Extreme-strike asymptotics for general Gaussian stochastic volatility models
We consider a stochastic volatility asset price model in which the volatility is the absolute value of a continuous Gaussian process with arbitrary prescribed mean and covariance. By exhibiting a Karhunen-Loève expansion for the integrated variance, and using sharp estimates of the density of a general second-chaos variable, we derive asymptotics for the asset price density for large or small values of the variable, and study the wing behavior of the implied volatility in these models. Our main result provides explicit expressions for the first five terms in the expansion of the implied volatility. The expressions for the leading three terms are simple, and based on three basic spectral-type statistics of the Gaussian process: the top eigenvalue of its covariance operator, the multiplicity of this eigenvalue, and the norm of the projection of the mean function on the top eigenspace. The fourth term requires knowledge of all eigen-elements. We present detailed numerics based on realistic liquidity assumptions in which classical and long-memory volatility models are calibrated based on our expansion.
JEL Classification: G13, C63, C02.
AMS 2010 Classification: 60G15, 91G20, 40E05.
Keywords: stochastic volatility, implied volatility, large strike, Karhunen-Loève expansion, chi-squared variates.
In this article, we characterize wing behavior of the implied volatility for uncorrelated Gaussian stochastic volatility models. This introduction contains a careful description of the problem’s background and of our motivations. Before going into details, we summarize some of the article’s specificities; all terminology in the next two paragraphs is referenced, defined, and/or illustrated in the remainder of this introduction.
We hold calibration of volatility smiles as a principal motivator. Cognizant of the fact that non-centered Gaussian volatility models can be designed in a flexible and parsimonious fashion, we adopt that class of models, imposing no further conditions on the marginal distribution of the volatility process itself, beyond pathwise continuity. The spectral structure of the integrated variance allows us to work at that level of generality. We find that the first five terms in the extreme-strike implied volatility asymptotics – which is typically amply sufficient in applications – can be determined explicitly thanks to three parameters characterizing the top of the spectral decomposition of the integrated variance, with the exception of a factor appearing in the coefficient of the 4th term in these asymptotics, which depends on higher-order eigen-elements. In order to prove such a precise statement while relying on a moderate amount of technicalities, we make use of the simplifying assumption that the stochastic volatility is independent of the asset price’s driving noise.
When considering the trade-off between this restriction and calibration considerations, we observe that our model flexibility combined with known explicit spectral expansions and numerical tools may allow practicioners to compute the said spectral parameters in a straightforward fashion based on smile features, while also allowing them to select their favorite Gaussian volatility model class. Specific examples of Gaussian volatility processes are non-centered Brownian motion, Brownian bridge, and Ornstein-Uhlenbeck processes. This last sub-class can be particularly appealing since it contains stationary volatilities, and includes the well-known Stein-Stein model. We also mention how any Gaussian model specification, including long-memory ones, can be handled, thanks to the numerical ability to determine its spectral elements. We understand that the assumption of the stochastic volatility model being uncorrelated implies the symmetry of the implied volatility on either side of the money, which in some applications, is not a desirable feature. Moreover, while in many option markets, liquidity considerations limit the ability to calibrate using the large-strike wing (see the calibration study on SPX options in [21, Section 5.4]), the ability to work with a correlated volatility model is nonetheless important as soon as one uses the result of the calibration to, say, price illiquid options such as out-of-the-money calls. Hence a fully functional general Gaussian model would require a method for estimating the volatility’s correlation with the asset using liquid options data. Such a study is beyond the scope of our article, since the case of general correlated Gaussian stochastic volatility models presents additional mathematical challenges which may require completely new methods and techniques. We will investigate them separately from this article. An important step toward a better understanding of the asymptotic behavior of the implied volatility in some correlated stochastic volatility models is found in the articles [14, 15].
Another problem which is mathematically interesting and important in practice is the asymptotics for implied volatility in small or large time to maturity. The techniques developed in the present paper are used in the subsequent paper  to study the small-time asymptotics of densities, option pricing functions, and the implied volatility in Gaussian self-similar stochastic volatility models.
