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#### Geometric Arbitrage Theory reformulates a generic asset model possibly allowing for arbitrage by packaging all assets and their forwards dynamics into a stochastic principal fibre bundle, with a connection whose parallel transport encodes discounting and portfolio rebalancing, and whose curvature measures, in this geometric language, the "instantaneous arbitrage capability" generated by the market itself. The cashflow bundle is the vector bundle associated to this stochastic principal fibre bundle for the natural choice of the vector space fibre. The cashflow bundle carries a stochastic covariant differentiation induced by the connection on the principal fibre bundle. The link between arbitrage theory and spectral theory of the connection Laplacian on the vector bundle is given by the zero eigenspace resulting in a parametrization of all risk neutral measures equivalent to the statistical one. This indicates that a market satisfies the no-free-lunch-with vanishing-risk condition if itTélécharger gratuit Geometric Arbitrage and Spectral Theory pdf

arXiv:1509.03264vl [q-fin.MF] 2 Sep 2015

Geometric Arbitrage and Spectral Theory

Simone Farinelli

Core Dynamics GmbH

Scheuchzerstrasse 43

CH-8006 Zurich

Email: simone@coredynamics. ch

September 11, 2015

Abstract

Geometric Arbitrage Theory reformulates a generic asset model pos¬

sibly allowing for arbitrage by packaging all assets and their forwards

dynamics into a stochastic principal fibre bundle, with a connection whose

parallel transport encodes discounting and portfolio rebalancing, and whose

curvature measures, in this geometric language, the "instantaneous arbi¬

trage capability” generated by the market itself. The cashflow bundle is

the vector bundle associated to this stochastic principal fibre bundle for

the natural choice of the vector space fibre. The cashflow bundle carries

a stochastic covariant differentiation induced by the connection on the

principal fibre bundle. The link between arbitrage theory and spectral

theory of the connection Laplacian on the vector bundle is given by the

zero eigenspace resulting in a parametrization of all risk neutral measures

equivalent to the statistical one. This indicates that a market satisfies

the no-free-lunch-with vanishing-risk condition if it is only if 0 is in the

spectrum.

Contents

1 Introduction

2 Geometric Arbitrage Theory Background

2.1 The Classical Market Model.

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

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

2.4 Arbitrage Theory in a Differential Geometric Framework.

2.4.1 Market Model as Principal Fibre Bundle.

2.4.2 Numeraire as Global Section of the Bundle of Gauges . .

2.4.3 Cashflows as Sections of the Associated Vector Bundle . .

2.4.4 Stochastic Parallel Transport.

2.4.5 Nelson V Differentiable Market Model.

a

3

3

6

7

8

8

1C

10

12

1

2.4.6 Arbitrage as Curvature

3 Spectral Theory

3.1 The Connection Laplacian Associated to the Market Model . . .

3.2 Arbitrage and Utility.

4 Conclusion

A Derivatives of Stochastic Processes

12

M

15

18

[13

E3

1 Introduction

This paper further develops a conceptual structure - called Geometric Arbitrage

Theory - to link arbitrage modeling in generic markets with spectral theory.

GAT rephrases classical stochastic finance in stochastic differential geomet¬

ric terms in order to characterize arbitrage. The main idea of the GAT approach

consists of modeling 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. 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 de¬

scribed by an appropriate principle fibre bundle. A principle like the invariance

of market laws under change of numeraire can be seen then as gauge invariance.

The vector bundle associated to the principal fibre bundle carries a covariant

differentiation induced by the connection. The connection Laplacian under the

Neumann boundary condition is a selfadjoint operator whose spectrum contains

0 if and only if the market model satisfies the no-free-lunch-with-vanishing-

risk condition. If 0 has simple multiplicity, then then market is complete, and

viceversa.

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 in¬

dex problem (' |Ma96j . | We06| ). Ilinski (see [I100J and [IIP 1 1 ) and Young i I 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] f and utilized tech¬

niques from differential geometry to reduce the complexity of asset models be¬

fore stochastic modeling.

This paper is structured as follows. Section 2 reviews classical stochastic

finance and Geometric Arbitrage Theory. Arbitrage is seen as curvature of a

principal fibre bundle representing the market which defines the quantity of

arbitrage associated to it. A guiding example is provided for a market whose

2

asset prices are Ito processes. Proof are omitted and can be found in IFal4j .

where Geometric Arbitrage Theory has been given a rigorous mathematical

foundation utilizing the formal background of stochastic differential geometry

as in Schwartz ( [Schw80l ). Elworthy ( [E1821 1. Emervf lEm89l ). Hackenbroch and

Thalmaier (' |HaTh94) h Stroock ( [StOOp and Hsu f |Hs02p . In Section 3 the

relationship between arbitrage and spectrum of the connection Laplacian, on

one hand, and between arbitrage and utility maximization on the other, are

investigated. Appendix A reviews Nelson’s stochastic derivatives. Section 4

concludes.

