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Keywords:

  • exponential Lévy models;
  • Blumenthal–Getoor index;
  • short-dated options;
  • implied volatility

Abstract

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. OPTION PRICE ASYMPTOTICS CLOSE TO THE MONEY
  5. 3. ASYMPTOTIC BEHAVIOR OF IMPLIED VOLATILITY
  6. 4. NUMERICAL RESULTS
  7. 5. A QUALITATIVE COMPARISON WITH MARKET SMILES
  8. 6. PROOFS
  9. APPENDIX
  10. REFERENCES

We analyze the behavior of the implied volatility smile for options close to expiry in the exponential Lévy class of asset price models with jumps. We introduce a new renormalization of the strike variable with the property that the implied volatility converges to a nonconstant limiting shape, which is a function of both the diffusion component of the process and the jump activity (Blumenthal–Getoor) index of the jump component. Our limiting implied volatility formula relates the jump activity of the underlying asset price process to the short-end of the implied volatility surface and sheds new light on the difference between finite and infinite variation jumps from the viewpoint of option prices: in the latter, the wings of the limiting smile are determined by the jump activity indices of the positive and negative jumps, whereas in the former, the wings have a constant model-independent slope. This result gives a theoretical justification for the preference of the infinite variation Lévy models over the finite variation ones in the calibration based on short-maturity option prices.

1. INTRODUCTION

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. OPTION PRICE ASYMPTOTICS CLOSE TO THE MONEY
  5. 3. ASYMPTOTIC BEHAVIOR OF IMPLIED VOLATILITY
  6. 4. NUMERICAL RESULTS
  7. 5. A QUALITATIVE COMPARISON WITH MARKET SMILES
  8. 6. PROOFS
  9. APPENDIX
  10. REFERENCES

In financial markets, the price of a vanilla call or put option on a risky asset with strike inline image and maturity t is often quoted in terms of the implied volatility inline image (see (3.3) in Section 'ASYMPTOTIC BEHAVIOR OF IMPLIED VOLATILITY' for the definition and Gatheral 2006 for more information on implied volatility). Similarly, given a risk-neutral pricing model, one can define a function inline image via the prices of the vanilla options under that model. The implied volatility is a central object in option markets and it is, therefore, not surprising that understanding the properties and computing the function inline image for widely used pricing models has been of considerable interest in the mathematical finance literature. Typically, for a given modeling framework, the implied volatility inline image is not available in closed form. Hence, the study of the asymptotic behavior in a variety of asymptotic regimes [e.g., fixed t and inline image (Lee 2004; Friz and Benaim 2009; Gulisashvili 2010); inline image with k constant (Tehranchi 2009) or proportional (Jacquier, Keller-Ressel, and Mijatović 2013) to t; inline image and k constant (Roper and Rutkowski 2009; Tankov 2010; Figueroa-López and Forde 2012) etc.] has attracted a lot of attention in the recent years.

In this paper, we assume that the returns of the risky asset inline image are modeled by a Lévy process X and study the relationship between the jump activity of X and the implied volatility at short maturities in the model S. Most existing approaches analyze either the at-the-money (ATM) case, when the implied volatility is determined exclusively by the diffusion component and converges to zero in the pure-jump models (Tankov 2010; Muhle-Karbe and Nutz 2011; Houdré, Gong, and Figueroa-López 2011), or the fixed-strike out-of-the-money (OTM) case, when the implied volatility for short maturities explodes in the presence of jumps (Roper 2009; Tankov 2010; Figueroa-López and Forde 2012). However, in option markets, (a) although the implied volatility for liquid strikes grows with decreasing t, it remains within a range of reasonable values and appears not to explode, and (b) the liquid strikes become concentrated around the money as the maturity gets shorter. For instance, in foreign exchange (FX) option markets, which are among the most liquid derivatives markets in the world, options with fixed values of the Black–Scholes delta are quoted for each maturity (see Section 'FX Option Quotes' and Andersen and Lipton 2013 for the conventions in FX option markets and the natural delta parameterization of the smile).

The market data in Figure 1.1, therefore, suggests that, in order to understand the behavior of the volatility surface at short maturities, one should look for a moving log-strike inline image, for inline image, such that (i) the corresponding implied volatility has a nontrivial limit inline image and (ii) the log-strike inline image converges to the ATM log-strike value as maturity t tends to zero (i.e., inline image if one assumes that inline image).

image

Figure 1.1. The liquid at-the-money, 10- and 25-delta strikes (left panel) and the corresponding implied volatilities (right panel) for the market-defined maturities t in the set inline image in the EURUSD option market (taken on the January 4, 2013) suggest the following: as maturity t becomes small, the relevant strikes inline image approach the at-the-money strike and the implied volatilities inline image remain bounded. These features of liquid strikes and implied volatilities are persistent in time and across option markets.

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This paper defines a new universal and model-free parameterization of the log-strike given by

  • display math

For fixed θ, the corresponding strike value tends to the ATM strike as inline image but is OTM for each short maturity inline image. We prove that under suitable assumptions the limiting implied volatility inline image takes the following form as a function of θ:

  • display math(1.1)

In this formula σ denotes the volatility of the Gaussian component of the underlying Lévy process X and inline image (resp. inline image) denotes the jump activity (Blumenthal–Getoor) index of the positive (resp. negative) jumps of X. More precisely, if the jump measure of X is denoted by ν, inline image and inline image are given by

  • display math

Unlike in the case of fixed strike, where short maturity smile explodes in the presence of jumps, our parameterization of the strike as a function of time yields a nonconstant formula for the limiting implied volatility, which depends on the balance between the size of the Gaussian volatility parameter and the activity of small jumps. It allows us to make the following observations about the relationship between the short-dated option prices and the characteristics of the underlying model:

  1. the formula for inline image depends on the jump measure of the log-spot process X only if the jumps are of infinite variation; put differently, if the jumps of X are of finite variation, then the absolute value of the slope of the limiting smile for large inline image is equal to one and, in particular, inline image does not depend on the structure of jumps;
  2. the limiting smile inline image is “V-shaped” in the absence of the diffusion component (i.e., when inline image) and is “U-shaped” otherwise.

