A partial rough path space for rough volatility

We develop a variant of rough path theory tailor-made for analyzing a class of financial asset price models known as rough volatility models. As an application, we prove a pathwise large deviation principle (LDP) for a certain class of rough volatility models, which in turn describes the limiting behavior of implied volatility for short maturity under those models. First, we introduce a partial rough path space and an integration map on it and then investigate several fundamental properties including local Lipschitz continuity of the integration map from the partial rough path space to a rough path space. Second, we construct a rough path lift of a rough volatility model. Finally, we prove an LDP on the partial rough path space, and the LDP for rough volatility then follows by the continuity of the solution map of rough differential equations.


Introduction
A rough volatility model is a stochastic volatility model for an asset price process with volatility being rough, meaning that the Hölder regularity of the volatility path is less than half.Recently, such models have been attracting attention in mathematical finance because of their unique consistency to market data.Indeed, rough volatility models are the only class of continuous price models that are consistent to a power law of implied volatility term structure typically observed in equity option markets, as shown by [19].One way to derive the power law under rough volatility models is to prove a large deviation principle (LDP) as done by many authors [11,4,3,13,14,31,33,34,38,35,39,32] using various methods.An introduction to LDP and some of its applications to finance and insurance problems can be found in [44,15].In the context of the implied volatility, a short-time LDP under local volatility models provides a validity proof for a precise approximation known as the BBF formula [6,1].The SABR formula, which is of daily use in financial practice, is also proved as a valid approximation under the SABR model by means of LDP [43].From these successes in classical (non-rough) volatility models, we expect LDP for rough volatil-ity models to provide in particular a useful implied volatility approximation formula for financial practice such as model calibration.
For the classical models that are described by standard stochastic differential equations (SDEs), an elegant way to prove an LDP is to apply the contraction principle in the framework of rough path analysis [17,18].Under rough volatility models, the volatility of an asset price has a lower Hölder regularity than the asset price process.The stochastic integrands are therefore not controlled by the stochastic integrators in the sense of [28].Hence, a rough volatility model is beyond the scope of rough path theory, which motivated [3] to develop a regularity structure for rough volatility.For classical SDEs, the Freidlin-Wentzell LDP can be obtained as a consequence of the continuity of the solution map (the Lyons-Itô map) that is the core of rough path theory.In [3], the LDP for rough volatility models is obtained using the continuity of Hairer's reconstruction map.Herein, we take an approach that is similar to that of [3] in spirit but differs somewhat.Instead of embedding a rough volatility model into the abstract framework of regularity structure, we develop a minimal extension of rough path theory to incorporate rough volatility models.Besides the relatively elementary construction, an advantage of our theory is that it ensures the continuity of the integration map between rough path spaces, which enables us to treat a more general model than [3].
We focus on a model of the following form: where is a -dimensional Brownian motion, ˆ is an -dimensional stochastic process of which components include ∫ 0 ( − )d with a deterministic 2 kernel .The stochastic integration is in the Itô sense.An example is the rough Bergomi model ( = is the Riemann-Liouville kernel (3.1), is exponential, and ( ) = in (1.1)) introduced by [2].When = or more generally has a similar singularity to with < 1/4, beyond the case of ( ) = 1 or ( ) = , no LDP is available in the literature so far, including [3].As mentioned above, the difference between classical SDEs and (1.1) is that the volatility process ˆ is not controlled by because of its lower regularity.From empirical evidence, we are particularly interested in the case where ˆ is correlated with and < 1/4 [26,5,24,7].Unfortunately, the application of existing rough path theory involves iterated integrals of ˆ while, as is well-known, the standard rough path lift of ( , ˆ ) that is amenable to LDP does not work when < 1/4; see e.g., [18].
Our idea, inspired by [3], is to consider a partial rough path space in which we lack the iterated integrals of ˆ but are still able to treat (1.1).More precisely, we define the space of a triplet of iterated integrals driven by (we do not consider iterated integrals driven by ˆ ) and rederive analytical results obtained in existing rough path theory.The notion of a partial rough path was introduced in [30] to prove the existence of global solutions for differential equations driven by a rough path with vector fields of linear growth.Our motivation is different and requires a space of higher-level paths.In contrast to [3], our method does not rely on the theory of regularity structure and enables us to treat not only the rough Bergomi model but also the following rough volatility models: -the rough SABR model [22,41,23,20]; -the mixed rough Bergomi model [8]; -rough local stochastic volatility [37]; -the two-factor fractional volatility model [25].
To the best of our knowledge, no LDP for these models is established so far in the literature.
To explain the idea of the partial rough path, here, we argue for how such a partial rough path space should be.Suppose that , ≧ 1, : [0, ] → R , ˆ : [0, ] → R , and : R → R are good enough.By the Taylor expansion, for < (which are close enough), we have where := ∫ 0 ( ˆ )d , , , are multi-indices, and we use the following notation: ..., ).Therefore, following the idea of rough path theory, we would be able to define a rough path integral for ˆ := ˆ − ˆ .By the linearity of the integration and the binomial theorem (see Section 8.1 in [10]), ( ) and X ( ) should satisfy the following formulas respectively: for any , , ∈ Z + and ≦ ≦ , and where, for , ∈ Z + , ≦ means for all ∈ {1, ..., }, ≦ , and Z + is the set of the nonnegative integers.Our partial rough space is a space for ˆ , ( ) and X ( ) , where the formulas (1.2) and (1.3) should play the role of Chen's identity.
In Section 2, we formulate such a partial rough path space and state some fundamental properties including the continuity of the integration map.In Section 3, we construct a rough path lift of our rough volatility model and state an LDP.Proofs are relegated to Section 4.
is a triplet of functions on Δ satisfying the following conditions for any ∈ , ( , ) ∈ , and ≦ ≦ .
Remark 2.3.Our modified Chen's relation is a particular form of the algebraic structure of branched rough paths studied in [29].However, because ˆ is not a controlled path of , the novel framework of ( , ) rough paths is essential for establishing the rough path integral stated in the Introduction.

