The Black-Scholes Equation in Presence of Arbitrage
aa r X i v : . [ q -f i n . P R ] J un The Black-Scholes Equation in Presence of Arbitrage
Simone FarinelliCore Dynamics GmbHScheuchzerstrasse 43CH-8006 ZurichEmail: [email protected] TakadaDepartment of Information Science, Toho UniversityMiyama 2-2-1, Funabashi-ShiJP-274-8510 ChibaEmail: [email protected] 28, 2019
Abstract
We apply Geometric Arbitrage Theory to obtain results in Mathematical Finance, which donot need stochastic differential geometry in their formulation. First, for a generic market dynamicsgiven by a multidimensional Itˆos process we specify and prove the equivalence between (NFLVR)and expected utility maximization. As a by-product we provide a geometric characterization ofthe (NUPBR) condition given by the zero curvature (ZC) condition. Finally, we extend the Black-Scholes PDE to markets allowing arbitrage.
Contents D Differentiable Market Model . . . . . . . . . . . . . . . . . . . . . . . . 112.4.4 Arbitrage as Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
This paper provides applications of a conceptual structure - called Geometric Arbitrage Theory - toprove results in financial mathematics which are comprehensible without the use of stochastic differentialgeometry and extend well known classical facts. We expect therefore to make GAT accessible to a widerpublic in the mathematical finance community.GAT rephrases classical stochastic finance in stochastic differential geometric terms in order tocharacterize arbitrage. The main idea of the GAT approach consists of modeling markets made ofbasic financial instruments together with their term structures as principal fibre bundles. Financialfeatures of this market - like no arbitrage and equilibrium - are then characterized in terms of standarddifferential geometric constructions - like curvature - associated to a natural connection in this fibrebundle. Principal fibre bundle theory has been heavily exploited in theoretical physics as the languagein which laws of nature can be best formulated by providing an invariant framework to describe physicalsystems and their dynamics. These ideas can be carried over to mathematical finance and economics.2 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 num´eraire can be seen then as gaugeinvariance.The fact that gauge theories are the natural language to describe economics was first proposed byMalaney 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 developedFlesaker and Hughston seminal work ([FlHu96]) and utilized techniques from differential geometry toreduce the complexity of asset models before stochastic modeling.This paper is structured as follows. Section 2 reviews classical stochastic finance and GeometricArbitrage Theory, summarizing [Fa15], where GAT has been given a rigorous mathematical foundationutilizing the formal background of stochastic differential geometry as in Schwartz ([Schw80]), Elworthy([El82]), Em´ery ([Em89]), Hackenbroch and Thalmaier ([HaTh94]), Stroock ([St00]) and Hsu ([Hs02]).Arbitrage is seen as curvature of a principal fibre bundle representing the market which defines thequantity of arbitrage associated to it. The zero curvature condition is a weaker condition than (NFLVR).It becomes equivalent under additional assumptions introduced for a guiding example, a market whoseasset prices are Itˆo processes. In general, the zero curvature condition is equivalent to the no-unbounded-profit-with-bounded-risk condition, as we prove in Section 3, where we analyze the relationship betweenarbitrage and expected utility maximization. In Section 4 GAT is applied to prove an extension ofthe Black Scholes PDE in the case of markets allowing for arbitrage. Appendix A reviews Nelson’sstochastic derivatives. Section 5 concludes.
In this section we explain the main concepts of Geometric Arbitrage Theory introduced in [Fa15], towhich we refer for proofs and examples.
In this subsection we will summarize the classical set up, which will be rephrased in section (2.4) indifferential geometric terms. We basically follow [HuKe04] and the ultimate reference [DeSc08].We assume continuous time trading and that the set of trading dates is [0 , + ∞ [. This assumptionis general enough to embed the cases of finite and infinite discrete times as well as the one with a finite3orizon in continuous time. Note that while it is true that in the real world trading occurs at discretetimes only, these are not known a priori and can be virtually any points in the time continuum. Thismotivates the technical effort of continuous time stochastic finance.The uncertainty is modelled by a filtered probability space (Ω , A , P ), where P is the statistical(physical) probability measure, A = {A t } t ∈ [0 , + ∞ [ an increasing family of sub- σ -algebras of A ∞ and(Ω , A ∞ , P ) is a probability space. The filtration A is assumed to satisfy the usual conditions, that is • right continuity: A t = T s>t A s for all t ∈ [0 , + ∞ [. • A contains all null sets of A ∞ .The market consists of finitely many assets indexed by j = 1 , . . . , N , whose nominal prices aregiven by the vector valued semimartingale S : [0 , + ∞ [ × Ω → R N denoted by ( S t ) t ∈ [0 , + ∞ [ adapted tothe filtration A . The stochastic process ( S jt ) t ∈ [0 , + ∞ [ describes the price at time t of the j th asset interms of unit of cash at time t = 0. More precisely, we assume the existence of a 0th asset, the cash , astrictly positive semimartingale, which evolves according to S t = exp( R t du r u ), where the predictablesemimartingale ( r t ) t ∈ [0 , + ∞ [ represents the continuous interest rate provided by the cash account: onealways knows in advance what the interest rate on the own bank account is, but this can change fromtime to time. The cash account is therefore considered the locally risk less asset in contrast to theother assets, the risky ones. In the following we will mainly utilize discounted prices , defined asˆ S jt := S jt /S t , 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 atleast one strictly positive asset, in our case the cash. If we want to renormalize the prices by choosinganother asset instead of the cash as reference, i.e. by making it to our num´eraire , then this asset musthave a strictly positive price process. More precisely, a generic num´eraire is an asset, whose nominalprice is represented by a strictly positive stochastic process ( B t ) t ∈ [0 , + ∞ [ , and which is a portfolio of theoriginal assets j = 0 , , , . . . , N . The discounted prices of the original assets are then represented interms of the num´eraire by the semimartingales ˆ S jt := S jt /B t .We assume that there are no transaction costs and that short sales are allowed. Remark that theabsence of transaction costs can be a serious limitation for a realistic model. The filtration A is notnecessarily generated by the price process ( S t ) t ∈ [0 , + ∞ [ : other sources of information than prices areallowed. All agents have access to the same information structure, that is to the filtration A .An admissible strategy x = ( x t ) t ∈ [0 , + ∞ [ is a predictable semimartingale for which the Itˆo integral R t x · S is almost surely t -uniformly bounded from below. Definition 1 ( Arbitrage ) . Let T ≤ + ∞ , the process ( S t ) [0 , + ∞ [ be a semimartingale and ( x t ) t ∈ [0 , + ∞ [ nd admissible strategy. We denote by ( x · S ) T := lim t → T R t x u · S u if such limit exits, and by K thesubset of L (Ω , A T , P ) containing all such ( x · S ) T ) . Then, we define • C := K − L (Ω , A T , P ) . • C := C ∩ L ∞ + (Ω , A T , P ) . • ¯ C : the closure of C in L ∞ with respect to the norm topology. • X V T := (cid:8) ( x · S ) T (cid:12)(cid:12) ( x · S ) = V , x admissible (cid:9) .We say that S satisfies • (NA), no arbitrage , if and only if C ∩ L ∞ (Ω , A T , P ) = { } . • (NFLVR), no-free-lunch-with-vanishing-risk , if and only if ¯ C ∩ L ∞ (Ω , A T , P ) = { } . • (NUPBR), no-unbounded-profit-with-bounded-risk , if and only if X V T is bounded in L forsome V > . The relationship between these three different types of arbitrage has been elucidated in [DeSc94] andin [Ka97] with the proof of the following result.
