Asymptotic analysis of stock price densities and implied volatilities in mixed stochastic models

In this paper, we obtain sharp asymptotic formulas with error estimates for the Mellin convolution of functions, and use these formulas to characterize the asymptotic behavior of marginal distribution densities of stock price processes in mixed stochastic models. Special examples of mixed models are jump-diffusion models and stochastic volatility models with jumps. We apply our general results to the Heston model with double exponential jumps, and make a detailed analysis of the asymptotic behavior of the stock price density, the call option pricing function, and the implied volatility in this model. We also obtain similar results for the Heston model with jumps distributed according to the NIG law.


Introduction
The random behavior of the stock price in a mixed model is described by a stochastic process X = X (1) X (2) , where X (1) and X (2) are strictly positive independent integrable processes on a complete filtered probability space (Ω, F , {F t }, P). Important examples of mixed models are jumpdiffusion models and stochastic volatility models with Lévy type jumps. More information on models with jumps can be found in [11] and [27].
In this paper, we obtain asymptotic formulas with error estimates for the distribution density of the stock price and the implied volatility in special mixed stochastic stock price models. Let us suppose that the distributions µ and D (2) t , respectively. Then the distribution µ t of the stock price X t also admits density D t , which can be represented by the Mellin convolution (the definition of the Mellin convolution is given below). The fact that the distribution density of the product of two independent random variables is the Mellin convolution of their densities was mentioned in [13].
In [5] (see also [9]), D. Arandelović obtained an asymptotic formula for the Mellin convolution of functions defined on the half-line (0, ∞). However, Arandelović's formula does not contain an ----------------------------------------E-mail addresses: gulisash@ohio.edu (A. Gulisashvili), josep.vives@ub.edu (J. Vives). error estimate. In Subsection 2.3 of the present paper, asymptotic formulas with error estimates are established for the Mellin convolution (see the formulas in Theorems 8 and 9). These formulas extend Arandelović's result. They are used in the paper to characterize the asymptotic behavior of the stock price density in special mixed stochastic stock price models. Note that asymptotic expansions of the Mellin convolution under different restictions than those imposed in Arandelović's work and in the present paper were obtained by R. A. Handelsman and J. S. Lew (see the presentation of their results in Section 3.4 of [28]).
One of the examples considered below is the Heston model with asymmetric double exponential jumps. Theorems 13 and 14 obtained in this paper deal with the case where the jump part of the mixed model dominates, while Theorems 15 and 16 concern the asymptotics of the stock price density in the case where the Heston part dominates. Weaker estimates were obtained earlier in [20]. In Section 7, we briefly discuss some other models.
In [23] (see also [24]), S. Kou introduced and studied a jump-diffusion model that is in fact a mixture of the Black-Scholes model with the double exponential jump model. An asymptotic formula (without an error estimate) for the distribution function of the stock price in the Kou model was obtained in [2], Example 7.6. In [29] and [16], an asymptotic formula with an error estimate was found for the call pricing function in the Kou model. In the present paper, we obtain a similar formula for a class of models, including the Heston model with double exponential jumps and the Kou model (see (92)). The error estimate in formula (92) is better than that in [16].
In our analysis of the stock price density in the Heston model perturbed by double exponential jumps, we use some of the results obtained in [23]. It is interesting to mention that the asymptotic behavior of the stock price density in the Heston model without jumps and that of the absolutely continuous part of the distribution associated with the double exponential jump part is similar (compare (31) with (78) and (32) with (79)). It follows that in the study of the asymptotic behavior of the stock price density in the mixture of the Heston model with the double exponential jump model, we have to take into account which part of the mixed model dominates the other. This dichotomy does not appear in the Kou model since the double exponential jump part always dominates the Black-Scholes part. Note that the similarity between the asymptotic behavior of the call pricing functions in the Heston model without jumps and in the Kou model was observed in [16] too.
