Change of drift in one-dimensional diffusions
aa r X i v : . [ q -f i n . M F ] J u l arXiv preprint Change of drift in one-dimensional diffusions
Sascha Desmettre · Gunther Leobacher · L.C.G. Rogers
July 15, 2020
Abstract
It is generally understood that a given one-dimensional diffusionmay be transformed by Cameron-Martin-Girsanov measure change into an-other one-dimensional diffusion with the same volatility but a different drift.But to achieve this we have to know that the change-of-measure local martin-gale that we write down is a true martingale. We provide a complete charac-terisation of when this happens.This enables us to discuss absence of arbitragein a generalized Heston model including the case where the Feller conditionfor the volatility process is violated.
Keywords
One-dimensional diffusions · change of measure · Heston model · Feller condition · free lunch with vanishing risk Mathematics Subject Classification (2010) · JEL Classification
G130
Our original goal in this paper was to understand how change of measure worksin the celebrated stochastic volatility model of Heston [11]. When this modelis specified, if the growth rate of the asset is not equal to the riskless rate thenwe need to change measure to a pricing measure in which the growth rate is S. Desmettre (corresponding author)Institute of Financial Mathematics and Applied Number Theory, Johannes Kepler UniversityLinz, Altenbergerstraße 69, 4040 Linz, Austria, E-mail: [email protected]. LeobacherDepartment of Mathematics and Scientific Computing, University of Graz, Heinrichstr. 36,8010 Graz, AustriaL.C.G. RogersStatistical Laboratory, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB,United Kingdom Sascha Desmettre et al. the riskless rate. The question then arises: ‘Can this be done?’ The answer wefound was ‘Not always’; and in cases where it cannot be done, then generalresults say that there must be arbitrage (in a suitable sense).We then realized that the question is closely related to changing the givendrift of a one-dimensional diffusion to a different drift, using change of measure.This uses the Cameron-Martin-Girsanov theorem, but as is well known thisvery general result cannot be applied without care, the main point being todecide whether the local martingale we write down to do the change of drift isactually a martingale. In general, this is hard to decide, but in the special casethat concerns us, where the drift is again a function of the diffusion, we areable to derive necessary and sufficient conditions for the change of measure to‘work’; we present this in Section 2, as an algorithm to be followed to decide forany particular situation, and we illustrate this with two interesting examples.It might appear that the result we give is a reprise of the main results of [17,16], but what we do here is actually rather different. A standing assumptionthroughout [17,16] is that if a boundary point of the diffusion can be reached infinite time, then the diffusion stops there. For the application we have in mind,in Section 3, this assumption does not hold; we want to consider CIR processeswhich reach zero and immediately return. Unusually, this behaviour can becompletely specified by an SDE with nice coefficients, but more generally adiffusion which can reflect from a boundary point cannot be specified by anSDE without explicitly involving a local time term, as in the Tanaka SDE forreflecting Brownian motion. While solutions of SDEs are very general regularone-dimensional diffusions, they are not the most general examples; as is wellknown, the most general regular one-dimensional diffusion is specified by itsscale function and speed measure, and for our purposes it is necessary towork in this setting. This requires us to identify the Markov-process formof the Cameron-Martin-Girsanov change-of-measure local martingale, and tounderstand its effect on the generator of the diffusion. All this is explained inSection 2; we are discussing here the transformation of Markov processes bymultiplicative functionals, a topic which goes back a very long way, to [14], [8]and references therein, and which is still of interest nowadays, see e.g. [18], [3].Our work shares common features with [12,13] and [15], who also workwith the scale and speed representation. The question they answer is: If a one-dimensional diffusion X is in natural scale, when is it a martingale, and nota local martingale? Our study determines when the change-of-measure localmartingale Z is a true martingale, which includes the problem of [12,13,15] asthe special case Z = X .In Section 3 we turn to the Heston model for the stock price S and thevolatility v , which defines their evolution in the ‘real-world’ probability P as: dS t /S t = µ ( v t ) dt + √ v t ( ρdW t + ρ ′ dW ′ t ) , (1.1) dv t = κ ( θ − v t ) dt + σ √ v t dW t . (1.2) The main result of [15] is also obtained by [7] using different methods.hange of drift in one-dimensional diffusions 3
Here, W and W ′ are independent Brownian motions, κ , θ and σ are strictlypositive constants, and ρ ∈ ( − ,
1) is the constant correlation between theBrownian motions driving stock and volatility. We write ρ ′ ≡ p − ρ . Thefunction µ is continuous. In Heston’s original paper and many other studies, µ is taken to be constant, as is the riskless rate of interest r . Here we will take r = 0 throughout in order to simplify notation; this loses no generality, as wecould equally consider S defined by (1.1) to be the discounted stock e − rt S t .In option pricing papers on Heston’s stochastic volatility model, it is typi-cally assumed that a risk-neutral measure ˜ P exists and that the dynamics arestated in the corresponding risk-neutral form; see, for example, the extensivetextbook [21] and the references therein. Yet, the question of existence of sucha risk-neutral measure is rarely investigated – save for the trivial case µ ≡ r .But absence of such a risk-neutral measure implies existence of free lunch withvanishing risk, that is, a form of arbitrage! A notable exception, where thisproblem is addressed, is [24], where the authors give a solution to this problemassuming the Feller condition, a condition which keeps the volatility processstrictly positive.However, the Feller condition is frequently violated in practice as has beenpointed out in [1] or [4] (consult in particular Table 6.3). Building on resultsin [16,17], this problem is addressed for several stochastic volatility models in[2], including the classical Heston model, by modifying the model so that thevolatility process is stopped as soon as it hits 0. While this solves the problemsincurred by a violated Feller condition mathematically, this approach is notconvincing from an economic point of view.In Section 3, we show that in the classical Heston model where the function µ is constant, then failure of the Feller condition implies that there is norisk-neutral measure. However, if the drift µ is not constant, but satisfies asimple integrability condition at 0, we show that there is an equivalent localmartingale measure (ELMM), still in the case where the Feller condition is notsatisfied. When the Feller condition is satisfied, we show that there is alwaysan ELMM.In the Appendix, as a gentle amusement, we directly construct a free lunchwith vanishing risk (FLVR), from which if follows by the celebrated Funda-mental Theorem of Asset Pricing (FTAP) of [5,6] that there is no equivalent σ -martingale measure and a fortiori no ELMM. This is a rare application ofthe FTAP!Does it really matter if the Feller condition fails, so that there is no ELMM?It does not; all that has happened is that we started off from a bad place,and what we should do is to immediately put ourselves into the risk-neutralmeasure (in effect, assume that µ ≡ Sascha Desmettre et al.