1.1. Background and heuristics
Studies in quantitative finance based on the Black-Scholes-Merton framework have shown awareness of the inadequacy of the constant volatility assumption, particularly after the crash of 1987, when practitioners began considering that extreme events were more likely than what a log-normal model will predict. Propositions to exploit this weakness in log-normal modeling systematically and quantitatively have grown ubiquitous to the point that implied volatility (IV), or the volatility level that market call option prices would imply if the Black-Scholes model were underlying, is now a bona fide and vigorous topic of investigation, both at the theoretical and practical level. The initial evidence against constant volatility simply came from observing that IV as a function of strike prices for liquid call options exhibited non-constance, typically illustrated as a convex curve, often with a minimum near the money as for index options, hence the term ‘volatility smile’.
Asset price models where the volatility is a stochastic process are known as stochastic volatility models; the term ‘uncorrelated’ is added to refer to the submodel class in which the volatility process is independent of the noise driving the asset price. In a sense, the existence of the smile for any uncorrelated stochastic volatility model was first proved mathematically by Renault and Touzi in . They established that the IV as a function of the strike price decreases on the interval where the call is in the money, increases on the interval where the call is out of the money, and attains its minimum where the call is at the money. Note that Renault and Touzi did not prove that the IV is locally convex near the money, but their work still established stochastic volatility models as a main model class for studying IV; these models continued steadily to provide inspiration for IV studies.
A current emphasis, which has become fertile mathematical ground, is on IV asymptotics, such as large/small-strike, large-maturity, or small-time-to-maturity behaviors. These are helpful to understand and select models based on smile shapes. Several techniques are used to derive IV asymptotics. For instance, by exploiting a method of moments and the representation of power payoffs as mixtures of a continuum of calls with varying strikes, in a rather model-free context, R. Lee proved in  that, for models with positive moment explosions, the squared IV’s large strike behavior is of order the log-moneyness times a constant which depends explicitly on supremum of the order of finite moments. A similar result holds for models with negative moment explosions, where the squared IV behaves like for small values of . More general formulas describing the asymptotic behavior of the IV in the ‘wings’ ( or ) were obtained in [4, 5, 6, 23, 24, 27, 18] (see also the book ).
From the standpoint of modeling, one of the advantages of Lee’s original result is the dependence of IV asymptotics merely on some simple statistics, namely as we mentioned, in the notation in , the maximal order of finite moments for the underlying , i.e.
This allows the author to draw appropriately strong conclusions about model calibration. A special class of models in which is positive and finite is that of Gaussian volatility models, which we introduce next.
1.2. Gaussian Stochastic volatility models
Let be a standard Brownian motion on a probability space , and let be a continuous Gaussian process on the same space that is independent of . We have , where is a continuous deterministic function on (the mean function) and is a continuous centered Gaussian process on independent of , with covariance . Suppose is a filtration such that is a Brownian motion with respect to , and the process is adapted to .
In the present paper, we study the following asset price model:
on the filtered probability space , where the filtration is such as above. It is also assumed that the short rate is constant. The initial condition for the asset price process will be denoted by . Note that the initial condition for the process may be a nonconstant random variable.
We will next provide a typical example of a filtration satisfying the conditions mentioned above. Let be the -algebra generated by the events of probability zero, and let and be the augmentations by the family of the filtrations generated by the processes and , respectively. Consider the filtration such that for every , . Then the process is a Brownian motion with respect to the filtration , and the process is adapted to . Note that if a.s., then is a sub--algebra of , while if is a random variable, then .
Note that it is not supposed in (1) that the process is a solution to a stochasic differential equation as is often assumed in classical stochastic volatility models. A well-known special example of a Gaussian stochastic volatility model is the Stein-Stein model introduced in , in which the volatility process is the mean-reverting Ornstein-Uhlenbeck process satisfying
where is the level of mean reversion, is the mean-reversion rate, and is level of uncertainty on the volatility; here is another Brownian motion, which may be correlated with . In the present paper, we adopt an analytic technique, encountered for instance in the analysis of the uncorrelated Stein-Stein model by this paper’s first author and E.M. Stein in  (see also ).