Theorem 2. (NFLVR) ⇔ (NA) + (NUPBR) . (1) We are going to introduce a more general representation of the market model introduced in section 2.1,which better suits to the arbitrage modeling task.
Definition 3. A gauge is an ordered pair of two A -adapted real valued semimartingales ( D, P ) , where D = ( D t ) t ≥ : [0 , + ∞ [ × Ω → R is called deflator and P = ( P t,s ) t,s : T × Ω → R , which is called termstructure , is considered as a stochastic process with respect to the time t , termed valuation date and T := { ( t, s ) ∈ [0 , + ∞ [ | s ≥ t } . The parameter s ≥ t is referred as maturity date . The followingproperties must be satisfied a.s. for all t, s such that s ≥ t ≥ :(i) P t,s > ,(ii) P t,t = 1 . emark 4. Deflators and term structures can be considered outside the context of fixed income. Anarbitrary financial instrument is mapped to a gauge ( D, P ) with the following economic interpretation: • Deflator: D t is the value of the financial instrument at time t expressed in terms of somenum´eraire. If we choose the cash account, the -th asset as num´eraire, then we can set D jt :=ˆ S jt = S jt S t ( j = 1 , . . . N ) . • Term structure: P t,s is the value at time t (expressed in units of deflator at time t ) of a syn-thetic zero coupon bond with maturity s delivering one unit of financial instrument at time s . Itrepresents a term structure of forward prices with respect to the chosen num´eraire.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 positivesemimartingale to obtain another suitable set of deflators. Of course term structures have to be modifiedaccordingly. The term ”deflator” is clearly inspired by actuarial mathematics. In the present context itrefers to a nominal asset value up division by a strictly positive semimartingale (which can be the stateprice deflator if this exists and it is made to the num´eraire). There is no need to assume that a deflatoris a positive process. However, if we want to make an asset to our num´eraire, then we have to makesure that the corresponding deflator is a strictly positive stochastic process. We want now to introduce transforms of deflators and term structures in order to group gauges con-taining the same (or less) stochastic information. That for, we will consider deterministic linear combi-nations of assets modelled by the same gauge (e. g. zero bonds of the same credit quality with differentmaturities).
Definition 5.
Let π : [0 , + ∞ [ −→ R be a deterministic cashflow intensity (possibly generalized) function.It induces a gauge transform ( D, P ) π ( D, P ) := (
D, P ) π := ( D π , P π ) by the formulae D πt := D t Z + ∞ dh π h P t,t + h P πt,s := R + ∞ dh π h P t,s + h R + ∞ dh π h P t,t + h . (2) Proposition 6.
Gauge transforms induced by cashflow vectors have the following property: (( D, P ) π ) ν = (( D, P ) ν ) π = ( D, P ) π ∗ ν , (3)6 here ∗ denotes the convolution product of two cashflow vectors or intensities respectively: ( π ∗ ν ) t := Z t dh π h ν t − h . (4)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 7.
The term structure can be written as a functional of the instantaneous forward rate f defined as f t,s := − ∂∂s log P t,s , P t,s = exp Å − Z st dhf t,h ã . (5) and r t := lim s → t + f t,s (6) is termed short rate . Remark 8.
Since ( P t,s ) t,s is a t -stochastic process (semimartingale) depending on a parameter s ≥ t ,the s -derivative can be defined deterministically, and the expressions above make sense pathwise in aboth classical and generalized sense. In a generalized sense we will always have a D ′ derivative for any ω ∈ Ω ; this corresponds to a classic s -continuous derivative if P t,s ( ω ) is a C -function of s for any fixed t ≥ and ω ∈ Ω . Remark 9.
The special choice of vanishing interest rate r ≡ or flat term structure P ≡ for allassets corresponds to the classical model, where only asset prices and their dynamics are relevant. Now we are in the position to rephrase the asset model presented in subsection 2.1 in terms of a naturalgeometric language. Given N base assets we want to construct a portfolio theory and study arbitrageand thus we cannot a priori assume the existence of a risk neutral measure or of a state price deflator. Interms of differential geometry, we will adopt the mathematician’s and not the physicist’s approach. Themarket model is seen as a principal fibre bundle of the (deflator, term structure) pairs, discounting andforeign exchange as a parallel transport, num´eraire as global section of the gauge bundle, arbitrage ascurvature. The no-free-lunch-with-vanishing-risk condition is proved to be equivalent to a zero curvaturecondition. 7 .4.1 Market Model as Principal Fibre Bundle Let us consider -in continuous time- a market with N assets and a num´eraire. A general portfolio attime t is described by the vector of nominals x ∈ X , for an open set X ⊂ R N . Following Definition 3,the asset model induces for j = 1 , . . . , N the gauge( D j , P j ) = (( D jt ) t ∈ [0 , + ∞ [ , ( P jt,s ) s ≥ t ) , (7)where D j denotes the deflator and P j the term structure. This can be written as P jt,s = exp Å − Z st f jt,u du ã , (8)where f j is the instantaneous forward rate process for the j -th asset and the corresponding short rateis given by r jt := lim u → + f jt,u . For a portfolio with nominals x ∈ X ⊂ R N we define D xt := N X j =1 x j D jt f xt,u := N X j =1 x j D jt P Nk =1 x k D kt f jt,u P xt,s := exp Å − Z st f xt,u du ã . (9)The short rate writes r xt := lim u → + f xt,u = N X j =1 x j D jt P Nk =1 x k D kt r jt . (10)The image space of all possible strategies reads M := { ( x, t ) ∈ X × [0 , + ∞ [ } . (11)In subsection 2.3 cashflow intensities and the corresponding gauge transforms were introduced. Theyhave the structure of an Abelian semigroup G := E ′ ([0 , + ∞ [ , R ) = { F ∈ D ′ ([0 , + ∞ [) | supp( F ) ⊂ [0 , + ∞ [ is compact } , (12)where the semigroup operation on distributions with compact support is the convolution (see [H¨o03],Chapter IV), which extends the convolution of regular functions as defined by formula (4). Definition 10.
