Asymptotics of the time-discretized log-normal SABR model: The implied volatility surface
AAsymptotics of the time-discretized log-normal SABRmodel: The implied volatility surface
Dan Pirjol , Lingjiong Zhu March 11, 2020
Abstract
We propose a novel time discretization for the log-normal SABR model which is apopular stochastic volatility model that is widely used in financial practice. Our timediscretization is a variant of the Euler-Maruyama scheme. We study its asymptoticproperties in the limit of a large number of time steps under a certain asymptoticregime which includes the case of finite maturity, small vol-of-vol and large initialvolatility with fixed product of vol-of-vol and initial volatility. We derive an almostsure limit and a large deviations result for the log-asset price in the limit of largenumber of time steps. We derive an exact representation of the implied volatilitysurface for arbitrary maturity and strike in this regime. Using this representation weobtain analytical expansions of the implied volatility for small maturity and extremestrikes, which reproduce at leading order known asymptotic results for the continuoustime model.
The method of the asymptotic expansions has been used in the literature to study theproperties of stochastic volatility models under a wide variety of strike-maturity regimes.The short maturity limit at fixed strike for the implied volatility was first derived for theSABR model in the celebrated work of [23]. This model is defined by the two-dimensionalstochastic differential equation: dS ( t ) = σ ( t ) S γ ( t ) dW ( t ) , (1.1) dσ ( t ) = ωσ ( t ) dZ ( t ) , where W ( t ) and Z ( t ) are two standard Brownian motions, with E [ dW ( t ) dZ ( t )] = (cid:37)dt . ω > γ ∈ [0 ,
1] is an exponent which controlsthe backbone of the implied volatility [23]. The correlation (cid:37) can take any values in (cid:37) ∈ [ − , +1], although the most important cases for applications have (cid:37) ≤ School of Business, Stevens Institute of Technology, Hoboken, NJ 07030, United States of America;[email protected] Department of Mathematics, Florida State University, 1017 Academic Way, Tallahassee, FL-32306,United States of America; [email protected] a r X i v : . [ q -f i n . M F ] M a r ther stochastic volatility models popular in financial practice such as the Heston model[10, 12] and in more general forms [24, 8]. Asymptotics in the large maturity regime wereobtained, both at fixed strike [31, 32, 45, 13] and in the joint strike-maturity regime [11],in a wide variety of stochastic volatility models. The option price asymptotics can betranslated into implied volatility asymptotics using the transfer results of Gao and Lee[16].The properties of the SABR model are well understood in continuous time. The mar-tingale properties of the model were studied in [27, 35]. Short maturity asymptotics for theasset price distribution, option prices and implied volatility were first derived at leadingorder by Hagan et al [23]. The expansion was extended to higher order in [24, 38, 33]. Theasymptotics was studied using operator expansion methods in [6]. For a survey of existingresults see [1]. A mean-reverting version of the model called λ -SABR was introduced in[24]; the asymptotics of options prices was recently studied in [17]. The simulation andpricing under the SABR model have been studied extensively. For 0 < γ < (cid:37) = 0, an exact representation for the conditional distribution of S T for given( σ T , (cid:82) T σ ( t ) dt ) was given by Islah [26]. This was further simplified by [2] who derived aone-dimensional integral representation for the option prices. Cai, Song and Chen [5] gavean exact simulation method for the SABR model for γ = 1 and for γ < , (cid:37) = 0, using aninversion of the Laplace transform for 1 / (cid:82) T σ ( t ) dt .The log-normal ( γ = 1) SABR model is an important limiting case. This can beregarded as a particular case of Hull-White stochastic volatility model [25] dS ( t ) = (cid:112) V ( t ) S ( t ) dW ( t ) , dV ( t ) = ξV ( t ) dZ ( t ) + ηV ( t ) dt , ξ > , η < ξ . (1.2)The process for the instantaneous variance V ( t ) = σ ( t ) is equivalent to dσ ( t ) = ξσ ( t ) dZ ( t )+( η − ξ ) σ ( t ) dt , which reduces to the volatility process in the SABR model when η = ξ .Option price and implied volatility asymptotics in the Hull-White model at large strike werestudied by Gulisashvili and Stein [18, 20, 21]. The large maturity asymptotics in the SABRmodel were studied by Forde and Pogudin [13] and by Lewis in the Hull-White model [32].The log-normal SABR model can be mapped to the Brownian motion on the Poincar´espace H [24, 33]. The model can be simulated exactly by conditional Monte Carlo meth-ods, exploiting the fact that log S T is normally distributed, conditional on a realization of( σ T , (cid:82) T σ ( t ) dt ). While an exact result for this distribution is known from Yor [47], thenumerical evaluation of the result is challenging, requiring very high precision in interme-diate steps [5]. The paper [5] presents an alternative approach involving the inversion of aLaplace transform. An integral representation of the pricing kernel has been presented in[33].To the best of our knowledge, all the asymptotic results in the literature are obtainedin the continuous time context. In many practical applications of these models, they aresimulated in discrete time, by application of time discretization schemes such as the Euler-Maruyama scheme. We study in this paper the asymptotics of the model (1.1) with γ = 12iscretized in time under an application of the Euler-Maruyama scheme to log S ( t ), andappropriate model parameters rescaling with n , the number of time steps.Consider a grid of time points { t i } ni =1 with uniform time step size τ . Application of theEuler-Maruyama discretization of (1.1) to log S ( t ) gives the stochastic recursion S i +1 = S i e σ i ∆ W i − σ i τ , (1.3) σ i = σ e ωZ i − ω t i , where ∆ W i := W ( t i +1 ) − W ( t i ) and Z i := Z ( t i ). This was called in [41] the Log-Euler-log-Euler scheme. Its asymptotic properties were studied in [41] in the limit n → ∞ at fixed β = ω τ n , ρ = σ √ τ , in the uncorrelated limit (cid:37) = 0. The main results obtained werean almost sure limit for the asset price lim n →∞ n log S n = − ρ a.s. (Proposition 19 [41]),a fluctuations result lim n →∞ log S n + ρ n √ n = N (0 , ρ + ρ β ) in distribution (Proposition20 [41]), and a closed form result for the Lyapunov exponents of the asset price moments λ ( ρ, β ; q ) = lim n →∞ n log E [( S n ) q ]. It was pointed out that the scheme (1.3) differs fromthe continuous time model in one notable respect: the asset price is a martingale forany correlation (cid:37) ∈ [ − , (cid:37) (cid:54) = 0. In addition we show that this scheme reproducesthe martingale properties of the continuous time model as n → ∞ : the asset price is amartingale provided that (cid:37) ≤
0, see Proposition 8. Numerical study shows that it producesa martingale defect for (cid:37) >
0. This scheme is defined in (2.2) and reduces to the Log-Euler-log-Euler scheme (1.3) in the uncorrelated limit (cid:37) = 0.We study in this paper the asymptotics of the new scheme in the limit of a large numberof time steps n → ∞ at fixed β = ω τ n , ρ = σ √ τ . We derive a large deviations propertyfor the log-price of the asset P ( n log S n ∈ · ) in this limit. The large deviations result istranslated into option prices and implied volatility asymptotics. The rate function turnsout to be independent of the time step τ . The limit considered includes the regime of finitematurity T = nτ = O (1), small vol of vol ω T → σ T → ∞ at fixed( ω T )( σ T ). We obtain the volatility surface of the model for arbitrary maturity T andstrike K under this regime in Theorem 16: σ BS ( x, T ) = σ Σ BS (cid:18) xσ T ; 2( σ T )( ω T ) (cid:19) , x = log KS , (1.4)where the equality above means the ratio LHS/RHS goes to one in the limit considered. Thefunction Σ BS ( y ; a ) is given in explicit form in Theorem 16. To the best of our knowledgethis is the first stochastic volatility model for which the entire volatility surface can beapproximated in closed form in a certain limit of the model parameters.The limiting result for the volatility surface given by Theorem 16 is compared withknown asymptotic expansions of the implied volatility in the log-normal SABR model in3able 1: Comparison of the known asymptotic results for the continuous-time model withthe discrete-time asymptotics obtained in this paper, showing the relevant references andsections where they are discussed. In all cases we reproduce the known asymptotic resultsfor the continuous-time model.asymptotics continuous time discrete time T → O ( T ): Hagan et al [23] Sec. 6.1 O ( T ) , O ( T ): Paulot [38] T → , x → O ( T k x n ) Lewis [31, 33] Sec. 6.2 n = 0 , k ≤ | x | → ∞ : Gulisashvili, Stein ( (cid:37) = 0) [19] Sec. 6.3the short-maturity [23, 24, 37], large-maturity [32, 31, 13] and extreme strikes [19] regimes.The asymptotic result of (1.4) reproduces all these expansions, after taking the small volof vol and large initial volatility limit. A summary of the known asymptotic results in thecontinous time model and their counterparts in the discrete-time model is given in Table 1.We comment also on the relation of our results to those obtained by Forde [14] andForde and Kumar [15] for the large maturity asymptotics in stochastic volatility models.In the simplest setting these papers study the class of models defined by dS ( t ) S ( t ) = σ ( Y ( t )) dW ( t ) , (1.5) dY ( t ) = − αY ( t ) dt + dW ( t ) , (1.6)where W ( t ) , W ( t ) are standard Brownian motions that may be correlated. The methodis based on proving large deviations for P ( A ( t ) ∈ · ) as t → ∞ for the time average of theintegrated variance A ( t ) = 1 t (cid:90) t σ ( Y ( s )) ds . (1.7)This is done using the Donsker-Varadhan type large deviations [9] for the occupation timeof the Y ( t ) process, and then applying the contraction principle. The rate function for thelarge deviation principle of P ( A ( t ) ∈ · ) is denoted in [15] as I f ( a ). Assuming for simplicityzero correlation as in [14], the log-asset price X ( t ) = log S ( t ) is related to A ( t ) as1 t X ( t ) = − A ( t ) + Z √ t (cid:112) A ( t ) , (1.8)with Z ∼ N (0 ,