Remark 2.4 (A comparison with [3]). The iterated integral
plays a key role also in [3] (see section 3.1 in [3], where ( ) = W in their notation).In [3], its derivative d d W appears in the structure space of regularity structure.Our ( , ) rough path consists of not only ( ) but also X ( ) .The latter is required to construct a rough path integral as an element of a rough path space, while in [3] the corresponding integral is constructed as merely a distribution and such terms as X ( ) are not necessary for that purpose.As mentioned in Introduction, the key to treat (1.1) with a general function is to construct ∫ ( ˆ , )d as an element of a rough path space.

( , ) rough path integration
Extending the rough path integration, here we introduce an integration with respect to an ( , ) rough path.
Denote by Ω -Hld the -Hölder rough path space, and denote by the metric function on Ω -Hld ; see [16], for example.Here, we state our first main result, the proof of which is given in Section 4.1.
∈ Ω, X( (ii) It holds that where the left-hand side is the first level of the ( , ) rough path integral and the right-hand side is the Itô integral.
Theorem 3.3.The sequence of the processes := ∫ ( X )dX ≧0 satisfies the LDP on (Ω -Hld , ) with speed −1 with good rate function where Proof.By Theorems 2.6 and 3.2 together with the contraction principle, we have the claim.

Rough differential equation driven by an ( , ) rough path integral and the Freidlin-Wentzell LDP
We now discuss the following type of rough differential equation (RDE) (in Lyons' sense; see Section 8.8 of [16], for example): where Proof.(i) is a standard result from rough path theory; see e.g., Theorem 1 in [40] or Chapter 8 in [16].(ii) follows from Proposition 3.1; see Chapter 9 in [16].
Theorem 3.5.Let ∈ 3 and ¯ := Φ( ), where Φ is the solution map of Theorem 3.4.Then the sequence of the processes { ¯ } ≧0 satisfies the LDP on Ω -Hld with speed −1 with good rate function Proof.Because the solution map Φ is continuous, Theorem 3.4 and the contraction theorem imply the claim.

Short-time asymptotics
We consider the case = (see (3.1)).By the scaling property of the Riemann-Liouville fractional Brownian motion ˆ and the standard Brownian motion , we have This implies where ( ˆ , ) = ( ˆ , ), of which the rough path lift is X of Theorem 3.2.Letting we have and we can derive an LDP for ˜ by an extended contraction principle [45].
Theorem 3.6.Let ∈ 3 .Then { ˜ } 0< ≦1 satisfies the LDP on Ω -Hld as → 0 with speed −2 with good rate function Proof.Denote by Φ the solution map of the RDE (3.2) with ¯ = ˜ .We are going to show that Φ is locally equicontinuous.Because for all ∈ Z + , the local Lipschitz constants of Φ can be taken uniformly in by Theorem 4 in [40].Therefore Φ is equicontinuous on bounded sets, and we conclude Φ ( ) → Φ 0 ( ) for any converging sequence → for any with ˜ ### ( ) < ∞.Then by Theorem 3.3 and an extended contraction principle [45][Theorem 2.1], we have the desired results.Remark 3.7.By the usual argument, adding a drift term to the above RDE is straightforward.The result then generalizes the existing LDP for the rough Bergomi model: in [11,3,31,38,35].To deal with the mixed rough Bergomi model [8] or the two-factor fractional volatility model [25], we need an extension with higher dimensional and that is also straightforward.
An LDP for the marginal distribution ˜ 1 follows from the contraction principle, and the corresponding one-dimensional rate function extends the one obtained by [11] as follows.

Proof. See Appendix B.
A short-time asymptotic formula of the implied volatility (regarding as a price or a log-price process) then follows from Theorem 3.8 as in [11].From the rate function of Theorem 3.8, we observe that the effect of the function to the short-time asymptotics is only through the constant ( 0 ).In particular, the local volatility function does not add any flexibility to the asymptotic shape of the implied volatility surface.
Therefore, (1) is well-defined.Furthermore, by (4.6), we have Thus we have proved the statement of Claim 1.
Then the second level of the ( , ) rough path integral (2) is well-defined and has the following inequality: where 1) .

Definition 4.3 ([27]
).Let { } be a sequence of R-valued semimartingales on [0, ] × Ω .We say that the sequence is uniformly exponentially tight (UET) if for every > 0 and every > 0 there is , such that lim sup where − • is the Itô integral of with respect to : For a one-dimensional Brownian motion , = −1/2 is an example of a UET sequence; see Lemma 2.4 of [27].
Then Jensen's inequality implies that ) .Then we have that and so taking > 0 small enough (2 2 < 1), Stirling's formula implies the claim.