Asymptotic formulas for the stock price density can be used to study the asymptotic behavior of option pricing functions and the implied volatility. In Section 6 of the present paper, we obtain asymptotic formulas with five explicit terms and error estimates for the implied volatility at extreme strikes in the Heston model with double exponential jumps. Similar formulas for the Heston model with NIG type jumps are established in Section 7. We use some of the methods developed in [15] and [18] to estimate the implied volatility. A little weaker asymptotic formulas for the implied volatility with four explicit terms were established for the Heston model without jumps in [15] and for the Kou model in [29] and [16]. These formulas can be extended to include five terms and an error estimate. We would also like to bring the reader's attention to the paper [3] concerning the asymptotic behavior of the implied volatility in exponential Lévy models.
We will next briefly overview the contents of the present paper. In Subsection 2.1, we define the Mellin convolution and introduce several related notions. Regularly varying functions play an important role in the paper. In Subsection 2.2, various definitions and facts from the theory of regularly varying functions are gathered, while Subsection 2.3 is devoted to Arandelović's theorem and its generalizations. In Section 3, known asymptotic formulas for marginal distribution densities of the stock price process in the Heston model are formulated. Section 4 is devoted to the Heston model with double exponential jumps. Here we obtain new results concerning the jump part of the perturbed Heston model. We prove that the absolutely continuous part of the marginal distribution of the exponential Lévy process associated with the perturbed Heston model is regularly varying, and provide asymptotic formulas characterizing its asymptotic behavior near infinity and near zero. Section 5 of the present paper deals with density approximations in the Heston model with double exponential jumps. We obtain sharp asymptotic formulas with error estimates for the distribution density of the stock price in the perturbed Heston model (see . The generalizations of Arandelović's theorem obtained in Theorems 8 and 9 are used in the proofs. In Section 6, sharp asymptotic formulas with error estimates are provided for the implied volatility in the Heston model with double exponential jumps (see Theorems 18 and 19) and for more general models. Finally, Section 7 discusses the implied volatility in the Heston model with jumps distributed according to the symmetric centered NIG law.
Let µ be a distribution on [0, ∞), and let η be a real number. The moment of order η of the distribution µ is defined as follows: It is not hard to see that if U is a distribution density, then M U (η) = m −η−1 (U ) for all real numbers η in the domain of M U .

Regularly varying functions
In the present subsection, several notions and results from the theory of regularly varying functions are discussed. These functions play an important role in the paper. A rich source of information about regularly varying functions is the book [9] by N. H. Bingham, C. M. Goldie, and J. L. Teugels.
as x → ∞. The class of all regularly varying functions with index ρ is denoted by R ρ . Functions from the class R 0 are called slowly varying functions.
The next result is known as the uniform convergence theorem for regularly varying functions.
3. The condition f ∈ R ρ with ρ < 0 implies that formula (7) holds uniformly in λ on each Another fundamental result in the theory of slowly varying functions is the representation theorem (see [9], Theorem 1.3.1).
Theorem 2. For a nonnegative measurable function l, the condition l ∈ R 0 is equivalent to the following: for some a > 0, where the functions c and ε are such that c(x) → c ∈ (0, ∞) as x → ∞ and ε(u) → 0 as u → ∞. Let l be a normalized slowly varying function. Then Theorem 2 shows that for some C ∈ R and a > 0, where the function ε is such that ε(u) → 0 as u → ∞. For a normalized slowly varying function l, we have (see [9], p. 15). If the function l is differentiable, then the equality in (10) holds everywhere on (a, ∞).
The following class was introduced by A. Zygmund.
Definition 5. A nonnegative measurable function l defined on (0, ∞) belongs to the Zygmund class Z if, for every α > 0, the function φ α (x) = x α l(x) is ultimately increasing and the function ψ α (x) = x −α l(x) is ultimately decreasing.
The next known statement gives a description of the Zygmund class.
Theorem 3 (see Theorem 1.5.5. in [9]). The class Z coincides with the class of normalized slowly varying functions.