1) following a standard normal distribution. This is similar to our Eq. (2.2)in the zero correlation limit (cid:37) = 0. An application of the contraction principle gives that4 ( t X ( t ) ∈ · ) satisfies a large deviation principle with rate function I ( x ) = inf a> (cid:32) I f ( a ) + 12 a (cid:18) x + 12 a (cid:19) (cid:33) . (1.9)Our approach differs from that of [14, 15] in two respects. First, the analog of the ratefunction of the integrated variance I f ( a ) is obtained exactly as in [42], without requiringthe Donsker-Varadhan large deviations results [9]. This allows a more explicit treatment,and for our particular model, we do not rely on numerical methods. Second, we do notrequire the large maturity limit, and our asymptotic regime can include arbitrary maturity.The paper is organized as follows. In Section 2 we introduce the new time discretizationscheme for the asset price. In Section 3 we study the n → ∞ asymptotics for the volatilityprocess. These results are used in Section 4 to study the asymptotics of the asset priceprocess. We derive an almost sure limit and large deviations for P ( n log S n ∈ · ). Theseresults are used in Section 5 to obtain option price asymptotics and the implied volatilityasymptotics in the n → ∞ limit. Section 6 studies in detail the implications of these resultsin various regimes of small maturity and extreme strikes. Finally Section 7 compares theasymptotic result against numerical benchmarks. An Appendix derives the rate functionsfor the uncorrelated model in explicit form, and obtains asymptotics in various regimes ofsmall/large arguments. Definition 1 (Modified Log-Euler, Log-Euler scheme) . Assume the timeline { t i } ni =0 withuniform time step t i +1 − t i = τ , and denote for simplicity S ( t i ) = S i , σ ( t i ) = σ i . Thediscretization scheme of the γ = 1 SABR model (1.1) is defined recursively by log S i +1 = log S i + (cid:37) ⊥ σ i ∆ W ⊥ i − σ i τ + (cid:37) ω ( σ i − σ ) , (2.1) with σ i = σ e ωZ i − ω t i and ∆ W ⊥ i = W ⊥ ( t i +1 ) − W ⊥ ( t i ) where W ⊥ ( t ) is a Brownianmotion independent of Z ( t ) , given by W ( t ) = (cid:37)Z ( t ) + (cid:112) − (cid:37) W ⊥ ( t ) . It follows from Definition 1 that the recursion can be written in closed form aslog S n = log S + (cid:37) ⊥ n − (cid:88) i =0 σ i ∆ W ⊥ i − n − (cid:88) i =0 σ i τ + (cid:37) ω ( σ n − σ ) (2.2)= log S + (cid:37) ⊥ (cid:112) V n Z − V n + (cid:37) ω ( σ n − σ ) , where V n = (cid:80) n − i =0 σ i τ and Z ∼ N (0 ,
1) follows a standard normal distribution, independentof σ i . The construction of this scheme is motivated by writing the SDE of the continuous5ime model in terms of X ( t ) = log S ( t ) as dX ( t ) = σ ( t ) dW ( t ) − σ ( t ) dt = (cid:37)σ ( t ) dZ ( t ) + (cid:37) ⊥ σ ( t ) dW ⊥ ( t ) − σ ( t ) dt (2.3)= (cid:37) ω dσ ( t ) + (cid:37) ⊥ σ ( t ) dW ⊥ ( t ) − σ ( t ) dt , where we denoted (cid:37) ⊥ := (cid:112) − (cid:37) . In this form we see that X ( t ) can be decomposed asthe sum of two independent processes X ( t ) = X ⊥ ( t ) + X (cid:107) ( t ) with dX ⊥ ( t ) = (cid:37) ⊥ σ ( t ) dW ( t ) ⊥ − σ ( t ) dt , X ⊥ = log S , (2.4)and X (cid:107) ( t ) = (cid:37) ω ( σ ( t ) − σ ). The discretization scheme (2.1) is obtained by applying Eulerdiscretization to X ⊥ ( t ) and keeping X (cid:107) ( t ) in closed form.We will consider the properties of the time-discretized model (2.2) in the n → ∞ limitat fixed β := 12 ω τ n , ρ := σ √ τ . (2.5)This limit covers the following asymptotic regimes of practical interest: • Finite maturity, low vol of vol and large initial volatility. This corresponds to t n = nτ = O (1) , ω = O ( n − ) , σ = O ( n ). Taking ω = ˜ ωn − , σ = ˜ σ √ n and fixed T = nτ , we have β = ˜ ω T, ρ = ˜ σ √ T , independent of n . • Small maturity regime and large initial volatility. This corresponds to σ = O ( n ) , τ = O ( n − ) and ω = O (1). This gives t n = nτ = O ( n − ). • Large maturity regime and low vol of vol. This corresponds to the situation when σ , τ are fixed, vol of vol ω = O (1 /n ), and the maturity t n = nτ = O ( n ).Under the first two regimes the time step goes to zero τ → n → ∞ limit, andthus they are appropriate for studying the continuous time limit of this model. The firstregime is studied in more detail in Sections 5 and 6. The asymptotic predictions underthis regime are compared with known asymptotic results in the continuos time limit. Inall cases where an analytical result is known for the continuous time case, we recover it inthe n → ∞ limit of the discrete time model. The time discretized asset price under the scheme (2.2) is the sum of two terms1 n log S n = 1 n log S ⊥ n + (cid:37) nω ( σ n − σ ) . (3.1)6onditional on a path of the volatility process { σ k } nk =1 , the asset price S n is log-normallydistributed. The distribution of S n is obtained by folding the log-normal distribution withthe distribution of the volatility process. It will be seen that the large deviations propertiesof n log S n are related to the large deviations for the joint volatility process { V i , σ i } ni =1 .We start by reviewing the known results for the n → ∞ asymptotics of the volatilityprocess. It was shown in [42] that for n → ∞ at fixed β, ρ , one has the almost sure limitlim n →∞ n V n = ρ a.s. (3.2)A similar argument gives the almost sure limit for the volatility processlim n →∞ σ n nω = v := ρ √ β a.s. (3.3)The large deviations for P ( n V n ∈ · ) were obtained in Proposition 6 of [42]. This re-sult is summarized for convenience in the Appendix A. The first term in (3.1) depends on { σ , σ , · · · , σ n − } through V n , while the second term depends on σ n . This introduces cor-relation among the two terms. This requires that we study the large deviations propertiesof the joint process ( V n , σ n ) as n → ∞ .We start by introducing some background concepts about large deviations theory. Werefer to Dembo and Zeitouni [7] for a more comprehensive exposition of large deviationsand its applications. A sequence ( P n ) n ∈ N of probability measures on a topological space X is said to satisfy a large deviation principle (LDP) with the rate function I : X → R ∪ {∞} if I is non-negative, lower semicontinuous and for any measurable set A , we have − inf x ∈ A o I ( x ) ≤ lim inf n →∞ n log P n ( A ) ≤ lim sup n →∞ n log P n ( A ) ≤ − inf x ∈ A I ( x ) . (3.4)Here, A o is the interior of A and A is its closure. The rate function I is said to be good iffor any m , the level set { x : I ( x ) ≤ m } is compact.The contraction principle (e.g. Theorem 4.2.1. [7]) plays a key role in our proofs. Forthe convenience of the readers, we state the contraction principle as follows. If P n satisfiesa large deviation principle on X with rate function I ( x ) and F : X → Y is a continuousmap, then the probability measures Q n := P n F − satisfies a large deviation principle on Y with rate function J ( y ) = inf x : F ( x )= y I ( x ).We start by noting that the last term in (3.1) satisfies a LDP. Lemma 2.
In the limit n → ∞ , P ( nω σ n ∈ · ) satisfies a LDP with rate function I ( x ) = 14 β log (cid:18) xv (cid:19) , (3.5) for x > , with v = ρ √ β , and I ( x ) = ∞ otherwise. roof. Start with nω σ n = σ nω e ωZ n − ω t n . We have ωZ n = ω √ τ nZ = √ β √ n Z with Z ∼ N (0 , ω t n = ω τ n = βn . Since Z ∼ N (0 , θ ∈ R ,lim n →∞ n log E (cid:104) e θn ( ωZ n − ω t n ) (cid:105) = 12 (2 β ) θ , (3.6)which by G¨artner-Ellis theorem [7] implies that P ( ωZ n − ω t n ∈ · ) satisfies a large devia-tion principle with rate function sup θ ∈ R { θx − (2 β ) θ } = β x . Note that σ nω = ρ √ τnn √ τ √ β = ρ √ β , and thus nω σ n = ρ √ β e ωZ n − ω t n . By the contraction principle [7], P ( nω σ n ∈ · )satisfies a LDP with rate function β log ( xρ/ √ β ).Next we prove a LDP for the joint distribution of ( V n , σ n ), for P (( n V n , nω σ n ) ∈ · ). Theproof proceeds in close analogy with Proposition 6 in [42]. Proposition 3.