The
Market Fibre Bundle is defined as the fibre bundle of gauges B := { ( D xt , P xt, · ) π | ( x, t ) ∈ M, π ∈ G ∗ } . (13)8he cashflow intensities defining invertible transforms constitute an Abelian group G ∗ := { π ∈ G | it exists ν ∈ G such that π ∗ ν = [0] } ⊂ E ′ ([0 , + ∞ [ , R ) . (14)From Proposition 6 we obtain Theorem 11.
The market fibre bundle B has the structure of a G ∗ -principal fibre bundle given by theaction B × G ∗ −→ B (( D, P ) , π ) ( D, P ) π = ( D π , P π ) (15) The group G ∗ acts freely and differentiably on B to the right. Let us consider the projection of B onto Mp : B ∼ = M × G ∗ −→ M ( x, t, g ) ( x, t ) (16)and its differential map at ( x, t, g ) ∈ B denoted by T ( x,t,g ) p , see for example, Definition 0.2.5 in ([Bl81]) T ( x,t,g ) p : T ( x,t,g ) B | {z } ∼ = R N × R × R [0 , + ∞ [ −→ T ( x,t ) M | {z } ∼ = R N × R . (17)The vertical directions are V ( x,t,g ) B := ker (cid:0) T ( x,t,g ) p (cid:1) ∼ = R [0 , + ∞ [ , (18)and the horizontal ones are H ( x,t,g ) B ∼ = R N +1 . (19)An Ehresmann connection on B is a projection T B → VB . More precisely, the vertical projection musthave the form Π v ( x,t,g ) : T ( x,t,g ) B −→ V ( x,t,g ) B ( δx, δt, δg ) (0 , , δg + Γ( x, t, g ) . ( δx, δt )) , (20)9nd the horizontal one must readΠ h ( x,t,g ) : T ( x,t,g ) B −→ H ( x,t,g ) B ( δx, δt, δg ) ( δx, δt, − Γ( x, t, g ) . ( δx, δt )) , (21)such that Π v + Π h = B . (22)Stochastic parallel transport on a principal fibre bundle along a semimartingale is a well defined con-struction (cf. [HaTh94], Chapter 7.4 and [Hs02] Chapter 2.3 for the frame bundle case) in terms ofStratonovic integral. Existence and uniqueness can be proved analogously to the deterministic case byformally substituting the deterministic time derivative ddt with the stochastic one D corresponding tothe Stratonovich integral.Following Ilinski’s idea ([Il01]), we motivate the choice of a particular connection by the fact that itallows to encode foreign exchange and discounting as parallel transport. Theorem 12.
With the choice of connection χ ( x, t, g ) . ( δx, δt ) := Ç D δxt D xt − r xt δt å g, (23) the parallel transport in B has the following financial interpretations: • Parallel transport along the nominal directions ( x -lines) corresponds to a multiplication by anexchange rate. • Parallel transport along the time direction ( t -line) corresponds to a division by a stochastic discountfactor. Recall that time derivatives needed to define the parallel transport along the time lines have tobe understood in Stratonovich’s sense. We see that the bundle is trivial, because it has a globaltrivialization, but the connection is not trivial.
Remark 13.
An Ehresmann connection on B is called principal Ehresmann connection if and onlyif the decomposition T ( x,t,g ) B = V ( x,t,g ) B ⊕ H ( t,x,g ) B is invariant under the action of G ∗ . Equivalently,the corresponding connection 1-form χ must be smooth with respect to x, t and g and G ∗ -invariant, hich is the case, since, for arbitrary ( x, t, g ) ∈ B and a ∈ G ∗ ( R a ∗ ) χ ( x, t, g ) . ( δx, δt ) = dds (cid:12)(cid:12)(cid:12) s =0 g exp Ç s Ç D δxt D xt − r xt δt å g å · a = dds (cid:12)(cid:12)(cid:12) s =0 g · a exp Ç s Ç D δxt D xt − r xt δt å g å = χ ( x, t, g · a ) . ( δx, δt ) , where R a denotes the (right) action of a ∈ G ∗ and R a ∗ is the differential of the mapping R a : G ∗ → G ∗ . D Differentiable Market Model
We continue to reformulate the classic asset model introduced in subsection 2.1 in terms of stochasticdifferential geometry.
Definition 14. A Nelson D differentiable market model for N assets is described by N gaugeswhich are Nelson D differentiable with respect to the time variable. More exactly, for all t ∈ [0 , + ∞ [ and s ≥ t there is an open time interval I ∋ t such that for the deflators D t := [ D t , . . . , D Nt ] † and theterm structures P t,s := [ P t,s , . . . , P Nt,s ] † , the latter seen as processes in t and parameter s , there exist a D t -derivative. The short rates are defined by r t := lim s → t − ∂∂s log P ts .A strategy is a curve γ : I → X in the portfolio space parameterized by the time. This means thatthe allocation at time t is given by the vector of nominals x t := γ ( t ) . We denote by ¯ γ the lift of γ to M , that is ¯ γ ( t ) := ( γ ( t ) , t ) . A strategy is said to be closed if it represented by a closed curve. A D -admissible strategy is predictable and D -differentiable. In general the allocation can depend on the state of the nature i.e. x t = x t ( ω ) for ω ∈ Ω. Proposition 15. A D -admissible strategy is self-financing if and only if D ( x t · D t ) = x t · D D t − D ∗ h x, D i t or D x t · D t = − D ∗ h x, D i t , (24) almost surely. For the reminder of this paper unless otherwise stated we will deal only with D differentiable marketmodels, D differentiable strategies, and, when necessary, with D differentiable state price deflators. AllItˆo processes are D differentiable, so that the class of considered admissible strategies is very large.11 .4.4 Arbitrage as Curvature The Lie algebra of G is g = R [0 , + ∞ [ (25)and therefore commutative. The g -valued connection 1-form writes as χ ( x, t, g )( δx, δt ) = Ç D δxt D xt − r xt δt å g, (26)or as a linear combination of basis differential forms as χ ( x, t, g ) = D xt N X j =1 D jt dx j − r xt dt ! g. (27)The g -valued curvature 2-form is defined as R := dχ + [ χ, χ ] , (28)meaning by this, that for all ( x, t, g ) ∈ B and for all ξ, η ∈ T ( x,t ) MR ( x, t, g )( ξ, η ) := dχ ( x, t, g )( ξ, η ) + [ χ ( x, t, g )( ξ ) , χ ( x, t, g )( η )] . (29)Remark that, being the Lie algebra commutative, the Lie bracket [ · , · ] vanishes. After some calculationswe obtain R ( x, t, g ) = gD xt N X j =1 D jt Ä r xt + D log( D xt ) − r jt − D log( D jt ) ä dx j ∧ dt, (30)summarized as Proposition 16 ( Curvature Formula ) . Let R be the curvature. Then, the following quality holds: R ( x, t, g ) = gdt ∧ d x [ D log( D xt ) + r xt ] . (31)We can prove following results which characterizes arbitrage as curvature. Theorem 17 ( 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 β = ( β t ) t ≥ such that deflators and short rates satisfy for ll portfolio nominals and all times the condition r xt = −D log( β t D xt ) . (32) (iii) There exists a positive semimartingale β = ( β t ) t ≥ such that deflators and term structures satisfyfor all portfolio nominals and all times the condition P xt,s = E t [ β s D xs ] β t D xt . (33)This motivates the following definition. Definition 18.
