Change of drift in one-dimensional diffusions

It is generally understood that a given one-dimensional diffusion may be transformed by Cameron-Martin-Girsanov measure change into another 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 martingale that we write down is a true martingale; we provide a complete characterization of when this happens. This is then used to discuss absence of arbitrage in a generalized Heston model including the case where the Feller condition for the volatility process is violated.

we need to change measure to a pricing measure in which the growth rate is the riskless rate. The question then arises: 'Can this be done?' The answer we found was 'Not always'; and in cases where it cannot be done, then general results say that there must be arbitrage (in a suitable sense).
We then realized that the question is closely related to changing the given drift 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 this very general result cannot be applied without care, the main point being to decide whether the local martingale we write down to do the change of drift is actually a martingale. In general, this is hard to decide, but in the special case that concerns us, where the drift is again a function of the diffusion, we are able 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 for any particular situation, and we illustrate this with two interesting examples.
In Section 3 we turn to the Heston model for the stock price S and the volatility v, which defines their evolution in the 'real-world' probability P as: Here, W and W ′ are independent Brownian motions, κ, θ and σ are strictly positive constants, and ρ ∈ (−1, 1) is the constant correlation between the Brownian motions driving stock and volatility. We write ρ ′ ≡ 1 − ρ 2 . The function 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 we could 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 typically assumed that a risk-neutral measureP exists and that the dynamics are stated in the corresponding risk-neutral form; see, for example, the extensive textbook [13] and the references therein. Yet, the question of existence of such a 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 with vanishing risk, that is, a form of arbitrage! A notable exception, where this problem is addressed, is [16], where the authors give a solution to this problem assuming the Feller condition, a condition which keeps the volatility process strictly positive.
However, the Feller condition is frequently violated in practice as has been pointed out in [1] or [3] (consult in particular Table 6.3).
Building on results in [9,10], this problem is addressed for several stochastic volatility models in [2], including the classical Heston model, by modifying the model so that the volatility process is stopped as soon as it hits 0. While this solves the problems incurred by a violated Feller condition mathematically, this approach is not completely convincing 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 no risk-neutral measure. However, if the drift µ is not constant, but satisfies a simple integrability condition at 0, we show that there is an equivalent local martingale measure (ELMM), still in the case where the Feller condition is not satisfied. When the Feller condition is satisfied, we show that there is always an ELMM.
In the Appendix, as a gentle amusement, we directly construct a free lunch with vanishing risk (FLVR), from which if follows by the celebrated Fundamental Theorem of Asset Pricing (FTAP) of [4,5] that there is no equivalent σ-martingale measure and a fortiori no ELMM. This is a rare application of the 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-neutral measure (in effect, assume that µ ≡ 0). We gain nothing by being overly introspective about the growth rate of a stock, about which we know next to nothing in any case.
2 Changing drift in a one-dimensional diffusion.
We are going to begin with a regular diffusion taking values in an interval I ⊆ R. We write I • for the interior of I, which could be equal to I. We also set a = inf I, b = sup I, the endpoints of I. The interval I may be the whole real line, it may contain endpoints or not; the only limitation on I we shall demand is that if an endpoint is contained in I then there is not sticky reflection there. This is because we want the diffusion X to solve an SDE where we shall also insist that the coefficients are such that there exists a pathwise-unique strong solution from all starting points. Such a characterisation of X is not compatible with sticky reflecting endpoints. The starting point x 0 ∈ I • is fixed but general. To avoid any issues, we shall assume throughout that σ, β are continuous, and σ > 0 on I • .
Let Ω = C(R + , I) be the canonical path space with the canonical process X t (ω) = ω(t) and the raw filtration F • t = σ(X s : s ≤ t). If P is the law of X on (Ω, F • ∞ ) we let (F t ) be the usual P -augmentation of (F • t ). We write F = F ∞ for brevity. Now suppose that λ is a previsible process such that t 0 λ 2 s ds < ∞ ∀t > 0, (2.2) and let the change-of-measure local martingale Z be defined by Provided Z is a martingale, we can now define a new probabilityP by the recipe dP dP Ft To determine whether or not Z is a martingale, define the stopping times T n ≡ inf{t : Z t > n}, n = 2, 3, . . . (2.5) which reduce Z, and notice that it is possible to define a probabilityP on F Tn by dP dP FT n = Z Tn .
