Asymptotic arbitrage in the Heston model

In the context of the Heston model, we establish a precise link between the set of equivalent martingale measures, the ergodicity of the underlying variance process and the concept of asymptotic arbitrage proposed in Kabanov-Kramkov and in Follmer-Schachermayer.

is Theorem 3.6, which identifies sufficient (and sometimes necessary) conditions on the set of equivalent martingale measures under which asymptotic arbitrage occur with linear speed. These conditions are different from those in Proposition 3.11 in which we study the role of the ergodicity of the variance process on the existence of asymptotic arbitrage with slower speed.

Notations and definitions
Let (Ω, F, F, P) be a filtered probability space where the filtration F = (F t ) t≥0 satisfies the usual conditions, S = e X model a risky security under an equivalent martingale measure, and let H denote the class of predictable, S-integrable admissible processes. We define for each t ≥ 0 the sets of strategies K t and of equivalent local martingales M e t (S) by We shall always assume that M e t (S) is not empty. Furthermore, for any set A in Ω, we shall denote by A c := Ω \ A its complement.
2.1. Asymptotic arbitrage. We are interested here in specific forms of arbitrage (asymptotic arbitrage), first introduced by Kabanov and Kramkov [14], and refined recently by Föllmer and Schachermayer [8]. The following definition of a (ε 1 , ε 2 ) arbitrage is taken from the latter [8, Proposition 2.1]: Definition 2.1. The process S admits an (ε 1 , ε 2 )-arbitrage if for (ε 1 , ε 2 ) ∈ (0, 1) 2 , there exists X t ∈ K t such that (i) X t ≥ −ε 2 P-almost surely; This means that the maximal loss of the trading strategy, yielding the wealth X t at time t, is bounded by ε 2 and, with probability 1 − ε 1 , the terminal wealth X t equals at least 1 − ε 2 . Let us consider the following slightly weaker version, which imposes less stringent restrictions on the maximal loss: Definition 2.2. Let (e 1 , e 2 ) ∈ (0, 1) 2 . We say that the process S admits a partial (e 1 , e 2 )-arbitrage (up to time t > 0) if for any (ε 1 , ε 2 ) ∈ (e 1 , 1) × (e 2 , 1), there exists X t ∈ K t such that (i) X t ≥ −ε 2 P-almost surely; Obviously, Definition 2.2 is equivalent to Definition 2.1 when e 1 = e 2 = 0. However, as we shall see later, (partial) asymptotic arbitrage may only appear for e 1 exponentially small, but necessarily equal to zero. When e 1 is not null, this alternative definition may also be seen as a mid-point characterisation between Definition 2.1 and Definition 2.3 below. The latter in particular characterises the notion of asymptotic exponential arbitrage with exponentially decaying failure probability, first proposed in [8] and studied later in [3] and [7]. Definition 2.3. The process S allows for asymptotic exponential arbitrage with exponentially decaying failure probability if there exist t 0 ∈ (0, ∞) and constants C, λ 1 , λ 2 > 0 such that for all t ≥ t 0 , there is Remark 2.4. We see that there is a relation between Definition 2.3 and Definition 2.2. If the process S allows for asymptotic exponential arbitrage with exponentially decaying failure probability, then it admits a partial (e 1 , e 2 )-arbitrage with e 1 = Ce −λ1t and e 2 = e −λ2t . Indeed, for any ε 2 ∈ (e −λ2t , 1) we have Definition 2.5. Let f : R * + → R * + be a smooth function such that lim t↑∞ f (t) = +∞. The process S is said to have an average squared market price of risk γ above the threshold tends to zero as t tends to infinity.