The market model satisfies the zero curvature (ZC) if and only if the curvaturevanishes a.s.
Therefore, we have following implication relying two different definitions of no-abitrage:
Corollary 19. (NFLVR) ⇒ (ZC) . (34)As an example to demonstrate how the most important geometric concepts of section 2 can be appliedwe consider an asset model whose dynamics is given by a multidimensional Itˆo-process. Let us considera market consisting of N + 1 assets labeled by j = 0 , , . . . , N , where the 0-th asset is the cash accountutilized as a num´eraire. Therefore, as explained in the introductory subsection 2.1, it suffices to modelthe price dynamics of the other assets j = 1 , . . . , N expressed in terms of the 0-th asset. As vectorvalued semimartingales for the discounted price process ˆ S : [0 , + ∞ [ × Ω → R N and the short rate r : [0 , + ∞ [ × Ω → R N , we chose the multidimensional Itˆo-processes given by d ˆ S t = ˆ S t ( α t dt + σ t dW t ) dr t = a t dt + b t dW t , (35)where • ( W t ) t ∈ [0 , + ∞ [ is a standard P -Brownian motion in R K , for some K ∈ N , and, • ( σ t ) t ∈ [0 , + ∞ [ , ( α t ) t ∈ [0 , + ∞ [ are R N × K -, and respectively, R N - valued locally bounded predictablestochastic processes, • ( b t ) t ∈ [0 , + ∞ [ , ( a t ) t ∈ [0 , + ∞ [ are R N × K -, and respectively, R N - valued locally bounded predictablestochastic processes. 13 roposition 20. Let the dynamics of a market model be specified by following Itˆo’s processes d ˆ S t = ˆ S t ( α t dt + σ t dW t ) dr t = a t dt + b t dW t , (36) as above and where we additionally assume that the coefficients • ( α t ) t , ( σ t ) t , and ( r t ) t are predictable, • ( σ t ) t has vanishing quadratic variation.Then, the market model satisfies the (ZC) condition if and only if Span( α t + r t ) = Range( σ t ) = Span( e ) , (37) where e := [1 , , . . . , † . Remark 21.
In the case of the classical model, where there are no term structures (i.e. r ≡ ), thecondition (37) reads as Span( α t ) = Range( σ t ) = Span( e ) .Proof. Let us consider the expression for Itˆo’s integral with respect to Stratonovich’s Z t σ u dW u = Z t σ u ◦ dW u − Z t d h σ, W i u , (38)and take Nelson’s derivative corresponding to the Stratonovich’s integral: D Z t σ u dW u = σ t D W t − D h σ, W i t . (39)Since D W t = W t t (40)and, having the process ( σ t ) t ∈ [0 , + ∞ [ vanishing quadratic covariation, h σ, W i t ≡ , (41)because | h σ, W i t | ≤ h σ, σ i t | {z } ≡ h W, W i t , (42)14e obtain D Z t σ u dW u = σ t W t t , (43)which, inserted into the asset dynamicsˆ S xt = x † ˆ S exp Ç Z t ( α u −
12 diag( σ u σ † u )) du + Z t σ u dW u å , (44)leads to D log ˆ S xt = x † ˆ S xt ˆ S t Å α t −
12 diag( σ t σ † t ) + σ t W t t ã . (45)By Proposition 16 the curvature vanishes if and only if for all x ∈ R N D log ˆ S xt + r xt = C t , (46)for a real valued stochastic process ( C t ) t ≥ . This means that D log ˆ S t + r t = C t e, (47)or α t + r t −
12 diag( σ t σ t † ) + σ t W t t = C t e. (48)Equation (48) is the formulation of the (ZC) condition for the market model (36). By taking on bothsides of (48) lim h → + E t − h [ · ], and utilizing the predictability assumption, we obtain α t + r t −
12 diag( σ t σ t † ) = β t e, (49)where β t := lim h → + E t − h [ C t ] is a predictable process. Therefore, equation (48) becomes σ t W t t = ( C t − β t ) e, (50)and, by taking on both sides lim h → + Var t − h ( · ), since VCM t − h ( W t ) = h I , we obtain, in virtue of thepredictability of ( σ t ) t , 14 diag( σ t σ t † ) = γ t e, (51)where γ t := t lim h → + Var t − h (cid:0) h ( C t − β t ) (cid:1) is a predictable process. Therefore, we see that both α t + r t and σ t W t t are multiples of e and thusSpan( α t + r t ) = Range( σ t ) = Span( e ) . (52)15onversely, if (52) holds true, then diag( σ t σ t † ) ∈ Span( e ) and (48) follows. The proof of the equivalencebetween the (ZC) condition and (37) is completed.We can reformulate the result of Proposition 20 as follows. Corollary 22.
Let { J t , . . . , J Bt } be an orthonormal basis of ker( σ t ) ⊂ R N and ( σ t ) t ≥ is continuouswith bounded variation. The (ZC) condition for the market model (36) is equivalent to ρ t := J † t ( α t + r t ) ≡ ∈ R B , (53) where J t := [ J t , . . . , J Bt ] . Next, we show the equivalence of the (ZC) condition with (NFLVR) in the case of Itˆo’s dynamics.
Proposition 23.
