Large deviation principles for stochastic volatility models with reflection and three faces of the Stein and Stein model
aa r X i v : . [ q -f i n . M F ] J un LARGE DEVIATION PRINCIPLES FOR STOCHASTIC VOLATILITY MODELSWITH REFLECTION AND THREE FACES OF THE STEIN AND STEIN MODEL
ARCHIL GULISASHVILIA
BSTRACT . We introduce stochastic volatility models, in which the volatility is describedby a time-dependent nonnegative function of a reflecting diffusion. The idea to use re-flecting diffusions as building blocks of the volatility came into being because of a certainvolatility misspecification in the classical Stein and Stein model. A version of this modelthat uses the reflecting Ornstein-Uhlenbeck process as the volatility process is a specialexample of a stochastic volatility model with reflection. The main results obtained in thepresent paper are sample path and small-noise large deviation principles for the log-priceprocess in a stochastic volatility model with reflection under rather mild restrictions. Weuse these results to study the asymptotic behavior of binary barrier options and call pricesin the small-noise regime.
AMS 2010 Classification : 60F10, 60J60, 91B25, 91G20
Keywords : stochastic volatility models with reflection, reflecting diffusions, large devia-tion principles, binary barrier options, call pricing functions.Dedicated to the memory of E. M. Stein.1. I
NTRODUCTION . T
HREE F ACES OF THE S TEIN AND S TEIN M ODEL
In this paper, we introduce stochastic volatility models with reflection and establishsample path and small-noise large deviation principles for the log-price process associ-ated with such a model. We also study small-noise asymptotic behavior of call pricingfunctions and the implied volatility.Sample path large deviation principles go back to the celebrated work of Varadhan(see [46]) and Freidlin and Wentzell (see [16]). We refer the reader to [10, 11, 47] formore information about large deviations. Sample path and small-noise large deviationprinciples have numerous applications in the theory of stochastic volatility models (see,e.g., [2, 8, 15, 23, 24, 35, 39]).The present paper is dedicated to the memory of Elias M. Stein (1931-2018), a promi-nent mathematician, who for many years had been one of the most influential leaders inthe field of harmonic analysis. His mathematical legacy and the impact of his researchare well illustrated in [1, 14]. In 1991, Elias Stein and Jeremy Stein published the paper[45], in which they introduced a stochastic volatility model (the Stein and Stein model)
Department of Mathematics, Ohio University, Athens OH 45701; e-mail: [email protected] hat is currently considered one of the classical stochastic volatility models of financialmathematics.The idea to use reflecting diffusions as building blocks of the volatility came into beingin the present paper because of a certain volatility misspecification in [45]. More detailswill be provided below. The volatility process in a model with reflection is describedby a time-dependent function of a reflecting diffusion on the half-line. In this section,we will discuss three versions (faces) of the Stein and Stein model. In the first version,the volatility is modeled by the Ornstein-Uhlenbeck process (this is the original versionintroduced in [45]), while in the second and the third version, the volatility process is theabsolute value of the Ornstein-Uhlenbeck process and the reflecting Ornstein-Uhlenbeckprocess, respectively. The third face of the Stein and Stein model is an example of a modelwith reflection.The choice of the arithmetic Ornstein-Uhlenbeck process as the volatility process in[45] caused certain mostly psychological problems, since a generally accepted paradigmis that the volatility has to be positive. However, according to [49], Subsection 3.3.1, and[33], negative volatility in the Stein and Stein model does not cause any conceptual orcomputational problems. The Stein and Stein model is uncorrelated, which means thatBrownian motions driving the asset price process and the volatility process are indepen-dent. In such models, marginal distributions of the asset price depend on the integratedvariance rather than on the volatility (see, e.g., [21]). Hence, one can use the absolutevalue of the Ornstein-Uhlenbeck process as the volatility process not changing the modelmuch. However, in [45], Stein and Stein claimed the following: “Before proceeding, weought to comment on our assumption that volatility is driven by an arithmetic process,which raises the possibility that σ can become negative. This formulation is equivalent toputting a reflecting barrier at σ = σ enters everywhereelse in squared fashion” (see [45], p. 729). The symbol σ in the previous quotation standsfor the volatility process in the Stein and Stein model. This misspecification of the volatil-ity, which did not affect the main results in [45], was observed by Ball and Roma (see [3]).They concluded that actually the absolute value of the Ornstein-Uhlenbeck process, andnot the reflecting process, is, in fact, the model for the stochastic volatility in the Stein andStein model (see [3], p. 592). See also a relevant discussion in Subsection 3.3.1 of [49].For the sake of shortness, throughout the rest of the present paper we will write “the S&Smodel” instead of “the Stein and Stein model”, and also use the abbreviation “the OUprocess” instead of “the Ornstein-Uhlenbeck process”.Our next goal is to formally introduce the three versions of the S&S model. We will firstcomment on the volatility processes associated with these versions. The volatility process Y ( ) in the original S&S model is the OU process. It satisfies the stochastic differentialequation dY ( ) t = q ( m − Y ( ) t ) dt + ξ dB t , t ∈ [ T ] , (1.1)where T > q ≥ m ≥
0, and ξ >
0. Theinitial condition for the process Y ( ) will be denoted by y , and it will be assumed that y ∈ R . The OU process can be represented explicitly as follows: Y ( ) t = e − qt y + (cid:0) − e − qt (cid:1) m + ξ e − qt Z t e qs dB s , t ∈ [ T ] (1.2) see, e.g., [21], Lemma 1.18). In (1.1) and (1.2), the symbol B stands for a Brownian motionon a filtered probability space ( Ω , F , { e F t } , P ) , where { e F t } is the augmentation of thefiltration generated by the process B (see [31], Definition 7.2 in Chapter 2, for the definitionof an augmented filtration). The OU process is mean reverting. This is one of the stylisticproperties of the volatility. The mean reversion property of the OU process with q > Y ( ) diffuses above the long-run mean m , then the coefficient q ( m − Y ( ) t ) in the drift term of (1.1) becomes negative, and sincethe drift term is associated with the direction of change of Y ( ) , it pulls the process Y ( ) towards its mean m . A similar observation can be made in the case where Y ( ) diffusesbelow m .Let us set Y ( ) = | Y ( ) | , and let Y ( ) be the OU process instantaneously reflected at zero.For the process Y ( ) , we assume that y ≥
0. Reflecting diffusions will be discussed inSection 2. The processes Y ( k ) , 1 ≤ k ≤
3, are adapted to the filtration { e F t } . They are usedas the volatility processes in the models that will be introduced next.Consider the following three stochastic volatility models (the three faces of the S&Smodel): dS ( k ) t = µ S ( k ) t dt + Y ( k ) t S ( k ) t d ( ρ W t + ρ B t ) , t ∈ [ T ] , k =
1, 2, 3. (1.3)In the previous models, µ ∈ R is the drift coefficient, while the processes Y ( k ) , 1 ≤ k ≤ W in (1.3) is a Brownianmotion defined on the probability space ( Ω , F , P ) , and it is assumed that the processes W and B are independent. We will denote by {F t } the augmentation of the filtrationgenerated by the processes W and B . The processes S ( k ) , 1 ≤ k ≤
3, describing thestochastic behavior of the asset price, are adapted to the filtration {F t } . The number ρ ∈ ( −
1, 1 ) appearing in (1.3) is the correlation parameter, and we use the standard notation ρ = p − ρ . The parameter ρ characterizes the correlation between the process Z = ρ W + ρ B driving the asset price and the process B driving the volatility.We have already mentioned above that the model in (1.3) with k = ρ = ρ =
0) of the S&S model wasdeveloped in the paper [40] of Schobel and Zhu.It is not hard to see that the process Y ( ) defined by (1.2) is a continuous Gaussian pro-cess with the mean function m ( t ) = e − qt y + (cid:0) − e − qt (cid:1) m , t ∈ [ T ] , and the covariancefunction C ( t , s ) = ξ ( q ) − [ e − q | t − s | − e − q ( t + s ) ] , t , s ∈ [ T ] . The second face of the S&Smodel is the model in (1.3) with k =
2. The volatility in this model is the absolute valueof a Gaussian process, more precisely, Y ( ) t = | Y ( ) t | , t ∈ [ T ] . More general stochasticvolatility models, in which the volatility follows the absolute value of a Gaussian process,were studied in [26] and [27] in the case where ρ =
