> <\body> <\hide-preamble> \; <\doc-data|> \; |<\author-affiliation> Léonard de Vinci Pôle Universitaire Research Center 92916 Paris La Défense Cedex France |>>||<\author-affiliation> LIX, CNRS École polytechnique 91128 Palaiseau Cedex France |>>> 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 and that can be expanded with respect to asymptotic scales of ``exp-log functions''. We next apply our algorithm to various well-known functions coming from the world of quantitative finance. In particular, we deduce asymptotic expansions for the inverses of the Gaussian and the Black\UScholes pricing functions. ||> 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 \Pvolatility smiles\Q which play a central role in the pricing of so-called 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 . 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 model-free setting. This ultimately leads to focusing not on the Black\UScholes formula itself but on its inverse . 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\UScholes 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 symmetry theory, Watanabe theory, heat kernel expansion theory and Minakshisundaran\UPleijel's formula, large deviation theory, etc. . 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\UScholes formula then transforms vanilla option price asymptotics into implicit volatility asymptotics, which is the quantity of interest. The problem of inverting Black\UScholes formula is challenging because of its non-analytic boundary behaviour. In fact, since the Black\UScholes model (as any other stochastic model) uses Brownian motion, it is not surprising that the asymptotics of the Black-Scholes formula involves logarithms. More precisely, after a suitable change of variables, the relation between vanilla option price and volatility can be expressed an asymptotic expansion <\eqnarray*> |>|+|x>+|x>+\,>>>> where ,\,\> are polynomials in . In particular, this means that <\eqnarray*> ||+|x>+\+|x>+|)>,>>>> 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\Pexp-log\Q functions. Such functions are built up from the rationals and an infinitely large variable \> using the field operations, exponentiation and logarithm. An example of an exp-log function is -x*log x|)>/log log +3|)>>. The theory of transseries makes it possible to cover asymptotic expansions of an even wider class of functions comprising many formal solutions to non-linear differential equations. Several algorithms also exist for the functional inversion of exp-log functions. However, the right-hand side \x+\+\*x+\*x> of() is usually not an exp-log function, so these algorithms cannot be applied directly. When considering > as a formal transseries, there are also methods for computing the formal inverse =y+\+\*y+\*y+\> of > . However, , the analytic meaning() is lost during such formal computations. In this paper, we will show how to invert asymptotic expansions of the form() 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 is a differential subfield of >>. It is well-known that Hardy fields provide a suitable setting for asymptotic analysis. In section, we will introduce the abstract notion of an \Peffective Hardy field\Q, which formalizes what we need in order to make this asymptotic calculus fully effective. Typical examples of effective Hardy fields are generated by exp-log functions. For instance, in Sections and, we will show that ,\>|)>> is effective Hardy field. Using the aforementioned work on expansions of exp-log functions, it is possible to construct various other effective Hardy fields. Let be a Hardy field. We say that \K>\\> with => is if for any K>, there exists a \> with |)>>. An element K> is said to be if there exists a 0> with /x=|)>>. For instance, if >, then =x> is steep and x/x> 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 \K> with =|)>>. If> is highly tangent to identity and =>, then we will prove in Section 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(). 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. Indeed, functional inverses frequently occur when analyzing asymptotic behavior using the saddle point method. Secondly, our general setup only requires amoderate \Pinvestment\Q in the terminology from Section. Finally, it is natural to prove the results from Section 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 exp-log 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 \Phighly tangent to identity\Q. Consider the differential ring >\\>>, where > denotes the ring of -fold continuously differentiable functions at infinity (\>) for each . We recall that a is a differential subfield of >>. Since any non zero element of Hardy fields is invertible, the sign of > is ultimately constant for \>. We define 0> if > is ultimately positive. It can be shown that this gives the structure of an ordered field. The well-known asymptotic relations >, >, > and > can be defined in terms of the ordering on: given K>, we write <\equation*> |>|>|g>|>|B\\>,\B*>>|>|>|g>|>|\\\>,\\*>>>>> and <\eqnarray*> |g>|>|g\f>>|g>|>|g.