2 Geometric Arbitrage Theory Background

In this section we explain the main concepts of Geometric Arbitrage Theory

introduced in |Fal4l . to which we refer for proofs and examples.

2.1 The Classical Market Model

In this subsection we will summarize the classical set up, which will be rephrased

in section (12.41) in differential geometric terms. We basically follow IlhiKeOll

and the ultimate reference lDeSc081 .

We assume continuous time trading and that the set of trading dates is

[0, +oo[. 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 (f2, A, P), where P

is the statistical (physical) probability measure, A = {At} te [o,+oo[ an increasing

family of sub-cr-algebras of Too and (fi, .Aoo,P) is a probability space. The

filtration A is assumed to satisfy the usual conditions, that is

• right continuity: At = P| s>t A s for all t G [0,+oo[.

• Ao contains all null sets of Aoo •

The market consists of finitely many assets indexed by j = 1,..., N, whose

nominal prices are given by the vector valued semimartingale S : [0, +oo[xf2 —»•

R n denoted by (S t )te[0,+oo[ adapted to the filtration A. The stochastic process

(St )te[o,+oo[ describes the price at time t of the j th asset in terms of unit of cash

at time t = 0. More precisely, we assume the existence of a Oth asset, the cash, a

strictly positive semimartingale, which evolves according to >Sf = exp (f dur°),

where the predictable semimartingale (r®) te [o i+00 [ represents the continuous in¬

terest 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 con¬

trast to the other assets, the risky ones. In the following we will mainly utilize

3

discounted prices, defined as Sj := S 3 /S 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 numeraire, then this asset must

have a strictly positive price process. More precisely, a generic numeraire is an

asset, whose nominal price is represented by a strictly positive stochastic process

(I?t)te[o,+oo[> anc ^ w hich is a portfolio of the original assets j = 0,1,2,... ,7V.

The discounted prices of the original assets are then represented in terms of the

numeraire by the semimartingales S° t := S 3 t /B t .

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 A is not necessarily generated by the price

process (£>t) te [o.+oo[ : other sources of information than prices are allowed. All

agents have access to the same information structure, that is to the filtration

A.

A strategy is a predictable stochastic process x : [0,+oo[xS2 —>• de¬

scribing the portfolio holdings. The stochastic process (a^)t£[o,-i-oo[ represents

the number of pieces of jth asset portfolio held by the portfolio as time goes by.

Remark that the Ito stochastic integral

/ x- dS = / x v ■ dS u . (1)

J o Jo

and the Stratonovich stochastic integral

f x o dS := f x ■ dS + — f d (x, S) = f x u ■ dS u + — f d(x, S) t

Jn Jo 2 Jr, Jn 2 J Q

( 2 )

are well defined for this choice of integrator ( S ) and integrand (a;), as long as the

strategy is admissible. We mean by this that a; is a predictable semimartingale

for which the Ito integral f Q x ■ dS is a.s. f-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

{^t}te[0,+oo[ defined by

V t := V t x := x t ■ S t . (3)

An admissible strategy x is said to be self-financing if and only if the portfolio

value at time t is given by

V t = V 0 + f x u ■ dS u .

Jo

(4)

4

This means that the portfolio gain is the Ito 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

dV t = x t ■ dS t . (5)

As pointed out in |BjHu05] , if we want to utilize the Stratonovich integral to

rephrase the self-financing condition, while maintaining its economical inter¬

pretation (which is necessary for the subsequent constructions of mathematical

finance), we write

V t = \o + [ x u o dS u f d(x, S) u (6)

J 0 ^ J 0

or, equivalently

dV t = xt o dS t -^d(x, S) t . (7)

An arbitrage strategy (or arbitrage for short) for the market model is

an admissible self-financing strategy x, for which one of the following condition

holds for some horizon T > 0:

• P[V 0 X < 0] = 1 and P[Vi f > 0] = 1,

• P[Vo < 0] = 1 and P[Vi f > 0] = 1 with P[V£ > 0] > 0.

In Chapter 9 of |DeSc08j the no arbitrage condition is given a topological charac¬

terization. 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 (£t)te[o,+oo[ be a semimartingale and (£t)te[ 0 ,+oo[ an d ad¬

missible strategy. We denote by (x • 5')+ 00 := lim t _>. +00 f* x u ■ dS u , if such limit

exists, and by Kq the subset of T°(fl, Aoo, P) containing all such (x ■ S )+ Qc .