We are going to begin with a regular diffusion taking values in an interval I ⊆ R . The killing measure is assumed to be zero. We write I ◦ for the interiorof I , which could be equal to I . We also set a = inf I , b = sup I , the endpointsof I . The interval I may be the whole real line, it may contain endpoints ornot. We write C for the space of continuous functions f : I → R with limitsat the endpoints.We let Ω = C ( R + , I ) be the canonical path space with the canonicalprocess X t ( ω ) = ω ( t ) and the raw filtration F ◦ t = σ ( X s : s ≤ t ). If P is thelaw of X on ( Ω, F ◦ ∞ ) we let ( F t ) be the universal completion of ( F ◦ t ). Wewrite F = F ∞ for brevity. We write s for the scale function of X , and m forits speed measure, so that the infinitesimal generator of X is G = 12 d dm ds ; (2.1)see, for example, Theorem VII.3.12 of [19] . If a boundary point is in I ( a =0 ∈ I , to fix ideas), there is a boundary condition there: dfds (0+) = 2 m ( { } ) G f (0) . (2.2)See Proposition VII.3.13 of [19], again noting the different scaling factor here.We omit discussion of the situation m ( { } ) = ∞ , corresponding to absorptionat the boundary, as this is a special case already dealt with in the earlier works,c.f. [16,17]. Moreover, for volatility models, which are our main application,absorbing boundaries are not reasonable from an economic point of view.Specifying the domain of functions on which G acts is important. We fixsome reference point ξ ∈ I ◦ and define the domain D of G to be the set of all f which are represented as f ( x ) = A + Z xξ s ( dw ) (cid:26) B + Z wξ g ( u ) m ( du ) (cid:27) (2.3)for some constants A , B , g ∈ C , and which satisfy the necessary boundaryconditions (2.2) in the case of boundary points in I . If f ∈ D is given as in(2.3), then it is immediate from (2.1) that G f = g , and it can be shown that f ( X t ) − Z t G f ( X u ) du is a local martingale. (2.4)Now fix some measurable h : I → R + satisfying for all c < d ∈ I ◦ that Z [ c,d ] h ( x ) m ( dx ) < ∞ (2.5) .. while noting the different scaling factor for the speed measure there. If a = 0 ∈ I , then we could take ξ = 0 in (2.3) and would then have that B = 0 in orderto match the boundary condition there.hange of drift in one-dimensional diffusions 5 and define A t = Z t h ( X u ) du. (2.6)We write H z ≡ inf { t > X t = z } .The most common use of the first point of the next proposition is when h ≡ Proposition 2.1
Suppose that the endpoint a is accessible: s ( a ) > −∞ . Then:1. The following are equivalent:(i) R a + ( s ( x ) − s ( a )) h ( x ) m ( dx ) < ∞ ;(ii) P [ A ( H a − ) < ∞ ] > .
2. The following are equivalent:(i) R a + h ( x ) m ( dx ) < ∞ ;(ii) P a [ A ( H a +) < ∞ ] = 1 ;(iii) P a [ A ( H a +) < ∞ ] > . Proof.
Writing Y t ≡ s ( X t ), we have that Y is a diffusion in natural scalewith speed measure m Y defined by m Y ( s ( x )) = m ( x ) . (2.7)The additive functional A is thus expressed equivalently as A t = Z t ( h ◦ s − )( Y u ) du. (2.8)Since Y is in natural scale with speed m Y , it can be represented as Y u = B ( τ u ) , τ u = inf { t : Λ t > u } , Λ Yt = Z s ( I ) ℓ ( t, y ) m Y ( dy ) , where ℓ is the local time of the Brownian motion B , see [20, Theorem V.74.1].For Point 1 first assume that for all c, d ∈ I ◦ we have 0 < R I h ( u ) m ( du ).Then we construct another diffusion Z with speed measure m Z defined by m Z ( C ) ≡ Z C ( h ◦ s − )( y ) dm Y ( y ) for all measurable C ⊆ s ( I ) ◦ , which is a regular diffusion by [20, Remark (ii) after (V.47.5)] and by ourassumption (2.5) on h , and Z u = B ( τ Zu ) , τ Zu = inf { t : Λ Zt > u } , Λ Zt = Z s ( I ) ℓ ( t, y ) m Z ( dy ) . W.l.o.g. we may assume s ( a ) = 0, and we write T = inf { t : B t = 0 } , H Y =inf { t : Y t = 0 } (= H a ), H Z = inf { t : Z t = 0 } , and we note that τ Y ( H Y ) = T = τ Z ( H Z ). The occupation time formula [20, eqn.(V.49.2)] also holds for Sascha Desmettre et al. positive measurable (instead of bounded measurable) functions, by monotoneconvergence. Therefore Z u h ( X s ) ds = Z s ( I ) ℓ ( τ u , y )( h ◦ s − )( y ) m Y ( dy ) = Z s ( I ) ℓ ( τ u , z ) m Z ( dz ) = Λ Zτ u , such that for all t < T (where A Y and A Z are strictly increasing) Λ Zt = R Λ Yt h ( X s ) ds , and therefore A ( H Xa − ) = A ( H Y − ) = Z H Y − h ( X s ) ds = H Z . Point 1 now follows immediately from Theorem V.51.2 of [20] applied tothe diffusion Z .Next consider the general case where R [ c,d ] h ( u ) m ( du ) may vanish. Wechoose a positive function f : I → R with R a + f ( u ) m ( du ) < ∞ . By the firstpart, P [ R H a − f ( X s ) ds < ∞ ] >