Consider the n → ∞ limit at fixed β = ω τ n and ρ = σ √ τ . Define V n = (cid:80) n − k =0 σ k τ . Then P (( n V n , nω σ n ) ∈ · ) satisfies a LDP with rate function I ( x, y ) = inf g ∈AC x,y [0 , (cid:90) (cid:0) g (cid:48) ( z ) (cid:1) dz , (3.7) for x, y ≥ , and I ( x, y ) = ∞ otherwise. We denoted AC x,y [0 ,
1] = (cid:26) g (cid:12)(cid:12)(cid:12)(cid:12) g ∈ AC [0 , , g (0) = 0 , (cid:90) e √ βg ( z ) dz = xρ , e √ βg (1) = y √ βρ (cid:27) , (3.8) where AC [0 , denotes the space of absolutely continuous functions from [0 , to R .Proof. Write σ k as σ k = σ e ωZ k − ω t k = σ e ω √ τ (cid:80) k − j =1 U j − ω kτ , (3.9)where U j := √ τ ( Z j − Z j − ) are i.i.d. N (0 ,
1) random variables. Using ω τ = βn , we get σ k = σ e √ βn (cid:80) k − j =1 U j − βn k . (3.10)Note that e − βn ≤ e − βn k ≤ ≤ k ≤ n −
1. Thus we can neglect the term − βn k in the exponent in the n → ∞ limit.By Mogulskii theorem [7], P ( n (cid:80) [ · n ] j =1 U j ∈ · ) satisfies a LDP on L ∞ [0 ,
1] with good ratefunction I ( g ) = (cid:90) Λ( g (cid:48) ( x )) dx , (3.11)8f g ∈ AC [0 ,
1] and g (0) = 0 and I ( g ) = + ∞ otherwise, whereΛ( x ) := sup θ ∈ R (cid:110) θx − log E [ e θU ] (cid:111) = 12 x . (3.12)Let g ( x ) := n (cid:80) [ xn ] j =1 U j . Then (cid:90) e √ βg ( x ) dx = n − (cid:88) k =0 (cid:90) ( k +1) /nk/n e √ βg ( x ) dx = 1 n n − (cid:88) k =0 e √ βn (cid:80) kj =1 U j . (3.13)We know that P ( nω σ n ∈ · ) satisfies a LDP as shown above in Proposition 2 with ratefunction I ( x ) given in Equation (3.5). This random variable is expressed as1 nω σ n = ρ √ β e √ βn (cid:80) n − j =1 U j − βn , (3.14)which is exponentially equivalent to ρ √ β e √ βg (1) as n → ∞ . We can apply the contractionprinciple [7] to conclude that P (( n (cid:80) n − k =0 σ k τ, nω σ n ) ∈ · ) satisfies a LDP, with rate function I ( x, y ) given by the constrained variational problem (3.7). I ( u, v ) We present in this Section the solution of the variational problem in Eq. (3.7) for the ratefunction I ( u, v ), which is given by I ( u, v ) = inf g (cid:90) [ g (cid:48) ( t )] dt , (3.15)where the infimum is taken over all functions g ∈ AC [0 ,
1] satisfying g (0) = 0 and (cid:90) e bg ( t ) dt = uρ , e bg (1) = vbρ , b := (cid:112) β . (3.16) Proposition 4.
The rate function can be expressed as I ( u, v ) = 116 β I (¯ u, ¯ v ) , (3.17) where ¯ u := uρ , ¯ v := vv , and v := ρ √ β . We distinguish three cases:(i) ¯ u/ ¯ v > . For this case the rate function is I (¯ u, ¯ v ) = β (cid:26) β − γγ + 1 + 4 γe β + γ (cid:27) , (3.18)9 here β is the solution of the equation sinh( β/ β/ u ¯ v , (3.19) and γ is given by γ = e β/ ve β/ − e β/ − ¯ v .(ii) ¯ u/ ¯ v < . For this case the rate function is I (¯ u, ¯ v ) = 4 λ (cid:26) ¯ u cos η − (cid:27) , (3.20) where λ is the solution of the equation sin λλ = ¯ u ¯ v , (3.21) and η is given by tan η = ¯ v cos λ − v sin λ .(iii) ¯ u = ¯ v . For this case the rate function is I (¯ u, ¯ u ) = 4 (¯ u − ¯ u . (3.22)At the point ¯ u = 1 , ¯ v = 1, the optimal path is constant g ( t ) = 0 and the rate functionvanishes, in agreement with the almost sure limit I ( ρ , v ) = 0. The function I ( u, v ) can be expressed in an alternative form as I ( u, v ) = 8 F (cid:16) vu (cid:17) + 4 1 + v u − π , (3.23)where the function F ( ρ ) is defined as F ( ρ ) = (cid:40) x − ρ cosh x + π , < ρ < , − y + ρ cos y + πy , ρ > . (3.24)In Eqn. (3.24), x , y are the solution of the equations ρ sinh x x = 1 , y + ρ sin y = π . (3.25)This function appears in the small- t expansion of the Hartman-Watson distribution, andits properties are studied in more detail in [43]. The function F ( ρ ) has a minimum at ρ = π , with F ( π ) = π . We will require the expansion of F ( ρ ) around ρ = 1 [43]. F ( ρ ) = π − − ( ρ −
1) + 32 ( ρ − −
65 ( ρ − + O (( ρ − ) . (3.26)Using this expansion we can derive an approximation for I ( u, v ) around its minimum at u = v = 1. 10 roposition 5. Denote (cid:15) = log u and η = log v . The expansion of the rate function I ( u, v ) around u = v = 1 up to and including quartic terms in (cid:15), η is I ( u, v ) = 12 (cid:15) − (cid:15)η + 16 η − (cid:15) + 365 (cid:15) η − (cid:15)η + 325 η + 109175 (cid:15) − (cid:15) η + 1004175 (cid:15) η − (cid:15)η + 1552525 η + · · · , (3.27) where the ellipses denote terms of the form (cid:15) a η b with a + b ≥ .Proof. Follows from (3.23) after using the expansion (3.26).The quadratic approximation I q ( u, v ) := 12 log u −
24 log u log v + 16 log v (3.28)gives a good approximation for ( u, v ) sufficiently close to (1 , I ( u, v ) From the contraction principle, we have J ( v ) = inf u I ( u, v ) = 14 β log ¯ v , (3.29) J ( u ) = inf v I ( u, v ) = 18 β J BS (¯ u ) , (3.30)where J ( u ) and J ( v ) are the rate functions of large deviations of P ( n V n ∈ · ) and P ( nω σ n ∈· ) computed respectively above in Lemma 2. Expressed in terms of I (¯ u, ¯ v ) these relationsread inf ¯ u I (¯ u, ¯ v ) = 4 log ¯ v , inf ¯ v I (¯ u, ¯ v ) = 2 J BS (¯ u ) . (3.31)The variables ¯ u > , ¯ v > u > , v >
0. They are rescaled such that I (¯ u, ¯ v ) = 0.The rate function J BS ( u ) was given in Proposition 6 in [42] and Corollary 13 of thesame paper. The result is reproduced in Proposition 26 for convenience. S n Using the results of the previous section, we study here the asymptotics of n log S n for thetime discretization scheme (2.2) in the n → ∞ limit at fixed β = ω τ n and ρ = σ √ τ .11 .1 Almost sure limit Proposition 6.
We have the almost sure limit lim n →∞ n log S n = − ρ a.s. (4.1) Proof.
From (2.2) we have1 n log S n = 1 n log S ⊥ n + (cid:37) ωn ( σ n − σ ) (4.2)= 1 n log S + (cid:37) ⊥ (cid:114) V n n · Z √ n − n V n + (cid:37) ωn ( σ n − σ ) . Taking the n → ∞ limit of this relation and using the almost sure limit lim n →∞ n V n = ρ ,see Proposition 1 in [42], we get lim n →∞ n log S ⊥ n = − ρ a.s.Using nω = (cid:113) βτ , we get lim n →∞ ωn σ n = lim n →∞ (cid:113) τ β σ e ωZ n − ω t n = ρ √ β a.s. andlim n →∞ ωn σ = ρ √ β a.s. The two terms in the difference have the same limit so thecontribution of the last term in (4.7) cancels. Remark 7.
The discretization scheme (2.2) and the almost sure limit of Proposition 6 areeasily extended by allowing a drift for the volatility process dσ ( t ) = ασ ( t ) dt + ωσ ( t ) dZ ( t ) , (4.3) which can be solved as σ i = σ e ωZ i +( α − ω ) t i . (4.4) Taking the large n limit at fixed α ∞ := ατ n , the result of Proposition 6 can be modified totake into account the drift term, as lim n →∞ n log S n = − ρ e α ∞ − α ∞ a.s. (4.5) The driftless Hull-White model [25] is defined by dS ( t ) = (cid:112) V ( t ) S ( t ) dW ( t ) , dV ( t ) = ξV ( t ) dZ ( t ) , and is equivalent to the process dσ ( t ) = ξσ ( t ) dZ ( t ) − ξ σ ( t ) dt .The appropriate asymptotic limit is n → ∞ at fixed β = ω τ n , ρ = σ √ τ with ω = ξ . In this limit α ∞ = − ξ τ n = O (1 /n ) which vanishes for n → ∞ . We concludethat our results apply also to the Hull-White model with the replacement ω → ξ . As n → ∞ , the asset price S n under the scheme (2.2) is asymptotically a martingale fornon-positive correlation (cid:37) ≤ roposition 8. For non-positive correlation (cid:37) ≤ , we have as n → ∞ at fixed nτ = T lim n →∞ E [ S n ] = S E (cid:104) e − (cid:37) V T + (cid:37) ω ( σ T − σ ) (cid:105) = S , (4.6) with V T := (cid:82) T σ ( t ) dt .Proof. From (2.2) we have S n = S exp (cid:18) (cid:37) ⊥ (cid:112) V n Z − V n + (cid:37)ω ( σ n − σ ) (cid:19) . (4.7)Conditioning on { σ i } ni =0 , the asset price is log-normally distributed. Taking the expectationover Z gives E [ S n ] = S E (cid:104) e − (cid:37) V n + (cid:37)ω ( σ n − σ ) (cid:105) . (4.8)First we prove the convergence in L norm V n = τ n − (cid:88) i =0 σ i → V ( T ) = (cid:90) T σ ( t ) dt , n → ∞ , nτ = T . (4.9)This follows by adapting the proof of Theorem 13 in [39] which proves a similar convergencestatement for the discrete sum of a geometric Brownian motion to an integral. By theMarkov inequality this implies V n → V ( T ) in probability, and thus e − (cid:37) V n + (cid:37) ω ( σ n − σ ) → e − (cid:37) V T + (cid:37) ω ( σ T − σ ) , (4.10)in probability. For (cid:37) ≤
0, the exponential is bounded from above by e − (cid:37) V n . By theLebesgue dominated convergence theorem we can exchange limit and expectation. Usingthe known result for the continuous time case [4, 27, 35], we get (4.6).Numerical testing shows that for positive correlation there is a martingale defect E [ S n ]
For any (cid:37) ≤ we have inf u,v> (cid:26) I ( u, v ) + a(cid:37) u − (cid:37) √ a ( v − (cid:27) = 0 , (4.11) where a := 4 βρ . Denote the point where the infimum is reached as ( u m , v m ) . This pointdepends on the product (cid:37) √ a and approaches (1 , as (cid:37) √ a → . roof. The asymptotic martingale property (4.6) implieslim n →∞ E (cid:104) e − (cid:37) V n + (cid:37)ω ( σ n − σ ) (cid:105) = lim n →∞ E (cid:104) e nF ( Vnn , ωn σ n ) (cid:105) = 1 , (4.12)with F ( x, y ) = − (cid:37) x + (cid:37) ( y − v ) . (4.13)By Varadhan’s lemma we havelim n →∞ n log E (cid:104) e nF ( Vnn , ωn σ n ) (cid:105) = − inf u,v {I ( u, v ) − F ( u, v ) } = 0 . (4.14)This is written equivalently as (4.11). P ( n log S n ∈ · ) We are now in a position to prove the large deviations property for P ( n log S n ∈ · ) in thecorrelated log-normal SABR model. Proposition 10.