>>>> The quasi-ordering > is total on >>: given K>>, we have g\g\f>. <\example> The set > of germs at infinity is the smallest subset of >> that contains> and the identity function, and which is closed under , , >, , and . For instance, -x*log x|)>/+\*log log x\>. In his founding work, Hardy showed that > forms aHardy field. <\example> More generally, given a Hardy field , its Liouville closure > is the smallest subset of >> that contains and that is stable under , , >, , , and integration. It is well known that > is again a Hardy field. Let be a Hardy field. Given K>, let us show that <\eqnarray*> g\g\1>|>|\g>>|g\g\1>|>|\g.>>>> Let us first assume that \g>, whence \f>, and let \\> and 0> be such that |\|>\A*|\|>> for all x>. Modulo a further increase of >, we may assume without loss of generality that the signs of > and > are constant for x>. Then, for all \x>, we have <\equation> >g*\ t|\|>=>|\|>*\ t\A*>|\|>*\ t=A*>f*\ t|\|>. Consequently, f+b> for suitable integration constants \>. If 1>, then this yields f>. If 1> and 1>, then we may take =\> in(), so that , and we again obtain f>. If 1> and 1>, then we clearly have 1\f>. This proves that \g\f\g\g\1>, which implies(). One proves \g\f\g\g\1> and() in a similar way. Let be a Hardy field. We say that is 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 K>, we have a way to test whether 1> and 1>, then this yields effective tests for the relations > and >. Indeed, given K> and K>>, we have g\f/g\1> and g\f/g\1>. <\example> Let us show that > is an effective Hardy field. The basic operations , , >, and > can clearly be carried out by algorithm, and it is also clear how to do the equality test. Now consider *x+\+P|)>/*x+\+Q|)>\K>> with ,\,P,Q,\,Q\\> and \0>, \0>. Then /Q|)>*x>. Consequently, =sign/Q|)>> and 1\p\q> ( 1\p\q>). <\example> We claim that > is an effective Hardy field. As above, the basic operations , , >, , > and the equality test are straightforward. Now any non zero element K>> can be written as a fraction *x+\+P|)>/*x+\+Q|)>\K>> with ,\,P,Q,\,Q\\> and \0>, \0>. Similarly, we may write \ /Q=*+\+A|)>/*+\+B|)>\K>> with ,\,A,B,\,B\\> and \0>, \0>. Then /B|)>*x*>. Consequently, =sign/B|)>> and 1\\> ( 1\\>). Here we used the lexicographical ordering on pairs: \> if and only if q> or and b>. <\example> Let be an effective Hardy field and let \K> be such that \0> and \1>. Then \0>, whence > is ultimately strictly increasing and invertible for composition. Let =\> be the inverse of > and assume that \\\K>. Then \=\:f\K|}>> is again an effective Hardy field. Indeed, since right composition preserves the field operations and the ordering, \> is effectively isomorphic to as an ordered field. The derivation on \> is given by \|)>=\\|)>\f|)>\\>. Let K>> and let ,\=\1> be such that >\1>, >\1>. We define the >, > and > by <\eqnarray*> g>|>|c\\>,>\*c>>>|g>|>|c\\>,>\*c>>>|g>|>|g\f.>>>> Let >=f/f> denote the logarithmic derivative of a function . Taking logarithms, and using() and(), we observe that <\eqnarray*> g>|>|\log \f>\g>>>|g>|>|\log \f>\g>>>|g>|>|\log \f>\g>,>>>> for all K>> and K>=K:h\1|}>>. An element \K>> is said to be if \> (whence >\\>>) for all K>>. If \1>, then this allows us to define a valuation with respect to >: we set >=lim >/\>|)>> for K>> and >=\>. Notice that the corresponding valuation group >=im v>> is a subgroup of >. In particular, >> is Archimedean. For K> and K>>, we notice that <\eqnarray*> g>|>|>\v>.>>>> Indeed, since g\f/g\1> and >=v>-v>>, it suffices to show this for . Now assume that v>\0>. Then >\*\>>, whence \log \+C\log \> for some constant \>. It follows that \\1>. If \x>, then we also notice that >>|)>=0>. Indeed, \x\log \\log x\\>=|)>\\x> and >\x\>|)>\\\>/\>\1>, whence >>|)>=lim \\>/\>=0>. Two examples of steep elements are in > and >> in ,\>|)>>. The aim of the remainder of this section is to generalize Example and prove in particular that ,\>|)>> is indeed an effective Hardy field. Let be an effective Hardy field and let \K>=K:h\1|}>> be such that >\\> for all K>. By what precedes, this implies that \\>\f> for all K>. We claim that K|)>> is again an effective Hardy field. Modulo the replacement of > by |\|>> (and > by >), we may assume without loss of generality that \0> and \1>. 