In the present paper, we discuss various asymptotic formulas with error estimates. Note that throughout the paper the statement φ 1 (x) = O(φ 2 (x)) as x → ∞, where φ 1 is a real function and φ 2 is a positive function, means that there exist c > 0 and x 0 > 0 such that The following known definition introduces slowly varying functions with remainder (see [17], see also [9]). Definition 6. Let l and g be nonnegative measurable functions on (0, ∞) with g(x) → 0 as x → ∞. The function l is called slowly varying with remainder g if for all λ > 1, as x → ∞.
We will denote the class of slowly varying functions with remainder g by R g 0 . It is not hard to see that R g 0 ⊂ R 0 . The uniform convergence theorem for slowly varying functions with remainder is as follows.
The next statement, which is stronger than Theorem 4, provides a growth estimate in the variable λ in the uniform convergence result for slowly varying functions with remainder.
Theorem 5. Fix δ = 0, and let f and g be positive functions on [0, ∞) such that g ∈ R 0 and f ∈ R g 0 . Suppose also that the functions f , g, 1 f , and 1 g are locally bounded on [0, ∞). Then there exists A > 0 such that for all λ > 0 and x ≥ 0.
The estimate in Theorem 5 is contained in part (b) of Theorem 3.8.6 in [9]. Note that the condition f ∈ R g 0 implies the following inclusion: f ∈ OΠ l where l = f g, and hence the conditions in Theorem 3.8.6 (b) hold (see [9] for the definition of the class OΠ l and for more details).
The structure of slowly varying functions with remainder is known. The next result is the representation theorem for slowly varying functions with remainder (see [17], see also [9]).
as x → ∞, where C ∈ R, and the O functions are locally integrable.
where ε is the function appearing in formula (9). Proof. Since l is a normalized slowly varying function, formula (9) holds. This implies that formula (12) holds with g = |ε|. Next, using Theorem 6, we establish Corollary 1.
Let f be a measurable function on (0, ∞), and assume the following two conditions hold: Then Note that there is no error estimate in formula (13). The next assertion provides such an estimate under certain additional restrictions.
Theorem 8. Suppose the Mellin transform M U of a measurable function U converges at least in the strip σ ≤ ℜ(z) ≤ τ where −∞ < σ < τ < ∞. Let f be a measurable function on (0, ∞), and assume the following conditions hold: The functions g and h in the previous formula satisfy g(x) → 0 and h(x) → 0 as x → ∞.

The function
Then Proof. Let us first assume that and put x 1 = max{x 0 , y 0 }, where x 0 is such as in condition 2. Then we have Since xv < x 1 in the first integral in (16), condition 3 implies the following: Fix ε > 0 such that ε < ρ − σ, and take into account that lg ∈ R 0 . Then the previous estimates imply that , and hence It follows that the integral I 1 can be incorporated into the error term in formula (14). Our next goal is to estimate the integral I 2 . It is clear that Fix ε > 0 such that ε < ρ − σ. Then, for large values of x, we have Since g ∈ R 0 and M U (σ) < ∞, we obtain It remains to estimate J 2 . Denote by l and g the functions l and g, extrapolated by positive constants from (x 1 , ∞) to [0, ∞). Then l ∈ R g 0 . Moreover, condition 2 and the definition of the functions l and g imply that the functions l, g, l −1 , and g −1 are locally bounded on [0, ∞). Now, using Theorem 5, we see that for every δ > 0 there exists a constant A > 0 such that for all v > 0 and x ≥ 0. Recalling the definition of the functions l and g, we see that for It follows from the estimate in (20) that for every δ > 0 there exists x 1 > 0 depending on δ and such that for all x > x 1 . It is not hard to see that for small enough values of δ, the last integral in (21) is finite. Here we use the fact that M U (s) < ∞ for all σ ≤ s ≤ τ and the inequalities σ < ρ < τ . Now, (21) implies that Next, taking into account formulas (18), (19), and (22), we obtain as x → ∞. Finally, it is easy to see that formulas (16), (17), and (23) imply formula (14). This establishes Theorem 8 in a special case where (15) holds. We will next prove Theorem 8 in the general case. Suppose the conditions in the formulation of Theorem 8 hold. Then there exists where η is a measurable function such that |η(x)| ≤ Ah(x), for all x > x 1 and for some constant Applying the special case of Theorem 8 established above to the function f 2 , we obtain as x → ∞. In addition, reasoning as in the proof of (17), we obtain as x → ∞. We will next estimate the function U M ⋆ f 3 , using the same ideas as in the estimate for the function U M ⋆ f 2 . However, we will use the function l = lh instead of the function l. Recall that l ∈ R g 0 . Moreover, h ∈ R |ε| 0 , where ε is the function appearing in formula (9) for the function h (see Corollary 1).