Returning to the question of the value of , for a Gaussian volatility model, it can sometimes be determined by simple calculations, which we illustrate here with an elementary example. Assume is a geometric Brownian motion with random volatility, i.e. a model as in (1) where (abusing notation) is taken the non-time-dependent where is a constant and is an independent unit-variance normal variate (not dependent on ). Thus, at time , with zero discount rate, . To simplify this example to the maximum, also assume that is centered; using the independence of and , we get that we may replace by in this example, since this does not change the law of (i.e. in the uncorrelated case, ’s non-positivity does not violate standard practice for volatility modeling). Then, using maturity , for any , the th moment, via a simple change of variable, equals
which by an elementary computation is finite, and equal to , if and only if
In the cases where the random volatility model above is non-centered and is correlated with , a similar calculation can be performed, at the essentially trivial expenses of invoking affine changes of variables, and the linear regression of one normal variate against another.
The above example illustrates heuristically that, by Lee’s moment formula, the computation of might be the quickest path to obtain the leading term in the large-strike expansion of the IV, for more complex Gaussian volatility models, namely ones where the volatility is time-dependent. However, computing is not necessarily an easy task, and appears, perhaps surprisingly, to have been performed rarely. For the Stein-Stein model, the value of can be computed using the sharp asymptotic formulas for the asset price density near zero and infinity, established in  for the uncorrelated Stein-Stein model, and in  for the correlated one. These two papers also provide asymptotic formulas with error estimates for the IV at extreme strikes in the Stein-Stein model. Beyond the Stein-Stein model, little was known about the extreme strike asymptotics of general Gaussian stochastic volatility models. In the present paper, we extend the above-mentioned results from  and  to such models.
1.3. Motivation and summary of main result
Adopting the perspective that an asymptotic expansion for the IV can be helpful for model selection and calibration, our objective is to provide an expansion for the IV in a Gaussian volatility model relying on a minimal number of parameters, which can then be chosen to adjust to observed smiles. The restriction of non-correlated volatility means that the asset price distribution is a mixture of geometric Brownian motions with time-dependent volatilities, whose mixing density at time is that of the square root of a variable in the second-chaos of a Wiener process. That second-chaos variable is none other than the integrated variance By relying on a general Hilbert-space structure theorem which applies to the second Wiener chaos, we prove that, for a wide class of non-centered Gaussian stochastic volatility processes with a possible degeneracy in the eigenstructure of the covariance of viewed as a linear operator on (i.e. when the top eigenvalue is allowed to have a multiplicity larger than ), the large-strike IV asymptotics can be expressed with three terms and an error estimate. These terms depend explicitly on and on the following three parameters: , , and the ratio where is the norm in of the orthogonal projection of the mean function on the first eigenspace of . We also push the expansion to five terms, and notice that the fifth term also only depends on , , and , while the fourth term depends on all other eignevalues and the action of on all other eigenfunctions. Specifically, with the IV as a function of strike , letting be the discounted log-moneyness, as , we prove
where the constants , , , , and depend explicitly on and , also depends explicitly on , while also depends explicitly on , depends explicitly also on both and , and has an additional rather complex dependence on all the eigen-elements through a factor ; this is all stated in Theorem 13 and formula (47). A similar asymptotic formula is obtained in the case where , using symmetry properties of uncorrelated stochastic volatility models (see (55)). The specific case of the Stein-Stein model is expanded upon in some detail.
1.4. Practical implications
The first-order constant is always strictly positive. The second-order term (the constant ) vanishes if and only if is orthogonal to the first eigenspace of , which occurs for instance when . The third-order and fifth-order terms vanish if and only if the top eigenvalue has multiplicity , which is typical (the case can be considered degenerate, and does not occur in common examples). The behavior of and as functions of is determined partly by how the top eigenvalue depends on , which can be non-trivial. In the present paper, we assume is fixed.