0, so P [ A ( H a − ) < ∞ ] > ⇔ P [ Z H a − ( h + f )( X s ) ds < ∞ ] > . On the other hand, by our assumption on f , Z a + ( s ( x ) − s ( a )) h ( x ) m ( dx ) < ∞ ⇔ Z a + ( s ( x ) − s ( a )) ( h + f )( x ) m ( dx ) < ∞ . Thus Point 1 for the general case follows from the special case.2. If we start Y at the boundary point 0 and run til the first time T Y at which it reaches 1, then τ Y ( T Y ) = T ≡ { t : B ( t ) = 1 } and some simplecalculus gives us A ( T Y ) = Z T Y ( h ◦ s − )( Y u ) du = Z T ( h ◦ s − )( B v ) dΛ v = Z s ( I ) ( h ◦ s − )( y ) ℓ ( T , y ) m Y ( dy )= Z I h ( x ) ℓ ( T , s ( x )) m ( dx ) . By the Ray-Knight theorem [20, Theorem VI.52.1] the process y ℓ ( T , − y ) is a BESQ(2) diffusion started at 0. So E [ ℓ ( T , − y )] is finite, continuous in y , positive for y >
0. Thus almost surely, ℓ ( T , − y ) is bounded for y in [0 , R a + h ( x ) m ( dx ) < ∞ it follows that A T ( H a +) is a.s finite – part (ii)of the statement – and this implies part (iii) a fortiori. Going from part (iii)to part (i), if it were the case that R a + h ( x ) m ( dx ) = ∞ , then since ℓ ( T , s ( x ))is a.s. bounded away from zero in a neighbourhood of 0, it has to be that A T ( H a +) is a.s. infinite; a contradiction. (cid:3) hange of drift in one-dimensional diffusions 7 Remark 2.2
Of course, there is an analogous statement for an accessible upperboundary point.Now suppose that Z is a non-negative continuous local martingale, Z = 1. Provided Z is a martingale, we can define a new probability ˜ P by the recipe d ˜ PdP (cid:12)(cid:12)(cid:12)(cid:12) F t = Z t , (2.9)To determine whether or not Z is a martingale, define the stopping times T n ≡ inf { t : Z t > n } , n = 2 , , . . . (2.10)which reduce Z , and notice that it is possible to define a probability ˜ P on F T n by d ˜ PdP (cid:12)(cid:12)(cid:12)(cid:12) F Tn = Z T n . (2.11)The definition of ˜ P extends to the field W n F T n . But does ˜ P extend to thewhole of F ? The answer is in this simple result (see [22], Theorem 1.3.5),whose proof we give for completeness. Theorem 2.3
The local martingale Z is a martingale if and only if for each t > P ( T n ≤ t ) → n → ∞ ) . (2.12) Proof.
We have 1 = EZ ( t ∧ T n )= E [ Z t : t < T n ] + E [ Z T n : T n ≤ t ]= E [ Z t : t < T n ] + ˜ P [ T n ≤ t ] . By Monotone Convergence, the first term on the right converges to E [ Z t ], socondition (2.12) is equivalent to the statement that E [ Z t ] = 1 for all t > Z is a martingale. (cid:3) When is the condition (2.12) for Z to be a martingale satisfied? To answerthis, we define the reverse measure transformation˜ Z t ≡ Z t , (2.13)a positive ˜ P -local martingale. Obviously, T n = inf { t : ˜ Z t < n − } , so what wehave to determine is this: Question 1:
Under ˜ P , does ˜ Z reach zero in finite time? Sascha Desmettre et al.
If not, then the change-of-measure local martingale is a martingale.Now we need to be more specific about the local martingales Z that weconsider. If the diffusion X was specified as the solution of an SDE dX t = σ ( X t ) dW t + β ( X t ) dt, X = x , (2.14)with a pathwise-unique strong solution and C coefficients σ > β , then weconsider local martingales Z of the form dZ t = c ( X t ) Z t dW t , Z = 1 , (2.15)where c is assumed C for convenience. The SDE (2.15) has the solution Z t = exp (cid:20)Z t c ( X u ) dW u − Z t c ( X u ) du (cid:21) (2.16)= ϕ ( X t ) exp (cid:20) − Z t G ϕϕ ( X u ) du (cid:21) , (2.17)where G is the generator of X , andlog ϕ ( x ) = Z xx cσ ( y ) dy. (2.18)The equivalence of (2.16) and (2.17) is a simple exercise with Itˆo’s formula,and is beside the point. The point is that the form (2.16) of Z requires thatthe diffusion X is specified as the solution of an SDE, the form (2.17) doesnot . So we shall proceed to assume that Z has the form (2.17) for some strictlypositive function ϕ ∈ D which satisfies ϕ ( x ) = 1. In this generality, it mayhappen that ϕ vanishes in an endpoint a of I . In that case the integral in (2.17)might diverge, but since Z is a non-negative local martingale, and thereforea supermartingale, the limit Z H a − ≡ lim t → H a − Z ( t ) exists and we may set Z t ≡ Z H a − while X t remains in a .The process Z defined by (2.17) is still a local martingale, since usingpartial integration on (2.17) gives dZ t = (cid:16) dϕ ( X t ) − G ϕ ( X t ) dt (cid:17) exp (cid:20) − Z t G ϕϕ ( X u ) du (cid:21) , and dϕ ( X t ) − G ϕ ( X t ) dt is the differential of a local martingale by (2.4).The next question is how the change of measure (2.17) (if it is a change ofmeasure) transforms the diffusion X . To answer this, we let ˜ D be the set ofall functions f such that f ϕ ∈ D .Then using Itˆo’s formula it is a simple exercise to show that that for any f ∈ ˜ D Z t (cid:26) f ( X t ) − Z t ˜ G f ( X u ) du (cid:27) is a local martingale, (2.19)where ˜ G f = 1 ϕ (cid:8) G ( f ϕ ) − f G ϕ (cid:9) . (2.20)The following result relates the form of ˜ G just found to the form (2.1). hange of drift in one-dimensional diffusions 9 Proposition 2.4 ˜ G = 12 d d ˜ m d ˜ s , (2.21) where the scale and speed of X under ˜ P take the simple forms d ˜ m = ϕ dm, d ˜ s = ϕ − ds. (2.22) Proof.