We have l(xv) Now, reasoning as in the proof of formula (23), we see that as x → ∞. Finally, taking into account formulas (24) -(27), we see that formula (14) holds. This completes the proof of Theorem 8.
A similar theorem characterizes the asymptotic behavior of the Mellin convolution near zero.
Theorem 9. Suppose the Mellin transform M U of a measurable function U converges at least in the strip σ ≤ ℜ(z) ≤ τ where −∞ < σ < τ < ∞. Let f be a measurable function on (0, ∞), and assume the following conditions hold: , and h ∈ Z. The functions g and h in the previous formula satisfy g(y) → 0 and h(y) → 0 as y → ∞.

The Heston model
In this section, we gather several known results for the Heston model, which is a popular stochastic volatility model. It will be assumed in the sequel that the interest rate r is equal to zero. The stock price process X and the variance process Y in the Heston model satisfy the following system of stochastic differential equations: where µ ∈ R, a ≥ 0, b ≥ 0, c > 0. In (28), W and Z are correlated standard Brownian motions such that d W, Z t = ρdt with ρ ∈ (−1, 1). In the Heston model, the distribution µ t of the stock price X t admits density D t . The initial conditions for the processes X and Y will be denoted by x 0 and y 0 , respectively. The Heston model was introduced and studied in [21].
We have and the following formulas hold for the density D as x → ∞, and as x → 0. The constants appearing in formulas (29) and (30) will be described below. Formulas (29) and (30) were obtained in [19] in the case where ρ = 0, and in [14] for −1 < ρ < 0. A more detailed discussion of those and similar results can be found in [18].
For general x 0 and µ, we obtain as x → ∞, and as x → 0. In (31) and (32), the constants B 1 and B 1 are defined as follows: The proof of (32) uses (29) and the following simple formulas: The proof of (32) is similar. It is based on formula (30). We will next provide explicit formulas for the constants appearing in formulas (29) and (30). Given s ≥ 1, define the explosion time for the moment of order s by and for any t > 0, let the s + be the upper critical moment defined by For the Heston model, the explosion time T * is explicitly known (see [4,22]). The critical moment, for fixed t, can then be determined from T * (s + (t)) = t. The previous equality shows that s + (t) ≥ 1 is the generalized inverse of the function T * (·). The lower critical moment is defined as follows: For fixed t > 0, the quantities In formula (29), the constants A 1 , A 2 , and A 3 are given by In addition, the constants A 1 , A 2 , and A 3 in formula (30) are as follows: Note that the constants described above depend on t.