0. There are also numerous examples f stochastic volatility models, where the volatility is a nonnegative function of a Volterratype Gaussian process (see, e.g., [15, 22, 23, 24, 5]). We have chosen only these referenceshere because all of them are related to the main subjects of the present paper, which aresample path and small noise large deviation principles for log-prices in stochastic volatil-ity models.The third face of the S&S model is the model in (1.3) with k =
3. The volatility process inthis case is the reflecting OU process Y ( ) . The third face of the S&S model is an interestingspecial example of a stochastic volatility model with reflection.The three faces of the S&S model have many dissimilar features. The transition densi-ties p and p of the processes Y ( ) and Y ( ) , which are known explicitly (see, e.g., [21],formulas (1.19) and (1.23)), are distinct. However, as it has been already mentioned, when ρ =
0, the marginal distributions of the asset price processes S ( ) and S ( ) are identical(see [21] for more details). The reflecting OU process Y ( ) is a Markov process (see, e.g.,Theorem 1.2.2 in [36]). In [38], Ricciardi and Sacerdote showed that the transition den-sity p of this process is the unique solution to a certain Volterra type integral equation,while in [32], Linetsky found a spectral representation of p . It is important to mentionhere that if the long-run mean m of the Ornstein-Uhlenbeck process is equal to zero, thenthe processes Y ( ) and Y ( ) are equal in law (see Remark 2.2 below for more information).Therefore, the densities p and p coincide if m =
0. This can also be shown by comparingknown expressions for p and p . As we mentioned above, an explicit representation forthe density p can be found in [21]. In [19], the transition function associated with theprocess Y ( ) was characterized (see (4.10) in [19]). It is clear that an explicit formula forthe density p can be obtained by differentiating the functions appearing in the formulafor the transition function. The formula described in the previous sentence was rediscov-ered in [48], Theorem 2.1. It follows from the above-mentioned results that if m =
0, then p = p . If m =
0, then the densities p and p are different. In [3], these densities werecompared numerically for a certain set of model parameters such that m = p than for p , while far from the barrier, the values of thedensities almost coincide. See also Remark 3.15 in Section 3.We will next briefly comment on the structure of the present paper. Section 2 of thepaper deals with general time-inhomogeneous reflecting diffusions on the half-line. InSection 3, stochastic volatility models with reflection are introduced, and the main resultsobtained in the present paper are formulated and discussed (Theorems 3.4 and 3.7, andalso Corollaries 3.5 and 3.9). The proof of the general sample path large deviation prin-ciple established in Theorem 3.4 is given in Subsection 3.1. In the proofs of the resultsformulated in Section 3, we borrow some ideas from [17] and [24]. The last section ofthe paper (Section 4) is devoted to large deviation style asymptotic formulas in the small-noise regime for binary barrier options and call pricing functions.2. T IME -I NHOMOGENEOUS R EFLECTING D IFFUSIONS
This section deals with reflecting diffusions and sample path large deviation principlesfor them. A good source of information about such diffusions is [18], Section 23, andthe book [36]. Various large deviation principles for reflecting diffusions were obtained n [2, 4, 8, 12, 13, 29, 42]. Our approach is based on the Skorokhod map, the contrac-tion principle, and a large deviation principle for solutions of multidimensional diffusionequations with predictable coefficients established in [6] by Chiarini and Fischer. Notethat some of the ideas exploited in [6] were used in [17] to study stochastic volatilitymodels generalizing the fractional Heston model.Let a and c be jointly continuous functions on [ T ] × [ ∞ ) , and let us assume thatthese functions are locally Lipschitz continuous in the second variable, uniformly in time,that is, for every r > L r > | a ( t , x ) − a ( t , y ) | + | c ( t , x ) − c ( t , y ) | ≤ L r | x − y | , (2.1)for all t ∈ [ T ] and x , y ∈ [ r ] . We also assume that a and c satisfy the sublinear growthcondition in the second variable, uniformly in time: | a ( t , x ) | + | c ( t , x ) | ≤ C ( + | x | ) , x ∈ [ ∞ ) , t ∈ [ T ] . (2.2)Our goal is to construct a nonnegative diffusion Y having a stochastic differential of theform dY t = a ( t , Y t ) dt + c ( t , Y t ) dB t , t ∈ [ T ] ,when it lies inside the open half-line ( ∞ ) , and is instantaneously reflecting when it hitszero. As in the previous chapter, we denote by { e F t } the augmentation of the filtrationgenerated by the Brownian motion B . The initial condition for the process Y will be de-noted by y , and it will be assumed that y ≥ Y as the solution to the equation Y t = y + Z t a ( s , Y s ) ds + Z t c ( s , Y s ) dB s + l t , t ∈ [ T ] , (2.3)where l is an auxiliary stochastic process satisfying the following conditions almost surely: l is a continuous nondecreasing process with l =
0, and only the zeros of Y t can be pointswhere l t increases. The latter condition is equivalent to the following: l t = Z t { Y s = } dl s , t ∈ [ T ] .Skorohod proved in [43, 44] that the equation in (2.3) with two unknowns ( Y , l ) has aunique solution. The instantaneously reflecting process Y is a continuous stochastic pro-cess, and it follows from (2.3) that the process Y is a semimartingale. Under the restriction c ( t , 0 ) = t ∈ [ T ] , the following equality holds: l t = L t , for all t ∈ [ T ] , where L is the local time at zero for the process Y (see, e.g., [36], Theorem 1.3.1). The definitionof the local time of a semimartingale can be found in [31], Section 3.3.7, see also [37], pp.209-210. Remark 2.1.
In this remark, we follow Section 23 in [18] (see also [36] , Exercise 1.3.1). Supposea ( t , 0 ) = and c ( t , 0 ) > , for all t ∈ [ T ] , and extend the functions a and c to the set [ T ] × ( − ∞ , 0 ) by a ( t , x ) = − a ( t , − x ) and c ( t , x ) = c ( t , − x ) . Denote by Z the solution to thefollowing stochastic differential equation on R :dZ t = a ( t , Z t ) dt + c ( t , Z t ) dB t , t ∈ [ T ] , ith Z = z ≥ . Then the process e Z = | Z | is the solution to the equationd e Z t = a ( t , e Z t ) dt + c ( t , e Z t ) d e B t + e l t , t ∈ [ T ] , (2.4) with e Z = z , and a new Brownian motion defined by e B t = R t sign Z s dB s , t ∈ [ T ] . Now, usingthe unique solvability of the equation in (2.4), we see that under the restrictions on the functionsa and c formulated above, the reflecting process e Z and the process | Z | have the same laws. Remark 2.2.
The reflecting OU process Y ( ) , which is the volatility process in the third faceof the S&S model corresponds to the case where a ( t , x ) = q ( m − x ) and c ( t , x ) = ξ , for all ( t , x ) ∈ [ T ] × R + . The process Y ( ) is a time-homogeneous nonnegative diffusion. It followsfrom Remark 2.1 that if m = , then the processes Y ( ) and Y ( ) are equal in law. This has alreadybeen mentioned in the introduction. Let ε be a small-noise parameter. For every ε ∈ [
0, 1 ] , a scaled version of the equationin (2.3) is given by Y ( ε ) t = y + Z t a ( s , Y ( ε ) s ) ds + √ ε Z t c ( s , Y ( ε ) s ) dB s + l ( ε ) t , t ∈ [ T ] . (2.5)We will next define the Skorokhod map (see [36] for more details). Let C [ T ] be thespace of continuous functions on [ T ] equipped with the norm || f || = max t ∈ [ T ] | f ( t ) | for f ∈ C [ T ] . The Skorokhod map Γ : C [ T ] C [ T ] is given by ( Γ f )( t ) = f ( t ) − min s ∈ [ t ] ( f ( s ) ∧ ) , t ∈ [ T ] . (2.6)The Skorokhod map is a continuous nonlinear mapping from the space C [ T ] into itself(see, e.g., [36], Lemma 1.1.1). Actually, Γ maps C [ T ] into C + [ T ] , where the lattersymbol stands for the space of all nonegative functions from C [ T ] . It is also true that ( Γ ( α f ))( t ) = α ( Γ f )( t ) , for any α ≥ f ∈ C [ T ] , and t ∈ [ T ] . Moreover, it is easy tosee that for all f ∈ C [ T ] and t ∈ [ T ] , | ( Γ f )( t ) | ≤ s ∈ [ t ] | f ( s ) | . (2.7)In addition, we have | ( Γ f )( t ) − ( Γ f )( t ) | ≤ max s ∈ [ t ] | f ( s ) − f ( s ) | ,for all f , f ∈ C [ T ] and t ∈ [ T ] .The Skorokhod map is related to the solution Y ( ε ) of the equation in (2.5) as follows.Denote U ( ε ) t = y + Z t a ( s , Y ε s ) ds + √ ε Z t c ( s , Y ε s ) dB s , t ∈ [ T ] .Then we have Y ( ε ) t = ( Γ U ( ε ) )( t ) , t ∈ [ T ] ,and moreover for every ε ∈ (
0, 1 ] , the process t U ( ε ) t is the solution to the followingstochastic differential equation: dU ( ε ) t = a ( t , ( Γ U ( ε ) )( t )) dt + √ ε c ( t , ( Γ U ( ε ) )( t )) dB t , t ∈ [ T ] , (2.8) ith U ( ε ) = y , ε ∈ (
0, 1 ] (see [36], p. 5). Next, using (2.6), we can rewrite (2.8) as follows: dU ( ε ) t = a ( t , U ( ε ) ( t ) − min s ∈ [ t ] ( U ( ε ) ( s ) ∧ )) dt + √ ε c ( t , U ( ε ) ( t ) − min s ∈ [ t ] ( U ( ε ) ( s ) ∧ )) dB t ,for all t ∈ [ T ] . Remark 2.3.