Remark (i) provides a theoretical basis for distinguishing between the models with jumps of finite and infinite variation in terms of the observed prices of vanilla options with short maturity. It is well known that, for any short maturity t, the market-implied smile inline image exhibits pronounced skewness and/or curvature, due, in particular, to the risk of large moves over short time horizons perceived by the investors. Hence, jumps are typically introduced into the risk-neutral pricing models with the aim to capture this risk and modulate the ATM skew of the implied volatility inline image at small t (see, e.g., equation (5.10) in Gatheral 2006). However, since this task can be accomplished by jumps of either finite or infinite variation, this requirement tells us little about the options implied jump activity of the underlying risk-neutral model. On the other hand, the formula for inline image implies that, if we need to control the tails (in the parameter θ) of the implied volatility for short maturities, we must use jumps of infinite variation. This finding complements the analysis in Carr and Wu (2003) of the pathwise structure of the risk-neutral process implied by the option prices on the S&P 500 index.

The formula for the limiting implied volatility given in (1.1) should be compared to the recent results (for Lévy models) on the limiting behavior of the implied volatility at a fixed OTM strike (i.e., inline image). In Roper (2009), Figueroa-López and Forde (2012), and Tankov (2010), it is shown that, in the presence of jumps, the implied volatility explodes at the rate inline image as maturity t tends to zero. Furthermore, this rate is independent of the jump structure and is insensitive to the presence of the diffusion component. Hence, in the fixed-strike OTM asymptotic regime little can be deduced about the relation between the jump structure and the diffusion component of asset returns as the maturity t decreases to zero, since this makes the implied volatility tend to infinity in a model-independent way. In the ATM case (i.e., inline image), the limit of the implied volatility as the maturity decreases is equal to the diffusion component of the Lévy triplet (Tankov 2010, proposition 5), making it zero for a pure-jump Lévy process. In the light of these results, the formula for inline image in (1.1) provides new insight into the relation between the jump structure and the diffusion component implied by the short-maturity smile. It should be noted that the extension of the formula inline image to a more general class of processes with jumps (e.g., jump diffusions, stochastic volatility processes with jumps, or even general semimartingales) is likely to hold under the appropriate assumptions. In particular, it is reasonable to expect that the model-independent parametrization of the strike inline image given above will lead to analogous limit results for the implied volatility as the maturity t tends to zero, as the “tangential” Lévy process at inline image to a general model (Muhle-Karbe and Nutz 2011) will control the limiting behavior of the smile.

In recent years, there has been a lot of interest in the literature on the statistics of stochastic process in the question of the estimation of the Blumenthal–Getoor index of models with jumps based on high-frequency data. For example, it is shown in Aït-Sahalia and Jacod (2009) that the jump activity (measured by the Blumenthal–Getoor index) estimated on high-frequency stock returns for two large US corporates is well beyond one, implying that the underlying model for stock returns should have jumps of infinite variation. Likewise, the formula in (1.1) suggests that jumps of infinite variation are needed in order to capture the correct tails (in θ) of the quoted short-dated option prices (cf. Section 'Blumenthal–Getoor index and the short-dated option prices'). In contrast to the high-frequency setting, a spectral estimation algorithm for the Blumenthal–Getoor index of a Lévy process based on low-frequency historical and options data was proposed in Belomestny (2010). Unlike formula (1.1), which would require option prices of arbitrarily short maturities for the estimation of the Blumenthal–Getoor index, the algorithm in Belomestny (2010) relates the distributional properties of the Lévy process to the index, thus enabling its estimation using options with a fixed maturity.

1.1. Structure of the Results

The formula in (1.1) follows from Corollary 3.3, which gives the expansion of the implied volatility inline image, where inline image, up to order inline image. This expansion is a consequence of (A) Theorem 3.1, which itself gives an expansion of the implied volatility for a general log-strike inline image that tends to zero as inline image, and (B) Theorem 2.1 and Proposition 2.3, which describe the asymptotic behavior of the option prices under Lévy processes with infinite and finite jump variations, respectively. Theorem 3.1 relates the asymptotic behavior of the vanilla option prices under a general semimartingale model to the asymptotic behavior of the implied volatility as the log-strike inline image tends to zero [it should be noted that the asymptotic regime inline image in Theorem 3.1 is not covered by the analysis in Gao and Lee (2011), see Remark (iv) after Theorem 3.1 for more details]. The asymptotic formula in Corollary 3.3 then follows by combining Theorem 3.1 with the asymptotic behavior of the vanilla option prices established in Theorem 2.1 (for the case of jumps of infinite variation) and Proposition 2.3 (for jumps of finite variation).

In a certain sense, Theorem 2.1 and Proposition 2.3 represent the main contributions of this paper. The asymptotic formulae for the call and put options, struck at inline image and inline image, respectively, have the same structure in both results: the leading order term is a sum of two contributions, one coming from the diffusion component of the process and the other from the jump measure. Which of the two summands dominates in the limit depends on the level of the parameter θ. This structure of the asymptotic formulae is also reflected in the expression for inline image, as it is clear from (1.1) that inline image if θ is between inline image and inline image, and inline image only depends on the jump measure otherwise. However, the proofs of Theorem 2.1 and Proposition 2.3 differ greatly: the finite variation case follows from the Itô–Tanaka formula, which can, in this case, be applied directly to the hockey-stick payoff function, while the case of jumps with infinite variation requires a detailed analysis of the asymptotic behavior of the option prices.

1.2. Structure of the Paper

Section 'OPTION PRICE ASYMPTOTICS CLOSE TO THE MONEY' defines the setting and states Theorem 2.1 and Proposition 2.3. In Section 'ASYMPTOTIC BEHAVIOR OF IMPLIED VOLATILITY', we state the asymptotic formulae for the implied volatility and derive the limit in (1.1). Section 'NUMERICAL RESULTS' presents numerical results that demonstrate the convergence of option prices and implied volatilities given in the previous two sections, in the context of the CGMY (Carr, Geman, Madan, and Yor 2002) model and the CGMY model with an additional diffusion component. In Section 'A QUALITATIVE COMPARISON WITH MARKET SMILES', we present a qualitative comparison, based on the observed market quotes in FX, between our theoretical predicted shape (1.1) of the short-maturity smile and the actual market smiles. Also, our θ-parameterization of the strike variable is compared to the parameterization in terms of the option delta, commonly used in FX markets. Section 'PROOFS' concludes the paper by proving Theorem 2.1, Proposition 2.3, Theorem 3.1, and Corollary 3.1 in that order. The Appendix contains a short technical lemma, which is applied in Section 'PROOFS'.