Take some continuous finite-variation test function ψ : I → R whichvanishes off some compact set. In what follows, we shall assume that I is open,so that there are no boundary conditions to deal with, and leave the checkingof what happens in the other cases to the reader. Using integration-by-parts,we develop: Z ψ ( x ) ϕ ( x ) ˜ G f ( x ) m ( dx )= Z ψ ( G ( f ϕ ) − f G ϕ ) dm = Z ψ (cid:26) ddm (cid:18) dds ( f ϕ ) (cid:19) − f d dmds ϕ (cid:27) dm = − Z dds ( f ϕ ) dψ + Z dϕds d ( ψf ) = − Z ϕ dfds dψ + Z ψ dϕds df = − Z ϕ dfds dψ + Z ψ dϕds dfds ds = − Z ϕ dfds dψ + Z ψ dfds dϕ = − Z d (cid:18) ψϕ (cid:19) ϕ dfds = Z ψϕ ddm (cid:18) ϕ dfds (cid:19) dm = Z ψϕ d fd ˜ md ˜ s dm . Since ψ is arbitrary, the result follows. (cid:3) From now on, we shall make the simplifying assumption
Assumption A: ϕ has a continuous density with respect to m .Since ϕ ∈ D by assumption, it is automatic that ϕ has a continuous den-sity with respect to s , but in general not with respect to m . Assumption Awould hold if both s and m had continuous densities with respect to Lebesguemeasure, for example, a situation covering many examples of interest.Next, the ˜ P -local martingale ˜ Z can be expressed as ˜ Z t = exp (cid:20) − log ϕ ( X t ) − Z t ϕ ˜ G (1 /ϕ )( X u ) du (cid:21) (2.23)= exp (cid:2) ˜ M t − h ˜ M i t (cid:3) (2.24)for some continuous ˜ P -local martingale ˜ M . If we make the Itˆo expansion oflog ˜ Z as given in (2.23), we find after some calculations and simplificationsthat the finite-variation part of log ˜ Z is − ˜ h ( X t ) dt, (2.25) The representation (2.23) follows from the equality ϕ − G ϕ = − ϕ ˜ G (1 /ϕ ), an immediateconsequence of (2.20).0 Sascha Desmettre et al. where ˜ h = dϕd ˜ m dϕds = 1 ϕ dϕd ˜ m dϕd ˜ s = 1 ϕ dϕdm dϕds . (2.26)Hence by comparing (2.24) and (2.25) we learn that d h ˜ M i t = ˜ h ( X t ) dt . (2.27)In particular, ˜ h is non-negative. So under Assumption A it is now clear thatto answer Question 1, we have to answer: Question 2:
Does ˜ A t ≡ Z t ˜ h ( X u ) du (2.28)reach infinity in finite time? Remark 2.5
When the diffusion is specified as the solution to an SDE, (2.28)appears at equation (9) in [17], equation (2.6) in [16]. If X is the solution toan SDE, then s ′ ( x ) = exp (cid:0) − Z xx β ( X u ) σ ( X u ) du (cid:1) m ′ ( x ) = 1 σ ( x ) s ′ ( x ) , so ˜ h = 1 ϕ dϕdm dϕds = (cid:18) ϕ ′ ϕ (cid:19) m ′ s ′ = (cid:18) ϕ ′ ϕ (cid:19) σ ( x ) = (log( ϕ ) ′ ) σ ( x ) = (cid:18) c ( x ) σ ( x ) (cid:19) σ ( x ) = c ( x ) . The derivation we have given applies more widely.Of course, Question 2 must be answered in the law ˜ P . If K ⊂ I ◦ is anycompact set, and ζ = inf { t : ˜ A t = ∞} < ∞ , then clearly X must exit K before ζ , because the integrand in (2.28) is bounded on K , by Assumption A.By considering an increasing sequence of compact K n increasing to I ◦ , we seethat if ˜ A reaches infinity in finite time, it has to be at a time when X reachesa boundary point of I . To understand this, we look at the diffusion Y = ˜ s ( X ), which is a diffusionin natural scale under ˜ P , taking values in the interval ˜ s ( I ), whose endpointsare ˜ a ≡ ˜ s ( a ) < ˜ b ≡ ˜ s ( b ). Two cases arise. Case 1: ˜ a and ˜ b are both infinite . Since Y is a continuous local martingale, andtherefore a time-change of Brownian motion , Y cannot reach either endpointin finite time, so the change-of-measure local martingale Z is a true martingale. See, e.g., [19, Chapter V, Theorem 1.7]hange of drift in one-dimensional diffusions 11
Case 2: one at least of ˜ a and ˜ b is finite. To fix ideas, let us suppose that ˜ a isfinite and ˜ b = ∞ , and see what happens at ˜ a ; the treatment at a finite upperboundary point is analogous.Firstly we have to ask whether Y reaches the lower boundary point ˜ a infinite time. According to Proposition 2.1 (with h ≡