Then, we define

• Co :=K 0 -L 0 + (n,Aoo,P).

• C := CoCL^iocP).

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

The market model satisfies

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

only if C H Aoo, P) = {0}, and

• the 2nd order no-arbitrage condition or no-free-lunch-with-vanishing-

risk (NFLVR) if and only if C Cl L°° (fl, Aoo,P) = {0}.

Delbaen and Schaclrermayer proved in 1994 (see [DeSc08l Chapter 9.4, in par¬

ticular the main Theorem 9.1.1)

5

Theorem 2 (Fundamental Theorem of Asset Pricing in Continuous

Time). Let (5't) t6 [o ! + 0 o[ and (St)te[0,+oo[ be bounded semimartingales. There

is an equivalent martingale measure P* for the discounted prices S 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 |UeSc08| . 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 IDeSc08l . Chapter 2) which holds true in discrete time for

finite probability spaces.

An equivalent alternative to the martingale measure approach for asset pric¬

ing purposes is given by the pricing kernel (state price deflator) method.

Definition 3. Let (St)t^[ o,+oo[ be a semimartingale describing the price process

for the assets of our market model. The positive semimartingale (/3i)tg[o,+oo[

is called pricing kernel (or state price deflator) for S if and only if

(PtSt)t£[o,+oo[ is a P -martingale.

As shown in lHuKe04l (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 (St)te[ 0 ,+oo[ and (St)te[o,+oo[ be bounded semimartingales.

The process S admits an equivalent martingale measure P* if and only if there

is a pricing kernel fd for S (or for S ).

2.2 Geometric Reformulation of the Market Model: Prim¬

itives

We are going to introduce a more general representation of the market model

introduced in section 1241 which better suits to the arbitrage modeling task.

Definition 5. A gauge is an ordered pair of two A-adapted real valued semi¬

martingales ( D,P), where D = (D t )t >o : [0,+oo[xH —>- R is called deflator

and P = ( Pt,s)t,s ' T x — > R, which is called term structure , is considered

as a stochastic process with respect to the time t, termed valuation date and

T := {(f, s) £ [0,+oo[ 2 |s > t}. The parameter s > t is referred as matu¬

rity date. The following properties must be satisfied a.s. for all t,s such that

s>t> 0:

(i) Pt, s > 0,

(a) Pt,t = i.

Remark 6. Deflators and term structures can be considered outside the context

of fixed income. An arbitrary financial instrument is mapped to a gauge (D,P)

with the following economic interpretation:

6

• Deflator: D t is the value of the financial instrument at time t expressed

in terms of some numeraire. If we choose the cash account, the O-th asset

/\ • c<j

as numeraire, then we can set D-j. := S° t = ^ (j = 1,... N).

• Term structure: Pt tS 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 numeraire.

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 numeraire). There is no need to assume that a deflator is

a positive process. However, if we want to make an asset to our numeraire,

then we have to make sure that the corresponding deflator is a strictly positive

stochastic process.

2.3 Geometric Reformulation of the Market Model: Port¬

folios

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 7. Letir : [0,+oo[ —> R be a deterministic cashflow intensity (possi¬

bly generalized) function. It induces a qauqe transform (D,P) M- ir(D,P ) :=

(A py := (D n , py by the formulae

D

7T

t

D t

dh 7 T h P ttt+h

/ 0 + °° dh 7 T h P t , s+h

/ 0 + °° dh TT h P t , t+ h

( 8 )

Proposition 8. Gauge transforms induced by cashflow vectors have the follow¬

ing property:

((a pyy = «A pyy = (a py*\ w

where * denotes

respectively:

the convolution product of two cashflow vectors or intensities

( 7 r* !/)*:= / dh7r h u t - h . ( 10 )

-'o

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.

7

Definition 9. The term structure can be written as a functional of the instan¬

taneous forward rate f defined as

ft,a ■= log P t ,s, Pt,s = exp

OS

( 11 )

and

r t ■■= lim / t)S

s—»•£+

( 12 )

is termed short rate.

Remark 10. Since ( Pt, s )t,s is at-stochastic process (semimartingale) depending

on a parameter s > t, the s-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 T>' derivative for any lo G fi; this

corresponds to a classic s-continuous derivative if Pt tS (iv) is a C 1 -function of s

for any fixed t > 0 and ui G fl.

Remark 11. The special choice of vanishing interest rate r = 0 or flat term

structure P = 1 for all assets corresponds to the classical model, where only

asset prices and their dynamics are relevant.