2. OPTION PRICE ASYMPTOTICS CLOSE TO THE MONEY

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. OPTION PRICE ASYMPTOTICS CLOSE TO THE MONEY
  5. 3. ASYMPTOTIC BEHAVIOR OF IMPLIED VOLATILITY
  6. 4. NUMERICAL RESULTS
  7. 5. A QUALITATIVE COMPARISON WITH MARKET SMILES
  8. 6. PROOFS
  9. APPENDIX
  10. REFERENCES

In this paper, we study the behavior of option prices close to maturity in an exponential Lévy model inline image, where X is a Lévy process with the characteristic triplet inline image. Throughout the paper we assume the following:

  • S is a true martingale (i.e., the interest rate and the dividend yield are assumed to be zero);
  • S is normalized to start at inline image (i.e., as usual the Lévy process X starts at inline image);
  • the tails of the Lévy measure ν admit exponential moments:
    • display math(2.1)

In particular, assumption (2.1) guarantees the finiteness of vanilla option prices for any maturity inline image. Section 'Lévy Processes with Jumps of Infinite Variation' describes the asymptotic behavior of option prices for short maturities in the case the process X has jumps of infinite variation. Section 'Lévy Processes with Jumps of Finite Variation' deals with the case where the pure-jump part of X has finite variation.

2.1. Lévy Processes with Jumps of Infinite Variation

Theorem 2.1 describes the asymptotic behavior of option prices in the case the tails of the Lévy measure of X around zero have asymptotic power-like behavior. This assumption does not exclude any exponential Lévy models that appear in the literature but yields sufficient analytical tractability to characterize a nontrivial limit as maturity tends to zero for the option prices around the ATM. Before stating the theorem, we recall standard notation used throughout the paper: functions inline image and inline image, where inline image for all small inline image, satisfy

  • display math(2.2a)
  • display math(2.2b)
  • display math(2.2c)

Furthermore, we denote inline image for any inline image.

Theorem 2.1. Let X be a Lévy process as described at the beginning of the section and assume that the following holds

  • display math(2.3)

for inline image and inline image. Let inline image be a deterministic function satisfying

  • display math

and

  • display math

Then, if inline image, we have

  • display math(2.4)

and, if inline image, it holds

  • display math(2.5)
Remarks 2.2.
  1. Theorem 2.1 implies that the price of a call (resp. put) option struck at inline image (resp. inline image) tends to zero at a rate strictly slower than t if the paths of the pure-jump part of X have infinite variation. In particular, combining the notation in (2.2a) and (2.2b), we get that the following equalities hold as inline image:
    • display math
  2. The proof of Theorem 2.1 is given in Section 'Proof of Theorem 2.1'.
2.1.1. Blumenthal–Getoor index and the short-dated option prices

Recall that for any Lévy process Y with a nontrivial Lévy measure inline image, the Blumenthal–Getoor index, introduced in Blumenthal and Getoor (1961), is defined as

  • display math(2.6)

The Blumenthal–Getoor index is a measure of the jump activity of the Lévy process Y, since the following holds: inline image, if and only if inline image almost surely, where inline image denotes the size of the jump of Y at time s. Furthermore, it is clear from (2.6) that inline image lies in the interval [0, 2].

In recent years, there has been renewed interest in the Blumenthal–Getoor index from the point of view of estimation of the jump structure of stochastic processes based on high-frequency financial data. For example, it was estimated in Aït-Sahalia and Jacod (2009) that the value of inline image is around 1.7 (i.e., the stock price process has jumps of infinite variation) based on high-frequency transactions (taken at 5 and 15 time intervals) for Intel and Microsoft stocks throughout 2006. Since the pricing measure is equivalent to the real-world measure, the Blumenthal–Getoor index of the process under the pricing measure is in this case also close to 1.7 (by theorem 7.23(b) in Jacod and Shiryaev 2003, chapter III, which relates the semimartingale characteristics of the price process under the two measures).

Let inline image and inline image be the pure-jump parts of the Lévy process X from Theorem 2.1. In other words, inline image (resp. inline image) is a Lévy process with the characteristic triplet inline image (resp. inline image), where inline image (resp. inline image). Then assumption (2.3) implies

  • display math

and relations (2.4) and (2.5) of Theorem 2.1 describe how the Blumenthal–Getoor indices of the positive and negative jumps of X influence the asymptotic behavior of option prices at short maturities. These results clearly depend on the asymptotic behavior of the log-strike inline image. In Section 'ASYMPTOTIC BEHAVIOR OF IMPLIED VOLATILITY', we will prescribe a specific parametric form of inline image (see (3.4)) and give explicit formulae for the asymptotic expansion and the limit of the implied volatility as maturity tends to zero in terms of the Blumenthal–Getoor indices of inline image and inline image (see Corollary 3.3 for details).

2.2. Lévy Processes with Jumps of Finite Variation

In this section, we study the option price asymptotics at short maturities in the case the process X has a (possibly trivial) Brownian component and a pure-jump part of finite variation.

PROPOSITION 2.3. Let X be a Lévy process as described at the beginning of Section 'OPTION PRICE ASYMPTOTICS CLOSE TO THE MONEY'. Assume further that the jump part of X has finite variation, i.e.,

  • display math

Let inline image be a deterministic function satisfying

  • display math

and

  • display math

Then, as inline image, it holds:

  • display math(2.7)

and

  • display math(2.8)
Remarks 2.4.
  1. Proposition 2.3 implies that, in the absence of a Brownian component, the call and put prices of options struck at inline image and inline image, respectively, tend to zero at the rate equal to t if X has paths of finite variation (cf. Remark (i) after Theorem 2.1).
  2. The Blumenthal–Getoor indices of the positive and negative jump processes inline image and inline image of X, defined in Section 'Blumenthal–Getoor index and the short-dated option prices', are both smaller or equal to one by the assumption in Proposition 2.3. Furthermore, unlike in the case of jumps of infinite variation, Proposition 2.3 implies that the asymptotic behavior of short-dated option prices (as maturity t tends to zero) does not depend, up to order inline image, on the indices inline image and inline image. Hence, the same will hold for the short-dated implied volatility (cf. Corollary 3.3).
  3. It should be stressed that the proof of Proposition 2.3, given in Section 'Proof of Proposition 2.3', is fundamentally different from that of Theorem 2.1, as it relies on the pathwise version of the Itô–Tanaka formula for the processes of finite variation, which cannot be applied in the context of Theorem 2.1.