Consider the n → ∞ limit at fixed β = ω τ n and ρ = σ √ τ . In thislimit P ( n log S n ∈ · ) satisfies a LDP with rate function I X ( x, (cid:37) ) = inf x = (cid:37) ⊥ √ uz − u + (cid:37) ( v − v ) (cid:18) I ( u, v ) + 12 z (cid:19) (4.15)= inf ( u,v ) ∈ R (cid:32) I ( u, v ) + 12 (cid:37) ⊥ u (cid:18) x + 12 u − (cid:37) ( v − v ) (cid:19) (cid:33) , for x ≥ , and I X ( x, (cid:37) ) = ∞ otherwise. We denoted here v = ρ √ β .Proof. From Proposition 3 we know that P (( n V n , nω σ n ) ∈ · ) satisfies a LDP with ratefunction I ( u, v ). We also have that P ( √ n Z ∈ · ) satisfies a LDP with rate function J ( x ) = 12 x . (4.16)This follows from the G¨artner-Ellis theorem (see e.g. [7]) by noting that for any θ ∈ R wehave E (cid:104) e θn Z √ n (cid:105) = e θ n so that Λ( θ ) := lim n →∞ n log E (cid:104) e θn Z √ n (cid:105) = θ , which implies that J ( x ) = sup θ ∈ R { θx − Λ( θ ) } = 12 x . (4.17)14riting (2.2) as 1 n log S n = − V n n + (cid:37) ⊥ (cid:114) V n n · Z √ n + (cid:37) ωn ( σ n − σ ) , (4.18)we get from the contraction principle (see e.g. [7]) that P ( n log S n ∈ · ) satisfies a LDPwith rate function I X ( x, (cid:37) ) = inf − u + (cid:37) ⊥ √ uz + (cid:37) ( v − v )= x {I ( u, v ) + J ( z ) } . (4.19)This completes the proof of (4.15). I X ( x, (cid:37) ) We give here a few properties of the rate function I X ( x, (cid:37) ) introduced in the previoussection. Proposition 11.
The rate function I X ( x, (cid:37) ) vanishes for x L = − ρ . That is, I X ( x L , (cid:37) ) = 0 . (4.20) Proof.
Eq. (4.20) follows by noting that for x = − ρ , the minimizer in the extremalproblem of Prop. 10 is reached at u = ρ , v = v . At this point the rate function vanishes. Remark 12.
The result (4.20) agrees with the almost sure limit for n log S n in Proposi-tion 6. The rate function I X ( x, (cid:37) ) has a scaling property and depends only on x/ρ and theproduct a := 4 βρ = 2 σ ω ( τ n ) I X ( x, (cid:37) ) = 18 β J X (cid:0) x/ρ ; 4 βρ , (cid:37) (cid:1) , (4.21)where J X ( y ; a, (cid:37) ) = inf u,v I ( u, v ) + a(cid:37) ⊥ u (cid:32) y + 12 u − (cid:37) ( v − (cid:114) a (cid:33) . (4.22)Expressed in terms of this function, the property (4.20) reads J X (cid:0) − ; a, (cid:37) (cid:1) = 0. Therate function has a calculable expansion around its minimum given by the following result. Proposition 13.
The leading term in the expansion of the rate function J X ( y ; a, (cid:37) ) aroundits minimum at y = − is J X ( y ; a, (cid:37) ) = 6 a a − √ a(cid:37) (cid:18) y + 12 (cid:19) + O (cid:32)(cid:18) y + 12 (cid:19) (cid:33) . (4.23)15 roof. The minimum condition is u∂ u Λ( u, v ) = 0 , v∂ v Λ( u, v ) = 0 . (4.24)The minimizer in the extremal problem (4.22) for this rate function can be expandedin powers of y + : x ∗ = log u ∗ = a (cid:18) y + 12 (cid:19) + a (cid:18) y + 12 (cid:19) + · · · , (4.25) y ∗ = log v ∗ = b (cid:18) y + 12 (cid:19) + b (cid:18) y + 12 (cid:19) + · · · . (4.26)Substituting the expansion of the rate function I ( u, v ) in Proposition 5 gives a sequence ofequations for the coefficients a i , b i . The first coefficients are a = − a + 3 √ a(cid:37) − √ a(cid:37) + a , b = − a − √ a(cid:37) )2(6 − √ a(cid:37) + a ) . (4.27)Substituting the expansion into J X ( y ; a, (cid:37) ) gives an expansion in y + with coefficientsexpressed in terms of a i , b i . The leading term is given in (4.23).We prove next a lower bound on the rate function, and an equality on its value at acertain point, which will play an important role in the n → ∞ asymptotics of the optionprices. Proposition 14.
Assume (cid:37) ≤ . The rate function J X ( y ; a, (cid:37) ) is bounded from below as J X ( y ; a, (cid:37) ) ≥ ay . (4.28) The lower bound is reached at y R = 12 (cid:0) − (cid:37) (cid:1) u m + (cid:37) (cid:114) a ( v m − , (4.29) where ( u m , v m ) are given by Corollary 9. At this point we have J X ( y R ; a, (cid:37) ) = 2 ay R . (4.30) Proof.
By Corollary 9, we have a lower bound on I ( u, v )12 I ( u, v ) ≥ − a(cid:37) u + (cid:37) √ a ( v − , u, v > . (4.31)16ubstituting into the expression (4.21) gives the lower bound J X ( y ; a, (cid:37) ) − ay ≥ inf u,v> (cid:26) a(cid:37) ⊥ u ( y − y R ) (cid:27) , (4.32)with y R defined in (4.29). This proves the lower bound (4.28).In order to prove the equality in (4.28) for y = y R , note that by Corollary 9 thereexist ( u m , v m ) such that the lower bound on I ( u, v ) above is reached. Substituting into J X ( y R ; a, (cid:37) ) we get that this is equal to 2 ay R , as stated. Expressed in terms of the ratefunction I X ( k, (cid:37) ), the relation (4.30) reads I X ( k R , (cid:37) ) = k R with k R = y R ρ .In the uncorrelated case (cid:37) = 0, the rate function I X ( x, (cid:37) ) simplifies further. For thiscase the extremal problem (4.22) can be solved in closed form, using the result for the ratefunction J BS ( x ) obtained in [40]. The result for this rate function is given in Corollary 29in Appendix A.2. When (cid:37) = 0, the rate function I X ( x,
0) satisfies the symmetry relation I X ( x, − I X ( − x,
0) = x , (4.33)see Proposition 30 in Appendix A.2. Expressed in terms of J X ( x ; a,
0) this reads J X ( x ; a, − J X ( − x ; a,
0) = 2 ax . (4.34)
We derive in this section option prices asymptotics in the time discretized log-normal SABRmodel discretized in time under the scheme (2.2). This result will be used to obtain theasymptotics of the implied volatility.
We consider the vanilla European call and put options: C ( n ) := E (cid:2) ( S n − K ) + (cid:3) , P ( n ) := E (cid:2) ( K − S n ) + (cid:3) , (5.1)where K > C ( n ) and P ( n ) to emphasize the dependenceon the number of time steps n . We study here the n → ∞ asymptotics of the option priceswith strike K = S e nk . The asymptotics will be shown to be different in the three regimes:1. The large-strike regime k > y R ρ . In this regime the call option is out-of-the-money(OTM) and lim n →∞ C ( n ) = 0;2. The intermediate strike regime − ρ ≤ k ≤ y R ρ ; In this regime the covered calloption is OTM and lim n →∞ ( S − C ( n )) = 0;17. The small-strike regime k < − ρ . In this regime the put option is OTM andlim n →∞ P ( n ) = 0;Here y R is given by Eq. (4.29). The asymptotics of the option prices are given by thefollowing result. Theorem 15.
