Option Pricing in the Moderate Deviations Regime

We consider call option prices in diffusion models close to expiry, in an asymptotic regime ("moderately out of the money") that interpolates between the well-studied cases of at-the-money options and out-of-the-money fixed-strike options. First and higher order small-time moderate deviation estimates of call prices and implied volatility are obtained. The expansions involve only simple expressions of the model parameters, and we show in detail how to calculate them for generic local and stochastic volatility models. Some numerical examples for the Heston model illustrate the accuracy of our results.

It has received little attention, and, to the best of our knowledge, none at all in the classical diffusion case. The aim of the present paper is to fill this gap. This "moderately out-of-the-money" regime in fact reflects the reality of quoted option prices: As seen in Figure 1.1, the range of strikes tends to concentrate "around-the-money" as time to expiry becomes small. At the same time, the regime offers excellent analytic tractability. To put our results into perspective, we recall some well-known facts on option price approximations close to expiry. We write ( , ) for the normalized call price as a function of log-moneyness = log( ∕ 0 ) ( 0 , (1.1) In general, ( , ) depends tacitly on 0 , the (fixed) spot value. 2 We start with the at-the-money (ATM) regime = 0. In the Black-Scholes model, writing ( , ) = BS (0, ; ) with volatility parameter > 0, we have the following ATM call price behavior From Muhle-Karbe and Nutz (2011), the same is actually true in a generic semimartingale model with diffusive component (with spot volatility 0 = √ 0 > 0), 2) and this translates to the generic ATM implied variance formula (even in presence of jumps, as long as 0 > 0) 2 imp (0, ) = 0 + (1), ↓ 0.
(We use the notation imp ( , ) for the Black-Scholes implied volatility with log-moneyness and maturity .) Higher order terms in will be model dependent. For instance, in the Heston case, with variance dynamics = − ( −̄) + √ , implied variance has the ATM expansion  .
This is corollary 4.4 in Forde, Jacquier, and Lee (2012), and we note that (0) has no easy interpretation in terms of the model parameters. Relaxing = 0 to = ( √ ) amounts to what we dub "almost ATM" (AATM) regime. 3 (In particular, ∼ is in the AATM regime if and only if > 1∕2.) Again for generic semimartingale models with diffusive component and spot volatility 0 > 0, it is easy to see from Caravenna and Corbetta (2016) and Muhle-Karbe and Nutz (2011) that the ATM asymptotics (1.2) imply the almost ATM asymptotics This fails when ceases to be ( √ ). Indeed, for = √ with constant factor > 0, we have, from Caravenna and Corbetta (2016) and Muhle-Karbe and Nutz (2011), where (− , 2 0 ) stands for a Gaussian random variable with mean − and variance 2 0 . This, too, holds true in the stated semimartingale generality. In any case, the proof is based on the Lévy case with nonzero diffusity 0 , and the result follows from comparison results, which imply that the difference is negligible to first order. For a thorough discussion of the regime = ( √ ) in the (local) diffusion case, see Pagliarani and Pascucci (2017).
Beyond this regime, call price asymptotics change considerably. For instance, take an additional slowly diverging factor log(1∕ ), Even in the Black-Scholes model, we now lose the √ -behavior of call prices described above and in fact BS ( , ; ) = 1 2 + 2 2 2 ( ), for some slowly varying function ( ), see Mijatović and Tankov (2016). On the other hand, in a genuine out-of-the-money (OTM) situation, with ≡ > 0 fixed, option values are exponentially small in diffusion models, and we are in the realm of large deviations theory.
Throughout the paper, we reserve the term OTM for fixed OTM log-strike > 0, to distinguish this regime from the moderately out-of-the-money regime that we now define. Our basic observation is that for the cases of > 1 2 , resp. = 0, are covered by the before-discussed AATM, resp. OTM, results. This leaves open a significant gap, namely, ∈ (0, 1 2 ), which we call moderately out-of-the-money (MOTM). We have a threefold interest in this MOTM regime, .
(1.5) (i) First, it is related to the reality of quoted (short-dated) option prices, where strikes of option price data with acceptable bid-ask spreads tend to accumulate "around the money," as illustrated in Figure 1.1. To account for this accumulation, we consider strikes that move closer to the money as expiry shrinks, and the simplest way to do so is to consider strikes of the order = ( ) for some > 0. There is no reason why quoted strikes should always be almost ATM ( > 1∕2), which effectively means an extreme concentration around the money; we are thus led to study the regime (1.5).