3. ASYMPTOTIC BEHAVIOR OF IMPLIED VOLATILITY

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. OPTION PRICE ASYMPTOTICS CLOSE TO THE MONEY
  5. 3. ASYMPTOTIC BEHAVIOR OF IMPLIED VOLATILITY
  6. 4. NUMERICAL RESULTS
  7. 5. A QUALITATIVE COMPARISON WITH MARKET SMILES
  8. 6. PROOFS
  9. APPENDIX
  10. REFERENCES

The value inline image of the European call option with strike inline image (for any inline image) and expiry t under a Black–Scholes model (with log-spot inline image of constant volatility inline image) is given by the Black–Scholes formula

  • display math(3.1)

and inline image is the standard normal cumulative distribution function. The price of a put option with the same strike and maturity is given by inline image. Let S be a positive martingale, with inline image, that models a risky security and denote by

  • display math(3.2)

the prices of call and put options on S struck at inline image with maturity t, respectively. The implied volatility in the model S for any log-strike inline image and maturity inline image is the unique positive number inline image that satisfies the following equation in σ:

  • display math(3.3)

Implied volatility is well defined since the function inline image is strictly increasing on the positive half-line and the right-hand side of (3.3) lies in the image of the Black–Scholes formula by a simple no-arbitrage argument. Put–call parity, which holds since S is a true martingale, implies the identity inline image.

In order to study the limiting behavior of the implied volatility close to the ATM strike inline image for short maturities, we define the following parameterization of the log-strike inline image:

  • display math(3.4)

We can now define the implied volatility inline image as a function of θ in the asymptotic maturity-strike regime inline image, given by (3.4), for a short maturity t:

  • display math(3.5)

The implied volatility inline image is of interest in the context of processes with jumps, because its limit inline image, as inline image, exists and is finite for each θ, depends on both the jump and the diffusion components of the process and can be computed explicitly in terms of the parameters. In order to find the asymptotic behavior of inline image, we first state Theorem 3.1, which relates the asymptotics of inline image to the asymptotic behavior of the OTM option price

  • display math(3.6)

under the model S as maturity t tends to zero.

Theorem 3.1. Let S be a martingale model for a risky security with inline image and inline image a log-strike given in (3.4) for a fixed inline image. Let inline image and inline image be deterministic functions such that inline image and inline image as inline image, where inline image and inline image are given in (3.2), and define inline image. Assume further that the OTM option price inline image, given in (3.6), satisfies:

  • display math(3.7)

Then the implied volatility inline image, defined in (3.5), can be expressed by

  • display math(3.8)

and

  • display math(3.9)

where inline image and inline image are defined by the formula

  • display math

Before proceeding with the application of Theorem 3.3 given in Corollary 3.1 below (see also Section 'Proof of Corollary 3.3'), we make the following remarks in order to place its statement in context.

Remarks 3.2.
  1. In the Black–Scholes model with volatility inline image, the following well-known expansion of the call option price in the inline image maturity-strike regime (3.4) holds (e.g., a straightforward calculation using Gao and Lee 2011, equation (3.11) yields the expansion):
    • display math(3.10)
    In particular, we have inline image as inline image and hence the assumption in (3.7) is satisfied in the Black–Scholes model.
  2. Note that the log-strike inline image in (3.4) satisfies the assumptions of Theorem 2.1. For any Lévy process X as in Theorem 2.1, formula (2.4) and Remark (i) above imply
    • display math(3.11)
    Since the minimum of the constants in front of inline image is clearly larger than 1/2, assumption (3.7) of Theorem 3.1 is satisfied. As we shall soon see, it is the balance (as a function of θ) between the two constants in (3.11) that determines the value of the limiting smile inline image.
  3. Let a Lévy process X be as in Proposition 2.3 (i.e., with jumps of finite variation). Formulae (2.7) and (3.10) imply that the call option price inline image under the model S has the following asymptotic behavior
    • display math(3.12)
    In particular, note that assumption (3.7) is satisfied and that, in the case of jumps with finite variation, the constant in front of inline image does not depend on the Lévy measure but solely on the diffusion component of the model.
  4. In Gao and Lee (2011), the authors present a general result, which translates the asymptotic behavior of the option prices, in a generic maturity-strike regime, to the asymptotics of the corresponding implied volatilities. Unfortunately the results in Gao and Lee (2011) do not apply to the regime inline image, for inline image in (3.4), since the essential standing assumption inline image in Gao and Lee (2011, equation (4.3)) is not satisfied in our setting by (3.11) and (3.12). We therefore have to establish Theorem 3.1, which is applicable in our context as remarked in (ii) and (iii) above.

The main asymptotic formula of the paper is given in the following corollary.

Corollary 3.3. Let X be a Lévy process with the jump measure ν and the Gaussian component inline image. Pick inline image, let inline image be the log-strike from (3.4) and let inline image be the implied volatility defined in (3.5). Then the following statements hold.

  1. Let X be a Lévy process satisfying the assumptions of Theorem 2.1. Then the implied volatility inline image takes the form
    • display math(3.13)
    where
    • display math(3.14)
    and the sign ± denotes either + or − throughout the formulae in (3.13) and (3.14). In particular, the limiting smile inline image exists for any inline image and takes the form
    • display math
  2. Let a Lévy process X be as in Proposition 2.3 and let inline image be equal to the following integrals
    • display math
    Then the implied volatility inline image for short maturity t is given by
    • display math(3.15)
    where
    • display math(3.16)
    and ± denotes either + or − throughout the formulae in (3.15) and (3.16). The limit of the implied volatility smile as maturity tends to zero, inline image, exists for inline image and is equal to
    • display math
Remarks 3.4.
  1. Recall display (2.3) in Theorem 2.1 and note that the assumptions inline image and inline image of Corollary 3.3 (a) mean that, as inline image, the tails around zero of the Lévy measure ν of X behave as inline image and inline image. Note further that, once we have identified the precise rate of the tail behavior of ν at zero, the constants inline image and inline image do not feature in the limiting formula inline image.
  2. The assumption inline image in Corollary 3.3 (b) ensures that the process X has positive jumps when inline image and negative jumps when inline image as we are only interested in the asymptotic behavior of the implied volatility in the presence of jumps.

In Section 'Proof of Theorem 3.1' we establish Theorem 3.1 and in Section 'Proof of Corollary 3.3' we derive Corollary 3.3 from Theorem 3.1.

