Large deviations for some fast stochastic volatility models by viscosity methods

We consider the short time behaviour of stochastic systems affected by a stochastic volatility evolving at a faster time scale. We study the asymptotics of a logarithmic functional of the process by methods of the theory of homogenisation and singular perturbations for fully nonlinear PDEs. We point out three regimes depending on how fast the volatility oscillates relative to the horizon length. We prove a large deviation principle for each regime and apply it to the asymptotics of option prices near maturity.


Introduction
In this paper we are interested in stochastic differential equations with two small parameters ε > 0 and δ > 0 of the form where W t is a standard r-dimensional Brownian motion, the functions φ(x, y), σ(x, y), b(y), τ (y) are Z m -periodic with respect to the variable y, and the matrix τ is non-degenerate. This is a model of systems where the variables Y t evolve at a much faster time scale s = t δ than the other variables X t . The second parameter ε is added in order to study the small time behavior of the system, in particular the time has been rescaled in (1.1) as t → εt. Passing to the limit as δ → 0, with ε fixed, is a classical singular perturbation problem, its solution leads to the elimination of the state variable Y t and to the definition of an averaged system defined in R n only. There is a large literature on the subject, see the monographs [32], [30], the memoir [3] and the references therein. Here we study the asymptotics as both parameters go to 0 and we expect different limit behaviors depending on the rate ε/δ. Therefore we put δ = ε α , with α > 1, and consider a functional of the trajectories of (1.1) of the form (1.2) v ε (t, x, y) := ε log E e h(Xt)/ε |(X., Y.) satisfy (1.1) , where h ∈ BC(R n ). The logarithmic form of this payoff is motivated by the applications to large deviations that we want to give. It is known that v ε solves the Cauchy problem with initial data v ε (0, x, y) = h(x) for a fully nonlinear parabolic equation. Letting ε → 0 in this PDE is a regular perturbation of a singular perturbation problem, for which we can rely on the techniques of [4], stemming from Partially supported by the Fondazione CaRiPaRo Project "Nonlinear Partial Differential Equations: models, analysis, and control-theoretic problems" and the European Project Marie Curie ITN "SADCO -Sensitivity Analysis for Deterministic Controller Design". 1 Evans' perturbed test function method for homogenisation [19] and its extensions to singular perturbations [1,2,3]. We show that under suitable assumptions the functions v ε (t, x, y) converge to a function v(t, x) characterised as the solution of the Cauchy problem for a first order Hamilton-Jacobi equation A significant part of the paper is devoted to the analysis of the effective Hamiltonian H, which is obtained by solving a suitable cell problem. As usual in the theory of homogenisation for fully nonlinear PDEs, this is an additive eigenvalue problem. It turns out to have different forms in the following three regimes depending on α:    α > 2 supercritical case, α = 2 critical case, α < 2 subcritical case.
More precisely, in the supercritical case the cell problem involves a linear elliptic operator andH has the explicit formulā where µ is the invariant probability measure on the m-dimensional torus T m of the stochastic process In the critical case the cell problem is a fully nonlinear elliptic PDE andH can be represented in various ways based, e.g., on stochastic control. Finally, in the subcritical case the cell problem is of first order and nonlinear, and a representation formula forH can be given in terms of deterministic control. In particular, under the condition τ σ T = 0 of non-correlations among the components of the white noise acting on the slow and the fast variables in (1.1), we havē H(x, p) = max y∈R m |σ T (x, y)p| 2 . Let us mention that an important step of the method is the comparison principle for the limit Cauchy problem (1.3), ensuring that the weak convergence of the relaxed semilimits is indeed uniform, as well as the uniqueness of the limit. It is known that this property of the effective Hamiltonian may require additional conditions [3]. Here we show that no extra assumptions are needed in the superand subcritical cases, whereas in the critical case the comparison principle holds if either the matrix σ is independent on x, or it is non-degenerate, or the noncorrelation condition τ σ T = 0 holds.
The main application of the convergence results is a large deviations analysis of (1.1) in the three different regimes. We prove that the measures associated to the process X t in (1.1) satisfy a Large Deviation Principle (briefly, LDP) with good rate function whereL is the effective Lagrangian associated toH via convex duality. In particular we get that for any open set B ⊆ R n . Following [22] we also apply this result to an estimate of option prices near maturity and an asymptotic formula for the implied volatility.