2.4 Arbitrage Theory in a Differential Geometric Frame¬

work

Now we are in the position to rephrase the asset model presented in subsection

m in terms of a natural geometric language. Given N base assets we want to

construct a portfolio theory and study arbitrage and thus 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, numeraire as global section of the gauge bundle, arbitrage

as curvature. The no-free-lunch-with-vanishing-risk condition is proved to be

equivalent to a zero curvature condition.

2.4.1 Market Model as Principal Fibre Bundle

Let us consider -in continuous time- a market with N assets and a numeraire.

A general portfolio at time t is described by the vector of nominals x G X,

for an open set X C IR^. Following Definition [5j the asset model induces for

j = 1, • • •, N the gauge

= {(D J t ) te[0t+oo[ ,(Pl s ) s > t )

(13)

where D J denotes the deflator and P J the term structure. This can be written

8

where J- 7 is the instantaneous forward rate process for the j-th asset and the

corresponding short rate is given by r J t \= lim„_>o+ flu- For a portfolio with

nominals x G X C IR^ we define

N

IK ■■=

3 =1

N

flu ■■= E

X 3 D l

=1 2^j=

j =i

r-jD J t

flu

TDX

-t + o •

:= exp

The short rate writes

(15)

“ lim fl u =

u^f 0+

N

x j D l

E iv

i=i

N

-i X

jD'l

7

r t ■

(16)

The image space of all possible strategies reads

M := {(a:,t) G X x [0,+oo[}.

(17)

In subsection 12.31 cashflow intensities and the corresponding gauge transforms

were introduced. They have the structure of an Abelian semigroup

G := £'([0, +oo[, IR) = {F € X>'([0, +oo[) | supp(F) C [0, +oo[ is compact},

. (18)

where the semigroup operation on distributions with compact support is the

convolution (see IHondl . Chapter IV), which extends the convolution of regular

functions as defined by formula m-

Definition 12. The Market Fibre Bundle is defined as the fibre bundle of

gauges

B := {{D” t x ,P” t , x )\(x,t) G M,n € G*}. (19)

The cashflow intensities defining invertible transforms constitute an Abelian

group

G* := {7r G G\ it exists v G G such that n * v = [0]} C £'([(), +oo[, R). (20)

From Proposition [5] we obtain

Theorem 13. The market fibre bundle B has the structure of a G*-principal

fibre bundle given by the action

B x G* —> B

((D,P),n)^(D,Py = {D\P”) (21)

The group G* acts freely and differentiably on B to the right.

2.4.2 Numeraire 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 £ Num to our numeraire, we have to renormalize all deflators by

an appropriate gauge transform 7T 1S!um ’ x so that:

9

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

^Num _Num

D x t = 1.

( 22 )

• All other assets’ and portfolios’ are expressed in terms of the numeraire:

D

„Num

= FXf

Df

Df

(23)

It is easily seen that the appropriate choice for the gauge transform 7 r Num making

the portfolio x Num to the numeraire is given by the global section of the bundle

of gauges defined by

Num.x

:= FX°f^ x

(24)

Of course such a gauge transform is well defined if and only if the numeraire

deflator is a positive semimartingale.

2.4.3 Cashflows as Sections of the Associated Vector Bundle

By choosing the fiber V := R[ 0 ’+°°[ and the representation p : G — > GL(V)

induced by the gauge transform definition, and therefore satisfying the homo¬

morphism relation p(gi * < 72 ) = p(si)p(S 2 ), we obtain the associated vector

bundle V. Its sections represents cashflow streams - expressed in terms of the

deflators - generated by portfolios of the base assets. If v = (vf )t x ,t)eM is the

deterministic cashflow stream, then its value at time t is equal to

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

deflator Df,

• the stochastic quantity vf Df. if the value is measured in terms of the

numeraire (e.g. the cash account for the choice D 3 t := Sf for all j =

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 B are naturally isomorphic to the sections

of the bundle B (See Theorem 3.2.2 in [B181] l. Since G* 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 [7]

2.4.4 Stochastic Parallel Transport

Let us consider the projection of B onto M

p : B = M x G* —> M

(x,t,g) i-> (x,f)

(25)

10

and its tangential map

T( x ,t,g)P : T(x,t,g)B^ *

• T{x,t)M

(26)

SR N xlRxlR[ 0 .+ ao [

SIR" xR.