Stochastic volatility models.
We consider here the Heston stochastic volatility model, namely the unique strong solution to the stochastic differential equations (2.1) below. As is well-known [12], there may not be a unique risk-neutral martingale measure for this . The following SDEs are therefore understood under one such risk-neutral measure Q.
where W 1 and W 2 are independent Q-Brownian motions, a, σ > 0, µ, b ∈ R and |ρ| < 1. The class of equivalent martingale measures Q can be considered in terms of the Radon-Nikodym derivatives is necessary for an equivalent local martingale measure to exist, and ensures that the discounted stock price is a local martingale; here r denotes the constant risk-free rate. Since Z is a positive local martingale with Z 0 = 1, it is a supermartingale, and a true martingale if and only if E(Z t ) = 1. For the Heston stochastic volatility model we obtain, for any λ ∈ R,

Main results
For any (α, β, δ) ∈ R 3 , we introduce the process (X α,β,δ t ) t≥0 defined (pathwise) by where V is the Feller diffusion for the variance in (2.1). We shall always assume that β and δ are not both null simultaneously. In that case, X is simply the Feller diffusion, and its density is known in closed form [13, Part 1, Chapter 6.3]. The large-time behaviour of X will play a key role in determining average squared market prices of risk, and the case β = δ = 0 will never occur, so this assumption does not entail any loss of generality here. Define the real interval D β,δ by Whenever βδ = 0, we define D β,δ by taking the limits of the interval (a closed bound becoming open if it becomes infinite), where we use the slight abuse of notation " if β < 0 and δ = 0. Let us further define the function Λ β,δ : D β,δ → R by In the case δ ̸ = 0 above, we further impose the condition a > σ for the definition of the function Λ β,δ .
Remark 3.1. It may be surprising at first that the function Λ β,δ related-in some sense defined precisely below-does not depend on α. This function actually describes the large-time behaviour of the process X α,β,δ . Since the variance process V is strictly positive almost surely (by the Feller condition imposed above), the term ∫ t 0 V s ds clearly dominates V t for any t, which explains why α bears no influence on Λ β,δ . The condition a > σ imposed above in the case δ ̸ = 0 should not surprise the reader since this is nothing else than the Feller condition, ensuring that the variance process never touches the origin almost surely.
We further define the Fenchel-Legendre transform Λ * β,δ : Notation. Whenever β = 0 or δ = 0, we shall drop the subscript and write respectively Λ δ or Λ β . The same rule shall apply for the Fenchel-Legendre transforms and their respective domains.

The large deviations case.
In this section, we prove asymptotic arbitrage results (with linear speed) for the stock price process; we shall in particular observe that the ergodicity of the variance process plays a key role. We first start with the following lemma (proved in Appendix A), which will be used heavily in the remaining of the paper. For precise definitions of large deviations principles (LDP), we refer the reader to the excellent monograph [6]; we shall use the non-standard terminology 'partial large deviations principles' if an LDP holds only on subsets of the real line.
In each case, the rate function is Λ * β,δ and the (partial) LDP holds with speed t −1 .
In [8, Theorem 1.4], Föllmer and Schachermayer proved that if the stock price process has an average market price of risk above a threshold then asymptotic arbitrage holds. Using the large deviations principle proved above, we first show that S does not always admit an average market price of risk for γ 1 (Proposition 3.3) or γ 2 (Proposition 3.4) above any threshold. This is in particular so when the variance process is not ergodic (b ≤ 0). This however-as proved in Theorem 3.6 below-does not preclude absence of asymptotic arbitrage. Proof. Note first that λ = 0 implies γ 1 ≡ 0 and hence P(t −1 ∫ t 0 γ 2 1 (s)ds < c) = 1 for all t > 0, so that the proposition is trivial. Assume from now on that λ ̸ = 0 and let c be an arbitrary strictly positive real number. The definition of zero as t tends to infinity, which in turn implies that P(t −1 ∫ t 0 γ 2 1 (s)ds < c) converges to 1 as t tends to infinity, and statement (i) in the proposition follows. When b > 0, consider the case c > aλ 2 /b; then is strictly positive and we end up with the same as in the case b ≤ 0 which proves statement (ii) in the proposition.
(v) λρ = 0; Remark 3.5. Note that the case λρ = 0 precisely corresponds to the case of a complete market.
Proof. Let c be an arbitrary strictly positive real number. Note first that if λρ = 0, and µ = r, then and Lemma A.1 implies that Λ * 0,1 is strictly positive, so that (v) follows. Assume now that λρ ̸ = 0 and , and (i) follows immediately from Lemma A.1. When λρ(µ − r) < 0, the interval converges to zero as t tends to infinity.
We can now move on to our main theorem, which proves a partial arbitrage for the stock price process S.
Remark 3.7. The threshold e 1 has the form e 1 ∼ e −λ1t for some λ 1 > 0, which links partial arbitrage to exponentially decaying failure probability characterised in Definition 2.3. Note that we could slightly relax the constraint on λ, making the latter time-dependent, because we only need to ensure that e 1 ∈ (0, 1).
Since we are only interested in large t, this is however not essential here. The sufficient condition on λ is not necessary: for λ = 0 and µ = r, Z t = 1 almost surely for all t ≥ 0, and P (Z t ≥ e −γt ) = 1 for any Proof. Let γ > 0 and define the set A λ,t := {Z t ≥ e −γt } ∈ F t . Since the processes W 2 and V are independent, the tower property for conditional expectation implies E( ) .
Markov's inequality therefore yields From the proof of Lemma 3.2, we know that t −1 log Λ α,β t (t) converges to Λ β (1), which implies that for any η > 0 there existst > 0 such that for any t >t, Therefore, for any t >t, ] .

