
Abstract
It is known that an adaptation of Newton's method allows for the computation of functional inverses of formal power series. We show that it is possible to successfully use a similar algorithm in a fairly general analytical framework. This is well suited for functions that are highly tangent to identity and that can be expanded with respect to asymptotic scales of “explog functions”. We next apply our algorithm to various wellknown functions coming from the world of quantitative finance. In particular, we deduce asymptotic expansions for the inverses of the Gaussian and the Black–Scholes functions.
Keywords: Asymptotic expansion, algorithm, pricing, Hardy field, explog function, Black–Scholes formula
One notoriously complex problem in finance is the pricing of derivative products that are frequently traded on financial markets. Practitioners have proposed various sophisticated models for the dynamics of financial assets. In particular, it has been necessary to account for the existence of Ushaped “volatility smiles” which play a central role in the pricing of socalled vanilla options. Some models seem more reasonable than others because they explain not only the volatility smile, but also have properties that are directly exploitable in practice, notably the existence of easily implementable pricing formulas involving mathematical parameters that are easy to calibrate.
Subsequently, the volatility smile has been studied in a fairly general way, with a minimum of hypotheses on the probabilistic distribution of the assets [2, 1, 21, 6, 12]. This has made it possible to isolate intrinsic behaviours that are shared by a large number of models in the study of volatility smiles.
The next step has been to study the volatility smile in a modelfree setting. This ultimately leads to focusing not on the Black–Scholes formula itself but on its inverse [26, 10, 8, 32]. A notable advantage of this approach is that it simplifies pricing problems. Indeed, in the case of vanilla options, such problems usually do not admit closed form solutions (except in the Black–Scholes model), so we need to resort to approximate solutions. Different techniques have been proposed to this purpose: perturbation methods with partial or stochastic differential equations, Lie symmmetry theory, Watanabe theory, heat kernel expansion theory and Minakshisundaran–Pleijel's formula, large deviation theory, etc. [23, 22, 16, 9, 13, 5]. Most of these techniques give the asymptotics of price for large or small values of certain parameters involved in the computation of option prices. The study of the inverse function of the Black–Scholes formula then transforms vanilla option price asymptotics into implicit volatility asymptotics, which is the quantity of interest.
The problem of inverting Black–Scholes formula is challenging because of its nonanalytic boundary behaviour. In fact, since the Black–Scholes model (as any other stochastic model) uses Brownian motion, it is not surprising that the asymptotics of the BlackScholes formula involves logarithms. More precisely, after a suitable change of variables, the relation between vanilla option price and volatility can be expressed via an asymptotic expansion
where are polynomials in [8, 10]. In particular, this means that
for every . We are interested in computing a similar expansion for in terms of .
In computer algebra, various techniques have been developed for asymptotic expansions in general asymptotic scales. For instance, several algorithms exist for the asymptotic expansion of “explog” functions [30, 11, 25, 18, 31]. Such functions are built up from the rationals and an infinitely large variable using the field operations, exponentiation and logarithm. An example of an explog function is . The theory of transseries [7, 17, 19] makes it possible to cover asymptotic expansions of an even wider class of functions comprising many formal solutions to nonlinear differential equations.
Several algorithms also exist for the functional inversion of explog functions [28, 29]. However, the righthand side of (1) is usually not an explog function, so these algorithms cannot be applied directly. When considering as a formal transseries, there are also methods for computing the formal inverse of [17, 19]. However, a priori, the analytic meaning (2) is lost during such formal computations. In this paper, we will show how to invert asymptotic expansions of the form (1) from the analytic point of view.
For each , let be the ring of fold continuously differentiable functions at infinity (). Then is a differential ring. We recall that a Hardy field is a differential subfield of . It is wellknown that Hardy fields [14, 15, 3] provide a suitable setting for asymptotic analysis. In section 2, we will introduce the abstract notion of an “effective Hardy field”, which formalizes what we need in order to make this asymptotic calculus fully effective. Typical examples of effective Hardy fields are generated by explog functions. For instance, in Sections 2.3 and 2.4, we will show that is effective Hardy field. Using the aforementioned work on expansions of explog functions, it is possible to construct various other effective Hardy fields.
Let be a Hardy field. We say that with is steep if for any , there exists a with . An element is said to be highly tangent to identity if there exists a with . For instance, if , then is steep and is highly tangent to identity, contrary to . Now assume that is an effective Hardy field. We say that a germ admits an effective asymptotic expansion over if for every we can compute an element with . If is highly tangent to identity and , then we will prove in Section 3 that admits a functional inverse that also admits an effective asymptotic expansion over . Applied to the case when , this gives an algorithm for inverting asymptotic expansions of the form (1). Our algorithm relies on two main ingredients: Taylor's formula for right composition with functions that are highly tangent to identity, and Newton's method for reducing functional inversion to functional composition.