(ii) The second reason is mathematical convenience. In contrast to the genuine OTM regime (large deviation regime) in which the rate function Λ( ) is notoriously difficult to analyze-often related to geodesic distance problems-MOTM naturally comes with a quadratic rate function and, most remarkably, higher order expansions are always explicitly computable in terms of the model parameters. The terminology moderately OTM (MOTM) is in fact in reference to moderate deviations theory, which effectively interpolates between the central limit and large deviations regimes. 5 This also identifies the AATM regime as bordering the central limit regime, where asymptotics are precisely those of the Black-Scholes model, which in turn is the rescaled Gaussian (in log-coordinates) limit of a general semimartingale model with diffusive component.
(iii) Finally, our third point is that MOTM expansions naturally involve quantities very familiar to practitioners, notably, spot (implied) volatility, implied volatility skew, and so on.
In the Black-Scholes model, it is easy to check that we have the MOTM asymptotics Loosely speaking, our main results (Theorems 2.3 and 2.5 below) assert that such relations (even of higher order) are true in great generality for diffusion models, and that all quantities are computable and then related to implied volatility expansions. We note in passing that, for Lévy models, the regime (1.5) has been studied by Mijatović and Tankov (2016); then, call prices decay algebraically rather than exponentially. For recent related results on fractional stochastic volatility models, see Forde and Zhang (2017) and Guennoun, Jacquier, and Roome (2014). Guillin (2003), who considers small-noise moderate deviations of diffusions, should also be mentioned here; however, in Guillin (2003), the dynamics depend on a "fast" random environment (with motivation from physics, and no obvious financial interpretation), and the nondegeneracy assumption (D) is not satisfied in our context. The recent related paper by Gao and Wang (2016) contains a first-order moderate deviation principle (MDP) for diffusions under classical regularity assumptions from SDE theory. The main difference to the bulk of our results is that we develop higher order expansions, until Section 6 where we revisit first-order MOTM estimates from a moment-generating function perspective. However, in this case the underlying models (e.g., Heston) fall immediately outside the scope of Gao and Wang (2016), because of the typical square-root structure of coefficients. (The matter is discussed in more detail at the beginning of Section 6.) To round off the introduction, we briefly recall some background on moderate deviations. Consider the classical setting of a centered i.i.d. sequence ( ) ≥1 with finite exponential moments. Then the empirical meanŝ∶= −1 ∑ =1 converge to zero (law of large numbers, LLN), and this is quantified by an LDP according to Cramér's classical theorem: ℙ[̂> ] = exp(− ( ) + ( )) decreases exponentially as → ∞ for fixed > 0, governed by a rate function ( ) = sup ∈ℝ ( − log [ 1 ]). On the other hand, by the CLT, √̂= −1∕2 ∑ =1 has a Gaussian limit law. Moderate deviations cover intermediate scalings, i.e., √̂w ith → 0 and → ∞. It turns out (theorem 3.7.1 in Dembo & Zeitouni, 1998) that, for any such sequence > 0, the family √̂s atisfies an LDP with speed 1∕ and qquadratic rate function. (A natural scaling family is given by = 2 −1 , with parameter > 0, so that one considers . Interpolation between LDP, with LLN scaling, and CLT scaling then amounts to considering 0 < < 1∕2.) This is sometimes called an MDP. Formally, an MDP is thus just a certain LDP with appropriate scaling and speed function. Still, the terminology is often useful because of the trichotomy CLT -MDP for a range of scalings, with quadratic rate function -genuine LDP, which occurs in many situations, not just for i.i.d. sequences of random variables. For references to some other classical moderate deviations results (on empirical measures); e.g., see sections 6.7 and 7.4 of Dembo and Zeitouni (1998). Several authors have investigated moderate deviations in actuarial risk theory; see, e.g., Fu and Shen (2017) and references therein. The rest of the paper is organized as follows. Section 2 contains our main results, which translate asymptotics for the transition density of the underlying into MOTM call price asymptotics. The corresponding proofs are presented in Section 3. Section 4 and the Appendix give the implied volatility expansion resulting from our call price approximations. Section 5 applies our main results to standard examples, namely, generic local volatility models (Section 5.1), generic stochastic volatility models (Section 5.2), and the Heston model (Section 5.3). As usual, the square-root degeneracy of the Heston model makes it difficult to apply results for general stochastic volatility models, so we verify the validity of our results-if formally applied to Heston-by a direct "affine" analysis. Finally, in Section 6 we present a second approach to MOTM estimates, which employs the Gärtner-Ellis theorem from large deviations theory. Throughout we take zero rates, a natural simplification in view of our short-time consideration. Also, w.l.o.g. we normalize spot to 0 = 1.