For fixed maturity , assuming that has lead multiplicity for instance, a practitioner will have the possibility of determining a value and a value to match the specific root-log-moneyness behavior of small- or large-strike IV; moreover in that case, choosing a constant mean function , one obtains where is the top eigenfunction of . Market prices may not be sufficiently liquid at extreme strikes to distinguish between more than two parameters; this is typical of calibration techniques for implied volatility curves for fixed maturity, such as the ‘stochastic volatility inspired’ (SVI) parametrization disseminated by J. Gatheral: see [19, 20] (see also  and the references therein). Our result shows that Gaussian volatility models with non-zero mean are sufficient for this flexibility, and provide equivalent asymptotics irrespective of the precise mean function and covariance eigenstructure, since modulo the disappearance of the third-order term in the unit top multiplicity case , only and are relevant. The fourth-order term in our expansion can provide additional precision in calibration. Its use is illustrated in Section 7.
Modelers wishing to stick to well-known classes of processes for may then adjust the value of by exploiting any available invariance properties for the desired class For example, if is standard Brownian motion, or the Brownian bridge, on , we have or respectively, and these values scale quadratically with respect to a multiplicative scaling constant for , beyond which an arbitrary mean value may be chosen. If is the mean-zero stationary OU process, we have where is the smallest positive solution of , in which case, for a fixed arbitrarily selected rate of mean reversion , a scaling of is then equivalent to selecting the variance of , while a constant mean value can then be selected independently. [10, Chapter 1] can be consulted for the eigenstructure of the covariance of Brownian motion and the Brownian bridge, which are classical results, and for a proof of the eigenstructure of the OU covariance (see also ). The top eigenfunctions in all three of these cases are known explicit trigonometric functions (see [10, Chapter 1]), and need to be referenced when selecting For the OU bridge, the eigenstructure of (equivalently known as the Karhunen-Loève expansion of ) was found in , while in , such an expansion was characterized for special Gaussian processes generated by independent pairs of exponential random variables. On the other hand, fractional Brownian motion and OU processes driven by fractional Brownian motion (also known as fOU processes) do not fall in the class of Gaussian processes for which the Karhunen-Loève expansion is known explicitly.
However, efficient numerical techniques allowing to compute the eigenfunctions and eigenvalues in these cases were developed by S. Corlay (see Chapter 2 in ). Corlay uses the trapezoidal Nyström method and the three-step Richardson-Romberg method to approximate the five highest Karhunen-Loève eigenvalues of various Gaussian processes; in principle, eigenvalues and eigenfunctions of arbitrarily high order can be obtained using his method. He starts with such estimates for Brownian motion, Brownian bridge, and Ornstein-Uhlenbeck process, for which explicit expressions for the eigenvalues are known. The resulting approximations are very close to the values obtained from the explicit formulas for the eigenvalues, which shows that the method used by Corlay is rather powerful. Corlay also estimates the five highest Karhunen-Loève eigenvalues of fractional Brownian motion on with the Hurst exponent . Of special interest to the context of the present paper is the largest Karhunen-Loève eigenvalue of fractional Brownian motion, for which Corlay obtains the approximation .
While we do not need this value, and instead use Corlay’s method to compute for several fOU processes, we are confident that the values we obtain for the various ’s we use have similar levels of accuracy to what is illustrated in . Corlay’s method is thus one of the main ingredients in the numerical part of our paper (see the discussion after (80) in Section 7). Fractional OU processes were proposed early on for option pricing, and recently analyzed in [9, 8]; these processes are versions of the volatility process in the Stein-Stein model. Therefore, the resulting stochastic volatility models may be called fractional Stein-Stein models. Section 7 illustrates how, in the case of the classical and fractional Stein-Stein models (OU and fOU processes), the explicit, semi-explicit, or numerically accessible Karhunen-Lòeve expansion of can be used in conjunction with the asymptotics (3) for calibrating parameters. We find that market liquidity considerations limit the theoretical range of applicability of calibration strategies, but that significant practical results are nonetheless available.