The vertical directions are

V( x ,t,g)B • ker f —

R[0’ + o°[,

(27)

and the horizontal ones are

(28)

A connection on B is a projection TB —> VB. More precisely, the vertical

projection must have the form

x,t,g) ■ T{x,t,g)B * V(x,t,g)B

(6x,6t,6g) (0, 0,Sg + T(x,t,g).(6x,6t)),

and the horizontal one must read

^-(x,t,g) ■ T( Xt t,g)& > 'H( Xt t g )B

(Sx,St,Sg) (5x,St, —T(x,t, g).(6x,6t)),

(29)

(30)

such that

rr + n^iB. (3i)

Stochastic parallel transport on a principal fibre bundle along a semimartingale

is a well defined construction (cf. |HaTh94l . Chapter 7.4 and lHst)2l Chap¬

ter 2.3 for the frame bundle case) in terms of Stratonovic integral. Existence

and uniqueness can be proved analogously to the deterministic case by formally

substituting the deterministic time derivative ^ with the stochastic one T> cor¬

responding to the Stratonovich integral.

Following Ilinski’s idea ( |I101j j. we motivate the choice of a particular con¬

nection by the fact that it allows to encode foreign exchange and discounting as

parallel transport.

Theorem 14. With the choice of connection

T{x,t,g).(5x,5t) := g

(32)

the parallel transport in B has the following financial interpretations:

• 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.

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.

11

2.4.5 Nelson V Differentiable Market Model

We continue to reformulate the classic asset model introduced in subsection 12 .II

in terms of stochastic differential geometry.

Definition 15. A Nelson V differentiable market model for N assets is

described by N gauges which are Nelson T> differentiable with respect to the

time variable. More exactly, for all t £ [0,+oo[ and s > t there is an open

time interval I 9 t such that for the deflators D t := [D \,..., D ^]' and the

term structures Pt iS := [Pf s ,...,P t N s y, the latter seen as processes in t and

parameter s, there exist a V t-derivative. The short rates are defined by rt :=

lim s _j.(- ^log P t s-

A strategy is a curve 7 : I —> X in the portfolio space parameterized by the

time. This means that the allocation at time t is given by the vector of nominals

Xt := 7 {t). We denote by 7 the lift of 7 to M, that is 7 (f) := ( 7 (f), t). A

strategy is said to be closed if it represented by a closed curve. A T>-admissible

strategy is predictable and T>-differentiable.

In general the allocation can depend on the state of the nature i.e. xt = Xt(u>)

for w £ ft.

Proposition 16. A D-admissible strategy is self-financing if and only if

V(x t ■ D t ) = x t ■ VD t - i (x, D) t or Vx t ■ D t = ~{x, D ) t , (33)

almost surely.

For the reminder of this paper unless otherwise stated we will deal only with

V differentiable market models, T> differentiable strategies, and, when necessary,

with V differentiable state price deflators. All fto processes are V differentiable,

so that the class of considered admissible strategies is very large.

2.4.6 Arbitrage as Curvature

The Lie algebra of G is

g = r[°’ +0 °[

and therefore commutative. The fz-valued connection 1-form writes as

( rjSx \

—jy£ ~ r t St J 9,

or as a linear combination of basis differential forms as

X(x, t,g) = D t dx i ~ r t dt ) g '

3 =1

The g-valued curvature 2-form is defined as

R := d\+ [x,x],

(34)

(35)

(36)

(37)

12

meaning by this, that for all ( x,t,g ) G B and for all £, 77 £ T( x>t )M

R(x,t,g){£,r)) := dx(x, t, g){f, rf) + [x(x,t, g){^),x{x,t, g)(rj)\. (38)

Remark that, being the Lie algebra commutative, the Lie bracket [•, •] vanishes.

After some calculations we obtain

N

R {x,t,g) = (rf +V\og(D x t ) - r J t -Vlog (£>£)) dxj A dt, (39)

t jSi

summarized as

Proposition 17 (Curvature Formula). Let R be the curvature. Then, the

following quality holds:

R{x, t, g) = gdt A d x [D log (Df) + rf]. (40)

We can prove following results which characterizes arbitrage as curvature.

Theorem 18 (No Arbitrage). The following assertions are equivalent:

(i) The market model satisfies the no-free-lunch-with-vanishing-risk condition.

(ii) There exists a positive semimartingale ft = 0 such that deflators and

short rates satisfy for all portfolio nominals and all times the condition

rf =-Vlog(p t Df). (41)

(Hi) There exists a positive semimartingale ft = (ftt)t >0 such that deflators

and term structures satisfy for all portfolio nominals and all times the

condition

TJX

r t,

S

ttWsDf]

Pt.Df

(42)

This motivates the following definition.

Definition 19. The market model satisfies the Oth order no-arbitrage con¬

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

Therefore, we have following implications relying the three different definitions

of no-abitrage:

Corollary 20.