MOTM OPTION PRICES VIA DENSIT Y ASYMPTOTICS
We consider a general stochastic volatility model, i.e., a positive martingale ( ) ≥0 with dynamics = , and started (w.l.o.g.) at 0 = 1. We assume that the stochastic volatility process ( ) ≥0 itself is an Itôdiffusion, started at some deterministic value 0 , called spot volatility. Recall that in any such stochastic volatility model, the local (or effective) volatility is defined by As is well known, the equivalent local volatility model =̃l oc ( ,̃) has the property that̃= (in law) for all fixed times. See Brunick and Shreve (2013) for precise technical conditions under which this holds true. 6 As a consequence, European option prices ( , ) match in both models. Recall also Dupire's formula in this context We now state our two crucial conditions. Assumption 2.1. For all > 0, has a continuous pdf  → ( , ), which behaves asymptotically as follows for small time: Assumption 2.2. For ↓ 0 and → 0 = 1, the local volatility function of ( ) ≥0 converges to spot volatility The latter assumption is fairly harmless (in diffusion models; see the beginning of Section 5.2). The first assumption is potentially (very) difficult to check, but fortunately we can rely on substantial recent progress in this direction; see Violante (2014a, 2014b) and Osajima (2015). We shall see in Section 5.2 that both assumptions indeed hold in generic stochastic volatility models. Let us also note the fundamental relation between spot volatility 0 (actually equal to implied spot volatility imp (0, 0) here) and the Hessian of the energy function Λ = Λ( ), (This is well known [see Durrleman, 2004] and also follows from Proposition 2.4 below.) Now we state our main result. We slightly generalize the log-strikes considered in (1.5), replacing the constant factor by an arbitrary slowly varying function .

(i) The call price satisfies the moderate deviation estimate
If we restrict to (0, 1 3 ), then the following moderate second-order expansion holds true exhibiting a qquadratic rate function  → 2 ∕2 2 0 , typical of moderate deviation problems. 7 In a nutshell, (2.5) says that inserting the time-dependent log-strike (2.4) into the fixed-strike OTM/LD approximation ( , ) = exp(−Λ( )∕ (1 + (1))) yields a correct result, upon Taylor expanding Λ. Mind, however, that this needs a proof using the specifics of our situation, in light of the fact that validity of a large deviation principle does not automatically imply an MDP.
The quantities Λ ′′ (0), Λ ′′′ (0), … appearing above are always computable from the initial values and the diffusion coefficients of the stochastic volatility model. This is in stark contrast to the OTM regime, where one needs the function Λ(⋅), which is in general not available in closed form (with some famous exceptions, like the SABR model). We quote the following result on -factor models from Osajima (2015) and refer to Section 5.2 for detailed calculations in a two-factor stochastic volatility model.
in the sense that −1 defines a Riemannian metric. Then where the coefficients are given by using the functions ) .
Proof. See Osajima (2015, theorem 1(1), with = 1). □ The following result presents a higher order expansion in the MOTM regime. It yields an asymptotically equivalent expression for call prices (and not just logarithmic asymptotics).
The passage from the derivatives of the energy function to ATM derivatives of the implied volatility in the short time limit is best conducted via the BBF formula that was proved in Berestycki et al. (2002). (That said, theses relations are also a direct consequence of our expansions, as is pointed out in Section 4.) In this regard, we have: Theorem 2.6. Suppose that Λ is a function with the properties required in Assumption 2.1, with Λ ′′ (0) = −2 0 = −1 0 , and that the Berestycki-Busca-Florent formula 2 imp (0, ) = 2 ∕2Λ( ) holds. Then the small-time ATM implied variance skew and curvature, respectively, relate to Λ via Proof. By the BBF formula and our smoothness assumptions on Λ, This implies (2.8) and (2.9). □ Proposition 2.4 combined with Theorem 2.6 allows to compute skew and curvature (and higher derivatives of the implied volatility smile, if desired) directly from the coefficients of a general stochastic volatility model. Related formulae for "general" (even non-Markovian) models also appear in the work of Durrleman (theorem 3.1.1. in Durrleman, 2004; see also Durrleman, 2010). While not written in the setting of general Markovian diffusion models, and hence not in terms of the energy function Λ, they inevitably give the same results if applied to given parametric stochastic volatility models (see section 3.1 in Durrleman, 2004). However, Durrleman's work comes with some (seemingly) uncheckable assumptions, the drawbacks of which are discussed in section 2.6 of Durrleman (2004).