The Heston model with double exponential jumps
Let N be a standard Poisson process with intensity λ > 0, and consider a compound Poisson process defined by where V i are positive identically distributed random variables such that the distribution density g of the random variable U i = log V i is double exponential. This means that where η 1 > 1, η 2 > 0, and p and q are positive numbers such that p + q = 1. The condition η 1 > 1 is necessary and sufficient for the random variable J t to have finite expectation. S. Kou introduced and studied a perturbation of the Black-Scholes model based on the jump process described above (see [23], see also [24]). In [20], we considered a similar perturbation of the Heston model. The stock price process and the variance process Y in the perturbed Heston model satisfy the following system of stochastic differential equations: It is assumed in (33) that the compound Poisson process J is independent of standard Brownian motions W and Z. The initial conditions for the processes X and Y will be denoted by x 0 and y 0 , respectively. It is not hard to see that The validity of the equality in (34) follows from the Doléans-Dade formula (see, for example, [26]). The Heston model with double exponential jumps is a mixed stochastic model. Indeed, using formula (34), we can split the process X into the product of the following processes: and Note that X (1) = X, where X is the stock price process in the Heston model described by (28).
Note also that for every t ≥ 0, we have E X (2) t < ∞ (see Remark 2 below).
Let us put T t = Nt i=1 U i . The distribution µ t of the random variable T t is given by where y ∈ (−∞, ∞) and δ 0 is the delta-measure at y = 0. The functions G 1 and G 2 in the previous formula are defined by with and The numbers π n in the previous formulas depend on t. They are defined by π 0 = e −λt and π n = e −λt (λt) n (n!) −1 for all n ≥ 1. In addition, the numbers P n,k and Q n,k are given by for all 1 ≤ k ≤ n − 1, and We also have P n,n = p n and Q n,n = q n . Formula (37) can be derived using Proposition B.1 in [23] (see the derivation in [20], or in Section 10.8 of [18]).
It follows from (37) that the distribution µ (2) t of the random variable X (2) t satisfies dµ (2) t (x) = e −λt dδ 1 (x) + H(t, x)dx, x > 0. (42) In (42), δ 1 is the delta-measure at x = 1, the function H is defined by where and The next assertion provides useful approximations to the coefficients a k and b k appearing in (39) and (40).
Theorem 10. There exist positive constants c 1 and c 2 , independent of k, and such that and where a k = exp Proof. For i ≥ 1 and m ≥ i + 1, put It follows from (39) and (41) that for all k ≥ 1, where for all i ≥ 1. We have Therefore, (48), (49), and (50) imply the following: where β 1 is a positive constant. For i ≥ 2 and j ≥ 1, we have where β 2 is a positive constant. In the proof of the previous estimates, we used Striling's formula. Now, it is not hard to see that the last double series in (51) converges, and it follows from (51) that the estimate in (46) is valid. The proof of the estimate in (47) is similar. This completes the proof of Theorem 10. We will next further simplify formulas (46) and (47). Set and consider the following sequences: and Corollary 2. The following formula holds: and where α 1 and α 2 are some positive constants.
Proof. We will need Stirling's formula in the asymptotic form: as n → ∞. It is not hard to see, using (46) and (55) that as k → ∞. This establishes (53). The proof of (54) is similar.

Properties of the functions H 1 and H 2
In the present subsection, we study the asymptotic behavior of the functions H 1 and H 2 defined in (44) and (45). It will be shown first that these functions are of slow variation with remainder. Proof. Let us fix t > 0. Since the function x → H 1 (t, x) is increasing on x > 1, the function φ α (t, x) = x α H 1 (t, x), where α > 0, is also increasing. It remains to prove that the function ψ α (t, x) = x −α H 1 (t, x) is ultimately decreasing. We have for all x > x α , which in its turn is equivalent to the condition as x → ∞. Now it is clear that it suffices to prove (56).
Using the definition of the function G 1 , we obtain where the coefficients a k are defined by (39). It is not hard to see using (46) that for all k ≥ 2, with some c > 0. Hence for every ε > 0 there exists a positive integer k ε such that ka k ≤ εa k−1 for all k > k ε . It follows from (57) that It is clear that for fixed ε the first term on the right-hand side of (59) tends to 0 as x → ∞. Now, it is not hard to see that condition (56) holds. This completes the proof of Lemma 1 for the function x → H 1 (t, x). The proof for the function x → H 2 (t, 1 x ) is similar. Remark 2. It follows from (42), the fact that Z ⊂ R 0 , and Lemma 1 that the process t → X (2) t is an integrable process.