2nd order no-arbitrage => 1st order no-arbitrage => Oth order no-arbitrage

(NFLVR)

(NA)

(ZC)

(43)

13

As an example to demonstrate how the most important geometric concepts of

section [2] can be applied we consider an asset model whose dynamics is given

by a multidimensional multidimensional Ito-process. Let us consider a market

consisting of N + 1 assets labeled by j = 0,1,...,TV, where the 0-th asset

is the cash account utilized as a numeraire. Therefore, as explained in the

introductory subsection 12. 11 it suffices to model the price dynamics of the other

assets j = 1,,N expressed in terms of the 0-th asset. As vector valued

semimartingales for the discounted price process S : [0, +oo[xft —> R v and the

short rate r : [0,+oo[xf2 —> R^, we chose the multidimensional Ito-processes

given by

dS t = S t (a t dt + <r t dW t ),

dr t = atdt + btdWt,

(44)

where

• (Wt^g^+ool is a standard P-Brownian motion in R A , for some K G N,

and,

• ( cr t)te[o,+oo[, (at)te[o,+oo[ are R , NxK -, and respectively, R A - valued locally

bounded predictable stochastic processes,

• (6 t )te[0,+oo[) ( a *)te[o,+oo[ are R NxL and respectively, R jV - valued locally

bounded predictable stochastic processes.

Proposition 21. Let the dynamics of a market model be specified by (44\ )-

Then, the market model satisfies the 0th no-arbitrage condition if and only if

a t~\ (°1 w )t+ r t e Range(a t ). (45)

If the volatility term is deterministic, i.e ert(u;) = at, this condition becomes

a t +r t G Range(a t ). (46)

Remark 22. In the case of the classical model, where there are no term struc¬

tures (i.e. r = 0), the condition (4f>\ reads as at € Range(at).

Proposition 23. For the market model whose dynamics is specified by the

no-free-lunch-with-vanishing risk condition (no 2nd order arbitrage) is equiva¬

lent with the zero curvature condition (no 0th order arbitrage) if

E

< +oo,

(47)

for all x £ R w . This is the Novikov condition for the instantaneous Sharpe

Ratio —.

14

3 Spectral Theory

3.1 The Connection Laplacian Associated to the Market

Model

Ilinki’s connection x on the market principal fibre bundle B induces a covariant

differentiation V on the associated vector bundle V. More exactly, we have

Proposition 24. Let X = Jff^oXj g|- be a vector field over M and f = (f t )t

a section of the cashflow bundle V. Then

j=o J

where

Kq (x) = —r*

Di (49)

Kj(x) = ^ (1 <j<N).

Proof. The construction of a covariant differentiation on the associated vector

bundle starting from a connection A on a principle fibre bundle is a generic

procedure in differential geometry. The Ilinski connection % is a Lie algebra

Q = [R[ 0 -+°°[ valued 1-form on M, and, we can decompose the connection as

X(x,g) = gK(x), were K(pc) := o Kj(x)dxj. The tangential map T e p : Q —>

£([r[o,+o°[) 0 £ re p resen tation p : G —> GL(R[ 0 ’ +o °[) maps elements of the

Lie algebra on endomorpliisms for the bundle V. Given a local cashflow section

ft = / 0 + °° ds fsSs-t, in V |u an d a local vector field X in TM\jj the connection

V has a local representation

r+oo

V.Y /= / ds(dfs(e).v s + f s ui{X).v s ), (50)

J 0

where v s := d s -t and ui is an element of T*U\u (^) C(V\u) i.e. an endomorphism

valued 1—form defined as

u(x) := T e .x{x,e) =

as

p (exp(eA(x, e)e).

£ = 0

Since the derivative of the exponential map is the identity and

p{ 7r) = 7T * • e GL(V X ) T e p.t = t * • g C(V X ),

it follows that

oj( x) = A(x, e) * ■ = K{x)8 * •,

(51)

(52)

(53)

15

and, therefore

V.Y / =

r+oo

/ ds [df 8 (X)v 8 + f s K.X5 * t; s ] =

do

r +oo

= / ds[d/ s (A) + / s A7X]<; s =

J o

= df t (X) + f t K.X =

i=o J

( 54 )

□

We now continue by introducing the connection Laplacian on an appropriate

Hilbert space

Definition 25. The space of the sections of the cashflow bundle can be made

into a scalar product space by introducing, for stochastic sections f = f(t, x, ui) =

(/*(<,£, w)) se[0 , + oo[ and g = g(t,x,u) = (g s (t,x,uj)) se[ o, +00 [

C f,g ) := / dP f d N x

Jn J x

r+oo

(f,g)(uj,t,x) := / dsf 8 (u>, t, x)g s (uj, t, x).