Our first motivation for the study of systems of the form (1.1) comes from financial models with stochastic volatility. In such models the vector X t represents the log-prices of n assets (under a risk-neutral probability measure) whose volatility σ is affected by a process Y t driven by another Brownian motion, which is often negatively correlated with the one driving the stock prices (this is the empirically observed leverage effect, i.e., asset prices tend to go down as volatility goes up). Fouque, Papanicolaou, and Sircar argued in [25] that the bursty behaviour of volatility observed in financial markets can be described by introducing a faster time scale for a mean-reverting process Y t by means of the small parameter δ in (1.1). Several extensions, applications to a variety of financial problems, and rigorous justifications of the asymptotics can be found in [26,27,9,10,28], see also the references therein. On the other hand, Avellaneda et al. [5] used the theory of large deviations to give asymptotic estimates for the Black-Scholes implied volatility of option prices near maturity in models with constant volatility. In the recent paper [22], Feng, Fouque, and Kumar study the large deviations for system of the form (1.1) in the one-dimensional case n = m = 1, assuming that Y t is an Ornstein-Uhlenbeck process and the coefficients in the equation for X t do not depend on X t . In their model ε represents a short maturity for the options, 1/δ is the rate of mean reversion of Y t , and the asymptotic analysis is performed for δ = ε α in the regimes α = 2 and α = 4. Their methods are based on the approach to large deviations developed in [23]. A related paper is [21] where the Heston model was studied in the regime δ = ε 2 by methods different from [22].
Although sharing some motivations with [22] our results are quite different: we treat vector-valued processes under rather general conditions and discuss all the regimes depending on the parameter α; our methods are also different, mostly from the theory of viscosity solutions for fully nonlinear PDEs and from the theory of homogenisation and singular perturbations for such equations. Our assumption of periodicity with respect to the y variables may sound restrictive for the financial applications. It is made mostly for technical simplicity and can be relaxed to the ergodicity of the process Y t as in [9,10]: this will be treated in a paper in preparation.
Large deviation principles have a large literature for diffusions with vanishing noise; some of them were extended to two-scale systems with small noise in the slow variables, see [34], [37], and more recently [33], [18], and [35]. Our methods can be also applied to this different scaling. The paper by Spiliopoulos [35] also states some results for the scaling of (1.1) under the assumptions of periodicity and n = m = 1, but its methods based on weak convergence are completely different from ours. A related paper on homogenisation of a fully nonlinear PDE with vanishing viscosity is [13].
The paper is organized as follows. In Section 2 we give the precise assumptions and describe the parabolic PDEs satisfied by v ε in the different regimes. In sections 3, 4, 5 we analyse the cell problem and the properties of the effective Hamiltonian in the critical (α = 2), supercritical ( α > 2), and subcritical case (α < 2), respectively. Section 6 is devoted to the convergence result for each regime of the functions (1.2) to the unique viscosity solution of the limit problem (1.3) withH identified in the previous sections, see Theorems 6.1 and 6.2. In section 7 we prove the Large Deviation Principle for all the regimes, Theorem 7.1. Finally, in Section 8 we give some applications to option pricing.
2. The fast stochastic volatility problem 2.1. The stochastic volatility model. We consider fast-mean reverting stochastic volatility system that can be written in the form where ε > 0, α > 1 and W t is an r-dimensional standard Brownian motion. We assume φ : R n × R m → R n , σ : R n × R m → M n,r are bounded continuous functions, Lipschitz continuous in (x, y) and periodic in y, where M n,r denotes the set of n × r matrices. Moreover b : R m → R m , τ : R m → M m,r are locally Lipschitz continuous functions, periodic in y. These assumptions will hold throughout the paper. We will use the symbol S k to denote the set of k × k symmetric matrices. In the following we will assume the uniform nondegeneracy of the diffusion driving the fast variable Y t , i.e for some θ > 0 In order to study small time behavior of the system (2.1), we rescale time t → εt for 0 < ε ≪ 1, so that the typical maturity will be of order of ǫ. Denoting the rescaled processes by X ε t and Y ε t we get Next we consider the functional where g ∈ BC(R n ). We denote with BC(R n ) the space of bounded continuous functions in R n . The partial differential equation associated to the functions u ε is where b and τ are computed in y, φ and σ are computed in (x, y). The equation is complemented with the initial condition: Remark 1. Note that, since we assume the periodicity in y of the coefficients of the equation b, σ, τ, φ, we have that the solution u ε of the equation (2.5) is periodic in y itself.