The remainder of this article is structured as follows. Section 2 sets up a convenient second-chaos representation for the model’s integrated volatility. In Section 3, we generalize some results from [7, 28, 38], concerning the asymptotic behavior of densities of infinite linear combinations of chi-squared random variables, and derive precise asymptotics for the density of the mixing distribution. Section 4 converts these asymptotics into sharp asymptotic formulas for the density of the asset price , thanks to the analytic tools developed in [25, 22]. In Section 5, we characterize the wing behavior of the implied volatility in Gaussian stochastic volatility models. We find sharp asymptotic formulas for the implied volatility with five explicit terms and an error estimate. The special case of the uncorrelated Stein-Stein model is studied in more detail in Section 6. Finally, our practical study of calibration strategies, with numerics, is in Section 7.
2. General setup and second-chaos expansion of the integrated variance
Let be an almost-surely continuous Gaussian process on a filtered complete probability space with mean and covariance functions denoted by and
respectively, and suppose the restrictions imposed in (1) are satisfied.
Define the centered version of : , , and fix a time horizon . It is not hard to see that for all . Since the Gaussian process is almost surely continuous, the mean function is a continuous function on , and the covariance function is a continuous function of two variables on . Indeed, the continuity of the process implies its continuity in probability on . Hence, the process is continuous in the mean-square sense (see, e.g., , Lemma 1 on p. 5, or invoke the equivalence of norms on Wiener chaos, see ). Mean-square continuity of implies the continuity of the mean function on . In addition, the autocorrelation function of the process , that is, the function , , is continuous (see, e.g., , Lemma 4.2). Finally, since , the covariance function is continuous on . We refer the interested reader to  for more information on the continuity problems for general Gaussian processes.
In our analysis, it will be convenient to refer to the Karhunen-Loève expansion of . We will next provide certain details concerning the Karhunen-Loève expansion and introduce notation that will be used throughout the paper.
Consider the covariance operator defined by
The operator is a nonnegative compact self-adjoint operator on . The non-zero eigenvalues of the operator are of finite multiplicity, and we assume that they are rearranged so that
In particular, is the top eigenvalue, and is its multiplicity. It is known that the series converges. The system of eigenfunctions , corresponding to the system , is orthonormal, and each function is continuous on . The number always belongs to the spectrum of the covariance operator, and it may happen so that is an eigenvalue of . The spectral subspace associated with may be infinite-dimensional, and we choose a basis in this subspace. Then is a complete orthonormal system in . Note that the eigenvalues and eigenfunctions of depend on .
The classical Karhunen-Loève theorem (see, e.g., , Section 26.1) states that there exists an i.i.d. sequence of standard normal variates such that
The number of positive eigenvalues may be finite. We will assume throughout the paper that the set of positive eigenvalues is infinite; this is the case for all illustrative examples we use, such as the OU and fOU processes. It is easy to understand how the parameters used in the paper change if the number of positive eigenvalues is finite.
Using (4), we obtain
It is worth pointing out that the previous expression for the integrated variance in a Gaussian model with centered volatility is in fact the most general form of a random variable in the second Wiener chaos with half-bounded support, with mean adjusted to ensure almost-sure positivity of the integrated variance. This is established using a classical structure theorem on separable Hilbert spaces, as explained in [31, Section 2.7.4]. In other words (also see [31, Section 2.7.3] for additional details), any prescribed mean-adjusted integrated variance in the second chaos is of the form
for some standard Wiener process and some function . Moreover one can find a centered Gaussian process such that and one can compute the coefficients in the Karhunen-Loève representation (5) as the eigenvalues of the covariance of .