Lemma 2. For every t > 0, the functions x → H 1 (t, x) and x → H 2 (t, x −1 ) belong to the class R g 0 , where the function g is given by Proof. The function x → H 1 (t, x) is an increasing differentiable function from the Zygmund class. Therefore, it suffices to prove that there exists c > 0 such that for all x > x 0 (see Corollary 1 and (10)). It is easy to see that the estimate in (60) is equivalent to the following: Next, using (58) in (62), we obtain The proof of Lemma 2 in the case of the function x → H 1 (t, x) is thus completed. For the function x → H 2 t, 1 x , the proof is similar. Recall that G 1 (t, ·) = ∞ k=0 a k u k (see (38)), where the coefficients a k are given by (39). Define two auxiliary functions G 1 (t, ·) and G 1 (t, ·) as follows: where u ≥ 0 and the sequence d is given by (52). Then (53) implies that The functions G 1 and G 1 are defined as sums of certain power series. Our next goal is to compare these functions with some standard functions. Analyzing the structure of the coefficients d k , we guess that the following family of functions may be useful: where It is clear that d 0 = s. Moreover, using Stirling's formula, we see that as k → ∞. Next, comparing the coefficients d k and d k , we see that if we set then for some δ 1 > 0 and all k ≥ 0. Finally, it follows from (53) and (65) that the coefficients a k and d k satisfy the following condition: for some δ 2 > 0 and all k ≥ 0.
Theorem 11. There exists a positive constant c such that for all u > 0. In (71), the values of the parameters r and s are chosen according to (64).
It follows from Theorem 11 that in order to understand the asymptotic behavior of the function G 1 (t, u) as u → ∞, we have to study how the fractional integrals appearing in (71) behave for large values of u. Using (63) and (67), we obtain and Indeed, the third integral in (74) is O e − 1 2 r √ u , while the second integral can be estimated, using the integration by parts twice. Similarly, as u → ∞. In addition, For small values of y, we have (1 − y)(2 − y) Using Watson's lemma (see [10], p. 103), we obtain as y → ∞. Similarly, as y → ∞. Now we are ready to formulate and prove one of the main results of the present paper.
Theorem 12. Let H 1 (t, ·) and H 2 (t, ·) be the functions defined by (44) and (45), respectively. Then the following asymptotic formulas hold: as x → ∞, and Proof. Formula (78) follows from (71) and formulas (72) -(77). Here we take into account that the values of the parameters r and s are given by (64). The proof of formula (79) uses the same ideas, and we leave it as an exercise for the interested reader.

Asymptotic behavior of stock price densities in the Heston model with double exponential jumps
In this section, we study the asymptotic behavior of the marginal densities in the perturbed Heston model. Our first goal is to characterize the asymptotics of the density of the absolutely continuous part H of the distribution µ (2) t of the random variable X (see (42)). It is clear from Theorem 12 that the following assertion holds.
Corollary 3. For every t > 0, as x → ∞, and We will next consider the case of the perturbed Heston model where the jump part dominates.
Theorem 13. Fix t > 0, and suppose 1 + η 1 < A 3 . Then the following asymptotic formula holds for the stock price density D t in the Heston model with double exponential jumps: is the density of the random variable X (1) t defined by (35).