Throughout the paper, we denote by C δ [ T ] a subset of the space C [ T ] consistingof all the functions f , for which f ( ) = δ . By L [ T ] will be denoted the space of Lebesguesquare-integrable over [ T ] functions equipped with the norm || f || L [ T ] = (cid:26) Z T f ( t ) dt (cid:27) , f ∈ L [ T ] . The symbol H [ T ] will stand for the Cameron-Martin space associated with Brownian motion,that is, the space of all absolutely continuous functions f on [ T ] such that f ( ) = and R T ˙ f ( t ) dt < ∞ . The norm in the space H [ T ] is defined by || f || H [ T ] = (cid:26) Z T ˙ f ( t ) dt (cid:27) , f ∈ H [ T ] . By H δ [ T ] will be denoted the set consisting of all absolutely continuous functions f on [ T ] such that f ( ) = δ and R T ˙ f ( t ) dt < ∞ . Our next goal is to prove a sample path large deviation principle for the process T ( ε ) = ( √ ε W , √ ε B , U ( ε ) ) , ε ∈ (
0, 1 ] , (2.9)with state space C [ T ] × C y [ T ] . Let us consider the following system of stochasticdifferential equations: dT ( ε ) ,1 t = √ ε dW t dT ( ε ) ,2 t = √ ε dB t dT ( ε ) ,3 t = a ( t , ( Γ T ( ε ) ,3 )( t )) dt + √ ε c ( t , ( Γ T ( ε ) ,3 )( t )) dB t . (2.10)Chiarini and Fisher obtained in [6] a sample path large deviation principle for solutionsof diffusion equations with locally Lipschitz continuous predictable coefficients satisfyinga sublinear growth condition (see Theorem 3.1 in [6]). It is not hard to see that the systemdefined in (2.10) satisfies the conditions in Theorem 3.1 in [6]. Indeed, the coefficients inthe third equation in (2.10) have the following form: a ( t , ( Γ f )( t )) and c ( t , ( Γ f )( t )) , for all f ∈ C [ T ] and t ∈ [ T ] . The predictability and the continuity can be established usingthe definition of the mapping Γ and the continuity of Γ on the space C [ T ] . The localLipschitz continuity and the sublinear growth condition follow from the restrictions onthe functions a and c formulated in the beginning of this section, and the simple propertiesof the Skorohod map mentioned above. Therefore, Assumptions A1 and A2 on p. 13 of[6] hold, and hence Theorem 3.1 in [6] can be applied to the process T ε . We will next makeseveral remarks and then formulate a sample path LDP for the process ε T ε . he controlled equation associated with the system above is three-dimensional. It de-pends on two controls f , f ∈ L [ T ] , and has the following form: ϕ ( t ) = R t f ( s ) ds ϕ ( t ) = R t f ( s ) ds ϕ ( t ) = y + R t a ( s , ( Γ ϕ )( s )) ds + R t c ( s , ( Γ ϕ )( s )) f ( s ) ds (2.11)(see (2.4) in [6]). For fixed f , f ∈ L [ T ] , the system in (2.11) has a unique solution. Theunique solvability of the third equation in (2.11) was established in [6] (see the proof ofthe validity of Assumption H4 on p. 14 of [6]). Note that f ( t ) = ˙ ϕ ( t ) and f ( t ) = ˙ ϕ ( t ) ,for all t ∈ [ T ] .Let g ∈ L [ T ] , and let ϕ g ∈ C y [ T ] be the unique solution to the equation ϕ ( t ) = y + Z t a ( s , ( Γ ϕ )( s )) ds + Z t c ( s , ( Γ ϕ )( s )) g ( s ) ds . (2.12)(see Assumption H4 in [6]). Note that in our setting, Assumption H4 holds true (see [6],Section 3). Actually, it is clear from (2.12) that ϕ g ∈ H y [ T ] . Remark 2.4.
Under the conditions in (2.1) and (2.2), assumption H5 in [6] also holds. Thelatter assumption is as follows: The mapping G : L [ T ] C y [ T ] defined by Gg = ϕ g isa continuous mapping from V r into the space C [ T ] , where V r ⊂ L [ T ] is the closed ball ofradius r > centered at the origin equipped with the weak topology. Since such a ball is a compactset in the weak topology of L [ T ] , its image in C [ T ] under the mapping G is a compact subsetof C [ T ] . Let us define the following functional on the space C [ T ] : I ( ϕ , ϕ , ϕ ) = Z T ˙ ϕ ( t ) dt + Z T ˙ ϕ ( t ) dt , (2.13)provided that ϕ , ϕ ∈ H [ T ] , and ϕ ∈ C y [ T ] is the unique solution to the equation ϕ ( t ) = y + Z t a ( s , ( Γ ϕ )( s )) ds + Z t c ( s , ( Γ ϕ )( s )) ˙ ϕ ( s ) ds . (2.14)Otherwise, we set I ( ϕ , ϕ , ϕ ) = ∞ . The equality in (2.13) can be rewritten as follows:For all ϕ , ϕ ∈ H [ T ] , I ( ϕ , ϕ , G ˙ ϕ ) = Z T ˙ ϕ ( t ) dt + Z T ˙ ϕ ( t ) dt ,and I ( ϕ , ϕ , ϕ ) = ∞ , otherwise.Recall that a rate function on a topological space X is a lower semi-continuous mapping I : X 7→ [ ∞ ] such that for all y ∈ [ ∞ ) , the level set L y = { x ∈ X : I ( x ) ≤ y } is aclosed subset of X . It is assumed that I is not identically infinite. A rate function I iscalled a good rate function if for every y ∈ [ ∞ ) , the set L y is a compact subset of X .The next assertion can be established by applying Theorem 3.1 in [6]. Theorem 2.5.
Suppose that the functions a and c are locally Lipschitz continuous and the sub-linear growth condition holds for them. Then the process ε T ε defined in (2.9) satisfies thesample path large deviation principle with speed ε − and good rate function I given by (2.13). he validity of the large deviation principle means that for every Borel measurable subset A of thespace C [ T ] , the following estimates hold: − inf ( ϕ , ϕ , ϕ ) ∈A ◦ I ( ϕ , ϕ , ϕ ) ≤ lim inf ε ↓ ε log P ( T ε ∈ A ) ≤ lim sup ε ↓ ε log P ( T ε ∈ A ) ≤ − inf ( ϕ , ϕ , ϕ ) ∈ ¯ A I ( ϕ , ϕ , ϕ ) . The symbols A ◦ and ¯ A in the previous estimates stand for the interior and the closure of the set A , respectively. Remark 2.6.
Actually, in Theorem 3.1 in [6] the Laplace principle is established. However, sinceI is a good rate function, the Laplace principle is equivalent to the LDP.
Remark 2.7.
It also follows from the results in [6] that the sample path large deviation principleholds for the process ε U ( ε ) with speed ε − and good rate function J given on C [ T ] byJ ( ϕ ) =
12 inf f ∈ H [ T ] : ϕ = G ˙ f Z T ˙ f ( t ) dt , (2.15) if the equation ϕ = G ˙ f is solvable for f , and J ( ϕ ) = ∞ , otherwise. It follows from (2.14) that˙ ϕ ( t ) = a ( t , ( Γ ϕ )( t )) + c ( t , ( Γ ϕ )( t )) ˙ ϕ ( t ) ,and hence, if the function c is strictly positive, we have˙ ϕ ( t ) = ˙ ϕ ( t ) − a ( t , ( Γ ϕ )( t )) c ( t , ( Γ ϕ )( t )) .Therefore, the following assertion holds true. Corollary 2.8.
Suppose the conditions in Theorem 2.5 hold. Suppose also that the function c isstrictly positive. Then the good rate function I can be represented as follows: For all ϕ ∈ H [ T ] and ϕ ∈ H y [ T ] , I ( ϕ , M ϕ , ϕ ) = Z T ˙ ϕ ( t ) dt + Z T N ϕ ( s ) ds , where M ϕ ( t ) = Z t ˙ ϕ ( s ) − a ( s , ( Γ ϕ )( s )) c ( s , ( Γ ϕ )( s )) dsand N ϕ ( t ) = ˙ ϕ ( t ) − a ( t , ( Γ ϕ )( t )) c ( t , ( Γ ϕ )( t )) . It is also true that I ( ϕ , ϕ , ϕ ) = ∞ , otherwise. Our next goal is to prove a large deviation principle for the process F ε = ( √ ε W , √ ε B , Y ( ε ) ) , ε ∈ (
0, 1 ] . (2.16)The next definition will be important in the remaining part of the paper. Definition 2.9.
The mapping f b f of the space H [ T ] into the space C y [ T ] is defined by b f ( t ) = ( Γ ( G ˙ f ))( t ) , t ∈ [ T ] . (2.17) n Definition 2.9, Γ is the Skorokhod map, while G is defined in Remark 2.4. Theorem 2.10.
Suppose that the functions a and c are locally Lipschitz continuous and the sub-linear growth condition holds for them. Then the process ε F ( ε ) defined in (2.16) satisfies thesample path large deviation principle with speed ε − and good rate function e I given on C [ T ] by e I ( ψ , ψ , b ψ ) = Z T ˙ ψ ( t ) dt + Z T ˙ ψ ( t ) dt , for all ψ , ψ ∈ H [ T ] , where the mapping f b f is defined in (2.17). In the rest of the cases, e I ( ψ , ψ , ψ ) = ∞ .Proof. It is clear that F ε = V ( U ε ) , where V : C [ T ] C [ T ] is defined by V ( f , f , f ) = ( f , f , ( Γ f )) .Next, using the continuity of the Skorokhod map on the space C [ T ] , Theorem 2.5, andthe contraction principle, we see that the process ε F ( ε ) satisfies the sample path LDPwith speed ε − and good rate function e I defined on C [ T ] by e I ( ψ , ψ , ψ ) = Z T ˙ ψ ( t ) dt + Z T ˙ ψ ( t ) dt ,if there exists a function ϕ ∈ H y [ T ] such that ψ ( t ) = ( Γ ϕ )( t ) , t ∈ [ T ] , and simulta-neously ϕ = G ψ . We also have e I ( ψ , ψ , ψ ) = ∞ , otherwise. It is easy to see that if thefunction ϕ mentioned above exists, then ψ = ( Γ ( G ψ )) = b ψ .This completes the proof of Theorem 2.10.The next statement can be obtained from Theorem 2.10. Corollary 2.11.