2.2.
The log-tranform and its HJB equation. We introduce the logarithmic transformation method (see [24]). Assume that where u ε is defined in (2.4), x ∈ R n , y ∈ R m , and t ≥ 0. By (2.5) and some computations one sees that the equation associated to v ε is , where b and τ are computed in y, φ and σ are computed in (x, y). In general, the functions u ǫ are not smooth but one can check that v ǫ is a viscosity solutions of (2.7) (see in particular Chapter VI and VII of [24]).
In the following proposition we characterize the value function v ε as the unique continuous viscosity solution to a suitable parabolic problem with initial data for each of the three regimes. A general reference for these issue is [24]. The equation (2.7) satisfied by v ǫ involves a quadratic nonlinearity in the gradient. This case was studied by Da Lio and Ley in [15], where the reader can find a proof of the next result.
Then v ε is the unique bounded continuous viscosity solution of the Cauchy problem ii) Let α < 2 and define H ε (x, y, p, q, X, Y, Z) := |σ T p| 2 + |τ T q| 2 + 2(τ σ T p) · q + ε tr(σσ T X) + φ · p Then v ε is the unique bounded continous viscosity solution of the Cauchy problem (2.9) Our goal is to study the limit as ε → 0 of the functions v ε described in Proposition 2.1. Following the viscosity solution apoproach to singular perturbation problems (see [3], [2]), we define a limit or effective Hamiltonian H and we characterize the limit of v ε as the unique solution of an appropriate Cauchy problem with Hamiltonian H. The first step in the procedure is the identification of the limit Hamiltonian. In order to define this operator, we make the ansatz that the function v ε admits the formal asymptotic expansion and plug it into the equation. In the following sections we show that the limit Hamiltonian is different in the three different regimes: the critical case (α = 2), the supercritical case (when α > 2), and the subcritical case (when α < 2). Numerical experiments in [36] indicate that the first order approximation in the expansion (2.10) is sufficiently accurate to find option prices in a fast mean-reversion case of the volatility process.
3. The critical case: α = 2 3.1. The effective Hamiltonian. We plug in the equation . We want to eliminate the corrector w and the dependence on y in this equation and remain with a left hand side of the form v 0 t −H(x, D x v 0 ). Therefore we freezex and p = D x v 0 (x) and define the effective HamiltonianH(x,p) as the unique constant such that the following stationary PDE in R m , called cell problem, has a viscosity solution w: yy w(y)) = 0, where σ is computed in (x, y) and τ, b in y. This is an additive eigenvalue problem that arises the theory of ergodic control and has a wide literature. Under our standing assumptions we have the following result.
Proposition 3.1. For any fixed (x,p), there exists a uniqueH(x,p) for which the equation (3.2) has a periodic viscosity solution w. Moreover w ∈ C 2,α for some 0 < α < 1 and satisfies for some C > 0 independent ofp and ∀x,p ∈ R n To prove Proposition 3.1, we need the following lemma.
Then there exists C > 0 independent ofp such that for allx,p ∈ R n it holds Proof. The proof uses the Bernstein method, following the derivation of similar estimates in [20]. We carry out the computations in the case τ, σ, b are C 1 . When τ, σ, b are Lipschitz the result can be proved by smooth approximation. Denote by w δ := w δ (y;x,p) the solution of (3.4). By comparison with constant sub-and supersolutions we get the uniform bound Define the function z as follows Should z attains its maximum at some point y 0 , then at y 0 . . , m, where we are adopting the summation convention, and (3.9). Thus at y 0 Then and C > 0 depends only on the L ∞ norm of σ, b, τ and on the derivatives of σ, b and τ . Therefore Thanks to the uniform ellipticity of τ and using equation (3.4), we have Using (3.7), we get at y 0 Then (3.6) follows by dividing (3.11) by |Dw δ | 3 and noticing that the right member in (3.11) is polynomial of degree 4 in |p| and |Dw δ |.