4. NUMERICAL RESULTS

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. OPTION PRICE ASYMPTOTICS CLOSE TO THE MONEY
  5. 3. ASYMPTOTIC BEHAVIOR OF IMPLIED VOLATILITY
  6. 4. NUMERICAL RESULTS
  7. 5. A QUALITATIVE COMPARISON WITH MARKET SMILES
  8. 6. PROOFS
  9. APPENDIX
  10. REFERENCES

In this section, we present some numerical illustrations for the convergence results discussed in Section 'ASYMPTOTIC BEHAVIOR OF IMPLIED VOLATILITY'. We assume that the process X follows the widely used CGMY model (Carr, Geman, Madan, and Yor 2002) with Lévy density

  • display math(4.1)

For this process, the price of a European call option with payoff inline image at time t can be computed as

  • display math(4.2)

where inline image is the characteristic function of inline image and inline image (see, e.g., Carr and Madan 1998 or Tankov 2010). We compute the integral in (4.2) with an adaptive integration algorithm.

4.1. Testing the Algorithm

To ensure that the prices returned by our algorithm are correct, we first compare them to the values computed in Wang, Wan and Forsyth (2007) with their approximate “fixed point” algorithm (discretization of the partial differential equation). The following table shows that the values we obtain are very similar to those computed in Wang et al. (2007) with small discrepancies probably due to the discretization error of Wang et al. (2007).

SKTrcinline imageinline imageαValue (Wang et al. 2007)Our value
90980.250.0616.9729.977.080.644216.21257816.211904
90980.250.060.42191.24.371.01022.23070312.2306558
10100.250.119.28.81.84.37149724.3898433

4.2. Convergence of the ATM Options

In this section, we fix the parameters of the process at

  • display math(4.3)

and inline image. First, we analyze the rate of convergence to zero of the ATM options. It follows from the results in Muhle-Karbe and Nutz (2011) that the ATM option price satisfies

  • display math

where inline image is a stable random variable with the Lévy density inline image. Furthermore, it is known (see, e.g., Samorodnitsky and Taqqu 1994, property 1.2.17) that

  • display math

Figure 4.1 plots the dependence of the normalized option price inline image and the normalized “Bachelier” price inline image on inline image, i.e., on time to maturity expressed on the log-scale. The horizontal line in Figure 4.1 corresponds to the value of the constant C. The desired convergence is clearly visible.

image

Figure 4.1. Convergence of the renormalized price for ATM options: the parameters of the CGMY process are given in (4.3).

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4.3. Convergence of Option Prices with Variable Strike

In this section, we investigate numerically the convergence of the OTM option prices given in Theorem 2.1. The parameter values for the underlying process are given in (4.3). Note that in the case of the CGMY model with Lévy density (4.1), the limits in (2.3) of Theorem 2.1 take the form

  • display math

Figure 4.2 shows the dependence of the normalized option and “Bachelier” prices, respectively, given by

  • display math

on time to maturity in log-scale, where

  • display math

The horizontal dotted line shows the limiting value inline image predicted by Theorem 2.1.

image

Figure 4.2. Convergence of the renormalized price for OTM options: inline image with inline image and the parameters of the process X are given in (4.3).

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Similarly, Figure 4.3 plots the dependence on time to maturity (on the log-scale) of the normalized option price

  • display math

As in Figure 4.2, the limiting horizontal dotted line is given by inline image.

image

Figure 4.3. Convergence of the renormalized price for OTM options: inline image with inline image and the parameters of the process X are given in (4.3).

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4.4. Convergence of the Implied Volatilities to the Limiting Smile

In this section, we illustrate the convergence of the implied volatility (expressed as a function of the renormalized strike θ) to the limit inline image given in Corollary 3.3. In order to test the formula both with and without the diffusion component, we fix two models: the first model is a pure-jump CGMY Lévy process with the following parameter values

  • display math

which corresponds to the unit annualized volatility of about 14%. The second model is the same CGMY process with an additional diffusion component of volatility inline image.

Recall that the limiting formula for positive θ is inline image. Figure 4.4 plots the right wing of the implied volatility smile (as a function of θ) for different times to maturity when the diffusion component is present (left graph) and the diffusion component is absent (right graph), together with the limiting shape inline image. The convergence to the limit is visible in both graphs but slow, because the error terms in Corollary 3.3 are logarithmic in time. Nevertheless, the following observations can be made already at “not such small” times:

  • The smile is remarkably stable in time, when it is expressed as function of the renormalized variable θ. In particular, the slope of the wings predicted by Corollary 3.3 is achieved rather quickly.
  • The distinction between the U-shaped smile in the presence of a diffusion component and the V-shaped smile in the pure-jump case, is clearly visible.
image

Figure 4.4. Convergence of the implied volatilities. Left: a diffusion component is present. Right: no diffusion component.

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4.5. Approximation of the Implied Volatility for Small Times to Maturity

In this section, we illustrate the approximation of the implied volatility at small times by the asymptotic formula (3.13). We take the same parameters of the CGMY process as in Section 'Convergence of the Implied Volatilities to the Limiting Smile' and consider the case inline image (when the diffusion component is present, in the region where the pure-jump component dominates, the asymptotic formula is the same, and in the diffusion-dominated region, there are no additional terms added to the constant limit). Figure 4.5 illustrates the quality of the approximation for inline image day and inline image days.

image

Figure 4.5. Approximation of the implied volatilities by the asymptotic formula (3.13). Left: inline image day. Right: inline image days.

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5. A QUALITATIVE COMPARISON WITH MARKET SMILES

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. OPTION PRICE ASYMPTOTICS CLOSE TO THE MONEY
  5. 3. ASYMPTOTIC BEHAVIOR OF IMPLIED VOLATILITY
  6. 4. NUMERICAL RESULTS
  7. 5. A QUALITATIVE COMPARISON WITH MARKET SMILES
  8. 6. PROOFS
  9. APPENDIX
  10. REFERENCES

5.1. Aim

In this section, we aim to compare our theoretical insights into the shape of the short-maturity smile with the actual market smiles and to assess, using observed market quotes, the parameterizations of the implied volatility smiles in terms of the theta, delta, and strike. We base our qualitative analysis on the option prices for the two most liquid currency pairs: USDJPY and EURUSD. Figures 5.1 and 5.2 depict the implied volatilities corresponding to the five option prices for each currency pair, expressed in terms of the aforementioned parameterizations (midday quotes for two recent dates for each currency pair are used). The plotted implied volatilities give the market prices for the options with the following strikes: ATM, 25-delta call and put, 10-delta call and put, and maturities ranging from 1 day to 2 months. The options on the two currency pairs, which correspond to these strikes and maturities, are chosen as the basis of our analysis as they are the most liquid options in FX markets.

image

Figure 5.1. USDJPY option prices: January 4, 2013 (above) and November 5, 2012 (below).