Suppose the conditions in Theorem 2.10 hold. Suppose also that the function c isstrictly positive. Then the good rate function e I can be represented as follows. Let ψ ∈ H [ T ] and ϕ ∈ H y [ T ] . Then e I ( ψ , M ϕ , ( Γ ϕ ) − y ) = Z T ˙ ψ ( t ) dt + Z T N ϕ ( s ) ds , where M ϕ ( t ) = Z t ˙ ϕ ( s ) − a ( s , ( Γ ϕ )( s )) c ( s , ( Γ ϕ )( s )) ds (2.18) and N ϕ ( t ) = ˙ ϕ ( t ) − a ( t , ( Γ ϕ )( t )) c ( t , ( Γ ϕ )( t )) . In the rest of the cases, e I ( ψ , ψ , ψ ) = ∞ .Proof. By taking into account Theorem 2.10, we see that it suffices to prove the equality d M ϕ = ( Γ ϕ ) . (2.19)By differentiating the functions in (2.18) with respect to t , we obtain˙ ϕ ( t ) = a ( t , ( Γ ϕ )( t )) + c ( t , ( Γ ϕ )( t ))[ M ϕ ] ′ ( t ) , t ∈ [ T ] .We also have ϕ ( ) = y . The previous equalities mean that ϕ = G ( M ϕ ) . Therefore ( Γ G )( M ϕ ) = ( Γ ϕ ) , and it follows that the equality in (2.19) holds. he proof of Corollary 2.11 is thus completed.3. S TOCHASTIC V OLATILITY M ODELS WITH R EFLECTION
In this section, we introduce stochastic volatility models with reflection, and establishlarge deviation principles for log-price processes in such models. Let ( Ω , F , P ) be a prob-ability space carrying two independent standard Brownian motions W and B , and let Y be a time-inhomogeneous reflecting diffusion satisfying the equation in (2.3). It will beassumed that the conditions in (2.1) and (2.2) hold for the coefficients a and c . Considera stochastic volatility model, in which the asset price process S t , t ∈ [ T ] , satisfies thefollowing stochastic differential equation: dS t = S t b ( t , Y t ) dt + S t σ ( t , Y t )( ¯ ρ dW t + ρ dB t ) , S = s >
0, 0 ≤ t ≤ T . (3.1)In (3.1), s is the initial price, T > ρ ∈ ( −
1, 1 ) is the correlationcoefficient, and ¯ ρ = p − ρ . The functions b and σ are continuous functions on [ T ] × R . The equation in (3.1) is considered on a filtered probability space ( Ω , F , {F t } ≤ t ≤ T , P ) ,where {F t } ≤ t ≤ T is the augmentation of the filtration generated by the processes W and B . We will also use the augmentation of the filtration generated by the process B , anddenote it by { e F t } ≤ t ≤ T . It is clear that the process Y is adapted to the filtration { e F t } ≤ t ≤ T .It will be explained next what restriction we impose on the functions b and σ appearingin (3.1). This restriction is rather mild. The following definitions will be needed. Definition 3.1.
A locally bounded function ω : [ ∞ ) [ ∞ ) is called a modulus of continuityon [ ∞ ) , if ω ( ) = and lim u → ω ( u ) = . Definition 3.2.
Let ω be a modulus of continuity on [ ∞ ) . A function λ defined on [ T ] × R is called locally ω -continuous, if for every δ > there exists a number L ( δ ) > such that for allx , y ∈ B ( δ ) , the following inequality holds: | λ ( x ) − λ ( y ) | ≤ L ( δ ) ω ( || x − y || ) . (3.2) In (3.2), the symbol || · || stands for the Euclidean norm on [ T ] × R , and B ( δ ) denotes the closedball in the space [ T ] × R centered at (
0, 0 ) and of radius δ . We will next formulate the restriction that we impose on the functions b and ω . Assumption C.
The functions b and σ are locally ω -continuous on [ T ] × R with respectto some modulus of continuity ω . Moreover, the function σ is nonnegative and not iden-tically zero on [ T ] × R .If all the conditions formulated above hold, we call the model described by the equationin (3.1) a stochastic volatility model with reflection. Remark 3.3.
The third face of the S&S model is an example of a model with reflection. For thismodel, the process Y in (3.1) is the reflecting OU process Y ( ) . Here we have b ( t , x ) = µ for all ( t , x ) ∈ [ T ] × R , while σ ( t , x ) = x for all ( t , x ) ∈ [ T ] × [ ∞ ) , and σ ( t , x ) = for all ( t , x ) ∈ [ T ] × ( − ∞ , 0 ) (compare the model in (1.3) with k = and the model in (3.1)). Inaddition, a ( t , x ) = q ( m − x ) and c ( t , x ) = ξ , for all ( t , x ) ∈ [ T ] × R (see Remark 2.2). If µ = r, then the third face of the S&S model is a risk-neutral model (see Remark 4.4). he unique solution to the equation in (3.1) is the Dol´eans-Dade exponential S t = s exp (cid:26) Z t b ( s , Y s ) ds − Z t σ ( s , Y s ) ds + Z t σ ( s , Y s )( ¯ ρ dW s + ρ dB s ) (cid:27) , 0 ≤ t ≤ T ,(see, e.g., [37]). Therefore, the log-price process X t = log S t satisfies X t = x + Z t b ( s , Y s ) ds − Z t σ ( s , Y s ) ds + Z t σ ( s , Y s )( ¯ ρ dW s + ρ dB s ) , 0 ≤ t ≤ T ,where x = log s .We will work with the following scaled version of the model in (3.1): dS ( ε ) t = S ( ε ) t b ( t , Y ( ε ) t ) dt + √ ε S ( ε ) t σ (cid:16) t , Y ( ε ) t (cid:17) ( ¯ ρ dW t + ρ dB t ) ,where 0 ≤ t ≤ T , and Y ( ε ) is the process satisfying the equation in (2.5). The asset priceprocess in the scaled model is given by; S ( ε ) t = s exp (cid:26) Z t b ( s , Y ( ε ) s ) ds − ε Z t σ ( s , Y ( ε ) s ) ds + √ ε Z t σ ( s , Y ( ε ) s )( ¯ ρ dW s + ρ dB s ) (cid:27) ,(3.3)where 0 ≤ t ≤ T , while the log-price process is as follows: X ( ε ) t = x + Z t b ( s , Y ( ε ) s ) ds − ε Z t σ ( s , Y ( ε ) s ) ds + √ ε Z t σ ( s , Y ( ε ) s )( ¯ ρ dW s + ρ dB s ) , (3.4)where 0 ≤ t ≤ T .Our next goal is to formulate and prove large deviation principles for the process ε X ( ε ) − x . Analyzing the representation for the process X ( ε ) given in (3.4), we seewhy it was important to establish an LDP for the process F ε defined in (2.16). It is clearthat the components of the process F ε are building blocks of the process X ( ε ) , and our aimis to use the extended contraction principle (see [10]) to establish large deviation prin-ciples for the process X ( ε ) − x . Some of the techniques used in such proofs weredeveloped in [15, 22, 23, 24, 5] in the case, where the volatility is modeled by a functionof a Gaussian process, and in [17] for certain non-Gaussian models. In this section, weborrow some ideas employed in the proof of the sample path LDP in Theorem 4.2 of [24](see Subsection 5.6 of [24]). However, there are also significant differences between thetwo proofs, because the mappings f b f used in [24] and in the present paper are verydifferent. Recall that in this paper, b f ( t ) = ( Γ ( G ˙ f ))( t ) , t ∈ [ T ] , where Γ is the Skorokhodmap, and G is defined in Remark 2.4, while in [24], b f ( t ) = R t K ( t , s ) ˙ f ( s ) ds , t ∈ [ T ] ,where K is a Volterra type kernel that is Lebesgue square integrable over [ T ] . For thesake of convenience, we have decided to steal the notation b f from [24], since certain partsof the proofs in Subsection 5.6 of [24] and in the present section do not depend on a specialstructure of the mapping f b f .We will next formulate several theorems. They resemble the LDPs obtained in [24].First, we introduce some notation. Consider a measurable functional Φ : C [ T ] × C [ T ] C [ T ] efined as follows: For l , f ∈ H [ T ] and h = b f ∈ C y [ T ] , Φ ( l , f , h )( t ) = Z t b ( s , b f ( s )) ds + ¯ ρ Z t σ ( s , b f ( s )) ˙ l ( s ) ds + ρ Z t σ ( s , b f ( s )) ˙ f ( s ) ds , (3.5)where 0 ≤ t ≤ T . For all the remaining triples ( l , f , h ) , we set Φ ( l , f , h )( t ) = t ∈ [ T ] .Let g ∈ C [ T ] , and define e Q T ( g ) = inf l , f ∈ H [ T ] (cid:20) (cid:18) Z T ˙ l ( s ) ds + Z T ˙ f ( s ) ds (cid:19) : Φ ( l , f , b f )( t ) = g ( t ) , t ∈ [ T ] (cid:21) ,if the equation appearing on the right-hand side of the previous formula is solvable for l and f . If there is no solution, then we set e Q T ( g ) = ∞ . It is not hard to see that if theequation Φ ( l , f , b f )( t ) = g ( t ) is solvable, then g ∈ H [ T ] .The next two assertions contain sample path large deviation principles for the log-priceprocess in a time-inhomogeneous stochastic volatility model with reflection. At the firstglance, these assertions look exactly as the large deviation principles formulated in The-orems 4.2 and 4.3 in [24]. However, there is a significant difference between the LDPsobtained in [24] and in the present paper. This difference arises because of the contrastingforms of the mapping f b f in [24] and in this paper. Theorem 3.4.