The n → ∞ asymptotics of the option prices are given by k − I X ( k, (cid:37) ) = lim n →∞ n log C ( n ) for k > y R ρ , lim n →∞ n log( S − C ( n )) for − ρ ≤ k ≤ y R ρ , lim n →∞ n log P ( n ) for k < − ρ , (5.2) where I X ( k, (cid:37) ) is the rate function given by Proposition 10.Proof. Conditioning on ( V n , σ n ), the asset price S n is log-normally distributed and can bewritten as (with Z ∼ N (0 ,
1) independent of σ n , V n )) S n = F n e (cid:37) ⊥ √ V n Z − (cid:37) ⊥ V n , F n = S e − (cid:37) V n + (cid:37)ω ( σ n − σ ) . (5.3)The option prices can be written as expectations over ( V n , σ n ) of the Black-Scholes formula C ( n ) = E [ F n N ( d )] − K E [ N ( d )] , (5.4) P ( n ) = K E [ N ( − d )] − E [ F n N ( − d )] , (5.5)where d , are random variables d , := 1 (cid:37) ⊥ √ V n log F n K ± (cid:37) ⊥ (cid:112) V n (5.6)= 1 (cid:37) ⊥ √ V n (cid:18) log S K − (cid:37) V n + (cid:37) ω ( σ n − σ ) ± (cid:37) ⊥ V n (cid:19) . We are interested in the n → ∞ asymptotics of the option prices for strike K = S e nk .We give the proof of the n → ∞ asymptotics for the OTM call option; the other twocases follow analogously. The proof follows by upper and lower bounds on C ( n ).(i) Neglecting the second term in C ( n ) gives the upper bound C ( n ) ≤ E [ F n N ( d )] = S E (cid:104) e − (cid:37) V n + (cid:37) ω ( σ n − σ ) N ( d ) (cid:105) . (5.7)Using N ( d , ) = P ( d , > Z ) with Z ∼ N (0 ,
1) independent of σ n , V n , we havelim sup n →∞ n log C ( n ) ≤ lim sup n →∞ n log E (cid:20) e nF ( Vnn , σnωn ) P (cid:18) d √ n > Z √ n (cid:19)(cid:21) (5.8)= − inf D ( u,v ) ≥ z (cid:18) I ( u, v ) + 12 z + 12 (cid:37) u − (cid:37) ( v − v ) (cid:19) := I up C ( k ) , F ( x, y ) := − (cid:37) x + (cid:37) ( y − v ). The constraint is defined in terms of D ( u, v ) := 1 (cid:37) ⊥ √ u (cid:18) − k + 12 (1 − (cid:37) ) u + (cid:37) ( v − v ) (cid:19) , (5.9) D ( u, v ) := 1 (cid:37) ⊥ √ u (cid:18) − k − u + (cid:37) ( v − v ) (cid:19) . (5.10)They satisfy D ( u, v ) = D ( u, v ) + (cid:37) ⊥ √ u .(ii) We prove also a matching lower bound. For any (cid:15) > C ( n ) = E (cid:104)(cid:16) S n − S e nk (cid:17) S n ≥ S e nk (cid:105) (5.11) ≥ E (cid:104)(cid:16) S n − S e nk (cid:17) S n ≥ S e nk + n(cid:15) (cid:105) ≥ ( e n(cid:15) − S e nk P (cid:16) S n ≥ S e nk + n(cid:15) (cid:17) = ( e n(cid:15) − S e nk P (cid:18) d √ n ≥ Z √ n (cid:19) . This gives lim inf n →∞ n log C ( n ) ≥ k + (cid:15) + lim n →∞ n log P (cid:18) d √ n ≥ Z √ n (cid:19) (5.12)= k + (cid:15) − inf D ( u,v ) ≥ z (cid:18) I ( u, v ) + 12 z (cid:19) . Since this inequality holds for any (cid:15) >
0, we getlim inf n →∞ n log C ( n ) ≥ −I low C ( k ) := k − inf D ( u,v ) ≥ z (cid:18) I ( u, v ) + 12 z (cid:19) . (5.13)The bounds have different behavior depending on k , as I up C ( k ) = (cid:40) , k < y R ρ , I X ( k, (cid:37) ) − k , k > y R ρ , I low C ( k ) = (cid:40) , k < − ρ , I X ( k, (cid:37) ) − k , k > − ρ . (5.14)This follows from a study of the global minimum of the functions in the constrainedextremal problems for the bounds in Eq. (5.8) and (5.13) in relation to the constraints.Consider first the upper bound. By Corollary 9 the function in Eq. (5.8) has a globalminimum equal to zero at ( u m ρ , v m ρ ) and z = 0. For sufficiently small k < y R ρ ,this point is within the region allowed by the constraint D ( u, v ) ≥ z . This proves theupper line in Eq. (5.14). For k > y R ρ , the global minimum is excluded by the condition D ( u, v ) ≥ z . Convexity of I ( u, v ) implies that any local minimum in Eq. (5.8) is also the19lobal minimum, so the inf is reached on the boundary of the region D ( u, v ) = z , but notin the interior of the region.The function appearing in the lower bound I low C ( k ) has a global minimum of zero at u = ρ , v = v , z = 0, which is allowed by the constraint D ( u, v ) ≥ z only for k < − ρ . For k > − ρ the infimum in Eq. (5.13) is reached on the boundary of the region D ( u, v ) = z .On the respective boundaries, the two bounds coincide − inf D ( u,v )= z (cid:18) I ( u, v ) + 12 z + 12 (cid:37) u − (cid:37) ( v − v ) (cid:19) = k − inf D ( u,v )= z (cid:18) I ( u, v ) + 12 z (cid:19) (5.15)= k − I X ( k, (cid:37) ) . This proves the lower line equations in Eq. (5.14). This completes the proof of the resultfor the OTM call. The proofs for the other two cases are similar.
Using the option prices asymptotics of Theorem 15 one can obtain the asymptotics of theimplied volatility in the log-normal SABR model under the discretization scheme (2.2).
Theorem 16.
Consider the SABR model with correlation (cid:37) ≤ discretized in time with n points under the scheme (2.2). In the limit n → ∞ at fixed ρ = σ τ, β = ω τ n , theimplied volatility for maturity T := t n and log-strike x := log( K/S ) is given by σ BS ( x, T ) = σ Σ BS (cid:18) xσ T ; a (cid:19) , (5.16) where the equality in (5.16) means the LHS/RHS goes to one in the limit, and Σ BS ( y ; a ) := (cid:12)(cid:12)(cid:12)(cid:113) a J X ( y ; a, (cid:37) ) − y − (cid:113) a J X ( y ; a, (cid:37) ) (cid:12)(cid:12)(cid:12) for y > y R and y < − , (cid:113) a J X ( y ; a, (cid:37) ) − y + (cid:113) a J X ( y ; a, (cid:37) ) for − ≤ y ≤ y R , (5.17) where J X ( y ; a, (cid:37) ) is the rate function defined in (4.21) and a := 2( σ T )( ω T ) .Proof. The result is a standard transfer relation of the option price asymptotics to impliedvolatility. Similar results are obtained in Corollary 2.14 of the Forde and Jacquier paper[11] for the Heston model. A more general treatment of these transfer results is given inGao and Lee [16]. The different cases of the option price asymptotics in regions (1) and(3) of Theorem 15 correspond to the regime (+) in Section 4.1 of [16], and the region (2)corresponds to the regime (-). 20he n → ∞ limit of the implied volatility for maturity t n = nτ and log-strike limit x = log( K/S ) = nk with constant k islim n →∞ σ ( x, t n ) τ (5.18)= (cid:16) I X ( k, (cid:37) ) − k ) − (cid:112) I X ( k, (cid:37) )( I X ( k, (cid:37) ) − k ) (cid:17) for k > k R and k < k L (cid:16) I X ( k, (cid:37) ) − k ) + 4 (cid:112) I X ( k, (cid:37) )( I X ( k, (cid:37) ) − k ) (cid:17) for k L ≤ k ≤ k R = (cid:16)(cid:112) I X ( k, (cid:37) ) − (cid:112) I X ( k, (cid:37) ) − k (cid:17) for k > k R and k < k L (cid:16)(cid:112) I X ( k, (cid:37) ) + (cid:112) I X ( k, (cid:37) ) − k (cid:17) for k L ≤ k ≤ k R , where I X ( k, (cid:37) ) is the rate function given by Proposition 10 and k L = − ρ , k R = y R ρ .We note that although the result was derived in discrete time, the asymptotic impliedvolatility does not depend on the time step τ , but depends only on the product T = τ n = t n .The result (5.18) can be written equivalently as (5.16).We note that the asymptotic implied volatility of Theorem 16 has a scaling property,as it depends only on the two variables xσ T and a = 2( σ T )( ω T ). Remark 17.
The result of Theorem 16 reveals the existence of three regions of log-strikeseparated by x L = − σ T and x R = y R σ T . At the switch points we have σ BS ( x L , T ) = σ , σ BS ( x R , T ) = (cid:112) y R σ . (5.19) -2 -1 0 1 20.80.911.11.2 a=1 r=0 x -2 -1 0 1 20.80.911.11.2 a=1 r=-0.5 x Figure 1: Plot of the asymptotic implied volatility Σ BS ( x ; a, (cid:37) ) in the log-normal discrete-time SABR model for parameters a = 1 . (cid:37) = 0 . (cid:37) = − . his is illustrated in Figure 1 which shows the implied volatility function Σ BS ( y ; a ) for a = 1 . and correlation (cid:37) = 0 (left) and (cid:37) = − . (right). The three regions of Theorem 16are shown with different colors. This is different from the SABR formula (6.15) which doesnot distinguish between these regions. Remark 18.
The result of Theorem 16 is similar to the large maturity asymptotics for theHeston model derived by Forde and Jacquier [11]. However we note also a difference. Intheir result for the Heston model, the asymptotic implied volatility does not depend on σ ,the initial condition for the volatility. This is because their rate function does not dependon σ . On the other hand, with our scaling σ appears through the scaling variable ρ , whichintroduces dependence on σ in the asymptotic implied volatility. Remark 19.