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 |)>=P*\+\+P\K|]>>. If \0>, then for each p>, we have /P\\>, so that *\\P*\>. Hence \0> implies |)>\P*\>. This also shows that |)>=0\P=\=P=0>, which provides us with an effective zero test for |]>>, as well as for. Given a rational function |)>/Q|)>=*\+\+P|)>/*\+\+Q|)>\L> with \0> and \0>, we also have |)>/Q|)>\/Q|)>*\>. Consequently, =sign/Q|)>> and 1> if and only if q> or and \Q>. Similarly, 1> if and only if q> or and \Q>. <\example> Starting with > as in Example, applying the above argument twice shows that both |)>> and ,\>|)>=\,\>|)>> are effective Hardy fields. Applying Example for =log x>, we also obtain that ,x,x|)>> is an effective Hardy field. <\remark> In order to compute with more general exp-log germs in >, one also needs to show that fields such as ,\*x>|)>> form effective Hardy fields. One even more difficult problem is to provide an effective zero test for exp-log constants, constants formed from the rationals, using , , >, , and . Provided that Schanuel's conjecture holds, such an algorithm was given by Richardson. His algorithm always returns correct results, but might not terminate if one explicitly hits a counterexample to the conjecture. Given a zero-test for exp-log constants, it can be shown that > forms an effective Hardy field. Let be a Hardy field. Given K>=\K:\\1|}>>, there exists a unique \\> with \1>, which is called the of , and denoted by =lim f>. We say that is if K> for all K>. If is effective and >\K> is computable, then we say that admits an . An for is a multiplicative subgroup \K>> such that > is totally ordered for> and such that there exists a mapping :K>\\> with \f> for all K>>. We call > the 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 K>>, we call =|)>*\> the of , and notice that \>. If > and > are both computable, then the same clearly holds for >. <\example> In Example, we have given a method for the explicit computation of an equivalent in >*x>=:c\\>,k\\|}>> for any \>>. This both shows that > admits an effective limit map and that it admits >> as an effective asymptotic scale. Similarly, Example shows that the same holds for >, in which case the asymptotic scale becomes >*x>>. More generally, let be an effective Hardy field and let ,\> be as in Section. Assume that admits an effective limit map and that > is an effective asymptotic scale. For each K|)>>>, we have shown how to compute an equivalent g*\\\*\> with K>> and \>. Since \> for any \> and 0>, the group *\>> is totally ordered for >. This shows that |)>> admits both an effective limit map and an effective asymptotic scale *\>>. <\example> Let be an effective Hardy field and let > be as in Example. If admits an effective limit map, then so does \>, since \=lim f> for all K>>. If admits an effective asymptotic scale >, then \> admits \\> as an effective asymptotic scale, with \|)>=\\\> for all K>>. Let be a Hardy field which contains the identity function , as well as a steep element \K,\>=\K>:\\1|}>>. If \x>, then also assume that =x>. An element K> is said to be if there exists a 0> with /x=|)>>. Equivalently, this means that is of the form > with >|)>\v>>. If =x>, then this is the case when \\>> for some \-1>. If \x>, then we rather should have \\>> for some \0>. In particular, in both cases we have \1> and even >|)>\0>. 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 below). Moreover, under suitable hypothesis, there are algorithms for computing asymptotic expansions of compositions and functional inverses. <\lemma> Let \T-x>. Then for any germ \>> with \\> and \1>, we have <\eqnarray*> |)>-x>|||)>.>>>> <\proof> Without loss of generality, we may assume that \0>. For any \>, we claim that \|)>\\>. Indeed, given \0>, let > be such that > has constant sign and |\|>\\> for x>. Assume also that |)>> is defined for x>. Then <\eqnarray*> |)>-\|\|>>|>|>\*\ t|\|>\\**\,>>>> for all x>. We conclude that \|)>-\\\>, by letting> tend to zero. The assumption that \1> implies that |)>\1>, whence \x+\> is strictly increasing for sufficiently large . This shows that > indeed admits an inverse function > at infinity. Let 0> be such that |\|>\A*\> for sufficiently large . Setting =x-2*A*\> and =x+2*A*\>, our claim implies <\equation*> ||)>>||+\|)>>|>|+A*\|)>>|>|+2*A*\>||>||)>>||+\|)>>|>|-A*\|)>>|>|-2*A*\>||>>>> for sufficiently large . Since > is strictly increasing, it follows that \\\r>. In other words, -x|\|>\2*A*\> for sufficiently large . <\lemma> Let K> and \T>. Then for any germs ,\\>> with \f> and \\>, we have <\eqnarray*> \|)>>||.