Suppose the functions a and c are locally Lipschitz continuous and the sublineargrowth condition holds for them. Suppose also that Assumption C holds for the functions b and σ .Then the process ε X ( ε ) − x with state space C [ T ] satisfies the sample path large deviationprinciple with speed ε − and good rate function e Q T . The validity of the large deviation principlemeans that for every Borel measurable subset A of C [ T ] , the following estimates hold: − inf g ∈A ◦ e Q T ( g ) ≤ lim inf ε ↓ ε log P (cid:16) X ( ε ) − x ∈ A (cid:17) ≤ lim sup ε ↓ ε log P (cid:16) X ( ε ) − x ∈ A (cid:17) ≤ − inf g ∈ ¯ A e Q T ( g ) . The symbols A ◦ and ¯ A in the previous estimates stand for the interior and the closure of the set A , respectively. Corollary 3.5.
Suppose the conditions in Theorem 3.4 hold. Suppose also that the volatilityfunction σ is strictly positive on [ T ] × R . Then, for all g ∈ H [ T ] , e Q T ( g ) = inf f ∈ H [ T ] Z T " ˙ g ( s ) − b ( s , b f ( s )) − ρσ ( s , b f ( s )) ˙ f ( s ) ¯ ρσ ( s , b f ( s )) ds + Z T ˙ f ( s ) ds .(3.6) Remark 3.6.
Under the conditions in Corollary 3.5, the function e Q T : H [ T ] R is contin-uous. Indeed, since e Q T is a rate function on C [ T ] , it is lower semicontinuous on that space.It follows that e Q T is also lower semicontinuous on the space H [ T ] , since the latter space iscontinuously embedded into the space C [ T ] . The upper semicontinuity of the function e Q T onthe space H [ T ] follows from the fact that this function can be represented as the infimum ofa family of functions, which are continuous on the space H [ T ] (see (3.6)). The continuity ofthose functions on H [ T ] can be established as in Lemma 6.2 in [24] . e will next show how to derive Corollary 3.5 from Theorem 3.4. Suppose the condi-tions in Corollary 3.5 hold, and let f , g ∈ H [ T ] . Then, the equation Φ ( l , f , b f )( t ) = g ( t ) , t ∈ [ T ] , (3.7)is solvable for l ∈ H [ T ] . Moreover, for any such solution, we have˙ l ( s ) = ˙ g ( s ) − b ( s , b f ( s )) − ρσ ( s , b f ( s )) ˙ f ( s ) ¯ ρσ ( s , b f ( s )) , s ∈ [ T ] .The previous statement can be established by differentiating the functions in (3.7) withrespect to t , and solving the resulting equation for ˙ l . Now, it is clear how to finish theproof of Corollary 3.5.Our next goal is to formulate small-noise large deviation principles for the log-priceprocess ε X ( ε ) T − x with state space R , in a time-inhomogeneous stochastic volatilitymodel with reflection.Let y ∈ R , f ∈ H [ T ] , and put Ψ ( y , f , b f ) = Z T [ b ( s , b f ( s )) + ρσ ( s , b f ( s )) ˙ f ( s )] ds + ¯ ρ (cid:26) Z T σ ( s , b f ( s )) ds (cid:27) y .Define a function on R as follows: e I T ( x ) = inf y ∈ R , f ∈ H [ T ] (cid:20) (cid:18) y + Z T ˙ f ( s ) ds (cid:19) : Ψ ( y , f , b f ) = x (cid:21) , (3.8)if the equation Ψ ( y , f , b f ) = x is solvable, and e I T ( x ) = ∞ , otherwise. Theorem 3.7.
Suppose the functions a and c are locally Lipschitz continuous and the sublineargrowth condition holds for them. Suppose also that Assumption C holds for the functions b and σ .Then the process ε X ( ε ) T − x satisfies the small-noise large deviation principle with speed ε − and good rate function e I T given by (3.8). The validity of the large deviation principle means thatfor every Borel measurable subset A of R , the following estimates hold: − inf x ∈ A ◦ e I T ( x ) ≤ lim inf ε ↓ ε log P (cid:16) X ( ε ) T − x ∈ A (cid:17) ≤ lim sup ε ↓ ε log P (cid:16) X ( ε ) T − x ∈ A (cid:17) ≤ − inf x ∈ ¯ A e I T ( x ) . The symbols A ◦ and ¯ A in the previous estimates stand for the interior and the closure of the set A,respectively.
Theorem 3.7 follows from Theorem 3.4. The previous statement can be established byusing the same reasoning as in the derivation of Theorem 4.11 from Theorem 4.2 in Section4 of [24]. Note that the proof in [24] does not depend on a special form of the mapping f b f . Remark 3.8.
A set
A ⊂ C [ T ] is called a set of continuity for the rate function e Q T (see Theorem3.4), if inf g ∈A ◦ e Q T ( g ) = inf g ∈ ¯ A e Q T ( g ) . or such a set, Theorem 3.4 implies that lim ε ↓ ε log P (cid:16) X ( ε ) − x ∈ A (cid:17) = − inf g ∈A e Q T ( g ) . (3.9) A similar definition of a set of continuity can be given for the rate function e I T in Theorem 3.7, andan equality similar to that in (3.9) can be established. Corollary 3.9.
Suppose the conditions in Theorem 3.7 hold. Suppose also that the volatilityfunction σ is strictly positive on [ T ] × R . Then, for every x ∈ R , e I T ( x ) = inf f ∈ H [ T ] (cid:16) x − R T [ b ( s , b f ( s )) + ρσ ( s , b f ( s )) ˙ f ( s )] ds (cid:17) ρ R T σ ( s , b f ( s )) ds + Z T ˙ f ( s ) ds . (3.10)Corollary 3.9 can be obtained from Theorem 3.7 as follows. Let x ∈ R and f ∈ H [ T ] .Then, under the conditions in Corollary 3.9, the equation Ψ ( y , f , b f ) = x can be solved for y ∈ R , and for every such solution we have y = (cid:16) x − R T [ b ( s , b f ( s )) + ρσ ( s , b f ( s )) ˙ f ( s )] ds (cid:17) ρ R T σ ( s , b f ( s )) ds .Now, it is clear that Corollary 3.9 holds. Remark 3.10.
The good rate function function e I T given by (3.10) is continuous on R . The proofis similar to that in Remark 3.6. Remark 3.11.
It is not hard to see that Corollary 3.9 also holds if for every f ∈ H [ T ] , Z T σ ( s , b f ( s )) ds =
0. (3.11)
By the continuity of the functions in (3.11), the equality in (3.11) is equivalent to the followingcondition: For every f ∈ H [ T ] , there exists s ∈ [ T ] such that σ ( s , b f ( s )) = . The point sin the previous sentence may depend on f . We will next analyze the condition in (3.11).
Lemma 3.12.
The following are true:(i) Suppose y > . Suppose also that σ ( y ) = . Then, for every f ∈ H [ T ] , the conditionin (3.11) holds.(ii) Let y = , and suppose the functions a and c appearing in the model for the reflecting volatilityprocess are such that the function g defined by g ( t ) = − a ( t ,0 ) c ( t ,0 ) , t ∈ [ T ] , is Lebesgue squareintegrable over [ T ] . Suppose also that σ ( s , 0 ) = , for all s ∈ [ T ] . Then, the function f givenby f ( t ) = R t g ( u ) du, t ∈ [ T ] , is such that f ∈ H [ T ] , and moreover σ ( s , b f ( s )) = for alls ∈ [ T ] .Proof. Let the conditions in part (i) of Lemma 3.12 hold. We will reason by contradiction.Suppose for some f ∈ H [ T ] , we have b f ( t ) = t ∈ [ T ] . Since b f = Γ ( G ˙ f ) , and G ˙ f ( ) = y , we have b f ( ) = y >
0. The previous formula contradicts our originalassumption. This establishes part (i) of Lemma 3.12. uppose the conditions in part (ii) of Lemma 3.12 hold. Then we have ˙ f = g ∈ L [ T ] .It follows that 0 = Z t a ( s , ( Γ )( s )) ds + Z t c ( s , ( Γ )( s )) ˙ f ( s ) ds , t ∈ [ T ] . (3.12)Since for every function g ∈ L [ T ] , the equation in (2.12) is uniquely solvable, andwe denoted its unique solution by Gg (see Remark 2.4), the following equality can bederived from (3.12): G ˙ f ( t ) = t ∈ [ T ] . Recall that b f = Γ ( G ˙ f ) (see (2.17)). It followsthat b f ( s ) =
0, for all s ∈ [ T ] . Now part (ii) of Lemma 3.12 follows from the condition σ ( s , 0 ) = s ∈ [ T ] .This completes the proof of Lemma 3.12. Corollary 3.13.
The following is true for the third face of the Stein and Stein model, that is, themodel in (1.3) with k = . Suppose y > . Then, for every f ∈ H [ T ] , the function b f is givenby b f = Γ ϕ f , where ϕ f ∈ H y [ T ] is the unique solution to the equation ϕ f ( t ) = y + qmt − q Z t Γ ϕ f ( s ) ds + ξ f ( t ) , t ∈ [ T ] . (3.13) Moreover, the large deviation principle in Theorem 3.7 holds with the rate function e I T given by e I T ( x ) = inf f ∈ H [ T ] (cid:16) x − µ T − ρ R T ( Γ ϕ f )( s ) ˙ f ( s ) ds (cid:17) ρ R T ( Γ ϕ f )( s ) ds + Z T ˙ f ( s ) ds . Proof.