For zero correlation (cid:37) = 0 , the implied volatility given by Theorem 16 issymmetric in log-strike (see the left panel in Fig. 1 for an illustration) σ BS ( − x, T ) = σ BS ( x, T ) . (5.20) This follows from the symmetry relation (4.33) for the rate function in the zero correlationlimit I X ( k, − I X ( − k,
0) = k . This agrees with the well-known result of [44] that theimplied volatility in an uncorrelated stochastic volatility model is a symmetric function oflog-strike. The leading quadratic term in the expansion of the rate function around its minimumat y = − gives a linear approximation for Σ BS ( y ; a ) around y = − Σ BS ( y ; a ) = 1 − (1 − √ c ) (cid:18) y + 12 (cid:19) + O (cid:32)(cid:18) y + 12 (cid:19) (cid:33) , (5.21)with c = a − √ a (cid:37) . This linear approximation is shown as the dashed line in Fig. 1. We study in this section the limits of the asymptotic implied volatility of Theorem 16 inseveral regimes of short maturity T → | x | → ∞ at fixed maturity.The results are compared with existing results in the literature. Consider the T → a → √ ay = √ ζ , with ζ = ωσ x . Assuming that the limit exists, define J ( ζ ; (cid:37) ) := lim a → , √ ay = √ ζ J X ( y ; a, (cid:37) ) . (6.1)22et us take this limit in the rate function J X ( y ; a, (cid:37) ), expressed as the extremal problem(4.22). This gives J ( ζ ; (cid:37) ) = inf u,v> (cid:26) I ( u, v ) + 2 (cid:37) ⊥ u ( ζ − (cid:37) ( v − (cid:27) . (6.2)For ζ = 0 the minimizer is u ∗ = 1 , v ∗ = 1; at this point the rate function vanishes J (0; (cid:37) ) = 0. Proposition 20.
The solution of the extremal problem (6.2) for the rate function J ( ζ ; (cid:37) ) has the expansion around ζ = 0 J ( ζ ; (cid:37) ) = 2 ζ − (cid:37)ζ + (cid:18) −
23 + 52 (cid:37) (cid:19) ζ + O ( ζ ) . (6.3) In the uncorrelated case (cid:37) = 0 the extremal problem (6.2) can be solved exactly J ( ζ ; 0) = 2 log (cid:16)(cid:112) ζ + 1 + | ζ | (cid:17) . (6.4) Proof.
The infimum condition in (6.2) can be expressed as the vanishing of the partialderivatives of the function of ( u, v ) on the right hand side. This gives two equations forthe extremal point ( u ∗ , v ∗ ) u∂ u I ( u, v ) = 4 (cid:37) ⊥ u ( ζ − (cid:37) ( v − , (6.5) v∂ v I ( u, v ) = 8 (cid:37)(cid:37) ⊥ u v ( ζ − (cid:37) ( v − . Introducing (cid:15) = log u ∗ , η = log v ∗ , the minimizers can be expanded in ζ as (cid:15) = a ζ + a ζ + O ( ζ ) , η = b ζ + b ζ + O ( ζ ) . (6.6)Using the expansion for the rate function I ( u, v ), the two equations in (6.5) can be expandedalso in ζ . Requiring the equality of the terms of each order in ζ gives successive equationsfor a i , b j which can be solved recursively. The first two coefficients are a = b = (cid:37) , a = 23 − (cid:37) , b = 12 (1 − (cid:37) ) . (6.7)The coefficients a , , b , are sufficient to determine the expansion of the rate function J ( ζ ; (cid:37) ) to order O ( ζ ), with the result quoted in (6.3).We give next the proof of (6.4) for the uncorrelated case. As a → ay we have | y | → ∞ , such that we use the y > y R branch of the function Σ BS ( y ; a ) in Theorem 16.23he rate function J X ( y ; a,
0) is given by Corollary 29. The equation for ξ in this resultbecomes in this limit sinh (cid:18) ξ (cid:19) = 12 ay = ζ , (6.8)which determines ξ up to a sign as ξ = ± log (cid:16) ζ + 2 (cid:112) ζ (1 + ζ ) (cid:17) = ± (cid:16)(cid:112) ζ + 1 + (cid:112) ζ (cid:17) . (6.9)The rate function islim a → ,ay =2 ζ J X ( y ; a,
0) = 12 ξ = ξ tanh( ξ/
2) + 2 ζ ξ sinh ξ = 2 log (cid:16)(cid:112) ζ + 1 + (cid:112) ζ (cid:17) . (6.10) Remark 21.
The expansion (6.3) agrees with the first three terms in the Taylor expansionof the function ˜ J ( ζ, (cid:37) ) := 2 log (cid:112) (cid:37)ζ + ζ + ζ + (cid:37) (cid:37) . (6.11)Numerical testing shows that the rate function J ( ζ ; (cid:37) ) is reproduced to very goodprecision by this function; however we could not prove their equality analytically, exceptfor the uncorrelated case (cid:37) = 0, when (6.11) reduces to (6.4).The asymptotic implied volatility in the small-maturity limit can be expressed in termsof the rate function J ( ζ ; (cid:37) ) given by the limit (6.1). Proposition 22.
Assume that the limit (6.1) exists and is given by the rate function J ( ζ ; (cid:37) ) . Then the implied volatility in the T → limit of the SABR model with correlation (cid:37) and ω → , σ → ∞ at fixed σ ω is lim σ →∞ σ σ BS ( x, T ) = √ ζ (cid:112) J ( ζ ; (cid:37) ) , with ζ = ωσ x. (6.12) Proof.
Start with the result for Σ BS ( y ; a ) from Theorem 15Σ BS ( y ; a ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:114) a J X ( y ; a, (cid:37) ) − y − (cid:114) a J X ( y ; a, (cid:37) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , y > y R and y < − , (6.13)We choose the y > y R , y < − branch since as a →
0, the product √ ay can be constantonly if y → ∞ . Expanding this result for a → BS ( y ; a ) = (cid:114) a J X ( y ; a, (cid:37) ) (cid:32) − (cid:115) − ya J X ( y ; a, (cid:37) ) (cid:33) = √ ay (cid:112) J X ( y ; a, (cid:37) ) + O ( a / y ) . a → √ ay = √ ζ giveslim a → Σ BS ( y ; a ) = √ ζ (cid:112) J ( ζ ; (cid:37) ) , (6.14)where J ( ζ ; (cid:37) ) is given by the limit (6.1).Substituting the expansion (6.3) into (6.12) reproduces the first three terms in theexpansion of the celebrated analytical formula for the implied volatility in the SABR modelin the short maturity asymptotic limit [23] σ BS ( x, T ) = σ ζD ( ζ ; (cid:37) ) (cid:18) (cid:18) (cid:37)ω σ + 124 (2 − (cid:37) ) ω (cid:19) T + O ( T ) (cid:19) . (6.15)with D ( ζ ; (cid:37) ) := log (cid:112) ζ(cid:37) + ζ + ζ + (cid:37) (cid:37) , (6.16)and the O ( T ) terms holds only at the at-the-money (ATM) point x = 0 [38]. Assuming J ( ζ ; (cid:37) ) = ˜ J ( ζ ; (cid:37) ) reproduces the first factor in (6.15).The result (6.15) is the leading order term in a short maturity expansion, and the nexttwo terms in this expansion have been subsequently derived by Henry-Labordere [24] andPaulot [37]. The ATM limit of this result is σ BS (0 ,
0) = σ .In Figure 2 we compare the asymptotic result (colored curves) with the SABR formula(6.15) (dashed black curve), for the model parameters σ = 0 . , ω = 1 , (cid:37) = − .
75, forseveral maturities T = 0 . − .