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image

Figure 5.2. EURUSD option prices: Januray 4, 2013 (above) and November 5, 2012 (below).

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5.2. FX Option Quotes

In FX derivative markets, the option prices are typically quoted in terms of the implied volatility σ for a given amount of the Black–Scholes call (resp. put) delta inline image (resp. inline image):

  • display math

where inline image is the standard normal cumulative distribution function, S0 is the current exchange rate, t is the maturity, and inline image is the interest rate differential between the two currencies. Note that the ATM strike corresponds to a 50-delta call strike and the 25-delta (resp. 10-delta) put strike is equal to the 75-delta (resp. 90-delta) call strike.1

The parameterization in terms of the delta is convenient for the traders as it expresses by its definition the amount of delta risk contained in the quoted option. However, in order to obtain the parameterization of the implied volatility smile in terms of the strike (the rightmost graphs in Figures 5.1 and 5.2), one has to solve the nonlinear equation in the strike K with the right-hand side given by the market quote for the implied volatility and the left-hand side equal to the formula for either inline image or inline image. By contrast, the theta parameterization of the implied volatility smile (the leftmost graphs in Figures 5.1 and 5.2) is given by the simple formula in (3.4), which relates explicitly the strikes of the quoted options to the values of the parameter θ.

5.3. Discussion

In the context of this paper, it is natural to ask how formula (3.4) for the strike behaves across different maturities and whether the theta parameterization of the smile (1.1) relates to the market data. We now briefly discuss these questions.

5.3.1. Stability across maturities

It is clear from the graphs in Figures 5.1 and 5.2 that the parameterization of the implied volatility smile in terms of delta is more stable across different maturities than the parameterization based on strike. As described in Section 'FX Option Quotes', the parameterization of the smile in terms of theta possesses a much simpler relationship to the parameterization based on strike than the one based on delta. And yet, Figures 5.1 and 5.2 suggest that the stability of the smile across maturities is similar to that exhibited by the delta parameterization.

5.3.2. Implied volatility formula as a function of θ

It can be observed in Figures 5.1 and 5.2 that the 1-day market-implied volatilities have a qualitatively similar shape to that predicted by the limiting formula in (1.1). In particular, it appears that the slope of the wings of the simile in the leftmost graphs in Figures 5.1 and 5.2, computed from the two extreme points in the graph, is close to one as predicted by the limiting formula in (1.1) for the finite variation case. Furthermore, it appears that on November 5, 2012, the smiles for both currency pairs were converging to a flat smile (in θ) close to the ATM (i.e., inline image), which, by formula (1.1), implies the presence of the diffusion component in the underlying model. Analogously, the market data on January 4, 2013 would appear to suggest that on that day no diffusion component was present.

It should be stressed, however, that the maturity t equal to 1 day is, in the context of the smile formula in (1.1), still far from the limit since the magnitude of the error is of order of inline image (see Corollary 3.3). This fact makes it difficult to quantify, based on the market data, the observations on the structure of the underlying model made in the paragraph above. In particular, it is not feasible to estimate the Blumenthal–Getoor indices of the positive and negative jumps of the underlying process, based on the smile formula in (1.1), if only 1-day options data are available. We stress that the main aim of our study is not to develop quantitative estimation algorithms from short maturity options, but provide explicit insights into the qualitative behavior of the short-maturity smile in jump models.

6. PROOFS

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. OPTION PRICE ASYMPTOTICS CLOSE TO THE MONEY
  5. 3. ASYMPTOTIC BEHAVIOR OF IMPLIED VOLATILITY
  6. 4. NUMERICAL RESULTS
  7. 5. A QUALITATIVE COMPARISON WITH MARKET SMILES
  8. 6. PROOFS
  9. APPENDIX
  10. REFERENCES

6.1. Proof of Theorem 2.1

By Lemma A.1, to prove Theorem 2.1, it is sufficient to show that

  • display math(6.1)

as inline image for the call case and

  • display math(6.2)

as inline image for the put case. Note that (6.2) follows from (6.1) by a substitution inline image. Therefore, from now on we concentrate on the proof of (6.1), assuming with no loss of generality that inline image.

Step 1.   In this first step, we assume that inline image and would like to prove

  • display math(6.3)

Fix inline image, and inline image with inline image, let inline image be a Lévy process with no diffusion part, Lévy measure inline image and the third component of the characteristic triplet

  • display math

Let inline image be a sequence of i.i.d. random variables with the probability distribution

  • display math

and inline image a standard Poisson process with intensity inline image. Furthermore, we assume that inline image, inline image, and inline image are independent. Then the following equality in law holds

  • display math(6.4)

and it follows that

  • display math(6.5)
  • display math(6.6)
  • display math(6.7)

As a preliminary computation, we deduce from the assumptions of the theorem that the following asymptotic behavior holds as inline image (recall definition (2.2a)):

  • display math(6.8)
  • display math(6.9)
  • display math(6.10)

To estimate the term in (6.5), we apply the argument inspired by lemma 2 in Rüschendorf and Woerner (2002). In the current notation this implies

  • display math(6.11)

where inline image is the inverse function of inline image defined by

  • display math

By Taylor's theorem, this function satisfies

  • display math

This implies that

  • display math

and, therefore, substituting this into (6.11),

  • display math(6.12)

From the assumptions of the theorem and (6.8)(6.10), there exists inline image such that inline image implies

  • display math

for some constant inline image. By similar arguments it can be shown that

  • display math(6.13)

Coming back to the estimation of (6.5), we first deal with the case inline image. In this case, the Cauchy–Schwartz inequality allows to conclude that

  • display math

because the first factor remains bounded by (6.8)(6.10).

Let us now focus on the case inline image. Let inline image. The expectation in (6.5) can be expressed as

  • display math

By Taylor's formula, we then get

  • display math

We now need to show that the second and the third terms do not contribute to the limit. Since by assumption inline image, we have that inline image as inline image, and, therefore, by (6.8)(6.10),

  • display math

The last term can be split into two terms, which are easy to estimate using (6.8)(6.10):

  • display math

because by assumption of the theorem, inline image. On the other hand,

  • display math

by (6.10) and (6.13). We have, therefore, shown that

  • display math

From (6.8), the assumption on inline image in Theorem 2.1 and the Lipschitz property of the function inline image, it follows that

  • display math

as well.