For our application to mathematical finance, it would have sufficed to work with the particular effective Hardy field . There are several reasons why we have chosen to prove our main result for general effective Hardy fields. First of all, the more general result may be useful in other areas such as combinatorics [27]. Indeed, functional inverses frequently occur when analyzing asymptotic behavior using the saddle point method. Secondly, our general setup only requires a moderate “investment” in the terminology from Section 2. Finally, it is natural to prove the results from Section 3 in this setup; the proofs would not become substantially shorter in the special case when .
This paper contains three main contributions. As far as we are aware, the application of modern asymptotic expansion algorithms to mathematical finance is new. Secondly, we introduce the framework of effective Hardy fields which we believe to be of general interest for effective asymptotic analysis. One major advantage of this framework is that it separates the potentially difficult question of constructing a suitable effective Hardy field from its actual use. The existing literature on explog functions and transseries can be put to use for such constructions. But for various other problems, it suffices to assume the effective Hardy field to be given as a blackbox. The third contribution of this paper is to show that this is particularly the case for the inversion of asymptotic expansions that are “highly tangent to identity”.
Acknowledgment. We are very grateful to Martino Grasselli for his encouragement to write and submit the article and for the careful reading of our work.
Consider the differential ring , where denotes the ring of fold continuously differentiable functions at infinity () for each . We recall that a Hardy field is a differential subfield of . Since any non zero element of Hardy fields is invertible, the sign of is ultimately constant for . We define if is ultimately positive. It can be shown that this gives the structure of an ordered field.
The wellknown asymptotic relations , , and can be defined in terms of the ordering on : given , we write
and
The quasiordering is total on : given , we have .
Example
Example
Let be a Hardy field. Given , let us show that
Let us first assume that , whence , and let and be such that for all . Modulo a further increase of , we may assume without loss of generality that the signs of and are constant for . Then, for all , we have
(5) 
Consequently, for suitable integration constants . If , then this yields . If and , then we may take in (5), so that , and we again obtain . If and , then we clearly have . This proves that , which implies (4). One proves and (3) in a similar way.
Let be a Hardy field. We say that is effective if its elements can be represented by instances of a concrete data structure and if we have algorithms for carrying out the basic operations , as well as effective tests for the relations , , and .
In particular, the effective inequality test for yields an equality test. Inversely, if we have an algorithm to compute signs of elements in , then this yields effective inequality tests for both and . Similarly, if, given , we have a way to test whether and , then this yields effective tests for the relations and . Indeed, given and , we have and .
Example
Example
Example
Let and let be such that , . We define the flatness relations , and by
Let denote the logarithmic derivative of a function . Taking logarithms, and using (3) and (4), we observe that
for all and .
An element is said to be steep if (whence ) for all . If , then this allows us to define a valuation with respect to : we set for and . Notice that the corresponding valuation group is a subgroup of . In particular, is archimedean. For and , we notice that
Indeed, since and , it suffices to show this for . Now assume that . Then , whence for some constant . It follows that . If , then we also notice that . Indeed, and , whence .
Two examples of steep elements are in and in . The aim of the remainder of this section is to generalize Example 4 and prove in particular that is indeed an effective Hardy field.
Let be an effective Hardy field and let be such that for all . By what precedes, this implies that for all . We claim that is again an effective Hardy field. Modulo the replacement of by (and by ), we may assume without loss of generality that and . We clearly have algorithms for the field operations of . Using the rule , it is also straightforward to compute derivatives of elements of .
Now consider a polynomial . If , then for each , we have , so that . Hence implies . This also shows that , which provides us with an effective zero test for , as well as for . Given a rational function with and , we also have . Consequently, and if and only if or and . Similarly, if and only if or and .
Example
Remark
Let be a Hardy field. Given , there exists a unique with , which is called the limit of , and denoted by . We say that is closed under limits if for all . If is effective and is computable, then we say that admits an effective limit map.
An asymptotic scale for is a multiplicative subgroup such that is totally ordered for and such that there exists a mapping with for all . We call the dominant monomial of and notice that is necessarily a group homomorphism. If is effective and is computable, then we call an effective asymptotic scale.
Assume that is closed under limits and that also admits an asymptotic scale . Given , we call the dominant term of , and notice that . If and are both computable, then the same clearly holds for .
Example
More generally, let be an effective Hardy field and let be as in Section 2.4. Assume that admits an effective limit map and that is an effective asymptotic scale. For each , we have shown how to compute an equivalent with and . Since for any and , the group is totally ordered for . This shows that admits both an effective limit map and an effective asymptotic scale .
Example
Let be a Hardy field which contains the identity function , as well as a steep element . If , then also assume that .
An element is said to be highly tangent to identity if there exists a with . Equivalently, this means that is of the form with . If , then this is the case when for some . If , then we rather should have for some . In particular, in both cases we have and even . We will denote by the subset of of all elements that are highly tangent to identity.
Since Hardy fields are not necessarily closed under composition and functional inversion, the set does not necessarily form a group. The main aim of this section is to show that a suitable completion of does form a group (Theorem 20 below). Moreover, under suitable hypothesis, there are algorithms for computing asymptotic expansions of compositions and functional inverses.