Proof. We use the methods of [6] based on the small discount approximation where F is defined in (3.5). Let w δ := w δ (y,x,p) ∈ C 2 (R m ) be a solution of (3.12). We show that δw δ (y) converges along a subsequence of δ → 0 to the constant H(x,p) and w δ (y) − w δ (0) converges to the corrector w. The hard part is proving equicontinuity estimates for δw δ . Different from [6,3], here the leading term in (3.2) is |τ T Dw| 2 rather than tr(τ τ T D 2 w). Then the Krylov-Safonov estimates for elliptic PDEs must be replaced by the Lipschitz estimates proved in Lemma 3.2.
In fact, thanks to (3.6), for some C > 0 independent ofp and for all y, z ∈ R m and δ > 0 and the equicontinuity follows. The equiboundness follows from (3.7). Then by Ascoli-Arzela theorem, there is a sequence δ n → 0 such that δ n w δn converges locally uniformly to a constant thanks to (3.13). We call itH. Similarly, we prove that v δ := w δ (y) − w δ (0) is equibounded and equicontinuous and thus converges locally uniformly along a subsequence to a function w. Then, from (3.12) we get Since v δ is equibounded δv δ → 0. Then from δw δ →H we get that w is a solution of (3.2). Finally, by the comparison principle for (3.12), it is standard to see that H is unique. Moreover the regularity theory for viscosity solutions of convex uniformly elliptic equations implies that w ∈ C 2,α for some 0 < α < 1.
Finally the corrector inherits (3.13) and satisfies for some C > 0 independent ofp and for allx,p ∈ R n max y∈R m |D y w(y;x,p)| ≤ C(1 + |p|).

3.2.
Properties and formulas forH. The next result lists some elementary properties of the effective HamiltonianH.
(d) There exists C > 0 independent of p such that, for all x,x, p ∈ R n , then, for all x,x, p,p ∈ R n , Remark 2. The meaning of assumption (3.16) is that the components of the Brownian motion W t influencing the slow variables X t are not correlated with the components acting on the slow variables Y t . In fact the condition is satisfied if the last m columns of σ and the first n columns of τ are indentically zero.
Proof. The results (a), (b), and (c) are obtained by standard methods in the theory of homogenisation, by means of comparison principles for the approximating equation (3.12), see, e.g., [19,1]. Let us show one inequality in (3.15) (the other being symmetric). Let w δ (y) := w δ (y;x, p) ans v δ (y) := w δ (y; x, p). Then v δ satisfies Thanks to Lemma 3.2 we estimate Dv δ , and then, using the Lipschitz continuity of σ, we get for some C > 0 By letting δ → 0 we get the inequality forH(x, p) −H(x, p) in (3.15), and by exchanging x andx we complete the proof.
If (3.16) holds, (3.18) simplifies to Then, as before, we obtain by comparison the second inequality in (3.17), and the first is got in a symmetric way.
Next we give some representation formulas for the effective HamiltonianH.
where β(·) is an admissible control process taking values in R r for the stochastic control system (ii) moreover where w = w(·;x,p) is the corrector defined in Proposition 3.1 and µ = µ(·;x,p) is the invariant probability measure of the process (3.29) with the feedback β(z) = −τ T (z)Dw(z); (iii) finally where Y t is the stochastic process defined by Proof. (i) The first formula comes from a control interpretation of the approximating δ-cell problem (3.4). We write it as the Hamilton-Jacobi-Bellman equation and we represent w δ as the value function of the infinite horizon discounted stochastic control problem (see, e.g., [24]) where Z t is defined by (3.22). Then (3.20) follows from the proof of Proposition 3.1. For the formula (3.21) we consider the t-cell problem This is also a HJB equation, whose solution is the value function v(t, z;x,p) = sup where Z t is defined by (3.22). Then a generalized Abelian-Tauberian theorem (see [2] for a general proof based only on the comparison principle for the Hamiltonian) states that (ii) The formula (3.23) is derived from a direct control interpretation of the cell problem (3.2). In fact, it is the HJB equation of the ergodic control problem of maximizing among admissible controls β(·) taking values in R r for the system (3.22), as before.