For the term in (6.6), the Lipschitz property of the function inline image, (6.8)(6.10) and the assumption of the theorem (i.e., the first assumption on inline image in Theorem 2.1 in the case inline image and the second one otherwise) imply the following estimate:

  • display math

On the other hand, integration by parts implies

  • display math

where inline image, which yields the second term in (2.4).

To treat the summand in (6.7), observe that by (6.8)(6.10), for inline image,

  • display math

Therefore, the summand in (6.7) is of order inline image and hence inline image.

Step 2.   We now treat the case when inline image. Let inline image be a spectrally negative Lévy process with zero mean and zero diffusion part and Y be a spectrally positive Lévy process such that inline image. Let inline image (where we take inline image is inline image) and inline image. As before, we fix inline image and let inline image be a Lévy process with no diffusion part, zero mean and Lévy measure inline image, let inline image, let inline image be a sequence of i.i.d. random variables with the probability distribution

  • display math

and finally inline image. Decomposing inline image similarly to (6.4) in terms of inline image and inline image, it is easy to show that the option price inline image admits an upper bound

  • display math

and a lower bound

  • display math

Similarly to (6.8)(6.10), we have

  • display math

and with the same logic as in (6.12), we have that

  • display math

It is now clear that one can choose inline image so that inline image is of order of inline image. Since inline image admits the same estimate, and inline image as inline image, we get that for some deterministic functions inline image and inline image which converge to 1 as inline image,

  • display math

Since inline image and inline image, from (6.3), we then get

  • display math

Finally, since we also have inline image and inline image, we get that inline image and inline image, which allows to complete the proof of Theorem 2.1.inline image

6.2. Proof of Proposition 2.3

We first concentrate on the proof of (2.8). Let inline image be the characteristic triplet of X with respect to the zero truncation function, meaning that

  • display math

where as usual for any inline image we define inline image.

Assume first that inline image. The left derivative of the function

  • display math

and hence Itô–Tanaka formula (Protter 2004, chapter IV, theorem 70) applied to the process inline image yields

  • display math

for any inline image, since, in this case, X has paths of finite variation. Since inline image is a Poisson point process with intensity measure inline image, and inline image for at most countably many time s in the interval [0, t] almost surely, taking expectations on both sides of the pathwise representation above and applying the compensation formula for point processes, we obtain

  • display math(6.14)

From theorem 43.20 in Sato (1999), we have that inline image almost surely as inline image. Recall also that by assumption inline image as inline image. Therefore, for any inline image, each path inline image satisfies the following inequalities

  • display math(6.15)

Therefore, it holds inline image almost surely. Since on the other hand we have

  • display math

the dominated convergence theorem implies

  • display math

To deal with the second term in (6.14), we deduce from (6.15) that inline image also satisfies the following inequalities for any inline image, all sufficiently small times inline image and all inline image:

  • display math

and

  • display math

The second terms in both sides of the above two inequalities is in fact always zero for sufficiently small t due to (6.15). Therefore, we get the following almost sure convergence:

  • display math

Since the function inline image is Lipschitz with a Lipschitz constant that does not depend on the path inline image, the dominated convergence theorem and the representation in (6.14) yield

  • display math

Assume now that inline image. Define

  • display math

and note that

  • display math(6.16)

The derivative of f with respect to x is given by

  • display math(6.17)

It can be computed explicitly as

  • display math(6.18)

where inline image denotes the standard normal cumulative distribution function. Note also for future use that

  • display math(6.19)

with inline image.

Applying Itô's formula to the process inline image as a function of Z with t fixed, yields

  • display math

since inline image for all inline image. By taking the expectation and applying (6.16) we find that

  • display math(6.20)

The first term on the right-hand side of (6.20) is equal to the first term on the right-hand side of (2.8). As in the case inline image, using the almost sure convergence inline image, the explicit form (6.18) of inline image and the assumption that inline image as inline image, we get that

  • display math

almost surely. Since inline image for all inline image by (6.17), the dominated convergence theorem yields

  • display math

To treat the last term in (6.20), we use the fact that for any inline image, each path inline image satisfies the inequalities

  • display math

for all sufficiently small t. Therefore, since inline image, the following inequalities hold

  • display math(6.21)

for any trajectory inline image, where inline image, and all sufficiently small t. The random variable under the expectation in the last term on the right-hand side of (6.20) can be expressed as follows:

  • display math(6.22)

The pathwise bounds in (6.21) can be used to estimate (6.22) from above and below. For each path inline image, we have the following bound for inline image and all sufficiently small t:

  • display math

The explicit form (6.18) of inline image implies that for all inline image and inline image we have

  • display math

Since inline image is bounded, the dominated convergence theorem yields

  • display math

as inline image. Formula (6.18) for inline image implies that for all inline image and inline image we have

  • display math

An analogous argument for inline image to the one above and the representation in (6.22) imply the almost sure convergence

  • display math

Finally, since inline image is Lipschitz in x, with the Lipschitz constant independent of t, the dominated convergence theorem implies

  • display math

This concludes the proof of (2.8). Note that in this proof, we did not use the condition in (2.1), but only the assumption inline image.

We now concentrate on the proof of (2.1). Since the Lévy process X satisfies (2.1), we can define the share measure inline image, via inline image, as in the proof of Theorem 3.1. Analogous to the equality in (6.24), we have

  • display math(6.23)

where inline image denotes the expectation under the share measure inline image. Furthermore, by Sato (1999, theorem 33.1), under the measure inline image, the process X is again a Lévy process with a characteristic triplet inline image, where inline image, and inline image is a positive inline image-martingale started at one. The Lévy measure inline image clearly satisfies

  • display math

Therefore, we can apply (2.8) to the process inline image under the measure inline image. Hence, the identity in (6.23) yields:

  • display math

where we used the Black–Scholes put–call symmetry given in (6.27), the fact inline image and the equality inline image. This establishes the formula in (2.7) and concludes the proof of Proposition 2.3.inline image

6.3. Proof of Theorem 3.1

We first assume that inline image. Equality (3.10) implies the following

  • display math

as inline image for any inline image. Define

  • display math

and note that inline image corresponds to the left-hand side of the above formula with the change of variable inline image. The expansion shows that inline image is regular as inline image and the following equality holds

  • display math

The expansion for the inverse mapping can be deduced from this expression as follows. To keep the formulae simple, we give the expansion up to inline image:

  • display math

Denote by inline image the unique positive solution of the equation inline image, where y equals inline image (see the statement of Theorem 3.1 for the definition of inline image) and x is any arbitrage-free call option price with maturity inline image and strike inline image. The uniqueness of the quantity inline image is equivalent to the fact that the implied volatility is a well-defined quantity.