>>>> <\proof> Since > is a steep element, there exists a constant 0> with >|\|>\A*>|\|>>. We also notice that >*\\1>. Indeed, this is immediate if =1/x>. If \x>, then \\> for some 0> and >*\\\>*\\|)>\1>, since \x>. Let us first show that g\f>, whenever x> and x>. Since x> implies \1>, the function|\|>> is ultimately decreasing. For sufficiently large , it follows that |\|>\|\|>> for |]>>, whence <\equation*> |)>-f|\|>\>|\|>*\ t\|\|>*\\A*>|\|>*\*|\|>. Since >*\\1>, this shows that g\f>. Let us next show that we also have g\f> in the case when x> and x> (so that \0>). Then Lemma implies =x+|)>>, whence g\x-B*\> for some \>>. Let \0>. By what precedes, there exists an > with |)>-f|\|>\\*|\|>> for all x>. Modulo afurther increase of >, we may also arrange that > is monotonic for x>. It follows that |)>-f|\|>\\*|\|>>, whence g\f>. Post-composing with , we again obtain g\f>. Let us finally assume that x>. Then the above arguments prove that \g\>. Consequently, g=\g|)>\=f>. The above arguments conclude the proof in the case when =f> and =\>. Let us next consider the case when we still have =f>, but \\> is general. Let 0> be such that |\|>\B*|\|>>. For sufficiently large , it follows that |)>> is comprised between |\|>|)>> and |\|>|)>>, which are both equivalent to >. This shows that |)>\f>. As to the general case, let 0> be such that |\|>\C*>. By what precedes, we have |)>|\|>\C*|)>|\|>\2*C*|\|>> for all sufficiently large . This shows that \|)>\f>. <\lemma> Let K> and T>. Let ,\\K> and \> be such that \T> and >\\\>. Then for any ,\\>> with \\> and \\>, we have <\eqnarray*> |)>\|)>>||\+\+!>*f>\+|\|>,*\|\|>|)>|)>.>>>> <\proof> Let us first consider the case when =\=0> and consider <\eqnarray*> >||*+\+!>*f>*>>|||g-\>>>> For sufficiently large , Taylor's formula with integral remainder yields <\eqnarray*> >||>!>*f>*-t|)>*\ t.>>>> For sufficiently large , the function >> is also monotonic, whence <\eqnarray*> |\|>>|>|*max>|\|>,>|)>|\|>|)>*-x|\|>.>>>> By Lemma, we have >\g\f>>, whence f>*\\>. This completes the proof in the case when =\=0>. As to the general case, we have <\equation*> +\|)>-f|)>|\|>\>+\>|\|>*\ t\max|)>|\|>,+\|)>|\|>|)>*|\|>, for all sufficiently large . Now Lemmas and imply \|)>=\\\g|)>\g\\\g\\> and similarly \|)>\f>. Consequently, <\eqnarray*> |)>\|)>-\|\|>>|>|\|)>|\|>+|)>-f\g|\|>+g-\|\|>>>||>|\|)>|\|>,*\|\|>,|\|>|)>>>||>|*\|\|>,|\|>|)>.>>>> This concludes the proof in the general case. <\lemma> For any K>, T> and \K>>, there exists an \> with >\\\>. <\proof> Let us first consider the case when =x>, so that >\-1>. For any K>>, we have =f>*f\\>*f\f/x>, whence >|)>\v>+1>. Consequently, <\eqnarray*> >>*|)>>|>|>+n+n*v>.>>>> It thus suffices to take >|)>-v>|)>/>+1|)>> in order to ensure that >>*|)>\v>|)>> and therefore >*\\>. Assume next that \x>, so that >\0>. We again have \\>*f> for all K>>, but this time, we rather obtain >|)>\v>>, since >>|)>=0>. Therefore, <\eqnarray*> >>*|)>>|>|>+n*v>.>>>> Taking >|)>-v>|)>/v>>, we again obtain the desired result. If is an effective Hardy field, then the above lemmas lead to the following algorithm for approximate composition: <\named-specified-algorithm|compose|)>>> K>, T> and \K>> with >|)>\v>|)>> K> with g=h+|)>> Moreover, for all ,\\>> with \\>, *f\\> and >/x|)>\0>, we have |)>\|)>=h+|)>> <|named-specified-algorithm> Let \> be minimal with >\\\> Return +!>*f>\> <\theorem> The algorithm > is correct. <\proof> The existence of is ensured by Lemma. Since is effective, we have an algorithm for doing the test >*\\>, which enables us to compute . Setting =\/f>, our assumption that >|)>\v>|)>> ensures that \T>. The result now follows from Lemma. <\remark> In addition, by considering both cases =x> and \x>, it can be verified that >=v>>, that T> implies T>, and that >\0> implies >\0>. A well-known way to solve functional equations of the form g=x> is Newton's method. We will now show that this method indeed yields a quadratic convergence in our setting. <\lemma> Let T> and \K> be such that g-x=|)>> and >|)>\0>. Let \T> be such that <\eqnarray*> >||g-x|f\g>+|)>.>>>> Then -x=|)>>. <\proof> Since x>, we notice that \1> and \g\1>. Let =g-=g-x|f\g>+|)>=|)>>. For all sufficiently large , we have <\eqnarray*> |)>>|||)>-f|)>*\+>-\>f*-\-t|)>*\t,>>>> whence, using the ultimate monotonicity of > on ,|]>>, <\eqnarray*> >||g-\g|)>*\+\g|\|>,\|\|>|)>*\|)>.>>>> Using Lemma, we also have \g\f> and \\f>, whence <\eqnarray*> >||g-\g|)>*\+*\|)>.>>>> Consequently, <\eqnarray*> -x>||g-x|)>-\g|)>*\+*\|)>>>|||\g|)>*\+|)>-\g|)>*\+*\|)>>>||||)>+*\|)>.>>>> Now x> implies \1\log x> and =\x>. Consequently, <\equation*> f\-x=|)>+*\|)>=|)>+/x|)>=|)>. This completes the proof. If is an effective Hardy field, then this lemma leads to the following algorithm for the computation of approximate functional inverses: <\named-specified-algorithm|invert|)>>> T> and \K>> with >|)>\0> T> with =g+|)>> Moreover, for any \>> with \x*\> and \1>, we have |)>=g+|)>> <|named-specified-algorithm> Let x> <\indent> Let |)>> If x*\> then return Let ,g,x*\|)>> Let g-/d> <\theorem> Let =/x>. The algorithm > is correct and terminates after at most |log>|)>/v>|)>|)>/log 2|\>+1> iterations of the main loop. <\proof> Let us first show that T> throughout the algorithm. This is clear at the start. At each iteration g-/d>, Remark implies >\v>> and >\v>>, whence >/d|)>\v>>, so that /d\T>. On termination, we have g+|)>> and |)>>, whence g-x=|)>>. Applying Lemma with > and g-x> in the roles of > and>, we obtain \f-x=g|)>-x=|)>>. Consequently, =g\\f|)>=g+\\f-x|)>|)>>=g+|)>>. Furthermore, |)>=\f|)>\f|]>=\\f|)>>=\|)>|)>>=\|)>|)>>=f\|)>|)>=f+|)>*x*\|)>=g+|)>>. As