Jo

The Hilbert space of integrable sections reads

H ■= {/ = f(t,X,Uj) = {f S (t, X, w))s€[0,+oo[| (/,/) < +°o} • (56)

A standard result functional analysis is, if we see the u> dependence as a

parameter

Proposition 26. The connection Laplacian A := V*V with domain of defini¬

tion given by the Neumann boundary condition

dom( A) := {/ € H\ f(u, ; ■) € H 2 (M , V) V v /(w, •, -)\ dM = 0 Vw € fl} (57)

is a selfadjoint operator on TL. Its spectrum lies in [0,+oo[.

The spectrum of the connection Laplacian under the Neumann boundary

condition contains information about arbitrage possibilities in the market. More

exactly,

Theorem 27. The market model satisfies the (NFLVR) condition if and only if

0 € spec(A). The harmonic sections parametrize the Radon-Nykodim derivative

for the change of measure.

r+oo

dt (/, g) (u>,t,x), where

(55)

16

Proof. The spectrum of the Laplacian under Neumann boundary conditions

contains 0 if and only if there exists a section / such that

V/ = 0.

By Proposition [M] this is equivalent with

djf

dxj

+ Kjft = 0 ,

for all j = 0,1,..., N. This means

2 ?log(/ t )-r*=0,

and, for j = 1,..., N,

dlogjft) _ Dj

dx> Df

Therefore, there exists a positive process /3 = (/3t)te[o,+oo[ such that

ft =

1

(58)

(59)

(60)

(61)

(62)

and

V\og(p t Df) + r? = 0 (63)

For fixed u g f! the Laplace operator has an elliptic symbol and by Weyl’s

theorem any harmonic f = ft = f(u,t,x) is a smooth function of ( t,x ). In

particular any path of / is cadlag with bounded variation, and, hence, (ft)t is

a semimartingale. By equation (1621) . being ( D t )t a semimartingale, it followas

that {Pt)t is a semimartingale as well. By Theorem [TS] this is equivalent to the

(NFLVR) condition. □

Remark 28. Any harmonic f = ft{x) defines a risk neutral measure by means

of the Radom-Nykodim derivative

dP* Pt D x t fo(x)

dP fa D x 0 f t (x ) ’

(which does not depend on x).

From formula (1641) we derive

Corollary 29. The market model satisfies is complete if and only if 0 € spec( A)

is an eigenvalue with simple multiplicity.

17

3.2 Arbitrage and Utility

Let us now consider a utility function, that is a real C-function of a real variable,

which is strictly monotone increasing (i.e. u! > 0) and concave (i.e. u" < 0).

Typically, a market participant would like to maximize the expected utility of its

wealth at some time horizon. Let us assume that he (or she) holds a portfolio

of synthetic zero bonds delivering at maturity base assets and that the time

horizon is infinitesimally near, that is that the utility of the instantaneous total

return has to be maximized. The portfolio values read as:

• At time t - h: Df_ h Pf_ ht+h .

• At time t: DfPf t+h .

• At time t + h: Df +h .

Proposition 30. The synthetic bond portfolio instantaneous return can be com¬

puted as :

Rett := lim Et-h

h— »0+

r^x _ t^x tdx

^t+h Lj t-h jr t-h,t+h

9 h D x P x

* lllJ t-h r t-h,tA-h

= Plog (Df)+r?

Proof. We can develop the instantaneous return as

(65)

lim E t _/,

h— >-0+

T^iX _ nx TJX

^t-\-h J -'t—h J ~t—h,t-\-h

9 h D x P x

£llLJt-h r t-h,t+h

= lim E t _ h

h- 5-0+

= lim —— VD

h-> o+ Df

D x _ d x

u t+h u t-h

2 h D x P x

* ni -'t-h jr t-h i t+h

l -Pi

t—h,t-\-h

2 hPf

t—h,t+h

exp ds ff_ h s ) - 1

2 h

= V log D : l

( 66 )

□

Remark 31. This portfolio of synthetic zero bonds in the theory corresponds

to a portfolio of futures in practice. If the short rate vanishes, then the future

corresponds to the original asset.

Definition 32 (Expected Utility of Synthetic Bond Portfolio Return).

Let t > s be fixed times. The expected utility maximization problem at time s

for the horizon T writes

sup E s

I — f-^h} h>8

exp / dt (2?log(Df*)+rf‘) D x /P t

S,T

(67)

where the supremum is taken over all V-differentiable self-financing strategies

*£ — {•Eu}‘w> 0 *

Now we can formulate the main result of this subsection.