An approximate expression for y is given by

  • display math

and hence we find

  • display math

Using the regularity of the coefficient a in the neighborhood of the point inline image, we can expand the inverse inline image around the point inline image as follows:

  • display math

In view of this expression, and using once again the regularity of the coefficients a and b, we can replace inline image with inline image in the second term, obtaining

  • display math

Hence, the following asymptotic equalities hold true:

  • display math

Making the substitution inline image in the above formula, we find an expansion for the implied volatility inline image given in (3.8). Now, inline image implies that inline image. Since all the coefficients in expansion (3.8) are regular, the additional term arising from this difference may be ignored in an expansion up to order inline image and (3.9) follows.

The formulae in the theorem in the case inline image will be established by applying the result for the positive log-strike under the share measure. More precisely, let inline image denote the original risk-neutral measure under which the process S is a positive martingale starting at one. For each time t, we define the share measure inline image on the σ-algebra inline image of events that can occur up to time t via its Radon–Nikodym derivative inline image and note that the following relationship holds for any log-strike inline image:

  • display math(6.24)

where inline image denotes the expectation under the share measure inline image of a call payoff with strike inline image, where the evolution of the risky asset is given by inline image. Note that inline image is a positive martingale starting at one under inline image and hence inline image represents an arbitrage-free call option price. Furthermore, the put–call symmetry formula in the Black–Scholes model (see (6.27)) and the equality in (6.24) mean that the implied volatility inline image defined by the put price inline image coincides with the implied volatility inline image defined by the call price inline image (see beginning of Section 'ASYMPTOTIC BEHAVIOR OF IMPLIED VOLATILITY' for the definition of inline image).

Note that, since inline image, we now have inline image and inline image, where inline image denotes inline image. In order to apply the formula in (3.8) to inline image, we have to ensure that assumption (3.7) is satisfied. Since (3.7) holds for inline image and inline image, the equality in (6.24) implies (3.7) for inline image. Therefore, formula (3.8) gives an asymptotic expansion of inline image in terms of inline image. Since equality (6.24) implies

  • display math

and the two leading order terms in (3.8) are regular in inline image, the asymptotic expansion in (3.8) also holds when inline image is replaced by inline image. The formula in (3.9) now follows by the same argument as in the case of the positive log-strike. This concludes the proof of the theorem.inline image

6.4. Proof of Corollary 3.3

(a) Assume first that inline image. Define inline image and note that (3.10), the definition of inline image in (3.4) and (2.4) of Theorem 2.1 imply

  • display math(6.25)

where inline image denotes the call option price with maturity t and strike inline image under the exponential Lévy model inline image. Assumption (3.7) of Theorem 3.1 is, therefore, satisfied by Remark (i) after Theorem 3.1. The formula for inline image takes the form

  • display math(6.26)

The formula in (3.9) of Theorem 3.1, together with (6.26) and the Taylor expansions in inline image as inline image

  • display math
  • display math

yield the formula in (3.13).

In the case inline image, the relation (6.25) is satisfied by inline image. This follows directly from the definition of inline image in (3.4) and Theorem 2.1 (see formula (2.4)). An analogous argument as the one above shows that in this case the assumptions of Theorem 3.1 are also satisfied. By the definition of inline image in Theorem 3.1, we find

  • display math

By Taylor's formula the following asymptotic relations hold as inline image:

  • display math

and

  • display math

Substituting these expressions into (3.9) establishes the formula in (3.13).

Assume now that inline image. Define inline image, where inline image is the put option price in the Black–Scholes model, and recall the well-known put–call symmetry

  • display math(6.27)

which holds since the laws of minus the log-spot under the share measure (i.e., the pricing measure where the risky asset is a numeraire) and the log-spot under the risk-neutral measure (i.e., the measure where the riskless asset is the numeraire) coincide. Analogous to the case above, (3.10) with the put–call symmetry, the definition of inline image in (3.4) and (2.5) of Theorem 2.1 imply

  • display math(6.28)

where inline image is the put option price under the exponential Lévy model inline image. Therefore, the assumptions of Theorem 3.1 are satisfied and inline image takes the form (6.26). Note that the right-hand side of (6.26) depends solely on the even powers of θ and hence the fact inline image does not influence the asymptotic behavior of inline image. The proof of formula (3.13) now follows in the same way as in the call case above.

In the case inline image, we define inline image. Under this assumption, the relation (6.28) is satisfied by (2.5) of Theorem 2.1 and the rest of the proof follows along the same lines as in the case inline image. This proves formula (3.13).

(b) The proof of part (b) of the corollary is based on Proposition 2.3 and Theorem 3.1. The steps are analogous to the ones in the proof of part (a):

  • display math

The details of the calculations are left to the reader.inline image

APPENDIX

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. OPTION PRICE ASYMPTOTICS CLOSE TO THE MONEY
  5. 3. ASYMPTOTIC BEHAVIOR OF IMPLIED VOLATILITY
  6. 4. NUMERICAL RESULTS
  7. 5. A QUALITATIVE COMPARISON WITH MARKET SMILES
  8. 6. PROOFS
  9. APPENDIX
  10. REFERENCES

Lemma A.1. Let X be a Lévy process satisfying (2.1) and inline image a deterministic function such that

  • display math

Then for any inline image we have

  • display math

Proof. Since inline image, it is clearly sufficient to prove the formula for the call in the case inline image. Let inline image and note the following: inline image for all inline image and inline image for all inline image. By Taylor's formula we have inline image for any inline image, and, considering inline image fixed, we find

  • display math

for some constant inline image. Under the assumption of the lemma, the right-hand side can be computed as

  • display math

A direct computation using the Lévy–Khintchine formula then shows that inline image as inline image. The put case is treated in a similar manner. inline image

REFERENCES

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. OPTION PRICE ASYMPTOTICS CLOSE TO THE MONEY
  5. 3. ASYMPTOTIC BEHAVIOR OF IMPLIED VOLATILITY
  6. 4. NUMERICAL RESULTS
  7. 5. A QUALITATIVE COMPARISON WITH MARKET SMILES
  8. 6. PROOFS
  9. APPENDIX
  10. REFERENCES
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  1. 1

    The implied volatilities in the markets are quoted for the OTM and ATM options only. For representational convenience, in Figures 5.1 and 5.2, we plot the smiles in terms of the call delta only. Note also that the definition of the delta used for option quotes in FX markets is different from the actual delta of the option, which includes an additional inline image factor.