For the market model whose dynamics is specified by Itˆo’s processes d ˆ S t = ˆ S t ( α t dt + σ t dW t ) dr t = a t dt + b t dW t , (54) where we additionally assume that ( σ t ) t ≥ is a continuous stochastic process with bounded variation,the no-free-lunch-with-vanishing risk condition is equivalent with the zero curvature condition.Proof. By Proposition 16 the zero curvature (ZC) condition R = 0 is equivalent with the existence ofa stochastic process ( C t ) t ≥ such that for all i = 1 , . . . , N the equation D log ˆ S it + r it = C t (55)holds. This means that D log ˆ S it = C t − r it log S it S i = Z t ( C u − r iu ) duS it = S i exp Ç Z t C u du å exp Ç − Z t r iu du å . (56)Therefore, there exist a positive stochastic process ( β t ) t ≥ , defined as β t := exp Ç − Z t C u du å (57)16uch that D log( β t D it ) + r it = 0 (58)for all i = 1 , . . . , N . By Theorem 17, if we can prove that ( β t ) t ≥ is a semi-martingale, then we haveproved (NFLVR). And this is the case as we see in the following steps:1. ( r it ) t is a semi-martingale, because it is an Itˆo’s process by definition.2. The equation C t = α it + r it − σ it σ it † + σ it W t t holds true for any i = 1 , . . . , N , and, hence ( C t ) t isa semi-martingale, because sums and products of semi-martingales are semi-martingales, see f.i.[Pr10] Chapter II.3. β t is a semi-martingale as integral of a semi-martingale, see f.i. [Pr10] Chapter II.5. Remark 24.
Condition (37) is always satisfied for the asset model with just one asset. Therefore,by Proposition 23, if the asset instantaneous volatility ( σ t ) t ≥ is a continuous stochastic process withbounded variation, then the no-free-lunch-with-vanishing risk follows. In [Fa15] it is proved that
Proposition 25.
For the market model whose dynamics is specified by Itˆo’s processes d ˆ S t = ˆ S t ( α t dt + σ t dW t ) dr t = a t dt + b t dW t , (59) If E ñ exp Ç Z T Å α xu | σ xu | ã du åô < + ∞ , (60) that is, the Novikov condition for the portfolio instantaneous Sharpe Ratio α xt σ xt , (61) holds for all x ∈ R N , where α xt := x † α t and σ xt := x † σ t are the conditional at time t expectation, and,respectively, volatility of the instantaneous portfolio log return, then the no-free-lunch-with-vanishing-risk condition and the zero-curvature condition are equivalent. Remark 26.
Combining Remark 24 with Proposition 25, we see that the asset model with just one assetsatisfies the no-free-lunch-with-vanishing-risk condition as soon as it satisfies the Novikov condition (60). emark 27. Note that Bessel processes are continuous and thus predictable, but do not have a vanishingquadratic variation and do not satisfy the Novikov condition. Hence, the asset models presented in [Fo15](Example 7.5) and [FoRu13] (page 59) do not fulfill the assumptions of Propositions 23 and 25. Theyare an example of dynamics satisfying (NUPBR) but not (NFLVR).
Let us now consider a utility function, that is a real C -function of a real variable, which is strictlymonotone increasing (i.e. u ′ >
0) and concave (i.e. u ′′ < • At time t − h : D xt − h P xt − h,t + h . • At time t : D xt P xt,t + h . • At time t + h : D xt + h .From now on we make the following Assumptions:(A1):
The market filtration ( A t ) t ≥ is the coarsest filtration for which ( D t ) t ≥ is adapted. (A2): The process ( D t ) t ≥ is Markov with respect to the filtration ( A t ) t ≥ . Proposition 28.
Under the assumptions ( A and ( A the synthetic bond portfolio instantaneousreturn can be computed as:Ret xt := lim h → + E t ñ D xt + h − D xt − h P xt − h,t + h hD xt − h P xt − h,t + h ô = D log( D xt ) + r xt . (62) Proof.
Under the assumptions ( A
1) and ( A
2) the conditional expectations with respect to the marketfiltration ( A t ) t ≥ are the same as those computed with respect to the present ( N t ) t ≥ , past ( P t ) t ≥ and18uture ( F t ) t ≥ filtrations (see Appendix A). Therefore, we can develop the instantaneous return aslim h → + E t ñ D xt + h − D xt − h P xt − h,t + h hD xt − h P xt − h,t + h ô == lim h → + E t ñ D xt + h − D xt − h hD xt − h P xt − h,t + h + 1 − P xt − h,t + h hP xt − h,t + h ô == 1 D xt D D xt + lim h → + exp Ä R t + ht − h ds f xt − h,s ä − h = D log D xt + r xt . (63) Remark 29.
This portfolio of synthetic zero bonds in the theory corresponds to a portfolio of futuresin practice. If the short rate vanishes, then the future corresponds to the original asset.
Definition 30 ( 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 x = { x h } h ≥ s E s ñ u Ç exp Ç Z Ts dt ( D log( D x t t ) + r x t t ) å D x s s P x s s,T åô , (64) where the supremum is taken over all D -differentiable self-financing strategies x = { x u } u ≥ . Now we can formulate the first result of this subsection.
Theorem 31.
Under the assumptions ( A and ( A the market curvature vanishes if and only ifthe expected utility maximization problem can be solved for all times and horizons for a chosen utilityfunction. This result can be seen as the natural generalization of the corresponding result in discrete time, asTheorem 3.5 in [F¨oSc04], see also [Ro94]. Compare with Bellini’s, Frittelli’s and Schachermayer’s resultsfor infinite dimensional optimization problems in continuous time, see Theorem 22 in [BeFr02] andTheorem 2.2 in [Scha01]. Nothing is said about the fulfilment of the no-free-lunch-with-vanishing-riskcondition: only the weaker zero curvature condition is equivalent to the maximization of the expectedutility at all times for all horizons.
Proof.