The definition ofP extends to the field n F Tn , and thence to the σ-field F T∞ = σ( n F Tn ). But doesP extend to the whole of F ? The answer is in this simple result (see [14], Theorem 1.3.5), whose proof we give for completeness. Proof. We have By Monotone Convergence, the first term on the right converges to E[Z t ], so condition (2.7) is equivalent to the statement that E[Z t ] = 1 for all t > 0, which is the condition that Z is a martingale.
If now we restrict to λ of the form λ t = c(X t ), where for convenience we assume c is continuous, then underP the canonical process X solves the SDE at least up to sup n T n . Here,W is aP -Brownian motion, and When is the condition (2.7) for Z to be a martingale satisfied? To answer this, we define the reverse measure transformatioñ a positiveP -local martingale. Obviously, T n = inf{t :Z t < n −1 }, so what we have to determine is this: Question 1: When X solves (2.8), doesZ reach zero in finite time?
If not, then the change-of-measure local martingale is a martingale. Noticing thatZ can be writteñ for some Brownian motion B, it is clear that Question 1 is equivalent to 0 c(X s ) 2 ds reach infinity in finite time? Of course, this question must be answered in the lawP , when X solves the SDE (2.8). If K ⊂ I • is any compact set, and ζ = inf{t : A t = ∞} < ∞, then clearly X must exit K before ζ, because c is bounded on K, being continuous. By considering an increasing sequence of compact K n increasing to I • , we see that if A reaches infinity in finite time, it has to be at a time when X reaches a boundary point of I.
To understand this, we look firstly at the scale functions of X, which satisfiess The diffusion Y =s(X t ) is a diffusion in natural scale, and is a local martingale, solving the SDE The speed measure is m(dy) =σ(y) −2 dy. The diffusion Y takes values in the intervals(I), whose endpoints areã ≡s(a) <b ≡s(b). Two cases arise.
Case 1:ã andb are both infinite. Since Y is a continuous local martingale, and therefore a time-change of Brownian motion, Y cannot reach either endpoint in finite time, so the change-of-measure local martingale Z is a true martingale.
Case 2: one at least ofã andb is finite. We suppose thatã = 0,b = ∞ with little loss of generality. It will become clear that the arguments pertaining tõ a apply mutatis mutandis tob if that also is finite. In this situation, Y is a non-negative local martingale, so must be almost surely convergent, and the only possible limit value is 0. Y reaches 0 in finite time with probability 1 if and only if (2.14) otherwise, the probability that Y reaches 0 in finite time is 0; see, for example, [12] by [12], Theorem V.51.2 once again.
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 endpointsã,b ofs(I) finite (see (2.12)) ? 2. Ifã (say) is finite, does Y reachã in finite time (see (2.14)) ? 3. If so, does A explode when Y reachesã (see (2.15)) ?
To summarize then, we have the following result.
which we assume has a pathwise-unique strong solution. We define the changeof-measure local martingale Z by where c : I → R is continuous. Suppose now thatX solves the SDE , and lets be the scale function ofX,ã =s(a), b =s(b). If all of the following three conditions are satisfied: 1. At least one of the endpointsã,b is finite; 2. At least one of the finite endpoints is reached in finite time (see (2.14)); 3. There is a finite endpoint which is reached in finite time and at which the additive functional A explodes (see (2.15)), then the change-of-measure local martingale Z is not a true martingale. Otherwise, it is.
If Z is a martingale, then the recipe (2.4) defines a new measureP on path space under which the canonical process solves the SDE (2.8). The law P is therefore absolutely continuous with respect to P , but not in general equivalent.
Example 1. A canonical example is when x 0 > 0, σ(x) ≡ 1, β(x) ≡ 0 and c(x) = 1/x. In this case, we are in Case 1 above, that is, bothã andb are infinite. By the preceding therefore, there is an absolutely continuous change of measure, taking Wiener measure P to the lawP of BES(3) started at 1/x 0 , which is absolutely continuous with respect to Wiener measure P , but not equivalent to it.