0. For sufficiently small maturity T , corresponding to smallvalues of the a = 2( ω T )( σ T ) parameter, the asymptotic result agrees very well with theshort maturity limit (6.15). We study here the T → Proposition 23. (i) The first few terms in the expansion of the ATM asymptotic impliedvolatility σ BS (0 , T ) in powers of a = 2 σ ω T are σ σ BS (0 , T ) = Σ BS (0; a ) = 1 + (cid:37) √ √ a + (cid:18) −
148 + 116 (cid:37) (cid:19) a + O ( a / ) (6.17)= 1 + 14 (cid:37)ωσ T −
124 (1 − (cid:37) ) σ ω T + O ( T ) . (ii) In the uncorrelated limit (cid:37) = 0 the expansion contains only integer powers of a Σ BS (0; a ) = 1 − a + 4323040 a − a − a + O ( a ) . (6.18)25 hich means that the implied volatility contains only even powers of Tσ BS (0 , T ) σ = 1 − σ ω T + 435760 σ ω T + O ( T ) . (6.19) Proof. (i) Taking x = 0 in Theorem 16 gives the ATM asymptotic implied volatilityΣ BS (0; a ) = 2 (cid:113) a J X (0; a, (cid:37) ). The rate function J X (0; a, (cid:37) ) is given by the extremal prob-lem (4.22). This can be expanded as a → u = v = 1. Denoting the minimizers u ∗ , v ∗ , this expansion reads (cid:15) = log u ∗ = a √ a + a a + O ( a / ) , η = log v ∗ = b √ a + b a + O ( a / ) . (6.20)Using the expansion (3.27) for the rate function I ( u, v ), the coefficients a i , b i can be deter-mined recursively. Substituting into (4.22) gives the expansion of the rate function J X (0; a, (cid:37) ) = 14 a + (cid:37) √ a / + 1384 ( − (cid:37) ) a + O ( a / ) . (6.21)Finally, substituting into Σ BS (0; a ) = 2 (cid:113) a J X (0; a, (cid:37) ) gives the expansion (6.17).(ii) The rate function J X ( y ; a,
0) is given in closed form by Eq. (A.10) J X (0; a,
0) = 2 λ (tan λ − λ ) + a sin 2 λ λ = 2 λ (2 tan λ − λ ) , (6.22)where λ is the solution of the equation λ cos λ = a . (6.23)The last equality in (6.22) follows by substituting a from (6.23).Expanding the solution for λ in powers of √ a and substituting into (6.22) gives J X (0; a,
0) = 14 a − a + 1960 a − a + O ( a ) . (6.24)This can be translated as before into an expansion for the asymptotic ATM implied volatil-ity. We can compare these expansions with the results in the literature. The O ( T ) termin (6.17) coincides with the first O ( T ) term in the short maturity expansion of the im-plied volatility (6.15). This expansion has been extended to O ( T ) by Paulot [38], wherethe O ( T ) term was evaluated partially numerically. A closed form result for the ATM26mplied volatility expansion in the log-normal ( γ = 1) SABR model to O ( T ) has beencommunicated to us by Alan Lewis [34]1 σ σ BS (0 , T ) = 1 + 124 σ ωT (cid:20) (cid:37) + ωσ (2 − (cid:37) ) (cid:21) (6.25)+ 11920 ω σ T (cid:20) ( −
80 + 240 (cid:37) ) + ωσ (cid:37) (240 − (cid:37) ) + ω σ ( −
12 + 60 (cid:37) − (cid:37) ) (cid:21) + O ( T ) . The short maturity expansion of the implied volatility in a wide class of stochasticvolatility models called MAP-like (Markov Additive Processes) which include γ = 1 SABR,is known ([33] page 505) to admit a double series expansion in ( x = log( K/S ) , T ).Recalling that the asymptotic limit considered in our paper corresponds to ω T (cid:28) , σ ω = O (1), it is easy to see that (6.17) is reproduced by taking this limit in (6.25).The existence of the limit ω T → , σ ω = O (1) constrains the form of the higherorder terms in the ATM implied volatility σ BS (0 , T ) σ = 1 + (cid:16) c (1)1 ωσ + c (0)1 ω (cid:17) T + ∞ (cid:88) k =2 σ k ( ω, σ , (cid:37) ) T k . (6.26)Assume that σ k ( ω, σ , (cid:37) ) T k is a polynomial in ( ω √ T ) , ( σ √ T ) of order 2 k , this must havethe form σ k ( ω, σ , (cid:37) ) = k (cid:88) j =0 c ( j ) k ( (cid:37) ) ω k − j σ j . (6.27)For example, a O ( T ) term of the form ωσ T is not allowed, as it diverges in the limitconsidered.In the limit considered, only the terms proportional to c ( k ) k contribute. The expansion ofProposition 23 determines these coefficients. For example we get c (4)4 ( (cid:37) ) = (43 − (cid:37) +315 (cid:37) ) for the coefficient of ( ωσ T ) , which has been confirmed by explicit computation[34]. We study here asymptotics in the extreme strikes region | x | → ∞ for the uncorrelated case (cid:37) = 0. Since the implied volatility is symmetric in x for this case, see Remark 19, it willbe sufficient to study asymptotics for large strike x → ∞ . Proposition 24.
In the large log-strike region x → ∞ , the asymptotic volatility (5.17) inthe uncorrelated log-normal SABR model (cid:37) = 0 has the expansion Σ BS ( x ; a,
0) = √ x − √ a log(2 x ) − a log(2 log(2 x )) + 1 √ a (6.28)+ 14 a √ x log (2 x ) + 12 a √ x log(2 x ) log(2 log(2 x )) + O (cid:18) log(2 log(2 x ))log(2 x ) (cid:19) . roof. We use the result of Proposition 31 for the large argument limit of the rate functionto obtain the asymptotics of Σ BS ( x ; a,
0) for x → ∞ Σ BS ( x ; a,
0) = Σ BS ( − x ; a,
0) = (cid:114) a J X ( x ; a, − (cid:114) − x + 1 a J X ( x ; a,
0) (6.29)= √ x (cid:16)(cid:112) r ( x, a ) − (cid:112) r ( x, a ) (cid:17) = √ x (cid:18) − (cid:112) r ( x, a ) + 12 r ( x, a ) + O ( r ( x, a )) (cid:19) , where r ( x, a ) = 14 ax log (2 x ) + 12 ax log(2 x ) log(2 log(2 x )) − ax log(2 x ) + O (cid:18) log log x ax (cid:19) . (6.30)Expanding the result gives (6.28). Corollary 25.
The large strike asymptotics of the implied variance for x → ∞ at fixed T is given by σ ( x, T ) T = σ Σ (cid:18) xσ T , a, (cid:19) T = 2 x (cid:18)(cid:114) r (cid:16) xσ T , a (cid:17) + (cid:114) r (cid:16) xσ T , a (cid:17)(cid:19) , (6.31) with a = 2( ω T )( σ T ) and r (cid:18) xσ T , a (cid:19) = 14 ω T x (cid:26)
12 log (cid:18) xσ T (cid:19) + log (cid:18) xσ T (cid:19) log (cid:18) (cid:18) xσ T (cid:19)(cid:19) − log (cid:18) xσ T (cid:19)(cid:27) . (6.32) Using the expansion ( √ r + √ r ) − = 1 − √ r + 2 r + O ( r / ) we get, keeping only the O ( √ r ) term, σ ( x, T ) T = 2 x (cid:18) − √ ω T x (cid:112) L + 2 L log(2 L ) − L + · · · (cid:19) (6.33)= 2 x − √ x √ ω T (cid:112) L + 2 L log(2 L ) − L + · · · , where we denoted L = log (cid:16) xσ T (cid:17) . The leading term in (6.33) agrees with the result expected from Lee’s moment formula.Recall that under the Log-Euler-log-Euler scheme, all moments E [( S n ) ε ] with ε > x →∞ σ ( x, T ) T = 2 x .28able 2: Scenarios for the numerical testing, from Table 8.6 in [33]. The model parametersare σ = 0 . , ω = 1 . , (cid:37) = − .
75. The table shows the parameter a = 2( ω T )( σ T ) of theasymptotic expansion, the point ( u m , v m ) determined by Corollary 9 and the right switchpoint y R given by (4.29). T a σ T ( u m , v m ) y R S = 1) σ ( K, T ) T = 2 log K − ω √ T (cid:112) K (log log K + log log log K ) + · · · . (6.34)The leading correction term to the Lee’s moment formula ∼ log log K agrees with (6.33). We compare in this section numerical benchmarks for implied volatility in the γ = 1 SABRmodel, with the asymptotic results of this paper.The benchmark option prices are taken from Table 8.6 in [33]. They were obtained usingthe transform method of [31, 33] with the model parameters σ = 0 . , ω = 1 . , (cid:37) = − . T = 0 . , , , σ BS ( x, T ) /σ of Theorem 16 (blue/red curve) is compared againstthe benchmark values in Figure 2 (black dots). The three regions of Theorem 16 are shownin different colors (red for the central region − σ T ≤ x ≤ y R σ T ). The figures showalso the leading order O ( T ) short-maturity asymptotics in the SABR model (6.15) as thedashed black curve.From these results we note the following observations:(i) For short maturities the agreement of the asymptotic result with the SABR asymp-totic formula (6.15), and with the numerical benchmark results is very good. The centralregion of log-strikes of Theorem 16 is very small, and it expands as the maturity increases.29able 3: Benchmark numerical values computed using the transform method (AL)[33] and the second order short maturity expansion O ( T ) from (6.25) for maturities T = 0 . , , , ,
50, comparing with the asymptotic results from Theorem 16. The modelparameters are ( σ , ω ) = (0 . , . , (1 . , .
1) and (cid:37) = 0. σ = 0 . , ω = 1 . σ = 1 . , ω = 0 . T AL O ( T ) expansion asymptotic AL O ( T ) expansion asymptotic0.25 0.20407 0.204068 0.19998 1.00018 1.00018 0.999971.0 0.21460 0.215083 0.19967 1.00041 1.00042 0.999582.0 0.22123 0.227 0.19870 0.999974 0.999998 0.998355.0 0.20451 0.24375 0.19286 0.993662 0.993734 0.9900250.0 0.07822 -2.925 0.11275 0.719669 -0.0015625 0.72071(ii) At larger maturities the short-maturity approximation (6.15) overestimates theactual implied volatility. While the asymptotic result reproduces the decreasing trend ofthe numerical result, it is an overestimate for longer maturities.As explained in the previous section, the asymptotic result holds in the limit σ /ω (cid:29) σ /ω = 0 . σ = 1 . , ω = 0 .
1. The agreement improves in the latter case, as expected.(iii) From Table 3 one observes that in the uncorrelated case (cid:37) = 0, the actual ATMimplied volatility has a non-monotonic dependence on maturity: starts at σ as T →
0, firstincreases with maturity, and then decreases as T → ∞ . On the other hand, the asymptoticresult has a monotonously decreasing trend.The discrepancy between the two results at short maturity can be traced back to theabsence of a O ( T ) term in the asymptotic expansion for the uncorrelated case, which isresponsible for the increasing trend of the numerical results for small maturity. Usingthe full O ( T ) expansion for the ATM implied volatility (6.25), which includes this term,reproduces well the benchmark results, as shown in Table 3. A The zero correlation case
The rate function I X ( x ; (cid:37) ) simplifies in the uncorrelated limit (cid:37) = 0, and can be expressedin closed form. This result can be used to derive the asymptotics of the rate function invarious limits of small/large arguments. We give in this Appendix these results and their30 x -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.750.511.52 T=1 x -0.75 -0.5 -0.25 0 0.25 0.5 0.750.511.52 T=2 x -0.75 -0.5 -0.25 0 0.25 0.5 0.750.511.52 T=5 x Figure 2: Plots of the asymptotic implied volatilities σ BS ( x, T ) /σ vs x = log( K/S )(colored curves) for the scenarios in Table 2, taken from Table 8.6 of [33]. The black dotsshow the benchmark values from [33], and the dashed black curve shows the short maturitySABR implied volatility. The different regions in Theorem 16 are shown in different colors(blue/red). 31roofs. A.1 The rate function J BS ( x ) The rate function appearing in the LDP for P ( n V n ∈ · ) can be extracted from Proposition6 in [42]. A simpler form is given in Corollary 13 of the same paper, in terms of the function J BS ( x ). We reproduce here this result for ease of reference. Proposition 26.