Corollary 3.13 follows from part (i) of Lemma 3.12, Remark 3.11, and Corollary3.9. We also take into account that for the third face of the S&S model, a ( t , u ) = q ( m − u ) , c ( t , u ) = ξ , and b ( t , u ) = µ , for all ( t , u ) ∈ [ T ] × R + . Moreover, we can assume that σ ( t , u ) = [ T ] × ( − ∞ , 0 ) and σ ( t , u ) = u on [ T ] × [ ∞ ) . The equation in (3.13)can be obtained from (2.12) with g = ˙ f , Remark 2.4, and (2.17).The proof of Corollary 3.13 is thus completed.The case where y = L = { f ∈ H [ T ] : σ ( s , b f ( s )) = s ∈ [ T ] } is not empty (see part (ii) of Lemma 3.12). For Gaussian stochastic volatility models, asimilar problem was encountered in [24] (see Lemma 4.10 in [24]). For the third versionof the S&S model, we have L = { f ∈ H [ T ] : ( Γ ϕ f )( s ) = s ∈ [ T ] } , (3.14)where the function ϕ f can be determined from (3.13). We also set L = H [ T ] \ L .The following assertion holds in the case where y =
0. The proof is similar to that ofLemma 4.10 in [24].
Corollary 3.14.
Suppose y = in Corollary 3.13. Then, for every f ∈ H [ T ] , the function b fis given by b f = Γ ϕ f , where ϕ f ∈ H [ T ] is the unique solution to the equation ϕ f ( t ) = qmt − q Z t Γ ϕ f ( s ) ds + ξ f ( t ) , t ∈ [ T ] . oreover, the large deviation principle in Theorem 3.7 holds with the rate function e I T given by e I T ( µ T ) =
12 min inf f ∈ L Z T ˙ f ( t ) dt , inf f ∈ L ρ (cid:16) R T ( Γ ϕ f )( s ) ˙ f ( s ) ds (cid:17) ¯ ρ R T ( Γ ϕ f )( s ) ds + Z T ˙ f ( s ) ds and e I T ( x ) = inf f ∈ L (cid:16) x − µ T − ρ R T b f ( s ) ˙ f ( s ) ds (cid:17) ρ R T b f ( s ) ds + Z T ˙ f ( s ) ds , for x = µ T. It may be difficult to find a simple explicit description of the set L . We will do it belowfor a special model that will be introduced next.Brownian motion with drift is defined by Y ( ) t = at + ξ B t , t ∈ [ T ] , (3.15)where a ≥ ξ >
0. Although the process in (3.15) is not a special case of the OUprocess, it can be informally obtained from the OU process by assuming that q = qm = a in (1.2). Therefore, one may say that the model, in which the volatility followsBrownian motion with drift, is an additional case of the S&S model. We can also generatethe third face of the previous model using the reflecting Brownian motion with drift asthe volatility process. Remark 3.15.
Let us denote by p the transition density associated with the absolute value ofBrownian motion with drift, and by p the transition density corresponding to the reflecting Brow-nian motion with drift. If a = , then p = p . This was mentioned in the introduction. Fora = , the functions p and p are different. Indeed, the transition density p is the sum of twoGaussian densities. As for p , an explicit formula is known for this density (see formula (91) in [9] ). Comparing the formulas for p and p , we see that p = p . We refer the reader to [20] and [34] for more information the reflecting Brownian motion with drift. It is not hard to see that in the model, where the volatility follows reflecting Brownianmotion with drift, we have ϕ f ( t ) = at + ξ f ( t ) , t ∈ [ T ] , and b f = Γ ϕ f , for all f ∈ H [ T ] .The following lemma provides a characterization of the set L . Lemma 3.16.
Let y = . Then a function f ∈ H [ T ] belongs to the set L defined in (3.14) ifand only if ˙ f ( t ) ≤ − a ξ − (3.16) almost everywhere on [ T ] with respect to the Lebesgue measure.Proof. It is not hard to see that for a function g ∈ C [ T ] , the condition ( Γ g )( t ) = t ∈ [ T ] if and only if g is a nonincreasing function on [ T ] . Indeed, it fol-lows from the definition of the Skorokhod map Γ that the previous condition is equivalentto the following equality: g ( t ) = min ≤ s ≤ t ( g ( s ) ∧ ) , t ∈ [ T ] . (3.17) ow, it is clear that the statement formulated in the beginning of the proof can be easilyderived from the equality in (3.17).Finally, by recalling the definition of the set L and observing that the condition in (3.16)characterizes the set of functions f ∈ H [ T ] , for which the function ϕ f ( t ) = at + ξ f ( t ) does not increase on [ T ] , we complete the proof of Lemma 3.16.Our next goal is to find a special representation for the rate function e I T under the as-sumption that y = Corollary 3.17.
Consider the model, where the volatility follows the reflecting Brownian motionwith drift, and let y = . Then, the following formulas hold: e I T ( µ T ) =
12 min a T ξ , inf f ∈ L ρ (cid:16) R T ( Γ ϕ f )( s ) ˙ f ( s ) ds (cid:17) ¯ ρ R T ( Γ ϕ f )( s ) ds + Z T ˙ f ( s ) ds and e I T ( x ) =
12 inf f ∈ L (cid:16) x − µ T − ρ R T ( Γ ϕ f )( s ) ˙ f ( s ) ds (cid:17) ¯ ρ R T ( Γ ϕ f )( s ) ds + Z T ˙ f ( s ) ds , for x = µ T.Proof.
Corollary 3.17 follows from Corollary 3.14. Indeed, it only suffices to prove thatinf f ∈ L Z T ˙ f ( t ) dt = a T ξ . (3.18)Using the characterization of the set L in (3.16) in Lemma 3.16, we see that for every f ∈ L , ˙ f ( t ) ≥ a ξ a.e. on [ T ] . Therefore, (3.18) holds.This completes the proof of Corollary 3.17. Remark 3.18.
The rate function in Corollary 3.13 is continuous on R , while the rate functionsin Corollaries 3.14 and 3.17 may be discontinuous at only one point x = µ T. This can be shownusing the same reasoning as in the proof of Lemma 4.17 in [24] . It remains to prove Theorem 3.4. This will be done in the next subsection.3.1.
Proof of Theorem 3.4.
We have already mentioned above that Theorem 3.4 looksexactly like Theorem 4.2 in [24]. However, there are two substantial differences hidden inthe formulations of those theorems. The first difference is in the structure of the mapping f b f . Recall that in the present paper, b f ( t ) = ( Γ ( G ˙ f ))( t ) , t ∈ [ T ] ,where Γ is the Skorokhod map, and G is defined in Remark 2.4, while in Theorem 4.2 in[24], b f ( t ) = Z t K ( t , s ) ˙ f ( s ) ds , t ∈ [ T ] ,where K is a Volterra type kernel that is Lebesgue square-integrable over [ T ] . Thesecond difference is that Theorem 3.4 uses the process ε Y ( ε ) as the volatility process, hile in Theorem 4.2 in [24], the process ε
7→ √ ε b B is employed instead. Recall that b B t = R t K ( t , s ) dB s , t ∈ [ T ] . In the present paper, the functional Φ (see (3.5)) and itsapproximation Φ m (see (5.34) in [24]) are defined on the space C [ T ] × C y [ T ] . Similarcomparisons can be made with the proofs of LDPs in [17].It follows that all the techniques employed in the proof of Theorem 4.2 in [24], whichdo not depend on the special form of the mapping f b f , or the special structure of thevolatility process, can be used in the proof of Theorem 3.4. It remains to make a carefulanalysis of the proof of Theorem 4.2 in [24] in order to identify the statements in the proof,which depend on the above-mentioned differences, and show that those statements holdin the environment of Theorem 3.4.We will next prove several auxiliary lemmas. Our first goal is to estimate the distribu-tion function of the random variablesup s ∈ [ T ] Y ( ε ) s = || Y ( ε ) || C [ T ] as ε → ∞ . Suppose y >
0, and define a subset of C [ T ] by A y = { ϕ ∈ C : || ϕ || C [ T ] ≥ y } .Then it is clear that the set A y is closed in the space C [ T ] .The next assertion follows from the large deviation principle in Remark 2.7. Lemma 3.19.
For every y > , lim sup ε → ε log P ( sup s ∈ [ T ] Y ( ε ) s ≥ y ) ≤ − inf ϕ ∈A − y J ( ϕ ) , (3.19) where J is defined in (2.15).Proof. It follows from (2.7) and the equality Y ( ε ) = Γ U ( ε ) that P ( sup s ∈ [ T ] Y ( ε ) s ≥ y ) ≤ P ( sup s ∈ [ T ] U ( ε ) s ≥ − y ) , y > A − y . Corollary 3.20.
The following estimate is valid: lim y → ∞ lim sup ε → ε log P ( sup s ∈ [ T ] Y ( ε ) s ≥ y ) = − ∞ . Proof.
Using Lemma 3.19, we see that it suffices to prove thatlim y → ∞ inf ϕ ∈A − y J ( ϕ ) = ∞ . (3.20)We will next reason by contradiction. Suppose the equality in (3.20) does not hold.Then, there exists a strictly increasing sequence y k > k ≥
1, such that lim k → ∞ y k = ∞ , andmoreover inf ϕ ∈A − yk J ( ϕ ) ≤ C , k ≥
1, (3.21)for some C >
0. Next, recalling the definition of J in (2.15), we see that the estimate in(3.21) can be rewritten as follows:inf { ϕ : || ϕ || C [ T ] ≥ − y k } inf { g ∈ L [ T ] : Gg = ϕ } Z T g ( t ) dt ≤ C , k ≥
1, (3.22) t follows from (3.22) that there exist two sequences { ϕ k } and { g k } such that Gg k = ϕ k , || ϕ k || C [ T ] ≥ − y k , k ≥
1, (3.23)and moreover, || g k || C [ T ] ≤ C , for all k ≥ C >
0. However, by the compact-ness statement in Remark 2.4, the set { ϕ k } is precompact in C [ T ] . Hence it is bounded,which contradicts (3.23). Therefore, the equality in (3.20) is valid.This completes the proof of Corollary 3.20.Let ψ ∈ C [ T ] . The modulus of continuity of ψ in C [ T ] is defined as follows: e ω δ ( ψ ) = sup t , s ∈ [ T ] : | t − s |≤ δ | ψ ( t ) − ψ ( s ) | , δ ∈ [ T ] . Lemma 3.21.