to the termination, consider the quantity <\equation*> \\>g-x|x>|)>\sup \\:g-x|x>=>|)>|}>. At the very start, we have =>|)>=v>|)>\0>. At every iteration \g-/d>, we have =g-g-x|f\g>+|)>>. Lemma therefore ensures that > doubles at least, whereas the algorithm terminates as soon as \v>|)>>. This happens after at most |log>|)>/v>|)>|)>/log 2|\>+1> iterations. We now extend the definition ofhigh tangency to identity to all germs. We say that a germ >> is if there exists a 0> with |)>> and =1+>. We denote by >> the set of such germs. We say that admits an over if for every \>, there exists an element \K> with =|)>>. If we have an algorithm for computing > as a function of , then we say that admits an over . <\proposition> Assume that >> and >> admit effective asymptotic expansions over. Then so does g>. If >>, then g\>>. <\proof> Given 1>, we may compute \K> and \T> with \f-\\\> and \g-\\\>. Assume that there exists an \\> with \>>. Then for all n>, we must have \\>> and >|)>\n\n>. Consequently, we may compute =,\,\|)>>, and g=+\|)>\+\|)>=\+|)>>. If \> for all \>, then we also have g\\> for all\>. If >>, then we also get ,\,\\T>, whence >g-x|)>=v>-x+|)>|)>\min>-x|)>,n|)>\v>>. Moreover, g|)>=g\f\g=|)> \g|)>=1+>, whence g\>>. <\proposition> Assume that >> admits an effective asymptotic expansion over . Then so does > and \>>. <\proof> Given 1>, we may compute \T> with \f-\\\>. Let =,\/x|)>>. Then =+\|)>=\+|)>>. Moreover, |)>=\f|)>=|)>\f|)>=1+>, whence \>>. Combining these two propositions, we have shown the following: <\theorem> The set of germs in >> that admit effective asymptotic expansions over forms a group for functional composition.> function> The Lambert function is defined to be the inverse function ofx*\>. Using our algorithm, we can compute the asymptotic expansion of the inverse function |)>> of x+log x>. This also yields the asymptotic expansion of > for large . Let |)>0>> be defined formally by\ <\eqnarray*> 0>a*X|)>>||0>\*,\,a|)>*X>>>> and let > be the Gaussian function: <\equation*> \=>\*dt,\=>>*\|2>>. For any \>, we have the well-known relation <\eqnarray*> >|||2>>|>*x>**!!|x>+>|)>|)>,>>>> where we used the notation <\eqnarray*> !!>|>|.>>>> The relation() shows that <\eqnarray*> >||+|x>+>|)>,>>>> with 0>, <\eqnarray*> >|>||)>|)>>>|>|>|>|)>>>|>|>|,*!!|)>0|)>>>>> 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 risk measures. The formula() gives itself an asymptotic expansion of a Gaussian Value-at-Risk >> in terms of its confidence level >. The expected shortfall of a portfolio with confidence level \> is the expected loss conditional that the loss is greater than the >-th percentile of the loss distribution. When the return of the portfolio is Gaussian with mean > and volatility >, the expected shortfall is <\equation*> ES>=\+\*\\|)>|1-\>. With |)>>, the relation () yields <\eqnarray*> |1-\>>||*!!|x>+>|)>>.>>>> \ So, for some constants \\>, we have <\eqnarray*> >-\|\>>|||x>+>|)>.>>>> By inverting (), we get an asymptotic expansion of >> in terms of\1>. Let |)>0>> \ be defined by =1> and, for 0>, =*\>. A well-known relation for > tells that for \> and 0>,\ <\eqnarray*> >||*\*|x>+>|)>|)>.>>>> Taking logarithms, we get <\eqnarray*> >||*log*x+|x>+>|)>,>>>> with =-\*,\,u|)>>. Considering the CEV process <\equation*> \ F=\*F>*\*W, where is a Brownian motion and denotes for instance a forward rate, the probability of absorption at zero before -time is given by <\equation*> |)>>*\,*F>>|\*t>|)>,|\=|)>>\|\>. Inverting () gives a confidence interval where the probability of absorption of is lower than>, asymptotically, for \1>. By definition, a call option is a contract which gives to the owner the value -K,0|)>> at afuture -date (known today) called maturity of the contract, where > denotes the value at date (unknown today) of an asset (like a stock) whose initial value is today, and is a constant called strike (known today). The initial price of this contract is denoted by >. In general, by no-arbitrage arguments, the option price > is always greater than the \Pintrinsic value\Q > and lower than the spot value : <\equation*> \C\S In the Black\UScholes model, the dynamics of |)>> is assumed to be log-normal: <\eqnarray*> >||*S*\W>>>> where |)>\>> is a Brownian motion and > is a constant parameter called volatility. In this framework, the well known Black\UScholes formula gives the price of any call option. It can be shown that <\equation*> C=BS|)>, where <\eqnarray*> |)>>|||)>-K*\|)>,d>=|)>\*T|2>|\*>.