The process Z t associated to each control is ergodic with a unique invariant measure µ on T m because it is a nondegenerate diffusion on T m , see, e.g., [3], so the limit in the payoff functional exists and it is the space average in dµ of the running payoff.
Since the HJB PDE (3.2) has a smooth solution w, it is known from a classical verification theorem that the feedback control that achieves the minimum in the Hamiltonian, i.e., β(z) = −τ T (z)Dw(z), is optimal. Then (3.23) holds with µ the invariant measure of the process (3.29) (iii) To prove (3.24), take v = v(t, x;x,p) a periodic solution of the t-cell problem and define the function f (t, y) = e v(t,y) . Then f solves the following equation By the Feynman-Kac formula, we have where Y t is defined by (3.25). Then v(t, y) = log E e  because e −w Le w = Lw + |τ T D y w| 2 gives e −w Lx ,p e w = e −w Le w + 2(τ σ Tp ) · D y w = Lw + |τ T D y w| 2 + 2(τ σ Tp ) · D y w.
We conclude that if w is a solution of (3.2), thenH is the first eigenvalue of the linear operator Lx ,p + Vx ,p , with eigenfunction g = e w . [31]. They prove the existence of a constantH such that there is a unique smooth solution w with prescribed growth of (3.2). Moreover they provide a representation formula forH as the convex conjugate of a suitable operator over a space of measures.

3.3.
Comparison principle forH. The comparison theorem among viscosity sub-and supersolutions of the limit PDE (3.32) v t −H(x, Dv) = 0 in (0, T ) × R n will be the crucial tool for proving that the convergence of v ε is not only in the weak sense of semilimits but in fact uniform, and the limit is unique. It is known from [3] that in general the regularity ofH with respect to x may be worse than that of H ε and the comparison principle may fail. Next result gives three alternative additional conditions ensuring the comparison.

Proof. In case (i)H is independent of x and it is continuous by Proposition 3.3.
Then the result follows from standard theory, see e.g. [11].
In case (ii) the bounds (3.14) and (3.33) givē ThenH is coercive and has the properties (a) and (b) and (c) of Proposition 3.3. The result follows from [15] once we prove that for some C > 0 and all x, y, p ∈ R n Takeq such thatL (x, p) =q · p −H(x,q). ThenL (x, p) −L(y, p) ≤H(y,q) −H(x,q) ≤ C(1 + |q| 2 )|x − y|, where we have used the property (d) of Proposition 3.3. We want to estimate |q|. If |q| > |p| ν , then 0 ≤L(x, p) =q · p −H(x,q) ≤q · p − ν|q| 2 < 0 and we reach a contradiction. Then By reversing the roles of x and y we get the full inequality (3.34).
In case (iii) we need the following semi-homogeneity of degree two ofH: This follows from the representation formula (3.24), because Jensen inequality gives µ log E e t 0 |σ T (x,Ys) p µ | 2 ds |Y 0 = y ≥ log E e µ t 0 |σ T (x,Ys) p µ | 2 ds |Y 0 = y and the conclusion is reached after dividing by t and letting t → ∞. The other ingredient of the proof is the first inequality in (3.17) that relates the regularity in x ofH with that of the pseudo-coercive Hamiltonian |σ T (x, y)p| 2 . With these two inequalities one can repeat the proof of the comparison principle for the pseudocoercive Hamiltonian by Barles and Perthame, see [12] for the stationary case and [7] for the evolutionary case. Let us give a sketch of the main points of the proof. We show that for µ < 1, µ sufficiently near to 1, it holds If this is true, then the inequality holds also for µ = 1, proving the Theorem. By contradiction, we assume that for every µ < 1, there exists (x, t) such that For ε, η small enough, Φ has a maximum point, that we denote with (x ′ , z ′ , t ′ , s ′ ). By standard arguments, we get |x −→ 0 as ε, η → 0. If either s ′ = 0 or t ′ = 0, it is easy to see that we get a contradiction with (3.36). So we consider the case ( Using the fact that u is a subsolution we get Since v is a supersolution andH satisfies (3.35), we get So, we multiply (3.38) by −µ and sum up to (3.37) to obtain Using (3.17) we get (3.40) Note that ∆(y) goes to zero for ǫ, η → 0 and for all δ fixed uniformly in y, and J(y) goes to zero for ǫ, η, δ → 0 uniformly in y. Then we can rewrite the rhs of (3.40) as Moreover, for all k 1 , k 2 > 0 and for all y ∈ R m it holds So, recalling (3.39), (3.40) and (3.41) we get If we choose k 1 , k 2 > 0 such that k 1 + k 2 < 1−µ µ then we obtain as ε, η, δ → 0, reaching a contradiction.