For every y > , lim δ → lim sup ε → ε log P ( e ω δ ( Y ( ε ) ) ≥ y ) = − ∞ . Proof.
It is known that for every function h ∈ C [ T ] , e ω δ ( Γ h ) ≤ e ω δ ( h ) for all δ ∈ [ T ] (see, e.g., Lemma 1.1.1 (2) in [36]). It follows thatlim sup ε → ε log P ( e ω δ ( Y ( ε ) ) ≥ y ) ≤ lim sup ε → ε log P ( e ω δ ( U ( ε ) ) ≥ y ) . (3.24)Set B y , δ = { ϕ ∈ C [ T ] : e ω δ ( ϕ ) ≥ y ) .It is not hard to prove that for every y > < δ < T , the set B y , δ is closed in C [ T ] .Next, using the LDP in Remark 2.7 and (3.24), we obtainlim sup ε → ε log P ( e ω δ ( Y ( ε ) ) ≥ y ) ≤ − inf ϕ ∈B y , δ J ( ϕ ) .It remains to prove that for every y > δ → inf ϕ ∈B y , δ J ( ϕ ) = ∞ . (3.25)We will next reason by contradiction. Suppose the equality in (3.25) does not hold.Then, it is not hard to prove, using the definition of the rate function J in (2.15), that thereexist sequences δ k > ϕ k ∈ C [ T ] , and g k ∈ L [ T ] , k ≥
1, such that the sequence { δ k } is strictly decreasing, lim k → ∞ δ k = ω δ k ( ϕ k ) ≥ y , for all k ≥
1, (3.26)and moreover Gg k = ϕ k and R T g k ( t ) dt ≤ C , for all k ≥ C >
0. It fol-lows from the compactness statement in Remark 2.4 that the set { ϕ k } is precompact in C [ T ] . By the Arzel`a-Ascoli theorem, this set is uniformly equicontinuos. The previousstatement contradicts (3.26). Therefore, the equality in (3.25) is valid.The proof of Lemma 3.21 is thus completed.Now, we are ready to identify the parts of the proof of Theorem 4.2 in Subsection 5.6of [24], which can not be directly transplanted into the proof of Theorem 3.4. We will useitalic font in the description of those parts below, and after every such description includea necessary justification. The drift term − ε R t σ ( s , Y ( ε ) ) ds can be removed from formula (3.4) not affecting the LDP see Section 5 of [22] for a similar situation). We can repeat the proof in Section 5 of [22] by choosing H = , and replacing the es-timates before (36) in [22] by the following: For every δ > η defined asin Section 5 of [22], but uniformly with respect to t ∈ [ T ] ,lim sup ε → ε log P ( sup t ∈ [ T ] Y ( ε ) t ≥ η − ( δε − T − )) = − ∞ . (3.27)It is not hard to see that if we prove (3.27), then we can remove the drift term mentionedabove exactly as in Section 5 of [22].We will next prove the equality in (3.27). Set τ ( ε ) = η − ( δε − T − ) . Then τ is a strictlydecreasing function on (
0, 1 ] . Moreover τ ( ε ) → ∞ as ε →
0, since η − ( u ) → ∞ as u → ∞ (see Section 5 of [22]). Fix γ > < ε ≤ γ . Then τ ( ε ) ≥ τ ( γ ) , and applying(3.19), we see thatlim sup ε → ε log P ( sup t ∈ [ T ] Y ( ε ) t ≥ η − ( δε − T − )) ≤ lim sup ε → ε log P ( sup t ∈ [ T ] Y ( ε ) t ≥ τ ( γ )) ≤ − inf ϕ ∈A − τ ( γ ) J ( ϕ ) ,for all γ >
0. Finally, by taking into account (3.20) and the fact that τ ( γ ) → ∞ as γ → Lemmas 5.23 and 5.24 in [24] hold in our setting.
Analyzing the proof of those lemmas in [24], we see that the only statement in the proofthat depends on the special structure of the mapping f b f is the following: For every α >
0, sup f ∈ H [ T ] : || ˙ f || L [ T ] ≤ α ω Tm ( b f ) → m → ∞ . The formula in (3.28) follows in our setting from the definition of b f , theboundedness of the Skorokhod map in C [ T ] , the compactness statement in Remark 2.4,and the Arzel`a-Ascoli theorem. Corollary 5.22 in [24] holds in our setting with √ ε b B replaced by Y ( ε ) . The previous statement follows from Lemma 3.21.
Lemma 5.25 in [24] holds in our setting with √ ε b B replaced by Y ( ε ) , and the random variables σ ε , ms and b ε , ms in formulas (5.42)-(5.44) [24] changed accordingly. In the proof of the equalities similar to those in (5.51) and (5.52) in [24], we use Corollary3.20 and Lemma 3.21, respectively. In our environment, the estimates similar to thosefor the first term on the right-hand side of (5.55) in [24] hold. This can be established byconsulting the proof of the fact that the process in (58) of [22] is a martingale, and also theproof of (61) and (62) in [22]. The rest of the proof of Lemma 5.25 in [24] can be adaptedto our environment with practically no changes. inally, by taking into account what was said above, we complete the proof of Theorem3.4. 4. A PPLICATIONS
Our first goal in the present section is to establish large deviation style formulas forbinary barrier options in the small-noise regime. For Gaussian models, such result wasobtained in [24]. Recall that the scaled asset price process S ( ε ) and the scaled log-priceprocess X ( ε ) are defined by (3.3) and (3.4), respectively. It will be assumed in the presentsection that the drift coefficient b in the model given by (3.1) satisfies b ( s , u ) = r , for all ( s , u ) ∈ [ T ] × R , where r ≥ K >
0, and let T > Definition 4.1.
Suppose the following inequality holds: s < K.(i) The up-and-in binary barrier option pays a fixed amount G of cash if the asset price processtouches the barrier at some time during the life of the option. The price function of such an optionin the small-noise regime is defined byV ( ε ) = Ge − rT P ( max t ∈ [ T ] S ( ε ) t ≥ K ) , ε ∈ ( T ] . (ii) The up-and-out binary barrier option pays a fixed amount G of cash if the asset price processnever touches the barrier during the life of the option. The small-noise price function in this caseis given by V ( ε ) = Ge − rT P ( max t ∈ [ T ] S ( ε ) t < K ) , ε ∈ ( T ] .Now, let K < s . In this case, the down-and-in and down-and-out binary options aredefined similarly to the definitions of the up-and-in and up-and-out options given above.The price functions of the down-and-in and down-and-out options are defined by V ( ε ) = Ge − rT P ( min t ∈ [ T ] S ( ε ) t ≤ K ) , ε ∈ ( T ] ,and V ( ε ) = Ge − rT P ( min t ∈ [ T ] S ( ε ) t > K ) , ε ∈ ( T ] ,respectively.Let s < K , and consider the following subsets of C : A ( ) T = { f ∈ C : f ( s ) + x ≥ log K for some s ∈ ( T ] } = { f ∈ C : f ( s ) + x = log K for some s ∈ ( T ] } and A ( ) T = { f ∈ C : f ( s ) + x < log K for all s ∈ ( T ] } . imilarly, for s > K , we set A ( ) T = { f ∈ C : f ( s ) + x ≤ log K for some s ∈ ( T ] } = { f ∈ C : f ( s ) + x = log K for some s ∈ ( T ] } and A ( ) T = { f ∈ C : f ( s ) + x > log K for all s ∈ ( T ] } .It is not hard to see that the sets A and A are closed in the space C , while the sets A and A are open.The next assertion provides large deviation style formulas for binary barrier options. Theorem 4.2.