Theorem 14. Fix t > 0, and suppose A 3 > η 2 − 1. Then the following asymptotic formula holds for the stock price density D t in the Heston model with double exponential jumps:  (6)). To get explicit formulas for the moments of the Heston density, we may use the equality and a known explicit formula for the moment generating function of the log-price in the Heston model (see, e.g., formula (3) in [12]).  (42)). However, we can still use a formula similar to formula (1) to estimate D t . We have Our next goal is to apply Theorem 8 to characterize the asymptotic behavior of the last term in (85). We put U (x) = D and h(x) = 0. Then, σ < ρ < τ . Indeed, the condition 1 + η 1 < A 3 is equivalent to σ < ρ. In addition, since A 3 > −1 (use the integrability of the function D (1) t and (30)), we have ρ < τ . Now, taking into account Remark 1 and Lemma 2, we see that the conditions in the formulation of Theorem 8 hold. It follows that as x → ∞. Finally, it is not hard to see that (29), (80), (85), (86), and the condition 1 + η 1 < A 3 imply formula (82).
This completes the proof of Theorem 13. The proof of Theorem 14 is similar to that of Theorem 13. It is based on Theorem 9, (30), (43), Lemma 2, (81), and (85). We leave filling in the details to the interested reader.
We will next explain how the density D t behaves in the case where the Heston part dominates.
Theorem 15. Fix t > 0, and suppose 1 + η 1 > A 3 . Then the following asymptotic formula holds for the stock price density D t in the Heston model with double exponential jumps: Theorem 16. Fix t > 0, and suppose A 3 < η 2 − 1. Then the following asymptotic formula holds for the stock price density D t in the Heston model with double exponential jumps: as x → 0.
Remark 5. Recall that the symbols m A3−1 (µ (2) t ) and m − A3−1 (µ (2) t ) in formulas (87) and (88) stand for the moments of the marginal distribution µ (2) t (see (6)). To compute the moments appearing in formulas (87) and (88), we can use a formula similar to formula (84) and an explicit formula for the moment generating function of the log-price in the exponential jump model with jump amplitudes distributed according to the asymmetric double exponential law (see, e.g., formula (1) with b = 0 and σ = 0 in [16]). 6 Smile asymptotics in the Heston model with double exponential jumps In order to create a risk-neutral environment, we assume that the following no-arbitrage condition holds for the parameters in the perturbed Heston model: Here we take into account that r = 0. Then the process X defined by (34) is a martingale (see [18], Section 10.8). Note that the proof uses the mean-correcting argument (see, e.g., Lemma 10.40 in [18], or [27], pp. 79-80). It will be assumed in the present section that condition (89) holds. The call and put pricing functions C and P in the Heston model with double exponential jumps are defined by C(T, K) = E ( X T − K) + and P (T, K) = E (K − X T ) + , respectively. In the previous formulas, T is the maturity and K is the strike price. The implied volatility I(T, K), T > 0, K > 0, in the Heston model with double exponential jumps is defined as follows. Given T and K, the implied volatility I(T, K) is equal to the value of the volatility σ = σ(T, K) in the Black-Scholes model such that C(T, K) = C BS (T, K, σ). Here the symbol C BS stands for the call pricing function in the Black-Scholes model. In the sequel, the maturity T will be fixed, and we will consider the functions C, P , and I as functions of only the strike price K.
The asymptotic behavior of the implied volatility I in the Heston model with double exponential jumps will be characterized utilizing the asymptotic formulas for the stock price densities provided in Theorems 13-16. We will start with the case of large strikes. Analyzing the formulas in Theorems 13 and 15, we see that it is important to understand how the implied volatility behaves if the stock price density D T satisfies the condition D T (x) = r 1 x −r3 exp{r 2 log x}(log x) r4 1 + O (log x) − 1 2 (90) as x → ∞, where r 1 > 0, r 2 ≥ 0, r 3 > 2, and r 4 ∈ R.
Remark 8. The asymptotic behavior of the implied volatility at large strikes in Kou's model was studied in [29] and [16]. Since Kou's model is the Black-Scholes model with double exponential jumps, the jump part always dominates. Indeed, the decay of the stock price density in the Black-Scholes model is log-normal, while the density of the exponential Lévy part of Kou's model decays as a regularly varying function. The authors of [16] obtain an asymptotic formula for the implied volatility with four terms an an error estimate of order O (log K) − 3