>>>> For simplicity, we have assumed that the interest rate is . If are fixed, then it is easy to see that the function <\eqnarray*> >|>||)>>>>> is non-decreasing and one to one from >> to ,S|)>>. Therefore, in an non Black\UScholes world and for a given call option price ,S|)>> observed on the market, there is a unique solution > (or simply >) of the equation <\eqnarray*> |)>>||>>> We call > the Black\UScholes implied volatility associated to and . For different reasons, it is interesting to invert the Black\UScholes function in (). For instance, using techniques from perturbation theory, sophisticated stochastic models (in a non Black\UScholes world) give only asymptotic expansions of an option price in terms of the maturity, whereas we really need aformula for the implied volatility . 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\UScholes model and under the conditions that 1> and S>, it can be proved that the asymptotic expansion of the \Ptime value\Q of the call price |)>>, defined by <\eqnarray*> |)>>|>||)>-,>>>> is given by <\eqnarray*> >*>|>*|)>>||*\>*|2>*a|8>|)>*v+*\>|)>,>>>> with \> arbitrarily large, <\equation*> u\log*|)>,v\2**T|u> <\equation*> a\!!,f\|j!*!!>. Therefore, setting <\equation*> x\=*|)>|2*\*T>,y\-log|)>, we have <\eqnarray*> ||*log*x+\+|x>+>|)>,>>>> for any integer , where <\eqnarray*> >|>|*\>|4*>>|)>>>|>|>|**a|8>|)>,\,|2>*a|8>|)>|)>0|)>>>>> Formula () is nothing but another expression for Black-Scholes formula. Hence we get an asymptotic expansion for *T> in terms of ||>>. Note that () is an asymptotic expansion of acall price in terms of for large . So, it gives also an asymptotic expansion of a call price when |log*||>|>> is large i.e. small strike or large strike.\ Notice also that there is another direct formula =\*|2>|)>> when , which gives an asymptotic expansion of in terms of *>. At the limit when 1>, the first author previously obtained a similar result. Setting this time <\equation*> CC\S-BS(S,K,T,\) and <\equation*> x\*T|8>,y\-log|)>, we have <\eqnarray*> >*\>*>|||>*|2>*c|8>|)>*>+>*\|)>,>>>> where <\equation*> c\!!*g,g\|j!*!!>. Therefore, we get <\eqnarray*> ||*log*x+\+|x>+>|)>,>>>> where <\eqnarray*> >|>|**a|8>|)>,\,|2>*c|8>|)>|)>0|)>>>>> In the Bachelier model, the dynamics of |)>> is assumed to be normal: <\eqnarray*> S>||*\ W,>>>> where |)>\>> is a Brownian motion and > is a constant parameter called normal volatility. In this framework, the price of a call option with strike and expiry is given by Bachelier's formula <\equation*> C=B|)>, with <\equation*> B|)>=*\*>|)>+\**\*>|)>. Denoting as before by <\equation*> TV=C- the time-value of the call option, we have <\equation*> TV=|4*>>*\,|2*\*T>|)> for K>. So, using (), we deduce an asymptotic expansion of > in terms of |\*>> for large strike or small maturity in the Bachelier model. By inverting the Gamma function as before, this gives an asymptotic expansion of the normal volatility in terms of the time-value. Therefore, by comparing with (), we obtain an equivalence between normal volatility and lognormal volatility in the cases when 1> or S>. For example, when 1>, the first terms of the expansion are given by <\eqnarray*> >||*|~>-\log*log*m|m-1>|)>|>*|~>*T+O*log|)>,>>>> where denotes the \Pmoneyness\Q > and |~>=|S>>. Note that at the money (), we have a closed form formula: <\eqnarray*> >|||T>>*S**|2>|)>-1|)>.>>>> In particular, taking the derivative in () and then letting gives <\eqnarray*> >||-*|~>|)>**|~>*T|)>+o,>>>> where > and > are the slopes of the smiles of volatility at the money: <\equation*> \=*\|\*K>|)>,\=*|~>|\*K>|)>. In particular when the volatility smile |~>> is flat at the money (=0>), we have: <\eqnarray*> >||*|~>**|~>*T|)>+o.>>>> This is a refinement of a classical result \-*|~>>. In general, () shows that if \*|~>>, then the slope of the volatility smile for the lognormal volatility is negative and this condition holds upto order1 in . . By inverting (), we get the implied lognormal volatility > from the time-value of the call-option in a model-free setting. In general, thanks to Tanaka's theorem, this time-value can be calculated by integrating the density function of the stochastic process between and the maturity of the option. On the other hand, this density function can be obtained using Minakshisundaram\UPleijel expansion - modulo conditions of regularity . So, in general this allows to translate an asymptotic expansion of for 1> in an asymptotic expansion of >. As an example, let us consider a local volatility model S=f*\ W>. Then Tanaka's formula reads: <\eqnarray*> >||*f*p,t|)>*dt>>>> In this particular case, the Minakshisundaram\UPleijel expansion gives <\eqnarray*> ,t|)>>|||)>|t>>|*t>*f|)>>*u|)>*t+o|)>|)>,>>>> with |)>=> u|f>> and > is explictly given by induction . The asymptotic expansion() can be integrated by part and thus give an asymptotic expansion for > thanks to(). Finally, by inverting (), we get asymptotic expansion of the lognormal implied volatility. The stochastic differential equation for the CEV diffusion model is <\eqnarray*> S>||*S>*\ W>>>> Let ;T|)>> be the probability density function to get state > at -time starting from at time. There is a closed form formula for in terms of the modified Bessel function : <\eqnarray*> ,T|)>>||*T>*\|2*\*T>>*>|)>/2>*I|\|>>*>|\*T>|)>>>>> with =-1|)>>> (in the most popular cases we have \>) and <\eqnarray*> >*>||>+k+1|)>>*|)>>.