The optimization problem (64) into a standard problem of stochastic optimal theory in contin-uous time which can be solved by means of a fundamental solution of the Hamilton-Jacobi-Bellmanpartial differential equation.However, there is a direct method, using Lagrange multipliers. First, remark that problem (30) isa concave optimization problem with convex domain and concave utility function and has therefore a19nique solution corresponding to a global maximum. The Lagrange principal function correspondingto the this maximum problem writesΦ( x, λ ) := E s ñ u Ç exp Ç Z Ts dt ( D log( D x t t ) + r x t t ) å D x s s P x s s,T å + − Z Ts dt λ t ( D x t · D t + 12 D ∗ h x, D i t ) ô . (65)Note that the Lagrange multiplier λ corresponding to the self financing condition is a stochastic process( λ t ) t ≥ . We assume that the Lagrange multiplier belongs to C , (see Appendix A). Since the strategy( x t ) t ≥ is a predictable process with continuous paths (being D -differentiable), then h x, D i t ≡ x, λ ) := E s ñ u Ç exp Ç Z Ts dt ( D log( D x t t ) + r x t t ) å D x s s P x s s,T å − Z Ts dt λ t ( D x t · D t ) ô . (66)To solve the maximization problem for Φ with respect to the processes ( x t ) and ( λ t ) we embed theoptimal solution into a one parameter family as x t ( ǫ ) := x t + ǫδx t λ t ( η ) := λ t + ηδλ t , (67)where ǫ and η are real parameters defined in a neighbourhood of 0 and δx t and δλ t are arbitraryvariations such that the boundary conditions x s ( ǫ ) ≡ x s x T ( ǫ ) ≡ x T , (68)are satisfied. The Lagrange principal equations associated to this maximization problem read ∂ Φ ∂ǫ (cid:12)(cid:12) ǫ = η :=0 = E s î u ′ Ä exp Ä R Ts dt ( D log( D x t t ) + r x t t ) ä D x s s P x s s,T ä ·· exp Ä R Ts dt ( D log( D x t t ) + r x t t ) ä D x s s P x s s,T ·· R Ts dt ∂∂x ( D log( D xt ) + r xt ) (cid:12)(cid:12)(cid:12) x = x t · δx t − R Ts dt λ t ( D δx t · D t ) ò = 0 ∂ Φ ∂η (cid:12)(cid:12)(cid:12) ǫ = η :=0 = − R Ts dt δλ t ( D x t · D t ) = 0 , (69)20ntegration by parts with respect to the time variable shows that − Z Ts dt λ t ( D δx t · D t ) = Z Ts dt D ( λ t D t ) · δx t , (70)which, inserted into the first equation of (69) leads to E s ñ Z Ts dt Ç M ∂∂x ( D log( D xt ) + r xt ) (cid:12)(cid:12)(cid:12)(cid:12) x = x t + D ( λ t D t ) å · δx t ô = 0 , (71)where M := u ′ Ç exp Ç Z Ts dt ( D log( D x t t ) + r x t t ) å D x s s P x s s,T å exp Ç Z Ts dt ( D log( D x t t ) + r x t t ) å D x s s P x s s,T (72)is a strictly positive random variable. Since the variation δx t is arbitrary we infer from (71) M ∂∂x ( D log( D xt ) + r xt ) (cid:12)(cid:12)(cid:12)(cid:12) x = x t + D ( λ t D t ) = 0 for any t ∈ [ s, T ] , (73)and, thus, for the choice t := s , it follows, being the initial condition x s ∈ R N arbitrary D log( D xt ) + r xt = − M D ( λ t D jt ) x j + C jt for all j = 1 , . . . , N, (74)for a stochastic process ( C jt ) t ≥ . Therefore − M D ( λ t D jt ) x j + C jt = − M D ( λ t D it ) x i + C it for all j = i, (75)which can hold true if and only if D ( λ t D jt ) = 0 C jt = C t (76)for all j = 1 , . . . , N . Hence, for the optimal Lagrange multiplier, D ( λ t D t ) = 0 ∈ R N , (77)and D log( D xt ) + r xt = C t for all j = 1 , . . . , N. (78)Therefore, by Proposition 16, the curvature must vanish, which means that the existence of a solutionto the maximization problem is equivalent to the vanishing of the curvature.21t turns out that the two weaker notions of arbitrage, the zero curvature and the no-unbounded-profit-with-bounded-risk are equivalent. Theorem 32. (ZC) ⇔ ( N U P BR ) . (79)Therefore, we have Corollary 33. (NFLVR) ⇔ (NA) + (ZC) . (80) Proof of Theorem 32.
By Proposition 2.1 (4) in [HuSc10] the (NUPBR) is equivalent with the existenceof a growth optimal portfolio. We apply the classic set up of portfolio optimization to the portfolioof futures under consideration, (which covers as a special case the portfolio of base assets). Since thevalue of the portfolio at time s is D x s s P x s s,T , (81)and the growth factor from s to T isexp Ç Z Ts dt ( D log( D x t t ) + r x t t ) å , (82)the solution of the expected utility maximization for s := 0 and arbitrary T with utility function u := logis the optimal growth portfolio. Therefore, the equivalence follows.The equivalence of expected utility maximization and (NFLVR) can be proved for a particular choiceof a Markov dynamics. Namely, if the asset dynamics follows an Itˆo’s process, Proposition 23 andTheorem 31 lead to Corollary 34.
For the market model whose dynamics is specified by an Itˆo’s process (36), the (NFLVR)condition holds true if and only if the expected utility maximization problem can be solved for all timesand horizons for a chosen utility function. Arbitrage and Derivative Pricing
For markets allowing for arbitrage we are in the position to derive the price dynamics of derivativeswhose underlying following an Itˆo’s process. It is a non linear PDE which coincides with the linearBlack-Scholes PDE as soon as the arbitrage vanishes.
Theorem 35.
Let us consider a market consisting in a bank account, an asset and a derivative whosediscounted prices X t and Φ( t, X t ) follow an Itˆo’s process. In particular dX t = X t ( α t dt + σ t dW t ) , (83) where ( α t ) t ∈ [0 , + ∞ [ and ( σ t ) t ∈ [0 , + ∞ [ are real valued predictable continuous processes, the latter withbounded variation. The derivative discounted price solves the PDE ∂ Φ ∂t + σ t X t ∂ Φ ∂x = ρ t Φ Ç Å ∂ Φ ∂x X t ã å , (84) where ρ t , defined in (53) measures the arbitrage allowed by the market.Proof. We prove this theorem in the context of Corollary 22 with vanishing short rate r t . By assumption,choosing N := 2 and B := 1, the market dynamics reads d ˆ S t = ˆ S t (¯ α t dt + ¯ σ t dW t ) , (85)where ˆ S t := X t Φ( t, X t ) , ¯ α t := α t β t , ¯ σ t := σ t τ t . (86)for appropriate real valued predictable processes ( β t ) t ∈ [0 , + ∞ [ and ( τ t ) t ∈ [0 , + ∞ [ characterizing the dy-namics of the derivative. We apply Itˆo’s Lemma to the second component of (85). By comparingdeterministic and stochastic terms we obtain ∂ Φ ∂t + ∂ Φ ∂x X t α t + σ t ∂ Φ ∂x X t = β t Φ ∂ Φ ∂x X t σ t = τ t Φ . (87)23he one dimensional ker(¯ σ t ) is spanned by J t := ( σ t + τ t ) − − τ t + σ t , (88)and the vector ¯ α t admits the decomposition¯ α t = λ t ¯ σ t + ρ t J t , (89)for reals λ t and ρ t = ¯ α t † J t . Now we can insert (89) into (87) and eliminate λ t , since the λ t terms cancelout. The first equation of (87) becomes ∂ Φ ∂t + σ t X t ∂ Φ ∂x = ρ t X t ∂ Φ ∂x Å σ t + τ t τ t ã . (90)The second equation of (87) can be written as σ t τ t = Φ X t ∂ Φ ∂x , (91)which, inserted into (90) gives (84). Remark 36.