3.2.
Case t/f (t) and f (t) tend to infinity as t tends to infinity. Let b > 0, in which case the variance process is ergodic and its stationary distribution π is a Gamma law with shape parameter a/σ and scale parameter σ/b; namely t −1 ∫ t 0 h(V s )ds converges to ∫ R h(x)π(dx) almost surely for any h ∈ L 1 (π) (see [16]). In this section, we consider a continuous function f : R * + → R + such that t/f (t) tends to infinity as t tends to infinity. We shall prove below that (under some conditions on the risk parameter λ) the ergodicity of the variance ensures that S allows an asymptotic arbitrage with sublinear speed f (t).
Proposition 3.9. The stock price process S in (2.1) has an average squared market price of risk γ 1 above the threshold aλ 2 /b with speed f (t). If furthermore a > σ and λρ(µ − r) ≤ 0, then there exists c 2 > 0 such that S has an average squared market price of risk γ 2 above the threshold c 2 with speed f (t).

Remark 3.10.
As the proof shows, we can actually be more precise regarding the threshold c 2 : • if µ = r, then c 2 = aλ 2 ρ 2 b(1−ρ 2 ) ; • if µ ̸ = r and ρλ < 0, then no further condition on c 2 is needed; • if µ ̸ = r and ρλ = 0, then It is rather interesting to compare this result with those of Proposition 3.3 and Proposition 3.4. Indeed, when b > 0, if f (t) ≡ t then the stock price process does not satisfy an average squared market price of risk γ 1 above the threshold aλ 2 /b. However, when t/f (t) tends to infinity, then S has an average squared market price of risk γ 1 above the threshold aλ 2 /b. When b > 0, λρ ̸ = 0 and µ = r, if f (t) ≡ t then the stock price process does not satisfy an average squared market price of risk γ 2 above the threshold , but does so above the threshold aλ 2 ρ 2 b(1−ρ 2 ) when t/f (t) tends to infinity. Finally, when b > 0, λρ = 0 and µ ̸ = r the stock price process never satisfies an average squared market price of risk γ 2 with speed f (t) ≡ t, but does above the threshold b(µ−r) 2 (1−ρ 2 )(a−σ) whenever t/f (t) tends to infinity.
Proof of Proposition 3.9. Let f be as stated in the proposition. For b > 0, the variance process is ergodic and its stationary distribution is a Gamma law with shape parameter a/σ and scale parameter σ/b (see [16]). In particular, t −1 ∫ t 0 V s ds converges in probability to a/b as t tends to infinity, and hence for any c 1 ∈ (0, aλ 2 /b), which proves the first part of the proposition.
