lambda}^{j}\left({T}^{*}M\right)$: exterior algebra over M;
- $\langle \xb7,\xb7\rangle $: Riemannian structure induced by g;
- ∇: lift of the Levi Civita connection;
- By means of interior and exterior multiplication, $\mathit{int}\left(v\right)\phi :=\phi (v,\xb7)$ and $\mathit{ext}\left(v\right)\phi :={v}^{\u266d}\wedge \phi $, we can define the following:$$\begin{array}{cc}\hfill \gamma & :\begin{array}{ccc}TM\hfill & \u27f6\hfill & Hom\left(W\right)\hfill \\ v\hfill & \u27fc\hfill & \gamma \left(v\right):=\mathit{ext}\left(v\right)-\mathit{int}\left(v\right).\hfill \end{array}\hfill \end{array}$$

**Definition**

**22.**

**Dirac operator**$Q:{C}^{\infty}(M,W)\to {C}^{\infty}(M,W)$ is defined by $Q:=\gamma \circ (\u266f\otimes \u22ae)\circ \nabla $

**Dirac Laplacian**.

**Definition**

**23**

**(Dirac Complex).**Let Q be the Dirac operator for the Dirac bundle W over the Riemannian manifold $(M,g)$ and $T\in \mathrm{Hom}\left(W\right)$. $(Q,T)$ is called a Dirac complex if and only if ${T}^{2}=\mathbf{1}$ and $QT=-TQ$. We introduce the following notation:

**Remark**

**18.**

- ${Q}_{\pm}:{C}^{\infty}(M,{W}_{\pm})\u27f6{C}^{\infty}(M,{W}_{\mp})$;
- $Q=\left[\begin{array}{cc}0& {Q}_{-}\\ {Q}_{+}& 0\end{array}\right]:{C}^{\infty}(M,\underset{W}{\underbrace{{W}_{+}\oplus {W}_{-}}})\u27f6{C}^{\infty}(M,\underset{W}{\underbrace{{W}_{+}\oplus {W}_{-}}})$;
- the sequence$$\begin{array}{ccccccccc}0& \stackrel{}{\to}& {C}^{\infty}(M,{W}_{+})& \stackrel{{Q}_{+}}{\to}& {C}^{\infty}(M,{W}_{-})& \stackrel{{Q}_{-}}{\to}& {C}^{\infty}(M,{W}_{+})& \stackrel{}{\to}& 0\end{array}$$is a complex, i.e., ${Q}_{-}{Q}_{+}=0$.

**Example**

**2.**

**(Exterior algebra bundle as a Dirac Bundle—Continuation).**The Dirac operator $Q=d+\delta $ is termed a Euler operator. We define the vector bundle isomorphism on the exterior algebra bundle T as $T\eta :={(-1)}^{j}\eta $ for $\eta \in {\Lambda}^{j}\left({T}^{*}M\right)$ and extend it by linearity to $\Lambda \left({T}^{*}M\right)$ in order to obtain the Dirac complex $(Q,T)$, termed the rolled-up De Rham complex.

**Definition**

**24**

**(Analytical Index).**Let $(Q,T)$ be a Dirac complex over a compact manifold. If $\partial M=\u2300$, then the analytical index of the complex is defined as follows:

**Theorem**

**14**

**(Atiyah–Patodi–Singer).**Let $(Q,T)$ be a Dirac complex over a compact manifold. If $\partial M=\u2300$, then

**Example**

**3.**

**(Exterior algebra bundle as a Dirac Bundle—Continuation).**In the boundaryless case we have (see [30] page 179 and [31] page 59)

- g: restriction of the Euclidean metric;
- ${W}_{n}:=\Lambda \left({T}^{*}M\right)\otimes {\mathcal{V}}_{n}$: twisted bundle of finite rank $(n+1){2}^{N+1}$;
- ${\langle {\eta}_{1}\otimes {v}_{1},{\eta}_{2}\otimes {v}_{2}\rangle}^{{W}_{n}}:={\langle {\eta}_{1},{\eta}_{2}\rangle}^{\Lambda \left({T}^{*}M\right)}{\langle {v}_{1},{v}_{2}\rangle}^{{\mathcal{V}}_{n}}$: Riemannian structure;
- ${\nabla}^{{W}_{n}}:={\nabla}^{\Lambda \left({T}^{*}M\right)}\otimes {\mathbf{1}}_{{\mathcal{V}}_{n}}+{\mathbf{1}}_{\Lambda \left({T}^{*}M\right)}\otimes {\nabla}^{{\mathcal{V}}_{n}}$: connection;
- ${\gamma}^{{W}_{n}}:={\gamma}^{\Lambda \left({T}^{*}M\right)}\otimes {\mathbf{1}}_{{\mathcal{V}}_{n}}$: real algebra bundle endomorphism $\gamma :\mathrm{Cl}(M,g)\u27f6\mathrm{Hom}\left({W}_{n}\right)$;
- ${T}^{{W}_{n}}:={T}^{\Lambda \left({T}^{*}M\right)}\otimes {\mathbf{1}}_{{\mathcal{V}}_{n}}\in \mathrm{Hom}\left({W}_{n}\right)$: a symmetry anticommuting with the Dirac operator.