Proof of Theorem 2.5. Taking logs in (3.5) yields Then (2.7) follows by Taylor expanding Λ. Note that ∕ = (1) for ≥ ⌊1∕ ⌋ + 1. ) + ( 2−4 − ), ↓ 0, which yields (4.1). □ We have no doubt that Corollary 4.1 is true for the whole MOTM regime, i.e., for all ∈ (0, 1 2 ), under very mild assumptions (Assumption A.1 in the Appendix). For any fixed ∈ (0, 1 2 ), one can compute the implied volatility expansion using the results of Gao and Lee (2014). However, for close to 1 2 , more and more terms are needed for the intermediate computations, and there does not seem to be a simple pattern that would allow for a general proof. The details are discussed in the Appendix, where we push the range of for which (4.1) is proven rigorously to 0 < < 3 7 ≈ 0.429. Note that the expansion in Theorem 2.5 becomes finer (i.e., contains more explicit terms) if is close to zero. Suppose, on the other hand, that is very close to 1 2 : Then the summands > ⌊1∕ ⌋ = 2 in (2.7), which are related to ATM derivatives of implied variance by Theorem 2.6 (see also paragraph (iii) in the Introduction), disappear into the (1)-term of (2.7).
Corollary 4.1 has some interesting consequences. Under the sheer assumption that implied volatility has a first-order Taylor expansion for small maturity and small log-strike of the form then of course in the MOTM regime, we have ≪ , and so the -term dominates the ( )-term, which in turn identifies the implied variance skew as On the other hand, Corollary 4.1 now implies that the right-hand side of (4.4) equals − 1 3 4 0 Λ ′′′ (0). We have thus arrived at an alternative proof of the skew representation (2.8) in terms of the energy function, without using the BBF formula. The curvature and higher order derivatives of the ATM smile can be dealt with similarly, if desired.

Generic local volatility models
Clearly, Assumption 2.2 is satisfied for any local volatility model, assuming continuity of the local volatility function. We now discuss Assumption 2.1 and show how to compute our MOTM expansions. First consider the time-homogeneous local volatility model where the deterministic function is 2 on (0, ∞). An expansion of the pdf (⋅, ) of has been worked out in Gatheral, Hsu, Laurence, Ouyang, and Wang (2012). They assume growth conditions on and its derivatives, which can be alleviated by the principle of not feeling the boundary (appendix A of Gatheral et al., 2012). Proposition 2.1 of Gatheral et al. (2012) states that (Recall that we normalize spot to 0 = 1 throughout.) This shows that Assumption 2.1 is satisfied, with .

Generic stochastic volatility models
We now discuss the results of Section 2 in generic stochastic volatility models. Rigorous conditions under which stochastic volatility models satisfy Assumption 2.1 can be found in Deuschel et al. (2014a) and Osajima (2015). The function Λ is given by the Riemannian metric associated to the model: 2Λ( ) is the squared geodesic distance from ( 0 = 1, 0 ) to {( , ) ∶ > 0} with = . Theorem 2.2 in Berestycki, Busca, and Florent (2004) gives conditions under which Assumption 2.2, concerning convergence of local volatility, is true. Now we describe how the expressions appearing in the expansions from Theorem 2.3 can be computed explicitly in a generic two-factor stochastic volatility model where ∶ ℝ → ℝ and ⟨ , ⟩ = . The Heston model ( ( ) ≡ 1) and the 3/2-model ( ( ) = ; see Lewis, 2000) are special cases. The infinitesimal generator of the stochastic process ( , ), neglecting first-order terms, can be written as , where 2 denotes the Hessian matrix of , and the coefficient matrix = ( ) is given by ) .