We now assume that µ ̸ = r. If a > σ we further know that (see proposition 4 in [1]) t −1 ∫ t 0 V −1 s ds converges in probability to b/(a − σ) as t tends to infinity. Therefore for any c ∈ (0, b/(a − σ)) we have Let c 2 , c ′ 1 , c ′ 2 be three strictly positive numbers such that with c ′ 1 = ρ 2 1−ρ 2 c 1 > 0. As long as c 1 ∈ (0, aλ 2 /b), the first probability tends to zero as t tends to infinity by (3.6). Now, when ρλ(µ − r) < 0, then since t/f (t) tends to infinity, the second probability tends to zero (as t tends to infinity) by (3.7) because c ′ 2 + 2ρλ(µ−r) 1−ρ 2 t f (t) tends to −∞ (and because the variance process is non-negative almost surely). No condition on c ′ 2 is needed here. When ρλ = 0, then the first line of the equation above simplifies to ) .
We now state and prove our final result, namely a strong asymptotic arbitrage statement for the stock price process when the speed is sublinear. Proposition 3.11. Fix γ > 0. Then, for t large enough, 2bγ/a], then S admits a partial (0, 1/2)-arbitrage with speed f (t); (2) if a > σ and λρ(µ − r) ≤ 0, then S admits a partial (0, 1/2)-arbitrage with speed f (t), when µ = r and ρ 2 ≤ 1/2; • if and only if λ ∈ R \ [− √ 2bγ/a, √ 2bγ/a] when µ = r and ρ 2 ≥ 1/2; • if µ ̸ = r and ρλ < 0; Proof. Recall that we are in the framework of Proposition 3.9, so that c 1 > 0 and c 2 > 0 are the thresholds for γ 1 and γ 2 above which S has an average squared market price of risk. In this proof, we follow steps similar to those in [8]. For any ε 1 > 0, fix 0 < γ <γ < c1 . Lett 0 > t 0 such that for any t ≥ t 0 we have f (t) ≥t 0 . Then for t ≥t 0 and using the fact that For Z 1 τ1 := exp , we then obtain We are now in position to construct contingent claim which satisfies the arbitrage estimates of the Theorem. We can introduce the random Assume now that a > σ, λρ(µ − r) ≤ 0, then S has an average squared market price of risk γ 2 above a threshold c 2 > 0. For any ε 1 > 0, let 0 < γ < γ ′ < c2 2 , and t 1 > Similarly to the first case, we can introduce the random variable Y t := ε 2 Q(B λ,t ) 1 1 B λ,t , and hence S satisfies a partial (0, 1/2)-arbitrage with speed f (t).
Appendix A. Large deviations results

Proof of Lemma 3.2.
Recall the standing assumption that β and δ are never null simultaneously. We first prove the lemma in the case b ̸ = 0. The moment generating function of the random variable X α,β,δ t /t is given by (see [2, proposition 2]), ) clearly tends to zero and lim t↑+∞ 1 t log Therefore, for u ∈ D β,δ where the interval D β,δ is given in (3.2). We can then immediately compute , for any u ∈ D o β,δ , and hence We also have, for any u ∈ D o β,δ , Therefore Λ β,δ is strictly convex on D β,δ , and the Gärtner-Ellis theorem (see [6]) only applies on subsets of ∂ u Λ β,δ (D o β,δ ). For any x ∈ ∂ u Λ β,δ (D o β,δ ), the equation ∂ u Λ β,δ (u) = x has a unique solution u * (x) and hence Λ * β,δ (x) := sup u∈D β,δ We now move on to the case b = 0. From [2, Corollary 1], the moment generating function of the random variable X α,β,δ t is given by