Lemma
Proof. Without loss of generality, we may assume that . For any , we claim that . Indeed, given , let be such that has constant sign and for . Assume also that is defined for . Then
for all . We conclude that , by letting tend to zero.
The assumption that implies that , whence is strictly increasing for sufficiently large . This shows that indeed admits an inverse function at infinity. Let be such that for sufficiently large . Setting and , our claim implies
Lemma
Proof. Since is a steep element, there exists a constant with . We also notice that . Indeed, this is immediate if . If , then for some and , since .
Let us first show that , whenever and . Since implies , the function is ultimately decreasing. For sufficiently large , it follows that for , whence
Since , this shows that .
Let us next show that we also have in the case when and (so that ). Then Lemma 10 implies , whence for some . Let . By what precedes, there exists an with for all . Modulo a further increase of , we may also arrange that is monotonic for . It follows that , whence . Postcomposing with , we again obtain .
Let us finally assume that . Then the above arguments prove that . Consequently, .
The above argments conclude the proof in the case when and . Let us next consider the case when we still have , but is general. Let be such that . For sufficiently large , it follows that is comprised between and , which are both equivalent to . This shows that .
Lemma
Proof. Let us first consider the case when and consider
For sufficiently large , Taylor's formula with integral remainder yields
For sufficiently large , the function is also monotonic, whence
By Lemma 11, we have , whence . This completes the proof in the case when .
As to the general case, we have
for all sufficiently large . Now Lemmas 10 and 11 imply and similarly . Consequently,
Lemma
Proof. Let us first consider the case when , so that . For any , we have , whence . Consequently,
It thus suffices to take in order to ensure that and therefore .
Assume next that , so that . We again have for all , but this time, we rather obtain , since . Therefore,
If is an effective Hardy field, then the above lemmas lead to the following algorithm for approximate composition:
Algorithm compose
Moreover, for all with , and , we have
Let be minimal with
Return
Theorem
Remark
A wellknown way to solve functional equations of the form is Newton's method [4]. We will now show that this method indeed yields a quadratic convergence in our setting.
Lemma
Then .
Proof. Since , we notice that and . Let . For all sufficiently large , we have
whence, using the ultimate monotonicity of on ,
Using Lemma 11, we also have and , whence
Consequently,
Now implies and . Consequently,
If is an effective Hardy field, then this lemma leads to the following algorithm for the computation of approximate functional inverses:
Algorithm invert
Moreover, for any with and , we have
Let
repeat
Let
If then return
Let
Let
Theorem
Proof. Let us first show that throughout the algorithm. This is clear at the start. At each iteration , Remark 15 implies and , whence , so that .
On termination, we have and , whence . Applying Lemma 10 with and in the roles of and , we obtain . Consequently, . Furthermore, .
As to the termination, consider the quantity
We now extend the definition of high tangency to identity to all germs. We say that a germ is highly tangent to identity if there exists a with and . We denote by the set of such germs. We say that admits an asymptotic expansion over if for every , there exists an element with . If we have an algorithm for computing as a function of , then we say that admits an effective asymptotic expansion over .
Proposition
Proof. Given , we may compute and with and . Assume that there exists an with . Then for all , we must have and . Consequently, we may compute , and . If for all , then we also have for all .
Proposition
Combining these two propositions, we have shown the following:
Theorem
Example
Example
and let be the Gaussian law:
Then, the wellknown relation
valid for any , shows that
with , , and for . Our algorithm now allows us to compute the asymptotic expansion of the inverse function of Gaussian law at . This is potentially of great interest in finance when it comes to calculate “valuesatrisk”. Such computations are imposed by regulators to manage market risks, among others.
Example
Taking logarithms, we get
with .
Example
(11) 
In the Black–Scholes model, the dynamics of is assumed to be lognormal:
where is a Brownian motion and is a constant parameter called volatility. In this framework, the well known Black–Scholes formula gives the price of any call option. It can be shown that with
and
To simplify, we have assumed that the interest rate is . If are fixed, then it is easy to see that the function
is nondecreasing and one to one from to . Therefore, in an a priori non Black–Scholes world and for a given call option price observed on the market, there is a unique solution (or simply ) of the equation
We call the Black–Scholes implied volatility associated to and . For different reasons, it is interesting to invert the Black–Scholes function in (15) [8]. For instance, very often, and using techniques like perturbation theory, sophisticated stochastic models (in a non Black–Scholes world) give only asymptotic expansions of an option price in terms of the maturity , whereas we really need a formula for the implied volatility [2, 16]. Indeed, call option prices are generally quoted in term of implied volatilities (and not as prices). This can be achieved in the following manner. In the Black–Scholes model and under the conditions that and , it can be proved that the asymptotic expansion of the “time value” of the call price , defined by
is given by
with arbitrarily large,
and for , [10]. Therefore, if we set
and
then, for any integer ,
with and for . Hence we get an asymptotic expansion for in terms of .
At the limit when , the first author previously obtained a similar result [10]. Setting this time
we have
where defined by
Therefore, we get
with , and for .
Example
For , our algorithm yields:
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