Example 2. An important example for the CIR process (1.2) followed by the volatility in the Heston model is the case where under P where again δ 1 > 0. This requires us to add a drift c(X t )dt to dW t in (2.19), where The scale functions is given bỹ There are three cases to understand: Here, 0 is accessible, and ∞ is not. The criterion (2.14) shows that 0 is reached in finite time, and the criterion (2.15) requires us to calculate so in this case there is never an absolutely continuous change of measure which achieves the desired drift, whatever δ 0 = δ 1 . 2. δ 1 = 2. In this case,s(x) = log x and so we are in Case 1 above; there is an absolutely continuous measure change that achieves the desired drift. 3. δ 1 > 2. This time,s(x) = −x −(δ1−2)/2 , so 0 is inaccessible, ∞ is not. However, the criterion (2.14) is infinite for approaching ∞, so X approaches but never reaches ∞ underP , and there is an absolutely continuous measure change which turns the dynamics of X into (2.20).
So to summarize, if we want to use a change of measure to change the dimension of a BESQ(δ 0 ) to δ 1 = δ 0 , this is 3 Arbitrage opportunities in the Heston model.
As is well known, the SDE (1.2) for the Heston volatility has a pathwise-unique strong solution from any non-negative starting point. The following fact about the strict positivity of a CIR process is also well-known; see for example [6].
Lemma 3.1 For the CIR process v specified by (1.2) the following are equivalent: By scaling time in the CIR SDE (1.2) to convert the volatility σ to the canonical value 2 appearing in the BESQ SDE (2.19), we see that the Feller condition is equivalent to the statement that the effective dimension of the CIR process is at least 2: is a martingale; (ii) Z T is a density forP ; (iii) The integrand γ satisfies

the generalized Heston model has an ELMM if
Proof of statement 1. We prove this by contradiction. So assume that there does exist an ELMM. By Lemma 3.3 there exists a martingale Z such that S t Z t t∈[0,T ] is a local martingale and with previsible a.s. pathwise square integrable processes γ, γ ′ satisfying (3.3).
Proof of statement 2. It is to be expected that if there is an ELMM then there will be many, so to prove the second statement we shall identify one. We choose to take the change-of-measure martingale to be We see that provided Z is a martingale the drift of S becomes 0, and the dynamics of v is unchanged. So we need show that Z is a true martingale, and for this we use Theorem 2.1 and the arguments of Section 2. As before at (2.10), we definẽ Here, dW ′ t = dW ′ t + c(v t ) dt. As we saw at (2.11), we have to prove that A t ≡ t 0 c(v s ) 2 ds remains finite for all time, and this is a question about the CIR process v. The scale function of v is given by The scale function s is therefore finite at 0, since δ < 2, and v will reach 0 in finite time. The criterion that A does not explode is (see (2.15)) that should be finite 2 , and this is condition (3.4).
Remark 3.5 Similar calculations as those in the preceding proof of the first statement appear in [7], where it is shown that there is no ELMM if the stock price process itself is a CIR process and the Feller condition does not hold. The significance of this result lies in the fact that by the famous fundamental theorem of asset pricing (FTAP) the non-existence of an ELMM implies the existence of a free lunch with vanishing risk, i.e. a weak form of arbitrage, see [4,5]. We give its explicit construction in the Appendix.
Finally, for completeness, we record this little result which tells us what happens in the case when the Feller condition holds. Proof. Recall our standing assumption that µ is continuous. We will use exactly the same change-of-measure martingale (3.8) as we used for the proof of Statement 2 of Theorem 3.4. Exactly as in that proof, we need to establish that A t ≡ t 0 c(v s ) 2 ds remains finite for all time. But we have and since the CIR process remains strictly positive for all t > 0 by Lemma 3.1, and does not explode, it follows immediately that if v 0 > 0 then A does not explode. It v 0 = 0, a separate argument is required, which we leave to the reader.