In [FaVa12], utilizing another measure of arbitrage ˜ ρ t , the PDE ∂ Φ ∂t + σ t X t ∂ Φ ∂x = −√ ρ t Φ ñ X t Φ Å ∂ Φ ∂x ã − ∂ Φ ∂x ô , (92) was derived. After some computations, it turns out that ˜ ρ t = − √ X t Φ (cid:0) ∂ Φ ∂x (cid:1) − X t Φ ∂ Φ ∂x + X t Φ (cid:0) ∂ Φ ∂x (cid:1) ! ρ t , (93) thus guaranteeing that both (87) and (92) are two representations of the same non linear Black-ScholesPDE for the price of a derivative in the presence of arbitrage. It is possible to reformulate Theorem 35 directly in terms of prices and not discounted prices.
Corollary 37.
Let us consider a market consisting in a bank account with constant instantaneous riskfree rate r , an asset and a derivative whose prices S t and Ψ( t, S t ) follow an Itˆo’s process. In particular dS t = S t ( α t dt + σ t dW t ) , (94)24 here ( α t ) t ∈ [0 , + ∞ [ and ( σ t ) t ∈ [0 , + ∞ [ are real valued predictable continuous processes, the latter withbounded variation. The derivative price solves the PDE ∂ Ψ ∂t + rS t ∂ Ψ ∂s + σ t S t ∂ Ψ ∂s − r Ψ = ρ t Ψ Ç Å ∂ Ψ ∂s S t ã å , (95) where ρ t , defined in (53) measures the arbitrage allowed by the market. Note that in the (ZC) case (95) becomes the celebrated linear Black-Scholes PDE well known fromtextbooks.
Proof.
In the equation (84) we insert Φ( t, x ) = Ψ( t, s ) e − rt x = e − rt s, (96)and, taking into account that ∂∂x = e rt ∂∂s ∂ ∂x = e rt ∂ ∂s ∂s∂t = rs, (97)we obtain, after some algebra equation (95). Corollary 38 ( Put Call Parity in the Presence of Arbitrage ) . Under the same assumptions asCorollary 37 the put call parity relationship holds true, even if the no arbitrage condition is not satisfied.More exactly, the prices C t of a call and, respectively, P t of a put option on an underlying S t , both withstrike price K at maturity T , satisfy the equality C t − P t = S t − Ke − r ( T − t ) . (98) Proof.
The derivative constituted by the long call and the short put has the same terminal conditionas the forward, namely Ψ(
T, S T ) = S T . They both solve the same PDE. Therefore, by the uniquenesstheorem of the solution of PDEs, the values of derivative and forward must coincide and the put callparity equation follows. Remark 39.
This result is in line with the representation of assets with the stochastic cashflow streamsthe generate: an asset built as a portfolio with a long call and a short put has the same stochasticcashflow stream as a long forward. Therefore, they must have the same price, wether there is arbitrageor not. .2 Approximate Solution of the Modified Black-Scholes PDE In this subsection we derive a dependence relation between a call option price, the price of its underlyingand the arbitrage measure ρ in an implicit form. For this purpose, we assume that the arbitrage measure ρ t ≡ ρ is constant during the period considered, typically between 0 and the derivative maturity T .As Vazquez and Farinelli [FaVa12] discussed empirically, arbitrage measure is relatively small so weconsider perturbations with respect to ρ and seek an approximate solution of the modified BSPDE(84). We note that the non linear term of the modified BSPDE (84) is multiplied by ρ linearly. Theorem 40.
For sufficiently small ρ > , an approximated solution of the modified Black-ScholesPDE (84) under the terminal condition Φ( T, X T ) = ( X T − K ) + , where K is the strike price at time T on the discounted value of the underlying with constant volatility σ , is given by Φ( t, X t ) = Ke log XtK − σ ( T − t ) u Å σ ( T − t ) , log X t K ã , (99) where u ( τ, y ) = u ( τ, y ) + ρU ( τ, y ) + ρ U ( τ, y ) (100) and u ( τ, y ) is the solution of ( ∂ τ − ∂ y ) u ( τ, y ) = 0 with the initial condition u (0 , y ) = max { e y − e − y , } ,and f ( v , v ) := 2 Kσ … v + v v + v G ( τ, y ; s, z ) := 12 p π ( τ − s ) exp Å − ( y − z ) τ − s ) ã U ( τ, y ) := Z τ ds Z ∞−∞ dz G ( τ, y ; s, z ) f ( u ( s, z ) , u ′ ( s, z )) U ( τ, y ) := Z τ ds Z ∞−∞ dz G ( τ, y ; s, z ) [ f. ( u ( s, z ) , u ′ ( s, z )) U ( s, z )+ f. ( u ( s, z ) , u ′ ( s, z )) U ′ ( s, z )] . (101) The prime ′ denotes the derivative with respect to the second argument and f. j is the derivative of thefunction f with respect to the j th variable.Proof. By means of the change of variables as x = Ke y , t = T − τ /σ and ∂∂t = − σ ∂∂τ , ∂∂x = 1 x ∂∂y , (102)26he modified BSPDE (84) and the terminal condition Φ( T, X T ) = ( X T − K ) + are rewritten for theunknown function v ( τ, y ) := K − Φ( t, x ) as ∂v ( τ, y ) ∂τ = ∂ v ( τ, y ) ∂y − ∂v ( τ, y ) ∂y + 2 ρKσ s v ( τ, y ) + Å ∂v ( τ, y ) ∂y ã v (0 , y ) = max { e y − , } . By introducing the new unknown function u = u ( τ, y ) defined as v ( τ, y ) = e y − τ u ( τ, y ), we obtain thecanonical form of diffusion equation ∂u∂τ = ∂ u∂y + ρf ( u ( τ, y ) , u ′ ( τ, y )) . (103)Here the terminal condition is changed to u (0 , y ) = max { e y − e − y , } . By introducing an unknownfunction B ( k, τ ), suppose that the solution of (103) has the form u ( τ, y ) = u ( τ, y ) + Z ∞−∞ √ π B ( k, τ ) e iky dk, (104)where u ( τ, y ) is the solution for the case ρ = 0, i.e., ( ∂ τ − ∂ y ) u ( τ, y ) = 0. Thus, u ( τ, y ) = Z ∞−∞ G ( τ, y ; 0 , z ) max { e z − e − z , } dz. Inserting the representation of u ( τ, y ) into (103) yields( ∂ τ − ∂ y ) u = Z ∞−∞ √ π n ∂B ( k, τ ) ∂τ + k B ( k, τ ) o e iky = ρf ( u, u ′ ) . Via Fourier transform, ∂B ( k, τ ) ∂τ = − k B ( k, τ ) + ρ ˜ f ( τ, k ) , (105)where ˜ f ( τ, k ) = Z ∞−∞ √ π f (cid:0) u ( τ, y ) , u ′ ( τ, y ) (cid:1) e − iky dy. We solve (105) via variation of parameters. By introducing new function ˜ B ( k, τ ), we assume that thesolution has the form B ( k, τ ) = e − k τ ˜ B ( k, τ ) . (106)Inserting this into (105) gives e − k τ ∂ ˜ B ( k, τ ) ∂τ = ρ ˜ f ( τ, k ) , B ( k, τ ) = ρ Z τ e k t ˜ f ( t, k ) dt. Consequently, the difference between the arbitrage solution u and the no arbitrage solution u is u ( τ, y ) − u ( τ, y ) = Z ∞−∞ √ π e iky B ( τ, k ) dk = Z ∞−∞ √ π e iky e − k τ ( ρ Z τ e k t ˜ f ( t, k ) dt ) dk = ρ π Z τ Z ∞−∞ (Z ∞−∞ e − k ( τ − s )+ ik ( y − z ) f (cid:0) u ( s, z ) , u ′ ( s, z ) (cid:1) dz ) dk ! ds = ρ Z τ (cid:16)Z ∞−∞ G ( τ, y ; s, z ) f (cid:0) u ( s, z ) , u ′ ( s, z ) (cid:1) dz (cid:17) ds =: ρF [ u ]( τ, y ) . (107)The non linear BSPDE (84) with the terminal condition is therefore equivalent to the functional equation G [ u ] := u − u − ρF [ u ] = 0 , (108)which can be solved by a Newton’s approximation scheme. The first element of the approximationsequence of the solution u is u . The second, u is the solution of the linearization of (108) at u G [ u ] + G ∗ [ u ] . ( u − u ) = 0 , (109)where the star denotes the Gˆateaux derivative. The solution reads u = u + ρ ( − ρF ∗ [ u ]) − .F [ u ]= u + ρ ( + ρF ∗ [ u ]) .F [ u ] + O ( ρ )= u + ρU + ρ F ∗ [ u ] .U + O ( ρ ) ( ρ → , (110)where U := F [ u ] corresponds to (101). We now compute the Gˆateaux derivative of F at u as F ∗ [ u ] .U ( τ, y )= Z τ ds Z ∞−∞ dz G ( τ, y ; s, z ) [ f. ( u ( s, z ) , u ′ ( s, z )) U ( s, z ) + f. ( u ( s, z ) , u ′ ( s, z )) U ′ ( s, z )] . (111)28e can now derive the second order approximate solution for u as u ( τ, y ) = u ( τ, y ) + ρ Z τ ds Z ∞−∞ dz G ( τ, y ; s, z ) f ( u ( s, z ) , u ′ ( s, z ))+ ρ Z τ ds Z ∞−∞ dz G ( τ, y ; s, z ) [ u ( s, z ) f. ( u ( s, z ) , u ′ ( s, z )) + u ′ ( s, z ) f. ( u ( s, z ) , u ′ ( s, z ))]+ O ( ρ ) ( ρ → . (112)By tracing back of the change of variables in (84) we can obtain the solution Φ( t, X t ) as in (99). We apply Geometric Arbitrage Theory to obtain results in Mathematical Finance, which do not needstochastic differential geometry in their formulation. First, we utilize the equivalence between theno-unbounded-profit-with-bounded-risk condition and the expected utility maximization to prove theequivalence between the (NUPBR) condition with the (ZC) condition. Then, we generalize the Black-Scholes PDE to markets allowing arbitrage, proving the extension of the put call parity to this contextand computing an approximated solution for the non linear PDE for a call option with arbitrage.
A Derivatives of Stochastic Processes
In stochastic differential geometry one would like to lift the constructions of stochastic analysis fromopen subsets of R N to N dimensional differentiable manifolds. To that aim, chart invariant definitionsare needed and hence a stochastic calculus satisfying the usual chain rule and not Itˆo’s Lemma isrequired, (cf. [HaTh94], Chapter 7, and the remark in Chapter 4 at the beginning of page 200). Thatis why we will be mainly concerned in this paper by stochastic integrals and derivatives meant in Stratonovich ’s sense and not in
Itˆo ’s.
Definition 41.
Let I be a real interval and Q = ( Q t ) t ∈ I be a vector valued stochastic process on theprobability space (Ω , A , P ) . The process Q determines three families of σ -subalgebras of the σ -algebra A :(i) ”Past” P t , generated by the preimages of Borel sets in R N by all mappings Q s : Ω → R N for < s < t .(ii) ”Future” F t , generated by the preimages of Borel sets in R N by all mappings Q s : Ω → R N for < t < s .(iii) ”Present” N t , generated by the preimages of Borel sets in R N by the mapping Q t : Ω → R N .Let Q = ( Q t ) t ∈ I be a S ( I ) process, i.e. a process with continuous sample paths and adapted to P and F , so that t Q t is a continuous mapping continuous mappings from I to L (Ω , A ) . Assuming thatthe following limits exist, Nelson’s stochastic derivatives are defined as DQ t := lim h → + E ï Q t + h − Q t h (cid:12)(cid:12)(cid:12)(cid:12) P t ò : forward derivative, D ∗ Q t := lim h → + E ï Q t − Q t − h h (cid:12)(cid:12)(cid:12)(cid:12) F t ò : backward derivative, D Q t := DQ t + D ∗ Q t : mean derivative . (113) Let S ( I ) the set of all S ( I ) -processes Q such that t Q t , t DQ t and t D ∗ Q t are continuousmappings from I to L (Ω , A ) . Let C ( I ) the completion of S ( I ) with respect to the norm k Q k := sup t ∈ I (cid:0) k Q t k L (Ω , A ) + k DQ t k L (Ω , A ) + k D ∗ Q t k L (Ω , A ) (cid:1) . (114) Remark 42.
The stochastic derivatives D , D ∗ and D correspond to Itˆo’s, to the anticipative and,respectively, to Stratonovich’s integral (cf. [Gl11]). The process space C ( I ) contains all Itˆo processes.If Q is a Markov process, then the sigma algebras P t (”past”) and F t (”future”) in the definitions offorward and backward derivatives can be substituted by the sigma algebra N t (”present”), see Chapter6.1 and 8.1 in ([Gl11]). Acknowledgements
We would like to extend our gratitude to Claudio Fontana, who highlighted that for a previous uncorrectversion of Proposition 23 Bessel’s processes, which satisfy (NUPBR) but not (NFLVR) as shown in [Fo15]and in [FoRu13], would have been a counterexample, thus leading to the current corrected version.
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