1, cf Remark 2.2 (ii)), thishappens if and only if Z a + { ˜ s ( x ) − ˜ s ( a ) } ˜ m ( dx ) < ∞ . (2.29)If Y does not reach ˜ a in finite time, then explosion of ˜ A in finite time isclearly impossible.However, if Y does reach ˜ a in finite time, then the additive functional˜ A may explode at that time. In the situation considered by [17], where thediffusion X stops at a if it ever gets there, then the criterion for explosion is(according to Proposition 2.1) Z a + { ˜ s ( x ) − ˜ s ( a ) } ˜ h ( x ) ˜ m ( dx ) = ∞ . (2.30)On the other hand, if the diffusion Y reflects off ˜ a , then explosion could happenat H a + even though there was no explosion at H a − , and the criterion now for˜ A to explode at H a + is Z a + ˜ h ( x ) ˜ m ( dx ) = ∞ , (2.31)by Proposition 2.1, part 2. For the applications of interest to us, this is therelevant criterion, as the CIR diffusions we deal with later all reflect off theboundary point.Notice that condition (2.31) can be equivalently expressed (due to the form(2.27) of ˜ h and (2.22)) as Z a + (cid:18) dϕds ( x ) dϕdm ( x ) (cid:19) m ( dx ) = ∞ . (2.32)Thus we see that in order to decide whether the local martingale Z is not a true martingale, we have to answer the three questions:1. Is at least one of the endpoints ˜ a , ˜ b of ˜ s ( I ) finite ?2. If ˜ a (say) is finite, does X reach a in finite time (see (2.29)) ?3. If so, does ˜ A explode when X reaches a (see (2.31)) ?To summarize then, we have the following result. Theorem 2.6
Let X be a diffusion on the interval I ⊂ R , X = x ∈ I , a ≡ inf I , b ≡ sup I , with scale function s and speed measure m . We definethe change-of-measure local martingale Z by Z t = ϕ ( X t ) exp (cid:20) − Z t G ϕϕ ( X u ) du (cid:21) , (2.33) where G is the generator of X and ϕ is strictly positive and C with ϕ ( x ) = 1 .We let ˜ P be the probability defined by the change-of-measure local martingale Z via (2.9) , defined on the σ -field F T ∞ , and we denote the scale function andspeed measure of X under ˜ P by ˜ s , ˜ m respectively. These are related to s , m asgiven at (2.22) . Denote ˜ a = ˜ s ( a ) , ˜ b = ˜ s ( b ) . Assume further Assumption A.If all of the following three conditions are satisfied:1. At least one of the endpoints ˜ a, ˜ b is finite;2. At least one of the finite endpoints is reached in finite time (see (2.29) );3. There is a finite endpoint which is reached in finite time and at which theadditive functional ˜ A explodes (see (2.31) or (2.30) ),then the change-of-measure local martingale Z is not a true martingale. Oth-erwise, it is.Remark 2.7 In [3] a similar question to ours is discussed, by giving a class ofabsolutely continuous measure changes using potentials. More specifically, wecharacterize the positive functions ϕ for which Z is a true martingale, whereas[3, Theorem 3.2] shows that if ϕ is a potential, then Z is a true martingale. Thekey differences are that [3] works in an SDE setting and that the statementof [3, Theorem 3.2] assumes X has a semimartingale local time, but the mostgeneral one-dimensional diffusion such as we work with does not have to be asemimartingale.If Z is a martingale, then the recipe (2.9) defines a new measure ˜ P onpath space under which the canonical process is again a regular diffusion. Thelaw ˜ P is therefore absolutely continuous with respect to P , but not in generalequivalent. Here are two interesting examples, where the process is given bythe SDE recipe (2.14) and the change-of-measure local martingale is of theform (2.15). Example 1.
A canonical example is when X solves (2.14) with σ ( x ) ≡ β ( x ) ≡
0, on I = [0 , ∞ ) with X = x >
0. That is, X is of the form X t = x + W H t , where W is a standard Brownian motion and H is the timewhen X hits { } . We want c ( x ) = 1 /x so that X solves the BES(3) SDE dX t = 1 X t dt + d ˜ W t X = x , under ˜ P . In this example, a = 0, b = ∞ , s ( x ) = x and m ′ ( x ) = 1. From(2.18) and (2.22) we find that ϕ ( x ) = x/x , and we may take ˜ s ( x ) = − x /x ,˜ m ′ ( x ) = x /x , and therefore ˜ a = −∞ , ˜ b = 0. According to our method wenext ask whether the finite boundary point ˜ b can be reached in finite time.By the integral test (2.29) (in the analogous form for an upper boundary) theprocess X approaches ∞ (or: ˜ s ( X ) approaches 0) under ˜ P , but never getsthere.Thus by Theorem 2.6 there is an absolutely continuous change of measure,taking Wiener measure P to the law ˜ P of BES(3) started at x , which is hange of drift in one-dimensional diffusions 13 absolutely continuous with respect to Wiener measure P . ˜ P is not equivalentto P , since Z t is not a.s. positive.Note that in this example we knew from the outset that Z is a true mar-tingale, but nevertheless, the application of our recipe is illuminating. Example 2.