The Heston model
This section contains an application of the results of Sections 2 and 4 to the familiar case of the Heston model, where many explicit "affine" computations are possible. At the beginning of Section 5.2, we recalled some general results implying our Assumptions 2.1 and 2.2. The Heston model is not covered by these results, but satisfies Assumptions 2.1 and 2.2 nevertheless, and thus Theorems 2.3 and 2.5 are applicable to Heston. We will explain how both assumptions can be verified rigorously by a dedicated analysis; full details would involve rather dull repetition of arguments that are found in the literature in a very similar form, and are therefore omitted. The model dynamics are wherē, , > 0, and ⟨ , ⟩ = with ∈ (−1, 1). According to Forde and Jacquier (2009), the first-order OTM (large deviations) behavior of the call prices is The locally uniform density asymptotics (2.2) hold, as seen from an easy modification of the arguments in Forde et al. (2012). There, the Fourier representation of the call price was analyzed by the saddle point method to obtain a refinement of (5.7). Proceeding analogously for the Fourier representation of the pdf of , we get the density approximation locally uniformly in , where is the characteristic function of = log , and * and are defined on p. 693 of Forde et al. (2012). (Note that Forde et al., 2012, use the notation Λ, Λ * instead of our Γ, Λ He .) From (5.9) and the fact that ( * (0)) = (0) = 1, we see that the factor ( ) from (2.2) converges to To verify Assumption 2.2 (convergence of local volatility), the Dupire formula (2.1) can be subjected to an analysis similar to De Marco, Friz, and Gerhold (2013) and Friz and Gerhold (2015). More precisely, ( , ) in the numerator of (2.1) is the pdf of , the analysis of which we have just described. Virtually the same saddle point approach can be applied to the numerator ( , ), yielding convergence of the quotient to 2 0 . We now calculate our MOTM asymptotic expansions for the Heston model. The Legendre transform Λ He is given by Λ He ( ) = sup { − Γ( )} with maximizer * = * ( ). From general facts on Legendre transforms, Λ ′′ He ( ) = 1 Γ ′′ ( * ( )) .

OTHER APPROACHES AT MOTM ASYMPTOTICS
In a recent paper, Gao and Wang, 2016 study small noise sample-path MDPs for SDE solutions, and specialize to the small-time regime (corollary 4.1.2 in Gao and Wang, 2016). Their asymptotic regime is in fact slightly more general than (2.4), allowing for (in our notation) any satisfying √ ≪ ≪ 1 as ↓ 0. (In the financial context, this offers no useful additional flexibility; it allows, e.g., switching between two regimes = 1 and = 2 infinitely often as ↓ 0.) However, Gao and Wang (2016) impose the assumptions of linearly bounded and locally Lipschitz coefficients. These are the typical assumptions for small-noise LDPs in the literature, but they are rarely satisfied in stochastic volatility models. In particular, their results are not directly applicable to the Heston model. The paper by Cai and Wang (2015) is also worth mentioning here: It presents moderate deviations for the CIR process (the Heston variance process) and a generalization, where the exponent in the dynamics is not necessarily 1∕2. The paper uses estimates tied to the (generalized) CIR stochastic differential equation.
In this section, we discuss a different approach at small-time moderate deviations. While yielding only first-order results, its conditions are usually easy to check for models with explicit characteristic function. Assumptions 2.1 and 2.2 are not in force here. Recall that in the classical setting of sequences of i.i.d. random variables, a moderate deviation analogue of Cramér's theorem can be deduced by applying the Gärtner-Ellis theorem to an appropriately rescaled sequence (see Dembo & Zeitouni 1998, section 3.7). The MD short-time behavior of diffusions can be subjected to a similar analysis. Consider the log-price = log with 0 = 0 and mgf (moment-generating function) (6.1) Assumption 6.1. For all ∈ (0, 1 2 ), the rescaled mgf satisfies We expect that this assumption holds for diffusion models in considerable generality. It is easy to check that (6.2) holds for the Heston model, either by its explicit characteristic function, or, more elegantly, from the associated Riccati equations; see Pinter (2017) for details. Thus, the results of the present section provide an alternative proof of the first order MOTM behavior (5.11) of Heston call prices.