>>>> The following well known asymptotic expansion for \> is due to Hankel: <\eqnarray*> ||>>|>||*z>>>*\1|8*z>>+\1|)>*\9|)>|2!*>>\\1|)>*\9|)>*\25|)>|3!*>>+\|)>.>>>> On the other hand, Tanaka's formula shows that the time-value <\equation*> TV=C- of the call-price > with strike and maturity is given by <\eqnarray*> >||*\*K>*p,u|)>*\ u>>>> For short maturities , the combination of() and() yields an asymptotic expansion of,T|)>>> that can be integrated by parts to any order with respect to . This leads to an asymptotic expansion of the time-value > of a call price. Using an asymptotic expansion of the inverse function of the Black-Scholes function seen in section, we deduce an asymptotic expansion of the implied lognormal volatility of the CEV model at any order in . implementation> We did an experimental implementation of our algorithm in the system. Each of the above examples comes down to the computation of the functional inverse of afunction > with an asymptotic expansion of the form <\eqnarray*> ||*log*x+\+|x>+>|)>.>>>> For , our algorithm yields: <\eqnarray*> ||*log-\+*log+\*\-\|)>*+*\*log+*\-+\|)>*\|)>*log+*\*\-*\-\|)>*\-\*\-\|)>*>+*\*log+**\-\|)>*\-\*\|)>*\-\*\-*\|)>*log+*-\*\+\|)>*\-*\-\|)>*\|)>*\-**\-2*\|)>*\+2*\*\|)>*\++\|)>*\-2*\*\*\-\*\+\*\-2*\*\|)>*log-*\-\|)>*\-\*\-\|)>*\+*-\*\+\|)>*\*\-**\-2*\|)>*\*\--\*\|)>*\-*\*\-\*\-2*\*\-\|)>*>+>|)>.>>>> In this paper, we have presented an algorithm for calculating asymptotic expansions of functional inverses of functions that are highly tangent to identity. In particular, we obtained asymptotic expansions of the implied volatility of an option call price at any order in a model-free setting. hen, we have shown how this can be applied to the CEV model. It is envisaged to apply these techniques to more sophisticated models such as the SABR model . More generally, it would be interesting to combine our approach with other algorithms for the calculation of heat kernel coefficients such as to get automatic asymptotic expansion at any order of the implied volatility of option call prices for a large class of financial models . <\bibliography|bib|tm-plain|> <\bib-list|37> I.G.AvramidiR.Schimming. , 150\U162. Vieweg+Teubner Verlag, Wiesbaden, 1996. I.G.Avramidi. . Birkhäuser, 2015. S.BenaimP.Friz. Smile asymptotics. ii. models with known moment generating functions. , 45(1):16\U23, 2008. H.Berestycki, I.FlorentJ.Busca. Asymptotics and calibration of local volatility models. , 2(1):61\U69, 2002. N.Bourbaki. . Éléments de Mathématiques (Chap. 5). Hermann, 2-nd, 1961. R.P.BrentH.T.Kung. |)>> algorithms for composition and reversion of power series. J.F.Traub, . Pittsburg, 1975. Proc. of a symposium on analytic computational complexity held by Carnegie-Mellon University. M.CraddockM.Grasselli. Lie symmetry methods for local volatility models. , 2016. S.De Marco, C.HillairetA.Jacquier. Shapes of implied volatility with positive mass at zero. , 8(1):709\U737, 2017. J.Écalle. . Hermann, collection: Actualités mathématiques, 1992. K.GaoR.Lee. Asymptotics of implied volatility to arbitrary order. , 18(2):349\U392, 2014. J.Gatheral, E.P.Hsu, P.Laurence, C.OuyangT.-H.Wang. Asymptotics of implied volatility in local volatility models. , 22(4):591\U620, 2012. P.K.F.Gatheral, A.Gulisashvili, A.JacquierJ.Teichmann. , 110. Springer-Verlag, 2015. C.Grunspan. A note on the equivalence between the normal and the lognormal implied volatility : a model free approach. , 2011. C.Grunspan. Asymptotic expansions of the lognormal implied volatility: a model free approach. 1112.1652, Arxiv, 2011. D.Gruntz. . , E.T.H. Zürich, Switzerland, 1996. A.Gulisashvili. Asymptotic formulas with error estimates for call pricing functions and the implied volatility at extreme strikes. , 1(1):609\U641, 2010. Kumar,D.,Lesniewski,A.Hagan, P.D.Woodward. Managing smile risk. , 84\U108, 2002. G.H.Hardy. . Cambridge Univ. Press, 1910. G.H.Hardy. Properties of logarithmico-exponential functions. , 10(2):54\U90, 1911. P.Henry-Labordère. A general asymptotic implied volatility for stochastic volatility models. cond-mat/050431, Arxiv, 2005. J.vander Hoeven. . , École polytechnique, Palaiseau, France, 1997. J.vander Hoeven. Generic asymptotic expansions. , 9(1):25\U44, 1998. J.vander Hoeven. , 1888. Springer-Verlag, 2006. J.vander Hoeven, G.Lecerf, B.Mourrain etal. Mathemagix. 