An important example for the CIR process (1.2) followed bythe volatility in the Heston model is the case where under P the diffusion X follows dX t = 2 q X + t dW t + δ dt, X = x > , (2.34)the squared-Bessel SDE of dimension δ >
0. See Chapter XI of [19] for adefinitive account. Suppose that we want to perform a measure change totransform the SDE to dX t = 2 q X + t d ˜ W t + δ dt, X = x > , (2.35)where again δ >
0. This requires us to add a drift c ( X t ) dt to dW t in (2.34),where c ( x ) = ( δ − δ ) / (2 √ x ) . (2.36)Simple calculations give us ϕ ( x ) = x ( δ − δ ) / , (2.37)taking x = 1 with no real loss of generality. The scale function ˜ s is given by˜ s ′ ( x ) = exp (cid:20) − Z x δ y dy (cid:21) = exp( − δ log x ) = x − δ / , so that (up to irrelevant constants)˜ s ( x ) = ( x (2 − δ ) / if δ = 2log x if δ = 2 . (2.38)There are three cases to understand:1. 0 < δ <
2. Here, ˜ a = 0 and ˜ b = ∞ . The criterion (2.29) shows that ˜ a isreached in finite time, and the criterion (2.31) requires us to calculate Z a + ˜ h ( x ) ˜ m ( dx ) = Z (cid:18) ϕ ′ ( x ) ϕ ( x ) (cid:19) s ′ ( x ) dx = ( δ − δ ) Z x − δ / dx = ∞ , (2.39)so in this case there is never an absolutely continuous change of measurewhich achieves the desired drift, whatever δ = δ .2. δ = 2. In this case, ˜ s ( x ) = log x , thus ˜ a = −∞ and ˜ b = ∞ . So the firstcheck of our recipe fails, and there is an absolutely continuous measurechange that achieves the desired drift. δ >
2. This time, ˜ s ( x ) = − x − ( δ − / , so ˜ b = 0, ˜ a = −∞ . However, thecriterion (2.29) is infinite for approaching ˜ b , so X approaches but neverreaches ∞ under ˜ P , and there is an absolutely continuous measure changewhich turns the dynamics of X into (2.35).So to summarize, if we want to use a change of measure to change the dimen-sion of a BESQ( δ ) to δ = δ , this is – never possible if δ < – always possible if δ ≥ As is well known, the SDE (1.2) for the Heston volatility has a pathwise-uniquestrong solution from any non-negative starting point. The following fact aboutthe strict positivity of a CIR process is also well-known; see for example [9].
Lemma 3.1
For the CIR process v specified by (1.2) the following are equiv-alent:(i) P ( ∀ t ∈ (0 , T ] : v ( t ) >
0) = 1 (ii) κθ ≥ σ (Feller condition) By scaling time in the CIR SDE (1.2) to convert the volatility σ to thecanonical value 2 appearing in the BESQ SDE (2.34), we see that the Fellercondition is equivalent to the statement that the effective dimension of theCIR process is at least 2: δ ≡ κθσ ≥ . (3.1) Definition 3.2
A probability measure ˜ P on F is an equivalent local martin-gale measure (ELMM) if(i) for all A ∈ F one has ˜ P ( A ) = 0 iff P ( A ) = 0;(ii) the process S is a local martingale under ˜ P .The following Lemma 3.3 is a direct consequence of standard results aboutELLMs in market models, which can be found, for example, in Lemma 5.4.2and Theorem 5.4.3 of [23]. Lemma 3.3
Let ˜ P be an ELMM for the generalized Heston model. Then thereexist previsible processes γ, γ ′ , both locally square-integrable, such that(i) the process Z with Z t ≡ e M t − [ M ] t with M t ≡ Z t γ t dW t + Z t γ ′ t dW ′ t , t ∈ [0 , T ] ; (3.2) is a martingale;(ii) Z T is a density for ˜ P ; hange of drift in one-dimensional diffusions 15 (iii) The integrand γ satisfies µ ( v t ) + √ v t ( ργ t + ρ ′ γ ′ t ) = 0 for a.e. t ∈ [0 , T ] (3.3) (iv) ( S t ) t ∈ [0 ,T ] is a local martingale w.r.t. ˜ P iff ( Z t S t ) t ∈ [0 ,T ] is a local martingalew.r.t. P . Theorem 3.4
Suppose that the Feller condition (3.1) fails: δ < . Then1. the generalized Heston model admits no ELMM if µ (0) = 0 ;2. the generalized Heston model has an ELMM if Z µ ( x ) x − δ/ dx < ∞ . (3.4) Proof of statement 1.
We prove this by contradiction. So assume thatthere does exist an ELMM. By Lemma 3.3 there exists a martingale Z suchthat (cid:0) S t Z t (cid:1) t ∈ [0 ,T ] is a local martingale and dZ t /Z t = γ t dW t + γ ′ t dW ′ t , ( t ∈ [0 , T ])with previsible locally square-integrable, and therefore a.s. pathwise squareintegrable processes γ, γ ′ satisfying (3.3). Using the continuity of µ , µ (0) = 0,and the fact that the γ, γ ′ are square-integrable, this implies Z T v t dt < ∞ P -a.s. (3.5)By Lemma 3.1 P ( ∀ t ∈ [0 , T ] : v t > <
1. Therefore, if we define τ ≡ inf { t ≥ v ( t ) = 0 } ∧ T we have P ( τ < T ) >
0, and, in particular P ( v τ = 0) > . (3.6)On the other hand, by Itˆo’s formula,log( v τ ) = log( v ) + Z τ σ √ v t dW t + Z τ (cid:16) κθ − σ v t − κ (cid:17) dt . (3.7)From (3.5) we get Z τ v t dt < ∞ P -a.s. , so that both integrals in (3.7) are finite a.s. and therefore P (log( v τ ) > −∞ ) = P ( v τ >
0) = 1. But this contradicts (3.6).
Proof of statement 2.