Heuristically, Assumption 6.1 can be derived from the density asymptotics in Assumption 2.1, which in turn hold in quite general diffusion settings (see Deuschel et al., 2014aDeuschel et al., , 2014b. In (6.3), we ignored that the density expansion (2.2) might not be valid globally in space; this might be made rigorous by estimating ( , ) by a Freidlin-Wentzell LD argument for sufficiently large. As for (6.4), we can expect concentration near ≈ 0, because Λ( ) increases with | |. Finally, (6.5), and thus (6.6), follows from a (rigorous) application of the Laplace method. If (6.6) is correct, then (6.2) clearly follows. The critical moment of is defined by It is obvious that is necessary for (6.2); i.e., + ( ) must grow faster than −1 as ↓ 0. In the Heston model, e.g., the critical moment is of order + ( ) ∼ ( )∕ ≫ −1 for small , as follows from inverting (6.2) in Keller-Ressel (2011). On the other hand, we do not expect our results to be of much use in the presence of jumps. Indeed, suppose that (6.1) is the mgf of an exponential Lévy model. Then, + ( ) ≡ + does not depend on , and is finite for most models used in practice. Therefore, (6.7) cannot hold, and so Assumption 6.1 is not satisfied. The Merton jump diffusion model is one of the few Lévy models of interest that have + = ∞, but it is easy to check that it does not satisfy (6.2), either.
After this discussion of Assumption 6.1, we now give an asymptotic estimate for the distribution function of (put differently, MOTM digital call prices) in Theorem 6.2. Then, we translate this result to MOTM call prices in Theorem 6.3. If desired, higher order terms in (6.2) will give refined asymptotics in Theorem 6.2, using Gulisashvili and Teichmann's (2015) recent refinement of the Gärtner-Ellis theorem. (Further work will be required to translate the resulting expansions into call price asymptotics.) For other asymptotic results on option prices using the Gärtner-Ellis theorem, see, e.g, Jacquier (2009, 2011).
Then, (6.2) is equivalent to As Γ is finite and differentiable on ℝ, the Gärtner-Ellis theorem (theorem 2.3.6 in Dembo & Zeitouni 1998) implies that ( ) ≥0 satisfies an LDP as ↓ 0, with rate and good rate function Λ , the Legendre transform of Γ . Trivially, Λ is qquadratic, too: . This is the same as (6.8). □ As in the LD/OTM regime, first-order cdf asymptotics translate readily into call price asymptotics. The proof of the following result is similar to Pham (2010, p. 30f; concerning the LD regime) and Caravenna and Corbetta (2016, theorem 1.5). In the MD/MOTM regime, one can replace the condition (1.19) of Caravenna and Corbetta (2016) by a mild condition on the moments of the model. (1 + (1)) ) , ↓ 0. (6.9) Proof. First assume (i). Let > 0 and definẽ= (1 + ) . Then (6.10) The first factor is For the second factor in (6.10), we apply (i) with̃.
Now let ↓ 0 to get the desired lower bound for ( , ).
The remaining upper bound of (ii) ⇐⇒ (i) is shown very similarly to the lower bound of the implication (i) ⇐⇒ (ii). □ In the light of the general MDP result by Gao and Wang (2016) quoted at the beginning of this section, it might be worth noting that Theorem 6.3 holds, with virtually the same proof, if the assumption = ( ) is replaced by √ ≪ ≪ 1.

ACKNOWLEDGMENTS
We gratefully acknowledge financial support from DFG Grant FR2943/2 (P. Friz) resp. the Austrian Science Fund (FWF) under Grant P 24880 (S. Gerhold, A. Pinter). We thank the anonymous referees, the editors, and participants of Global Derivatives 2015 (Amsterdam), the 9th World Congress of the Bachelier Finance Society (NYC 2016), and CSASC 2016 (Barcelona) for their valuable feedback.

ENDNOTES
1 As we focus on stochastic volatility models, which are in general incomplete, it is understood that call prices are computed w.r.t. some fixed pricing measure.
(iii) In (2.2), ( ) satisfies While proving Assumption A.1 for stochastic volatility models would go well beyond the scope of the present paper, there are good reasons to believe that it holds in reasonable generality. It does hold for local volatility models, which satisfy (i) according to proposition 2.1 of Gatheral et al. (2012). For stochastic volatility models, the approach of Deuschel et al. (2014a) suggests that the relative error term in (2.2) has a full expansion in (integer) powers of , which would imply (i).
Part (ii) is clear in local volatility models, just assuming smoothness of the local volatility function. In stochastic volatility models, it should be possible to refine the results of Berestycki et al. (2004) (convergence to 0 ) to a Taylor expansion.