**Theorem**

**15.**

**Remark**

**19.**

**Remark**

**20.**

**Remark**

**21.**

- ${\mathit{Index}}_{a}(P,{B}_{N})$ is an integral over M of an expression depending on the derivatives of Christoffel’s symbols for the connection ${\nabla}^{\mathcal{V}}$;
- ${\mathit{Index}}_{t}(P,{B}_{N})$ is a topological invariant of the manifold M;
- $\mathit{Boundary}\mathit{Term}(P,{B}_{N})=0$, because of the choice of the boundary condition and Green’s formula.

#### 4.2. Topological Obstruction to Arbitrage Induced by the Bochner–Weitzenböck Theorem

**Absolute boundary condition:**$$\begin{array}{cc}\hfill {B}_{\mathrm{abs}}^{0}\left(f\right)& {:=\left(\mathrm{int}\left(\nu \right)\left(f\right)\right)|}_{\partial M}\hfill \\ \hfill {B}_{\mathrm{abs}}^{1}\left(f\right)& :={B}_{\mathrm{abs}}^{0}\left(f\right)\oplus {B}_{\mathrm{abs}}^{0}\left({Q}^{\Lambda \left({T}^{*}M\right)}f\right),\hfill \end{array}$$$$\mathrm{int}\left(v\right)\left(f\right):=f(v,\xb7).$$**Relative boundary condition:**$$\begin{array}{cc}\hfill {B}_{\mathrm{rel}}^{0}\left(f\right)& {:=\left(\mathrm{ext}\left(\nu \right)\left(f\right)\right)|}_{\partial M}\hfill \\ \hfill {B}_{\mathrm{rel}}^{1}\left(f\right)& :={B}_{\mathrm{rel}}^{0}\left(f\right)\oplus {B}_{\mathrm{rel}}^{0}\left({Q}^{\Lambda \left({T}^{*}M\right)}f\right),\hfill \end{array}$$$$\mathrm{ext}\left(v\right)\left(f\right):={v}^{\#}\wedge f,$$

**Theorem**

**16.**

**Proof.**

- $\Rightarrow :$
- If NFLVR holds, then with Theorem 11 there exists an $f\in \mathrm{dom}\left({\Delta}_{{B}_{N}}^{\mathcal{V}}\right)$ such that ${\Delta}^{\mathcal{V}}f=0$, and ${\nabla}^{\mathcal{V}}=0$. Let us consider the section $\psi :=c\otimes f$, where $c\in \mathbb{R}$ is a constant. With Equation (137), we infer via integration by parts$$({{Q}^{W}}^{2}\psi ,\psi )=({{\nabla}^{W}}^{*}{\nabla}^{W}\psi ,\psi )=({\nabla}^{W}\psi ,{\nabla}^{W}\psi )={c}^{2}({\nabla}^{\mathcal{V}}f,{\nabla}^{\mathcal{V}}f)=0,$$
- $\Leftarrow :$
- If there is a $\psi \ne 0$ in $ker\left({Q}_{{B}_{\mathrm{abs}}^{0}\otimes {B}_{N}}^{W}{)}^{2}\right)$, then$$\psi =\underset{i,j}{\u2a01}{a}_{i,j}{c}_{i}\otimes {f}_{j},$$$${{Q}^{W}}^{2}{c}_{i}\otimes {f}_{j}=0,$$$$\begin{array}{cc}\hfill 0& =({{Q}^{W}}^{2}{c}_{i}\otimes {f}_{j},{c}_{i}\otimes {f}_{j})=({\nabla}^{W}({c}_{i}\otimes {f}_{j}),{\nabla}^{W}({c}_{i}\otimes {f}_{j}))\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =({\nabla}^{{\Lambda}^{i}\left({T}^{*}M\right)}{c}_{i},{\nabla}^{{\Lambda}^{i}\left({T}^{*}M\right)}{c}_{i})({\nabla}^{\mathcal{V}}{f}_{j},{\nabla}^{\mathcal{V}}{f}_{j}),\hfill \end{array}$$

**Remark**

**22.**

**Corollary**

**4.**

**Proof.**

**Remark**

**23.**

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Generalized Derivatives of Stochastic Processes

**Definition**

**A1.**

- (i)
- “Past” ${\mathcal{P}}_{t}$, generated by the preimages of Borel sets in ${\mathbf{R}}^{N}$ by all mappings ${Q}_{s}:\Omega \to {\mathbf{R}}^{N}$ for $0<s<t$;
- (ii)
- “Future” ${\mathcal{F}}_{t}$, generated by the preimages of Borel sets in ${\mathbf{R}}^{N}$ by all mappings ${Q}_{s}:\Omega \to {\mathbf{R}}^{N}$ for $0<t<s$;
- (iii)
- “Present” ${\mathcal{N}}_{t}$, generated by the preimages of Borel sets in ${\mathbf{R}}^{N}$ by the mapping ${Q}_{s}:\Omega \to {\mathbf{R}}^{N}$.

**Nelson’s stochastic derivatives**are defined as follows:

**Remark**

**A1.**

**Definition**

**A2.**

**Nelson’s generalized stochastic derivatives:**

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Can You Hear the Shape of a Market? Geometric Arbitrage and Spectral Theory. *Axioms* **2021**, *10*, 242.
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Can You Hear the Shape of a Market? Geometric Arbitrage and Spectral Theory. *Axioms*. 2021; 10(4):242.
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