Define A n := (cid:80) n − i =0 e sZ i +( m − s ) t i , with Z i a standard Brownian motionsampled on uniformly distributed times t i = τ i . Consider the n → ∞ limit at fixed b = s τ n and r = mτ n . In this limit P ( n A n ∈ · ) satisfies a LDP with rate function I BS ( · ) = b J BS ( · ) . For r = 0 , the rate function J BS ( x ) is given by J BS ( x ) = ξ − ξ tanh( ξ/ for x ≥ , λ (tan λ − λ ) for < x ≤ , for x = 1 , (A.1) where ξ > is the unique solution of the equation ξ sinh ξ = x , (A.2) and λ ∈ (0 , π ) is the unique solution of the equation λ sin(2 λ ) = x . (A.3)The sum V n = (cid:80) n − i =0 σ i τ is obtained by identifying V n σ τ → A n with the substitutions s → ω, m → ω . In the n → ∞ limit at fixed b = s τ n = 2 ω τ n , it is clear that r = mnτ = ωnτ →
0. This justifies the r = 0 limit used in (A.1).We will require also the derivative of the rate function J BS ( x ). This can be obtainedin closed form and is given by the following result. Corollary 27.
The derivative of the rate function J BS ( x ) is given by J (cid:48) BS ( x ) = (cid:40) ξ cosh ( ξ/ for x > , − λ cos λ for < x < , (A.4) where ξ is the solution of the equation (A.2) and λ is the solution of the equation (A.3). The case r = 0 covers the case considered in this paper. See Proposition 6 in [42] for the general r (cid:54) = 0case. .2 Closed form result for the rate function I X ( x ; (cid:37) = 0) The rate function I X ( x ; (cid:37) ) giving the large deviations for P ( n log S n ∈ · ) is given by thesolution of the extremal problem in Proposition 10 in the main text. In the zero correlationlimit (cid:37) = 0 this extremal problem simplifies to a one-dimensional extremal problem. Usingthe one dimensional projection relation inf v I ( u, v ) = 2 J BS ( u ), the extremal problem (4.22)simplifies as J X ( y ; a,
0) = inf u,v (cid:40) I ( u, v ) + au (cid:18) y + 12 u (cid:19) (cid:41) (A.5)= inf u (cid:40) J BS ( u ) + au (cid:18) y + 12 u (cid:19) (cid:41) = J BS ( u ∗ ) + au ∗ (cid:18) y + 12 u ∗ (cid:19) , where we denoted in the last line the optimal value of u in the extremal problem as u ∗ ( y ). Lemma 28.
The extremal value u ∗ ( y ) has the following properties:(i) u ∗ ( y ) > for | y | > ;(ii) < u ∗ ( y ) < for | y | < .Proof. The optimal value u ∗ ( y ) is given by the solution of the equation J (cid:48) BS ( u ) + au (cid:18) u − y (cid:19) = 0 . (A.6)This equation can be written equivalently as J (cid:48) BS ( u ) = a (cid:18) y u − (cid:19) . (A.7)The function on the right side is decreasing in u and is positive at y = 1 for | y | > , andnegative for | y | < . The function on the left side is increasing and vanishes at y = 1. Thisimplies that the two sides will become equal at a point u ∗ ( y ) which is larger than 1 in thefirst case, and lower than 1 in the second case. This proves the claim.We give next a closed form result for the rate function, which is useful for numericalevaluations and deriving asymptotic expansions. Corollary 29.
In the zero correlation limit (cid:37) = 0 , the rate function J X ( y ; a, has thefollowing explicit form.Case 1. | y | > . J X ( y ; a,
0) = 12 ξ − ξ tanh( ξ/
2) + a ξ sinh ξ (cid:18) y + 12 ξ sinh ξ (cid:19) , (A.8)33 here ξ ∈ (0 , ∞ ) satisfies the equation a ξ cosh ( ξ/
2) + 18 − y ξ sinh ξ = 0 . (A.9) Case 2. | y | < . J X ( y ; a,
0) = 2 λ (tan λ − λ ) + a λ sin 2 λ (cid:18) y + 12 sin 2 λ λ (cid:19) , (A.10) where λ ∈ (0 , π/ satisfies the equation − a λ cos λ + 18 − y (2 λ ) sin (2 λ ) = 0 . (A.11) Case 3. | y | = . J X (cid:18) −
12 ; a, (cid:19) = 0 , J X (cid:18)
12 ; a, (cid:19) = a. (A.12) Proof.
Case 1. | y | > . For this case u ∗ > J BS ( u ),with u ∗ = sinh ξξ . The value of ξ is determined by expressing (A.6) as an equation for ξ .This reproduces (A.9). The rate function is obtained by substituting u ∗ into (A.5). Thisreproduces (A.8).Case 2. | y | < . For this case 0 < u ∗ < J BS ( u ),with u ∗ = sin(2 λ )(2 λ ) . The equation for λ is obtained by substituting the expression for u ∗ into(A.6). This reproduces (A.11). Proceeding in a similar way we get the rate function bysubstituting u ∗ into (A.5), which reproduces (A.10). Proposition 30.
The rate function in the uncorrelated case satisfies the symmetry relation J X ( y ; a, − J X ( − y ; a,
0) = 2 ay . (A.13)
Proof.
This follows by noting that the optimizer u ∗ ( y ), the solution of the equation (A.6)is a symmetric function in y . If u ∗ ( y ) is a solution of this equation for a certain y , it willbe also a solution for − y . Thus we have J X ( y ; a, − J X ( − y ; a,
0) = au ∗ (cid:18) y + 12 u ∗ (cid:19) − au ∗ (cid:18) − y + 12 u ∗ (cid:19) = 2 ay . (A.14)34 .3 Asymptotic expansion for J X ( x ; a, We derive here the asymptotic expansion for the rate function J X ( x ; a,
0) for very largeargument | x | → ∞ . Proposition 31. As x → ∞ the rate function J X ( x ; a, has the expansion J X ( x ; a,
0) = 2 ax + 12 log (2 x ) + log(2 x ) log(2 log(2 x )) − log(2 x ) + O (log log x ) . (A.15) Proof.
We use the explicit form for the rate function J X ( x ; a,
0) given in Corollary 29.This has two branches, for | x | > and | x | < . We are interested in x (cid:29) x > for ease of reference. J X ( x ; a,
0) = 12 ξ − ξ tanh( ξ/
2) + a ξ sinh ξ (cid:18) x + 12 ξ sinh ξ (cid:19) , (A.16)where ξ is the solution of the equation14 a ξ cosh ( ξ/
2) + 18 − x ξ sinh ξ = 0 . (A.17)The first two terms in (A.16) correspond to J BS ( y ) in (4.21). The asymptotics of J BS ( y )for y → ∞ was obtained in Proposition 13 in [40]. We will follow a similar approach toobtain the asymptotics of J X ( x ; a,
0) for x → ∞ .The strategy will be to invert the equation (A.17) for x (cid:29) ξ ( x ) into (A.16). First we write the equation (A.17) for ξ as12 a sinh ξ cosh ( ξ/
2) + 14 sinh ξξ = x , (A.18)2 a sinh ( ξ/
2) + 14 sinh ξξ = x , (A.19)sinh ( ξ/ (cid:18) a + cosh ξξ (cid:19) = x . (A.20)We solve this equation for ξ as x → ∞ using asymptotic inversion, see for exampleSec.1.5 in [36]. We first write the equation (A.20) for ξ as e ξ (cid:16) − e − ξ (cid:17) (cid:18) a + 116 ξ e ξ (cid:16) e − ξ (cid:17) (cid:19) = x . (A.21)Take logs of both sides ξ + 2 log(1 − e − ξ ) + ξ + 2 log(1 + e − ξ ) − ξ ) + log (cid:18) ξ e − ξ a (1 + e − ξ ) (cid:19) = log x ,
35r equivalently2 ξ = log x − (cid:16) − e − ξ (cid:17) + 2 log(4 ξ ) − log (cid:18) ξ e − ξ a (1 + e − ξ ) (cid:19) . As x → ∞ , this is approximated as 2 ξ = log x + O (log ξ ). This estimate can beimproved by iteration, starting with this first order approximation and solving for ξ ( i +1) by inserting the previous iteration on the right-hand side. The first two iterations are2 ξ (1) = log x + O (log log x ) , (A.22)2 ξ (2) = log x + 2 log(2 log x ) + O (cid:18) log log x log x (cid:19) . (A.23)To this order, the dependence on a is of higher order. This means that we can approximatethe equation (A.20) for ξ with sinh ξξ = 2 x (by neglecting the a term) and we can read offthe solution from the Prop. 13 in [40] by replacing K/S → x .Thus we get the expansion of the first 2 terms in the rate function (A.16) (given by J BS ( y ∗ )) from Prop. 13 in [40] J BS ( y ∗ ) = 12 log (2 x ) + log(2 x ) log(2 log(2 x )) − log(2 x ) (A.24)+ 3 log (2 log(2 x )) − x )) + O (log − (2 x )) . The second term in (A.16) is a ξ sinh ξ (cid:18) x + 12 ξ sinh ξ (cid:19) = a x (cid:18) x + 12 2 x (cid:19) = 2 ax . (A.25)Adding them gives J X ( x ; a,
0) = 2 ax + 12 log (2 x ) + log(2 x ) log(2 log(2 x )) − log(2 x ) + O (log log x ) . This completes the proof of Eq. (A.15).
Acknowledgements
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