It is easy to see, using (7) that if the function belongs to the subspace of generated by the orthonormal system , then
For instance, the equality in (8) holds if is not an eigenvalue of the operator . In the case where the process is centered, we have
Let us denote the chi-squared distribution with the number of degrees of freedom and the parameter of noncentrality by (more information on such distributions can be found in  or in any probability textbook; the convention used here is that the mean of is ). Set
where the repeated chi-squared notation is used abusively to denote independent chi-square random variables. We will denote the distribution density of by .
3. Asymptotics of the mixing density
The asymptotic behavior of the distribution density of an infinite linear combination of independent central chi-squared random variables was characterized by Zolotarev (see formula (5) in ). In , Hoeffding found more general and sharp formulas. The results obtained by Zolotarev and Hoeffding were generalized to the case of noncentral chi-squared variables by Beran (see ). Note that Beran considered infinite sums of chi-squared variables with all the noncentrality parameters strictly greater than zero. Since there is a gap betweed the results of Zolotarev, Hoeffding, and Beran, we decided to include a discussion of a similar result, where there are no restrictions on the noncentrality parameters. Keeping in mind the series in (13), we will study the asymptotic behavior of the density of the following infinite sum:
where , , are integers, and for all . If for some , then the corresponding chi-squared random variable is central. It is also assumed that ,
and the chi-squared random variables in (14) are independent. We will denote by the distribution density of the random variable .
The distribution density of a chi-squared random variable will be denoted by . It is known that if , then
where is the modified Bessel function of the first kind (see, e.g., , Theorem 1.31). For , we have
(see, e.g., Lemma 1.27 in ). It is not hard to see that Let us also mention that
for all (see, e.g., 9.6.7 in ).
It is known that for , the moment generating function of a chi-squared random variable with is as follows:
In the formulation of the next result, we will use the following number:
where is defined as without the first term:
and it is not hard to see, by taking into account (15), that .
The next assertion is based on the results of Zolotarev, Hoeffding, and Beran.
Theorem 3 is a minor generalization of similar propositions obtained in  and . The difference between those propositions and our Theorem 3 is that  assumes that all the chi-squared variables in (14) are central, in Theorem 2 in  they are all assumed noncentral, while in our Theorem 3, we may have any combination of central and non-central chi-squared variables.
Theorem 2 in  provides an asymptotic formula for the complementary distribution function (tail) of an infinite linear combination of independent noncentral chi-square random variables. A sharper formula for the distribution density of such a linear combination can be extracted from the proof of Theorem 2 in  (see the very end of that proof).
Let us fix . We have
In the formulas above, the function is defined by
if , while if , then . Note that implies . Then, using calculations similar to those in the proof of Theorem 2 in , we find that when , is the leading term and is of order , while when , this term vanishes, and the next highest-order term is of order . This explains the different error estimates in the formulas in Theorem 3. We include two auxiliary statements below (Lemmas 5 and 6). They are needed to perform the above-mentioned calculations. This finishes the sketch of the proof of Theorem 3.
Under the assumptions in Theorem 3, the following holds:
Proof. We have , where with and for all . It follows that . Since , and the random varaible has the same structure as the random variable in (14), it suffices to show that for every ,
Next, we observe that (the previous formula follows from (19)). This reduces the case where to the already considered case where . It follows from the previous reasoning that (25) holds. This completes the proof of Lemma 6.
as , while if , then
Our next goal is to characterize the asymptotic behavior of the distribution density of the integrated variance . The following statement holds.
(i) If , then
as , where
(ii) If , then
as , where
In particular, if the process is centered, then (36) holds with
4. Asset price asymptotics
The model in (1) is described by a linear stochastic differential equation. Therefore, we have
The previous equality can be derived from the Doléans-Dade formula (see ). Since the processes and are independent, the following formula holds for the distribution density of the asset price :
In (41), is the distribution density of the random variable The function is called the mixing density. The proof of formula (41) can be found in  (see (3.5) in ). It is not hard to see that where the symbol stands for the density of the realized volatility .
Suppose first that the volatility process is such that . It follows from formula (34) that
as , where
Our next goal is to estimate the function . The asymptotic behavior as