4. The supercritical case: α > 2 As in Section 3, we prove the existence of an effective Hamiltonian giving the limit PDE and first we identify the cell problem that we wish to solve. Plugging the asymptotic expansion We consider the δ-cell problem for fixed (x,p,X) (4.1) δw δ (y) − |σ(x, y) Tp | 2 − b(y) · D y w δ (y) − tr(τ (y)τ (y) T D 2 yy w δ (y)) = 0 in R m , where w δ is the approximate corrector. The next result states that δw δ converges toH and it is smooth. Proposition 4.1. For any fixed (x,p) there exists a constantH(x,p) such that H(x,p) = lim δ→0 δw δ (y) uniformly, where w δ ∈ C 2 (R m ) is the unique periodic solution of (4.1). Moreover that is, the periodic solution of with T n µ(y) dy = 1.
Proof. The proof essentially follows the arguments presented in [6,3] of ergodic control theory in periodic enviroments.
Proof. For the proofs of (a), (b), (c), (d) we repeat the same arguments as in Proposition 3.3. Properties (f), (g) can be easily checked from the representation formula (4.2).
We now state the comparison principle among viscosity sub-and supersolutions of the limit PDE In this case, differently from the critical case, we do not need additional assumptions for th e comparison principle to hold. 5. The subcritical case: α < 2 5.1. The effective Hamiltonian. In this case, the asymptotic expansion we plug in the equation is x, y). Plugging (5.1) into the equation (2.7) we get . Therefore the cell problem we want to solve is finding, for any fixed (x,p), a unique constantH such that there is a viscosity solution w of the following equation (5.3)H(x,p) − 2(τ (y)σ(x, y) Tp ) · D y w(y) − |τ (y) T D y w(y)| 2 − |σ(x, y) Tp | 2 = 0.
Since 2(τ (y)σ T (x, y)p) · D y w = 2(σ T (x, y)p) · (τ T (y)D y w) , we can restate the cell problem as The following proposition deals with the existence and uniqueness ofH. For any fixed (x,p), there exists a unique constantH(x,p) such that the cell problem (5.3) admits a periodic viscosity solution w. Moreover w is Lipschitz continuous and there exists C > 0 independent ofx,p such that max y |Dw(y;x,p)| ≤ C(1 + |p|).
Proof. As for the other cases we introduce the following approximant problem, with δ > 0, Let w δ the unique periodic viscosity solution to (5.5). By standard comparison principle we get that |δw δ | ≤ max y∈R m |σ T (x, y)p| 2 ≤ C(1 + |p| 2 ) ∀y ∈ R m . Moreover, using the coercivity of the Hamiltonian (see [8,Prop II.4.1]), we get that w δ is Lipschitz continuous and there exists a constant C independent of δ andp such that max y∈R m |Dw δ | ≤ C(1 + |p|). So, we conclude as in the proof of Proposition 3.1.
We give some representation formulas for the effective HamiltonianH. where β(·) varies over measurable functions taking values in R r , y(·) is the trajectory of the control system ẏ(t) = 2τ (y(t))σ T (x, y(t))p − 2τ (y(t))β, t > 0, y(0) = y and the limit is uniform with respect to the initial position y of the system.
(ii) If, in addition, τ (y)σ T (x, y) = 0 for all x, y, then Proof. The formula (5.6) can be proved by writing (5.5) as a Bellman equation Then w δ is the value function of the infinite horizon discounted deterministic control problem appearing in (5.6) (see, e.g., [8,11]).
So, this gives immediately the inequality ≥ in (5.7). The other inequality is obtained by standard comparison principle arguments applied to the approximating problem (5.5).
Finally, in the case n = m = r = 1, ifp ≥ 0 we write explicitly the corrector as Note that w ∈ C 1 is periodic and does the job. A similar construction works for p < 0.