Part (iii) of Assumption A.1 is true for the Heston model (see (5.10)) and generic local volatility models (see (5.4)); the gist of the saddle point argument we applied for Heston, and the resulting expression (5.10), are not tied to that model. Theorem A.2. Under Assumption A.1, the statement of Corollary 4.1 holds for ∈ (0, 3 7 ).
To simplify notation in the following proof, we write ∼ ( ) for two functions and , if ( )∕ ( ) ∼ ( ) as ↓ 0 for some slowly varying function . We will use this for functions of algebraic growth order, which are then "almost" asymptotically equivalent. The index in "∼ " is a generic symbol and does not stand for any concrete slowly varying function.
Proof of Theorem A.2. We start by improving the call price expansion from Theorem 2.5, taking into account the asymptotic errors that were made in obtaining it. By part (ii) of Assumption A.1, the relative error in (3.1) is ( ). Part (ii) of Assumption A.1 shows the same for (3.2). The relative error in (3.5) is ( ), as seen from The only remaining source of error is the application of the Laplace method in (3.4). Here, it does not suffice to state the relative error; we have to do a little better than in the proof of Theorem 2.5. By the higher order extension of the Laplace method (theorem 3.8.1 in Olver, 1974), and because Λ( )∕ → ∞ as ↓ 0, we have the integral expansion with error term 2 ∕Λ( ) 2 ∼ 2(1−2 ) . Therefore, our MOTM call price approximation becomes where > 0 is arbitrarily small. The Taylor expansion Λ( ) = 1 2 Λ ′′ (0) 2 + 1 6 Λ ′′′ (0) 3 + ( 4 ) implies We now translate the refined call price expansion to implied volatility asymptotics. In the proof of Corollary 4.1, we used corollary 7.2 of Gao and Lee (2014) to achieve the transfer. This would suffice for the interval ∈ (0, 2 5 ), too, but for the larger interval ∈ (0, 3 7 ) we have to take a closer look at the (arbitrary order) asymptotic machinery developed in Gao and Lee (2014). Any unexplained terminology and notation in what follows is as in Gao and Lee (2014). Using proposition 5.6, lemma 5.8, and example 5.13 of Gao and Lee (2014) yields the following estimates in our MOTM regime: , (A.3) with integers , ≥ 1, = − log ( , ), an approximation̂of , dimensionless implied volatility ∶= 1∕2 imp ( , ), and error estimate Ψ. We suppress the time dependence of , ,̂, , and Ψ, in order to keep the notation of Gao and Lee (2014). Note that, in the MOTM regime, ∕ → 0 as ↓ 0.
Putting these formulas back into (A.6), we get For the second-order expansion of the implied volatility to be correct, the error term should be negligible compared to , which amounts to min{3(1−2 ),2 ,1− } = ( ). This is true if and only if min{3(1 − 2 ), 2 , 1 − } > , which is equivalent to our assumption ∈ (0, 3 7 ). □ For larger , closer to 1 2 , the whole analysis has to be refined. A more precise iteration scheme has to be chosen, so that the iteration scheme error in (A.4) gets smaller. Moreover, a better log-price approximation̂has to be taken into account, using even more terms of the Laplace expansion, in order to decrease the approximation error in (A.5). It should thus be possible to reduce the error in (A.7) to ( min{ (1−2 ),2 ,1− }− ), ↓ 0, where ∈ ℕ can be arbitrarily large. In this fashion, for any fixed , it should be straightforward to provide a proof of the second-order approximation (4.1) of the implied volatility for < 2 +1 . That is, we have a clear procedure for any > 2. For small , say = 3, 4, … , this can be implemented by hand, and larger values (say, = 17) are still feasible with the aid of Mathematica or similar software. In practice, as the calculations in each proof will be tied to that specific value of , very large remains out of reach. Here one would need a new idea to provide an argument for general , which would then prove (4.1) for all ∈ (0, 1 2 ). At this moment, despite some effort, the details of such a construction elude us. Still, we believe that Assumption A.1 suffices to treat the whole interval, i.e., that Theorem A.2 holds with 3 7 replaced by 1 2 . Note that the paper by Tehranchi (2016), which presents uniform (nonasymptotic) bounds on implied volatility, does not seem to be applicable here: For > 1 3 , the lower bound of proposition 4.6 in Tehranchi (2016) is not tight enough, as it yields a second-order term that is asymptotically larger than the second-order term in (4.1).