Under the conditions in Corollary 3.5 and the restrictions in the definitions ofbinary digital options (s < K, or K < s ), lim ε → ε log V k ( ε ) = − inf f ∈A ( k ) T e Q T ( f ) , 1 ≤ k ≤ where e Q T is the rate function given by (3.6). Theorem 4.2 can be established by imitating the proof of a similar result for Gaussianmodels (see the proof of Theorem 6.2 in [24]). The latter proof does not use a specialform of the mapping f b f , only the continuity of this mapping on the space C [ T ] isimportant. Note that in the proof of Theorem 4.2, we need to use the continuity of therate function e Q T from the space H [ T ] into the space R . For the Gaussian models, theprevious statement was established in [24], Lemma 6.3. For the models with reflection,the proof is similar.It is supposed in Theorem 4.2 that the conditions in Corollary 3.5 hold. These condi-tions include the assumption that the volatility function σ is strictly positive on [ T ] × R .If follows that Theorem 4.2 holds true for various models with reflection, in which thevolatility function is of exponential type. However, Theorem 4.2 can not be applied to thethird face of the S&S model, since the volatility function in this model, that is, the function σ ( x ) = x , x ≥
0, is such that σ ( ) =
0. For the S&S model with reflection, a correspond-ing large deviation principle is that in Theorem 3.4. However, we do not know, whetherthe rate function e Q T is continuous from the space H [ T ] into the space R , under therestrictions in Theorem 3.4.We will next turn our attention to the following problems. Suppose the drift functionin a stochastic volatility model with reflection (see (3.1)) is given by b ( t , u ) = r , for all ( t , u ) ∈ [ T ] × R + , where r ≥ σ satisfies the sublinear growth condition. Is the discounted asset price process t e − rt S t a martingale with respect to the filtration {F t } ? Can one get a large deviationstyle formula for the call pricing function in such a model? We will next show that for theS&S model with reflection, the answers to the previous questions are affirmative.For the sake of convenience, let us recall that the asset price process in the S&S modelwith reflection is as follows: S t = s exp (cid:26) rt − ξ Z t Y s ds + ξ Z t Y s ( ¯ ρ dW s + ρ dB s ) (cid:27) , 0 ≤ t ≤ T , (4.1) hile the scaled version of the asset price process is given by S ( ε ) t = s exp (cid:26) rt − ξ ε Z t ( Y ( ε ) s ) ds + ξ √ ε Z t Y ( ε ) s ( ¯ ρ dW s + ρ dB s ) (cid:27) , 0 ≤ t ≤ T . (4.2)In (4.1) and (4.2), s is the initial condition for the asset price. Moreover, Y ( ε ) t = ( Γ U ( ε ) )( t ) , t ∈ [ T ] , Y = y ≥
0, (4.3)where Γ is the Skorokhod map, while for every ε ∈ (
0, 1 ] , the process t U ( ε ) t is thesolution to the following stochastic integral equation: U ( ε ) t = y + q Z t ( m − ( Γ U ( ε ) )( s )) ds + √ εξ B t , t ∈ [ T ] (4.4)(see (2.8)). The process Y appearing in (4.1) is given by Y t = ( Γ U ( ) )( t ) , t ∈ [ T ] .For the S&S model with reflection, the call pricing function in the small-noise regime isdefined by C ( ε ) ( T , K ) = e − rT E h ( S ( ε ) T − K ) + i , ε ∈ (
0, 1 ] ,where T > K > u ∈ R , u + = max ( u , 0 ) . We assume that T and K are fixed, and study the asymptotic behavior ofthe call pricing function when ε → Theorem 4.3.
The following statements hold true for the S&S model with reflection:(i) The discounted asset price process t e − rt S t is a martingale with respect to the filtration {F t } .(ii) Suppose K > and y > . Then lim ε → ε log C ( ε ) ( T , K ) = − inf x : x ≥ log K − x e I T ( x ) , (4.5) where e I T is the rate function in Corollary 3.13.(iii) Suppose K > and y = . Suppose also that the call option is out-of-the-money, that is,K > s e rT . Then, the equality in (4.5) holds with the rate function e I T given in Corollary 3.14. Remark 4.4.
It follows from part (i) of Theorem 4.3 that P is a risk-neutral measure for the S&Smodel with reflection. Parts (ii) and (iii) provide large deviation style formulas for the call pricingfunction in the small-noise regime.Proof of Theorem 4.3. Set B ∗ t = max ≤ s ≤ t | B s | , t ∈ [ T ] . We will need the following lemma. Lemma 4.5.
For every ε ∈ (
0, 1 ] and t ∈ [ T ] , the following estimate holds P -a.s. on Ω : max ≤ s ≤ t Y ( ε ) t ≤ e qt y + m ( e qt − ) + √ εξ e qt B ∗ t . (4.6) Proof.
For all ε ∈ (
0, 1 ] and t ∈ [ T ] , denote Z ( ε ) t = max ≤ s ≤ t | U ( ε ) s | . Then, using (2.7) and(4.3), we get Y ( ε ) t ≤ Z ( ε ) t , ε ∈ (
0, 1 ] , t ∈ [ T ] . (4.7)It follows from (4.4) that | U ( ε ) t | ≤ y + qmt + q Z t Z ( ε ) s ds + √ εξ | B t | , nd therefore Z ( ε ) t ≤ y + qmt + √ εξ B ∗ t + q Z t Z ( ε ) s ds , (4.8)for all t ∈ [ T ] .Our next goal is to apply Gronwall’s inequality to (4.8). We will use the followingversion of Gronwall’s lemma (see [7], p.37). Let ϕ , ψ , and χ be real-valued continuousfunctions on the interval [ a , b ] . Suppose χ ( t ) >
0, and ϕ ( t ) ≤ ψ ( t ) + Z ta χ ( s ) ϕ ( s ) ds ,for all t ∈ [ a , b ] . Then ϕ ( t ) ≤ ψ ( t ) + Z ta χ ( s ) ψ ( s ) exp (cid:26) Z ts χ ( u ) du (cid:27) ds ,for all t ∈ [ a , b ] .Let [ a , b ] = [ T ] , ϕ ( t ) = Z ( ε ) t , χ ( t ) = q , and ψ ( t ) = y + qmt + √ εξ B ∗ t . Next, applyingGronwall’s lemma to (4.8), we obtain the following estimate: Z ( ε ) t ≤ y + qmt + √ εξ B ∗ t + q Z t ( y + qms + √ εξ B ∗ s ) exp { q ( t − s ) } ds = y + qmt + √ εξ B ∗ t + ( e qt − ) y + q m Z t s exp { q ( t − s ) } ds + ξ √ ε ( e qt − ) B ∗ t .Using the integration by parts formula in the integral on the previous line and simplifyingthe resulting expression, we see that Z ( ε ) t ≤ e qt y + m e qt − + ξ √ ε e qt B ∗ t .Finally, by taking into account (4.7), we get Y ( ε ) t ≤ e qt y + m ( e qt − ) + ξ √ ε e qt B ∗ t .Now, it is clear that (4.6) follows from the previous estimate.The proof of Lemma 4.5 is thus completed. Corollary 4.6.
There exists α > such that E " exp ( α sup ε ∈ ( ] max ≤ t ≤ T ( Y ( ε ) t ) ) < ∞ . (4.9) Proof.
Using the estimate in (4.6), we see that in order to prove (4.9), it suffices to showthat there exists β > E h exp n β ( B ∗ T ) oi < ∞ .The following estimate is known (see, e.g., [28], p. 31): P ( B ∗ T > y ) ≤ √ π T Z ∞ y exp (cid:26) − z T (cid:27) dz , (4.10) or all y >
0. The inequality in (4.10) can be established as follows. It is not hard to seethat { B ∗ T > y } ⊂ { max ≤ t ≤ T B t > y } ∪ { min ≤ t ≤ T B t < − y } = { max ≤ t ≤ T B t > y } ∪ { max ≤ t ≤ T ( − B t ) > y } .Since the process − B is also a Brownian motion, the previous inclusion and the reflectionprinciple imply (4.10). It follows from (4.10) that P ( B ∗ T > y ) = O (cid:18) exp (cid:26) − y T (cid:27)(cid:19) , (4.11)as y → ∞ .Choose β < T . Next, using (4.11) and the integration by parts formula, we obtain E h exp n β ( B ∗ T ) oi = − Z ∞ exp { β y } d P ( B ∗ T > y )= + β Z ∞ y P ( B ∗ T > y ) exp { β y } dy < ∞ .This completes the proof of Corollary 4.6.We will next return to the proof of Theorem 4.3. According to (4.1) the discounted assetprice process has the following form: e − rt S t = s exp (cid:26) − ξ Z t Y s ds + ξ Z t Y s ( ¯ ρ dW s + ρ dB s ) (cid:27) , 0 ≤ t ≤ T . (4.12)It follows from Corollary 4.6 that for some α > E (cid:20) exp (cid:26) α max ≤ t ≤ T ( Y t ) (cid:27)(cid:21) < ∞ .The previous inequality implies that the stochastic exponential on the right-hand side of(4.12) is a martingale (see, e.g., [21], Corollary 2.11).This completes the proof of part (i) of Theorem 4.3.We will next turn our attention to parts (ii) and (iii) of Theorem 4.3. The large deviationprinciples in Corollaries 3.13 and 3.14 will be used in the proofs. We will only sketch theseproofs since there exist well-known methods allowing to derive asymptotic formulas forcall pricing functions from large deviation principles. For such derivations, we refer thereader to [36], the proof on p. 36; [15], Corollary 4.13; [23], Section 7, pp. 1131-1133; or[24], part (i) of Theorem 5.2.The out-of-the-money condition K > s e rT appears in part (iii) of Theorem 4.3 becausewe do not know whether the rate function e I T defined in Corollary 3.14 is continuous at x = rT for y = R , and no extra restrictions are needed here. It follows fromthe previous remark that if y >
0, then the set [ log K − x , ∞ ) is a set of continuity for therate function e I T for all K >
0, while if y =
0, then the same set is a set of continuity for e I T for all K > s e rT . Note that the condition K > s e rT implies the inequality rT < log K − x .The first step in the proof of parts (ii) and (iii) is to establish a large deviation styleformula for the binary call option. The pricing function for such an option is defined by c ( ε ) ( T , K ) = e − rT P ( S ( ε ) T ≥ K ) = e − rT P (cid:16) X ( ε ) T − x ≥ ( log K ) − x (cid:17) , here K > T > K > y > ε → ε log c ( ε ) ( T , K ) = − inf x : x ≥ log K − x e I T ( x ) . (4.13)In addition, if y =
0, then the formula in (4.13) is valid when the option is out-of-themoney. The above-mentioned formulas follow from the large deviation principles inCorollaries 3.13 and 3.14. The proof also uses the continuity properties of the function e I T (see the discussion above).It remains to derive the formulas in parts (ii) and (iii) of Theorem 4.3 from the formulain (4.13). The lower large deviation estimate for the call pricing function can be obtainedfrom (4.13) and the continuity properties of the rate function. In addition, to prove theupper estimate we use H ¨older’s inequality, (4.13), the continuity properties of the ratefunction, part (i) of Theorem 4.3, and Corollary 4.6. More details can be found in [36],[15], or ([24]).This completes the proof of Theorem 4.3.R EFERENCES [1]
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