It is to be expected that if there is an ELMM thenthere will be many, so to prove the second statement we shall identify one.We choose to take the change-of-measure martingale to be dZ t Z t = − µ ( v t ) ρ ′ √ v t dW ′ t ≡ c ( v t ) dW ′ t , Z = 1 . (3.8) We see that provided Z is a martingale the drift of S becomes 0, and thedynamics of v is unchanged. So we need show that Z is a true martingale,and for this we use Theorem 2.3 and the arguments of Section 2. As before at(2.13), we define˜ Z t ≡ Z t = exp (cid:20) Z t − c ( v s ) d ˜ W ′ s − Z t c ( v s ) ds (cid:21) . (3.9)Here, d ˜ W ′ t = dW ′ t + c ( v t ) dt . Noticing that ˜ Z can be written˜ Z t = exp (cid:20) B ( Z t c ( v s ) ds ) − Z t c ( v s ) ds (cid:21) (3.10)for some Brownian motion B , it is clear that Question 1 from Section 2 is nowequivalent to Question 2’:
Does A t ≡ R t c ( X s ) ds reach infinity in finite time?This is a question about the CIR process v . The scale function of v is givenby ˜ s ′ ( v ) = exp (cid:20) − Z v κ ( θ − x ) σ x dx (cid:21) = v − δ/ e κv/σ . (3.11)The scale function ˜ s is therefore finite at 0, since δ <
2, and v will reach 0 infinite time. The criterion that A does not explode is (see Proposition 2.1) that Z c ( x ) ˜ m ( dx ) = Z c ( x ) dxσ ( x ) ˜ s ′ ( x ) ≍ Z µ ( x ) x − δ/ dx (3.12)should be finite , and this is condition (3.4). (cid:3) Remark 3.5
Similar calculations as those in the preceding proof of the firststatement appear in [10], where it is shown that there is no ELMM if thestock price process itself is a CIR process and the Feller condition does nothold.
Corollary 3.6
The classical Heston model with constant drift µ = 0 does notadmit an ELMM if the Feller condition is not satisfied. The significance of this result lies in the fact that by the famous fundamen-tal theorem of asset pricing (FTAP) the non-existence of an ELMM impliesthe existence of a free lunch with vanishing risk , i.e. a weak form of arbitrage,see [5,6]. We give its explicit construction in the Appendix.Finally, for completeness, we record this little result which tells us whathappens in the case when the Feller condition holds.
Theorem 3.7
Suppose that the Feller condition (3.1) holds: δ ≥ . Then thereis always an ELMM. The symbol ≍ means that the ratio of the two sides is bounded and bounded awayfrom 0.hange of drift in one-dimensional diffusions 17 Proof.
Recall our standing assumption that µ is continuous. We will useexactly the same change-of-measure martingale (3.8) as we used for the proofof Statement 2 of Theorem 3.4. Exactly as in that proof, we need to establishthat A t ≡ R t c ( v s ) ds remains finite for all time. But we have c ( v ) ≡ − µ ( v ) ρ ′ √ v , and since the CIR process remains strictly positive for all t > v > A doesnot explode. It v = 0, a separate argument is required, which we leave to thereader. (cid:3) We have provided a complete characterization of when the change-of-measurelocal martingale that transforms a one-dimensional diffusion to another onewith a different drift is a true martingale. We are able to decide this question bya simple three-step-algorithm (compare Theorem 2.6). This has practical im-plications for a generalized Heston model that allows for a volatility-dependentgrowth rate: We can show absence of arbitrage given that a simple integrabil-ity condition holds, even when the Feller condition is violated. This extendsthe results for the classical Heston model with constant growth rate differentfrom the riskless rate, for which we have shown that no ELMM exists and thusarbitrage opportunities are incurred in that case.
A Making an FLVR in the Heston model.
The main result of [5] is that for a locally bounded semimartingale the exis-tence of an equivalent local martingale measure is a condition equivalent tothe absence of a free lunch with vanishing risk. The following lemma followsreadily from the definition of free lunch with vanishing risk (FLVR) , see [5,Definition 2.8].
Lemma A.1
Suppose that there exists a sequence f n ≡ ( H n · S ) ∞ of admis-sible terminal wealths with the properties:1. the negative parts f − n tend uniformly to zero;2. the f n tend almost surely to some non-negative limit f ∞ which is not almostsurely zero.Then there exists a FLVR. Proof.
We need to construct a sequence ( K m ) of admissible strategies anda sequence ( g m ) of bounded measurable functions such that ( K m · S ) ∞ ≥ g m and a measurable function g ∞ which is non-negative, positive with positiveprobability, such that lim m k g m − g ∞ k ∞ = 0. W.l.o.g. we may assume that f n ≥ − n − for all n , by passing to a subse-quence if necessary. We have f n ∧ → f ∞ ∧ − ≤ − n − ≤ f n ∧ ≤ ≤ f ∞ ∧ ≤
1. Let A := { f ∞ ∧ > } . By assumption, P ( A ) >
0. By Egorov’s theorem, there exists a measurable set B ⊆ A with P ( B ) > n m ) such that f n m ∧ → f ∞ ∧ m → ∞ )uniformly on B .We now define K m := H n m , g m := ( f n m ∧ B − n − m Ω \ B and g ∞ :=( f ∞ ∧ B , which have the required properties. (cid:3) We shall here directly construct a sequence ( f n ) with above propertiesand thus a FLVR, and then it follows from the result of [5,6] that there is noequivalent σ -martingale measure, and a fortiori no equivalent local martingalemeasure.To fix ideas, we shall assume with no real loss of generality that r = 0,and that µ >