For the comparison principle it is useful to define and observe that the cell problem (5.3) is equivalent to the following equation Here are some properties ofH and H 0 . there exists C > 0 such that |H 0 (x, p)| ≤ C|p|, and Proof. For the proofs of (a), (b), (c) we repeat the same arguments as in Proposition 3.3. The properties of H 0 defined in (5.10) follow from standard theory, using comparison type argument in the approximating problem δv δ (y) − |τ T (y)D y v δ (y) + σ T (x, y)p| = 0 in R m .

5.2.
Comparison principle. We consider the limit PDE We now state the comparison principle for the effective HamiltonianH.
for all x ∈ R n and 0 ≤ t ≤ T .

The convergence result
In this Section we state the main result of the paper, namely, the convergence theorem for the singular perturbation problem. We will make use of the relaxed semi-limits which we define as follows. For the functions v ε introduced in Section 2.2 the relaxed upper semi-limitv = lim sup * We define analogously the lower semi-limit v = lim inf * ε→0 inf y v ε by replacing lim sup with lim inf and sup with inf. Since h is bounded the family v ε is equibounded and we havev ∈ BU SC([0, T ] × R n ) and v ∈ BLSC([0, T ] × R n Theorem 6.1. Assume α ≥ 2. Then i) The upper limitv (resp., the lower limit v) of v ε is a subsolution (resp., supersolution) of the effective equation either σ = σ(y) is independent of x and h ∈ BU C(R n ), or, for some ν > 0, |σ T (x, y)p| 2 > ν|p| 2 ∀ x, p ∈ R n , y ∈ R m , or, τ (y)σ T (x, y) = 0 for all x, y, then v ε converges uniformly as in ii).
Proof. i) The inequalities v(0, x) ≤ h(x) ≤v(0, x) follow from the definitions. The problem of taking the limit in the PDE is a regular perturbation of a singular perturbation problem, in the terminology of [4]. The result can be proved by the methods developed in [4] for such problems, with minor modifications. ii) By the definition of the semilimits v ≤v in [0, T ) × R n . The comparison principle Proposition 4.3 for the effective equation (6.1) gives the inequality ≤ and thereforev = v = v in [0, T ]× R n . Thanks to the properties of semilimits, we finally get that v ε converges locally uniformly to the unique bounded solution of (6.1).
iii) The proof is the same as for ii), but now we need the additional assumption (6.2) for the comparison principle Theorem 3.5.
Proof. The proof is the same as that of Theorem 6.1, by using the comparison principle Proposition 5.4.

Remark 6.
In the case α ≤ 2 we can give a convergence result analogous to Theorem 6.1 and Theorem 6.2 for a terminal cost h = h(x, y) depending also on the fast variable y, so that the payoffs is In this case we must find a suitable effective initial valueh depending only on the variable x; moreover the convergence cannot be up to time t = 0 but only on the compact subsets of (0, T ) × R n × R m to the unique viscosity solution of The proof follows the methods of [2], where an asymptotic problem for findingh is given and the relaxed semi-limits are modified at t = 0 to deal with the expected initial layer. For further details and proofs we refer to [29].

The large deviation principle
In this section we derive a large deviation principle for the process X ε t defined in (2.3). Throughout the section we suppose that σ is uniformly non degenerate, that is, for some ν > 0 and for all x, p ∈ R n (7.1) |σ T (x, y)p| 2 > ν|p| 2 .
For each x 0 ∈ R n and t > 0, define   .3) is continuous in the variable x (see, e.g., [16]) and is a nonnegative function such that I(x 0 ; x 0 , t) = 0.
(b) I satisfies the following growth condition for some C > 0 and all x, x 0 ∈ R n where ν is defined in (7.1). In fact, thanks to the property (3.14) stated in Proposition 3.3, we get that 1 4C Then we have from which we get (7.4).
(c) If σ does not depend on x, i.e.H =H(p), the rate function in (7.3) is (d) If σ does not depend of x and n = 1, I is a monotone nondecreasing function of x when x > x 0 . Analogously, I is a monotone nonincreasing function of x when x < x 0 .