Conclusion
We have provided a complete characterization of when the change-of-measure local martingale that transforms a one-dimensional diffusion to another one with a different drift is a true martingale. In particular, we are able to decide whether we face a martingale or not by a simple three-step-algorithm (compare Theorem 2.2). This has practical implications for a generalized Heston model that allows for a volatility-dependent growth rate: We can show absence of arbitrage given that a simple integrability condition holds, even when the Feller condition is violated. This extends the results for the classical Heston model with constant growth rate, for which we have shown that no ELMM exists and thus arbitrage opportunities are incurred in that case.
A Making an FLVR in the Heston model.
The main result of [4] is that for a locally bounded semimartingale the existence of an equivalent local martingale measure is a condition equivalent to the absence of a free lunch with vanishing risk. The following lemma follows readily from the definition of free lunch with vanishing risk (FLVR), see [4, Definition 2.8].
Lemma A.1 Suppose that there exists a sequence f n ≡ (H n · S) ∞ of admissible 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 almost surely zero.
Then there exists a FLVR.
Proof. We need to construct a sequence (K m ) of admissible strategies and a 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 positive probability, such that lim m g m − g ∞ ∞ = 0.
We shall here directly construct a sequence (f n ) with above properties and thus a FLVR, and then it follows from the result of [4,5] that there is no equivalent σ-martingale measure, and a fortiori no equivalent local martingale measure.
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 martingale measure and there is nothing interesting to say, and if µ < 0 the argument we give carries through by reversing signs in the appropriate places.
We firstly reduce the problem to a simpler canonical form, by modifying the SDE for v to We can always do this, because if we can construct a FLVR in this setting we can 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 3 . Once we have done this, we have that v is a BESQ process, or at least, a BESQ process run at a constant speed which may not be 1. Again, we change nothing that matters by rescaling the speed so that we are looking at an actual BESQ process where we have the correspondence δ = 4κθ/σ 2 . 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 BESQ processes, we refer to [11].
Looking at (1.1), it is rather obvious what the idea of the construction should be: we need to go into the asset when v is very small, because at such times the positive drift µ will dominate the tiny variance. Ideally, we could just hold the asset at the times when v is equal to zero, because then the martingale part of the gains-from-trade process would vanish and we would just get the drift contribution, but this does not work because the Lebesgue measure of the set of times when v = 0 is zero; see, for example, Proposition 1.5 on page 412 of [11]. So the next attempt is to hold the asset only at times when v t < ε for some very small ε > 0, which we hope will be an approximate arbitrage. 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 on BESQ processes, which turn out to be easier in terms of the scale and speed representation of v in terms of a standard Brownian motion. The scale function of v is easily verified to be If we then consider the diffusion in natural scale Y t = s(v t ) = v 1−δ/2 t , and apply 4 Itô's formula we find that 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 stick at 0 once it reaches 0. Of course, this does not happen, and this is because of a local time effect at zero -see [12], V.48.6 for more details. But (A.4) tells us that away from 0 the speed measure for Y is and the speed measure does not charge 0 because Y spends no time there. So we may create a weak solution to (A.2) starting from a standard Brownian motion B with local time process {L x t : x ∈ R, t ≥ 0} by the recipe for more details, see V.47, V.48 in [12].
The idea now is to make a portfolio ϕ(Y t )/S t so that the gains-from-trade process becomes where dŴ = ρdW + ρ ′ dW ′ ; see (1.1). 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 the Lebesgue integral in G t we see µ times The quadratic variation of the martingale part of G is by Doob's submartingale maximal inequality, and in view of (A.14) we have the bound P [M * (θ n ) > n −1 ] ≤ Cn 2 2 −n (A.18) for some finite constant C. Hence by Borel-Cantelli, for all but finitely many n we have M * θn ≤ n −1 and therefore θ n > A T . The negative part of G(A T ∧ θ n ) is no more than n −1 , and as we let n → ∞ the terminal value G(A T ∧ θ n ) converges to µL 0 (T ), which is of course non-negative, and positive with positive probability.
The FLVR is constructed.