0; if µ = 0 then we are already in an equivalent local martingalemeasure and there is nothing interesting to say, and if µ < v to dv t = σ √ v t dW t + κθ dt. (A.1)We can always do this, because if we can construct a FLVR in this setting wecan perform an absolutely-continuous change of measure to change the drift in(A.1) into the original drift in (1.2), and null events (and therefore an FLVR)will not be changed by this . Once we have done this, we have that v is aBESQ process, or at least, a BESQ process run at a constant speed whichmay not be 1. Again, we change nothing that matters by rescaling the speedso that we are looking at an actual BESQ process dv t = 2 √ v t dW t + δ dt (A.2)where we have the correspondence δ = 4 κθ/σ . Thus the Feller condition (3.1)is the statement that δ <
2, the familiar condition in terms of the dimension δ of the BESQ process that the process hits 0. For more background on BESQprocesses, we refer to [19].Looking at (1.1), it is rather obvious what the idea of the constructionshould be: we need to go into the asset when v is very small, because at suchtimes the positive drift µ will dominate the tiny variance. Ideally, we couldjust hold the asset at the times when v is equal to zero, because then themartingale part of the gains-from-trade process would vanish and we wouldjust get the drift contribution, but this does not work because the Lebesguemeasure of the set of times when v = 0 is zero; see, for example, Proposition1.5 on page 412 of [19]. So the next attempt is to hold the asset only at times Of course, this will change drift in (1.1), but an equivalent change of measure to W ′ canbe applied to reverse this.hange of drift in one-dimensional diffusions 19 when v t < ε for some very small ε >
0, which we hope will be an approximatearbitrage. As we shall see, this leads us to an FLVR.But in order to do this, we have to be able to do some calculations onBESQ processes, which turn out to be easier in terms of the scale and speedrepresentation of v in terms of a standard Brownian motion. The scale functionof v is easily verified to be s ( x ) ∝ x − δ/ . (A.3)If we then consider the diffusion in natural scale Y t = s ( v t ) = v − δ/ t , andapply Itˆo’s formula we find that dY t = (2 − δ ) v (1 − δ ) / t dW t = (2 − δ ) Y (1 − δ ) / (2 − δ ) t dW t , (A.4)at least while Y is strictly positive. Clearly (A.4) cannot hold for all time,otherwise Y would be a non-negative local martingale, and would have to stickat 0 once it reaches 0. Of course, this does not happen, and this is because ofa local time effect at zero - see [20], V.48.6 for more details.But (A.4) tells us that away from 0 the speed measure for Y is m ( dy ) = dy (2 − δ ) | y | δ − / (2 − δ ) , (A.5)and the speed measure does not charge 0 because Y spends no time there. Sowe may create a weak solution to (A.2) starting from a standard Brownianmotion B with local time process { ℓ xt : x ∈ R , t ≥ } by the recipe Λ t ≡ Z ℓ at m ( da ) (A.6)= Z t | B u | δ − / (2 − δ ) (2 − δ ) du, (A.7) τ t = inf { u : Λ u > t } , (A.8) Y t = | B ( τ t ) | , (A.9) v t = Y / (2 − δ ) t ; (A.10)for more details, see V.47, V.48 in [20].The idea now is to make a portfolio ϕ ( Y t ) /S t so that the gains-from-tradeprocess becomes G t ≡ Z t ϕ ( Y u ) ( √ v u d ˆ W + µ du ) , (A.11)where d ˆ W = ρdW + ρ ′ dW ′ ; see (1.1). As the scale function is not C , this is only valid in the region (0 , ∞ ) where s is C .0 Sascha Desmettre et al. We shall do this in such a way that the local martingale term in (A.11)is negligible, and the Lebesgue term is not. To explain, when we look at theLebesgue integral in G t we see µ times Z t ϕ ( Y u ) du = Z t ϕ ( | B ( τ s ) | ) ds = Z τ t ϕ ( | B u | ) dΛ u = Z τ t ϕ ( | B u | ) | B u | δ − / (2 − δ ) (2 − δ ) du. If we now choose ε > ϕ ( x ) = (2 ε ) − I [0 ,ε ] ( x ) · x − δ ) / (2 − δ ) (2 − δ ) , (A.12)we find that Z t ϕ ( Y u ) du = (2 ε ) − Z τ t I [0 ,ε ] ( | B u | ) du = 12 ε Z ε − ε ℓ x ( τ t ) dx → ℓ ( τ t ) ( ε ↓ . (A.13)The quadratic variation of the martingale part of G is h G i t = Z t ϕ ( Y u ) v u du = Z t ϕ ( Y u ) Y / (2 − δ ) u du = Z τ t ϕ ( | B u | ) | B u | / (2 − δ ) dΛ u = (2 − δ ) ε Z τ t I [0 ,ε ] ( | B u | ) | B u | du = (2 − δ ) ε Z ε − ε ℓ x ( τ t ) | x | dx (A.14) ∼ (2 − δ ) ℓ ( τ t ) ε ( ε ↓ . From (A.14) therefore lim ǫ ↓ h G i t = 0 a.s. (A.15)and E h h G i Λ ( t ) i = O ( ε ) . (A.16)Equations (A.13) and (A.14) are the main parts of what we need, all thatremains is to put the bits together.So we let M t denote the local martingale part of G t , fix some positive timehorizon T , and construct the FLVR. For this we consider a sequence ε = 2 − n of values of ε , and consider the portfolios ϕ given by (A.12) for the differentvalues of ε . We are only going to use this portfolio until the stopping timewhich is the minimum of t = Λ T and θ n ≡ inf { t : | M t | > n − } , (A.17) hange of drift in one-dimensional diffusions 21 after which everything stops. Now we have (with M ∗ t ≡ sup {| M u | : u ≤ t } ) P [ M ∗ ( Λ T ∧ θ n ) > n − ] ≤ n E [ M ∗ ( Λ T ∧ θ n ) ] ≤ n E [ M ( Λ T ∧ θ n ) ] , by Doob’s submartingale maximal inequality, and in view of (A.14) we havethe bound P [ M ∗ ( θ n ) > n − ] ≤ Cn − n (A.18)for some finite constant C . Hence by Borel-Cantelli, for all but finitely many n we have M ∗ θ n ≤ n − and therefore θ n > Λ T . The negative part of G ( Λ T ∧ θ n )is no more than n − , and as we let n → ∞ the terminal value G ( Λ T ∧ θ n )converges to µℓ ( T ), which is of course non-negative, and positive with positiveprobability.The FLVR is constructed. Conflict of interest
The authors declare that they have no conflict of interest.
Acknowledgments
S. Desmettre und G. Leobacher are supported by the Austrian Science Fund(FWF) projects F5507-N26 and F5508-N26, which are part of the SpecialResearch Program
Quasi-Monte Carlo Methods: Theory and Applications . S.Desmettre moreover appreciates support by the DFG-Research Training Group1932
Stochastic Models for Innovations in the Engineering Sciences . References
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