Remark 8. Thanks to Remark 7, if σ does not depend on x and n = 1, we have inf y>x I(y; x 0 , t) = I(x; x 0 , t) for x ≥ x 0 and (7.5) can be written in the following way lim ε→0 ε log P (X ε t > x) = −I(x; x 0 , t) when x > x 0 and analogously when x < x 0 lim ε→0 ε log P (X ε t < x) = −I(x; x 0 , t).
Remark 9. We note that the rate function I defined in (7.3) does not depend on the drift φ of the log-price X ε t and it depends only on the volatility σ and on the fast process Y ε t . In fact, this holds for the effective HamiltonianH by the representation formulas (3.20) for α = 2, (4.2) for α > 2 and (5.6) for α < 2, and hence it holds for the Legendre transformL.
Proof. We divide the proof in two steps, the first is the proof of the large deviation principle, while the second is the proof of the representation formula (7.3) for the good rate function.
Step. 1 (Large deviation principle) The proof of this step is similar to that of Theorem 2.1 of [22] with some minor changes. The idea is to apply Bryc's inverse Step. 2 (Representation formula for the good rate function) The solution v h to the effective equation in R n can be represented through the following formula ξ(s),ξ(s) ds | y ∈ R n , ξ ∈ AC(0, t), ξ(0) = x, ξ(t) = y , whereL is the effective Lagrangian defined in (7.2). We refer to [16] where it is shown that v h is continuous and is the solution of (7.13). We define (7.15) r(x; x 0 , t) = inf ξ(0)=x0,ξ(t)=x t 0L ξ(s),ξ(s) ds Thanks to (7.12) and (7.14), we can write Taking y = x, we obtain J h (x; x 0 , t) ≤ 0 and therefore J(x; x 0 , t) ≤ 0. Now we define a function h * ∈ BC(R) as follows: h * (y) = r(y; x 0 , t) ∧ r(x; x 0 , t).
Finally, the claim follows from the continuity of the function r(y; x 0 , t) in the variable y, that can be found, e.g., in [16], Section 4, Proposition 3.1 and Corollary 3.4.
Proof. By the definition of implied volatility where Φ is the Gaussian cumulative distribution function. Then the proof follows as in [22], using (8.5) and Corollary 8.1.
Appendix A.
We recall some standard notions from large deviation theory that we need in section 7. Throughout the section, µ ǫ will denote a family of probability measures defined on R n with its Borel σ-field B. For the definitions and theorems in a more general setting and for further details we refer to [17]. Given a family of probability measures {µ ǫ }, a large deviation principle characterizes the limiting behavior, as ǫ → 0, of {µ ǫ } in terms of a rate function through asymptotic upper and lower exponential bounds on the values that µ ǫ assigns to measurable subsets of R n .  The right-and left-hand sides of (A.1) are referred to as the upper and lower bounds, respectively. Definition A.3. A family of probability measures {µ ǫ } on R n is exponentially tight if for every α < ∞, there exists a compact set K α ⊂ R n such that lim sup ǫ→0 ǫ log µ ǫ (K c α ) < −α.
Moreover, for each Borel measurable function h : R n → R, define Λ ǫ h := ǫ log ǫ µ ǫ (dx) = Λ h provided the limit exists. Then, the so-called Bryc's inverse Varadhan Lemma permits to derive the large deviation principle as a consequence of exponential tightness of the measures µ ǫ and the existence of the limits (A.2) for every h ∈ BC(R n ). The statement is the following.
Lemma A.1. Suppose that the family {µ ǫ } is exponentially tight and that the limit in (A.2) exists for every h ∈ BC(R n ). Then {µ ǫ } satisfies the LDP with the good rate function Furthermore, for every h ∈ BC(R n ), Finally we recall the optional sampling theorem. For further details see [38].
Theorem A.2. Let M = {M t } t≥0 be a submartingale right-continuos and let τ be a stopping time, such that one of the following conditions is satisfied • τ is a.s. bounded, i.e. there exists T ∈ (0, ∞) such that τ ≤ T a.s.; • τ is a.s. finite and M τ ∧t ≤ Y for all t ≥ 0, where Y is an integrable variable (in particular |M τ ∧n | ≤ K for a constant K ∈ [0, ∞)) Then the variable M τ is integrable and If, instead, M is a supermartingale, then