2002. . R.Lee. The moment formula for implied volatility at extreme strikes. , 14(3):469\U480, 2004. Forde M.Jacquier A. Small-time asymptotics for an uncorrelated local-stochastic volatility model. , 18(6):517\U535, 2011. O.Osajima. The asymptotic expansion formula of implied volatility for dynamic sabr model and fx hybrid model. , 2007. D.Richardson. Zero tests for constants in simple scientific computation. , 1(1):21\U37, 2007. D.Richardson, B.Salvy, J.ShackellJ.vander Hoeven. Expansions of exp-log functions. Y.N.Lakhsman, , 309\U313. Zürich, Switzerland, July 1996. M.RoperM.M. Rutkowski. On the relationship between the call price surface and the implied volatility surface close to expiry. , 12(4):427\U441, 2009. B.Salvy. . , École Polytechnique, France, 1991. B.SalvyJ.Shackell. Asymptotic expansions of functional inverses. P.S.Wang, , 130\U137. New York, 1992. ACM Press. B.SalvyJ.Shackell. Symbolic asymptotics: multiseries of inverse functions. , 27(6):543\U563, 1999. J.Shackell. Growth estimates for exp-log functions. , 10:611\U632, 1990. J.Shackell. , 12. Springer-Verlag, 2004. M.R.Tehranchi. Uniform bounds for black\Uscholes implied volatility. , 7(1):893\U916, 2016. K.Yoshida. On the fundamental solution of the parabolic equation in a Riemannian space. , 1(1), 1953. > <\attachments> <\collection> <\associate|bib-bibliography> <\db-entry|+PXBVIVv9LqhElX|article|BF08> <|db-entry> P. > <\db-entry|+PXBVIVv9LqhElt|article|BBF02> <|db-entry> I. J. > <\db-entry|+HQXtNDqSXZLBBg|article|DMHJ13> <|db-entry> C. A. > <\db-entry|+PXBVIVv9LqhElg|article|Gul10> <|db-entry> > <\db-entry|+PXBVIVv9LqhElk|article|Lee05> <|db-entry> > <\db-entry|+bPGiDrVuGKP8cB|article|GL14> <|db-entry> R. > <\db-entry|+PXBVIVv9LqhElf|techreport|Gru11> <|db-entry> > <\db-entry|+PXBVIVv9LqhElo|article|RR09> <|db-entry> M. > <\db-entry|+PXBVIVv9LqhElp|article|Teh16> <|db-entry> > <\db-entry|+PXBVIVv9LqhEla|article|CG16> <|db-entry> M. > <\db-entry|+PXBVIVv9LqhEle|book|GGJT15> <|db-entry> A. A. J. > <\db-entry|+bPGiDrVuGKP8cC|article|HKLW02> <|db-entry> D. > <\db-entry|+PXBVIVv9LqhEli|techreport|Lab05> <|db-entry> > <\db-entry|+PXBVIVv9LqhElc|article|FJ11> <|db-entry> Jacquier > <\db-entry|+PXBVIVv9LqhElm|article|Osa07> <|db-entry> > <\db-entry|+Li3PZLCHp1NMdi|phdthesis|Gru96> <|db-entry> > <\db-entry|+Li3PZLCHp1NMmH|article|vdH:genae> <|db-entry> > <\db-entry|+63PGfIkmrfTqBB|inproceedings|vdH:issac96> <|db-entry> B. J. J. van der > > <\db-entry|+Li3PZLCHp1NMjv|article|Sh90> <|db-entry> > <\db-entry|+Li3PZLCHp1NMjz|book|Sh04> <|db-entry> > <\db-entry|+3a783HvX7LqvqJ|book|Ec92> <|db-entry> > <\db-entry|+Li3PZLCHp1NMmE|phdthesis|vdH:phd> <|db-entry> > <\db-entry|+Li3PZLCHp1NMmr|book|vdH:ln> <|db-entry> > <\db-entry|+63PGfIkmrfTqAl|inproceedings|SS92> <|db-entry> J. > > <\db-entry|+E00FWmNELVqHvw|article|SS99> <|db-entry> J. > <\db-entry|+Li3PZLCHp1NMaO|book|Bour61> <|db-entry> > <\db-entry|+63PGfIkmrfTq9W|book|Har10> <|db-entry> > <\db-entry|+Tg0iHv3bqcOqtY|article|Har11> <|db-entry> > <\db-entry|+Li3PZLCHp1NMjU|phdthesis|Sal:phd> <|db-entry> > <\db-entry|+Li3PZLCHp1NMj5|article|Rich07> <|db-entry> > <\db-entry|+63PGfIkmrfTq8c|inproceedings|BK75> <|db-entry> H. T. > |)>> algorithms for composition and reversion of power series> > <\db-entry|+HQXtNDqSXZLBBc|article|Gru11b> <|db-entry> > <\db-entry|+HQXtNDqSXZLBBZ|book|A15> <|db-entry> > <\db-entry|+HQXtNDqSXZLBBb|article|GHLOW12> <|db-entry> E. P. P. C. T.-H. > <\db-entry|+HQXtNDqSXZLBBe|article|Y53> <|db-entry> > <\db-entry|+3a783HvX7LqvqN|misc|vdH:mmx> <|db-entry> G. B. > > <\db-entry|+VqcQmXIyaUWNA4|inbook|AS96> <|db-entry> R. > > <\references> <\collection> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > <\auxiliary> <\collection> <\associate|bib> BF08 BBF02 DMHJ13 Gul10 Lee05 GL14 Gru11 RR09 Teh16 CG16 GGJT15 HKLW02 Lab05 FJ11 Osa07 GL14 Gru11 Gru96 vdH:genae vdH:issac96 Sh90 Sh04 Ec92 vdH:phd vdH:ln SS92 SS99 vdH:phd vdH:ln Bour61 Har10 Har11 Sal:phd Har10 Har11 Bour61 Rich07 vdH:genae BK75 GL14 RR09 BBF02 Lab05 Gru11 Gru11 Gru11b Gru11b A15 GHLOW12 Y53 vdH:mmx HKLW02 AS96 A15 <\associate|toc> |math-font-series||1Introduction> |.>>>>|> |math-font-series||2Effective Hardy fields> |.>>>>|> |2.1Hardy fields |.>>>>|> > |2.2Basic properties |.>>>>|> > |2.3Effective Hardy fields |.>>>>|> > |2.4Adjunction of steep exponentials |.>>>>|> > |2.5Limits and asymptotic scales |.>>>>|> > |math-font-series||3Composition and functional inversion> |.>>>>|> |3.1First order functional inversion |.>>>>|> > |3.2First order right composition |.>>>>|> > |3.3General composition |.>>>>|> > |3.4General functional inversion |.>>>>|> > |3.5Effective asymptotic expansions |.>>>>|> > |math-font-series||4Examples and applications to finance> |.>>>>|> |4.1Lambert |W> function |.>>>>|> > |4.2Gaussian function |.>>>>|> > |4.3Gaussian expected shortfall |.>>>>|> > |4.4Incomplete Gamma function |.>>>>|> > |4.5Black\UScholes formula |.>>>>|> > |4.6Bachelier's formula |.>>>>|> > |4.7Implied volatility of a local volatility process |.>>>>|> > |4.8Constant elasticity of variance model |.>>>>|> > |4.9Experimental |Mathemagix> implementation |.>>>>|> > |math-font-series||5Conclusion> |.>>>>|> |math-font-series||Bibliography> |.>>>>|>