18

Theorem 33. The market curvature vanishes if and only if all market agents

maximize their expected utility for all times and horizons.

This result can be seen as the natural generalization of the corresponding result

in discrete time, as Theorem 3.5 in |FoSc04| . see also }Ro94 . Compare with

Bellini’s, Frittelli’s and Schachermayer’s results for infinite dimensional opti¬

mization problems in continuous time, see Theorem 22 in |BeFr02| and Theorem

2.2 in [SchaOlj . Nothing is said about the fulfilment of the no-free-lunch-with-

vanishing-risk condition: only the weaker Oth order no-arbitrage condition is

equivalent to the maximization of the expected utility at all times for all hori¬

zons.

Proof. The optimization problem (1671) into a standard problem of stochastic op¬

timal theory in continuous time which can be solved by means of a fundamental

solution of the Hamilton-Jacobi-Bellman partial differential equation.

However, there is a direct method, using Lagrange multipliers. First, re¬

mark that problem (1321) is a convex/concave optimization problem with convex

domain and concave utility function and has therefore a unique solution corre¬

sponding to a global maximum. The Lagrange principal function corresponding

to the this maximum problem writes

$(x,A):= E s

exp (J\t (2? log (D?)+r?)\ Df°Pf; r

- \(Dx h ■ D h + - (x, D) h ).

( 68 )

The Lagrange principal equations associated to this maximization problem read

5 $

dx

u ' (exp (V dt (VlogiDf) + rf*)] D :r s °Pff\

exp ^ dt (27log(£>f‘) + rf )J D^Pfy

j T dt^-{V\og{D?)+r?)

(69)

= 0 ,

dQ 1

— = -Vx h ■ D h - - ( x , D) h =0 (Ae [s, T]),

where A denotes the Lagrange multiplier corresponding to the self financing

condition, the second equation. The first equation shows no contribution from

the constraint because -§^Vx h = V-§^x h = 0 and {x h ,D h ) = = 0.

Since the equation system must hold for all s and T such that T > s, it follows

— (2?l O g(D*‘)+rf*) = 0.

(70)

Therefore, it exist a positive process j3 = (/3t)t>o, (which is a priori not a semi

martingale), such that

V\og{p t D x t ') + rV =0, (71)

19

and thus by Proposition 1171 since for t = s this must hold true for any initial

condition x s £ R^, the curvature must vanish. Therefore: if the maximization

problem has a solution, then the equation system has a solution, implying by

Theorem [lH] the Oth oder no-arbitrage condition for the market. Conversely, if

the Oth oder no-arbitrage condition is satisfied, the equation system must have

a solution by Theorem [T8l

□

If the asset dynamics follows an Ito process, Proposition l23l and Theorem l33l

lead to

Corollary 34. For the market model whose dynamics is specified by an ltd’s

process 0 satisfying Novikov’s condition 0. the (NFLVR) condition holds

true if and only if all market agents maximize their expected utility for all times

and horizons.

4 Conclusion

By introducing an appropriate stochastic differential geometric formalism the

classical theory of stochastic finance can be embedded into a conceptual frame¬

work called Geometric Arbitrage Theory, where the market is modelled with a

principal fibre bundle with a connection and arbitrage corresponds to its curva¬

ture. The associated vector bundle, termed cashflow bundle, carries a covariant

differentiation induced by the connection. The presence of the eigenvalue 0 in

the spectrum of the connection Laplacian characterizes the fulfillment of no-

free-lunch-with-vanishing-risk condition for the market model.

A Derivatives of Stochastic Processes

In stochastic differential geometry one would like to lift the constructions of

stochastic analysis from open subsets of R jV to N dimensional differentiable

manifolds. To that aim, chart invariant definitions are needed and hence a

stochastic calculus satisfying the usual chain rule and not Ito’s Lemma is re¬

quired, (cf. |HaTh94l . Chapter 7, and the remark in Chapter 4 at the beginning

of page 200). That is why we will be mainly concerned in this paper by stochas¬

tic integrals and derivatives meant in Stratonovich’s sense and not in ltd’s.

Definition 35. Let I be a real interval and Q = (Qt)tei be a vector valued

stochastic process on the probability space (f i,A,P). The process Q determines

three families of a-subalgebras of the a-algebra A:

(i) ’’Past” Vt : generated by the preimages of Borel sets in RA by all mappings

Q s : Q, R N for 0 < s < t.

(ii) ’’Future” dFt, generated by the preimages of Borel sets in R w by all map¬

pings Q s : LI —> R w for 0 < t < s.

20

(in) ’’Present” Aft, generated by the preimages of Borel sets in R w by the

mapping Q s : LI —> R w .

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