Effective asymptotic analysis for finance

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 U-shaped “volatility smiles” 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 [3, 4, 8, 16, 25]. 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–Scholes formula itself but on its inverse [10, 14, 30, 36]. 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 symmetry theory, Watanabe theory, heat kernel expansion theory and Minakshisundaran–Pleijel's formula, large deviation theory, etc. [7, 12, 17, 20, 26, 27]. 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 non-analytic 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 Black-Scholes 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

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 “exp-log” functions [15, 22, 29, 34, 35]. 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

. The theory of transseries [9, 21, 23] 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 [32, 33]. However, the right-hand side

of (1) 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

[21, 23]. 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

. It is well-known that Hardy fields [5, 18, 19] 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 exp-log 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 exp-log 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 [31]. 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 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 “highly tangent to identity”.

Acknowledgment. We are very grateful to Martino Grasselli for his encouragement and for the careful reading of our work.

2Effective Hardy fields

2.1Hardy fields

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

. Since any non zero element

of Hardy fields is invertible, the sign of

is ultimately constant for

. We define

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

, we write

Example 1. The set

of exp-log germs at infinity is the smallest subset of

that contains

and the identity function, and which is closed under

and

. For instance,

. In his founding work [18, 19], Hardy showed that

forms a Hardy field.

Example 2. 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 [5].

2.2Basic properties

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

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.

2.3Effective Hardy fields

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 3. 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

with

and

. Then

. Consequently,

and

(resp.

Example 4. We claim that

is an effective Hardy field. As above, the basic operations

and the equality test are straightforward. Now any non zero element

can be written as a fraction

with

and

. Similarly, we may write

with

and

. Then

. Consequently,

and

(resp.

). Here we used the lexicographical ordering on pairs:

if and only if

and

Example 5. Let

be an effective Hardy field and let

be such that

and

. Then

, whence

is ultimately strictly increasing and invertible for composition. Let

be the inverse of

and assume that

. Then

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

2.4Adjunction of steep exponentials

Let

denote the logarithmic derivative of a function

. Taking logarithms, and using (3) and (4), we observe that

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

and

. 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

(and

), 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

and

. Similarly,

if and only if

and

Example 6. Starting with

as in Example 4, applying the above argument twice shows that both

and

are effective Hardy fields. Applying Example 5 for

, we also obtain that

is an effective Hardy field.

Remark 7. In order to compute with more general exp-log germs in

, one also needs to show that fields such as

form effective Hardy fields. One even more difficult problem is to provide an effective zero test for exp-log constants, i.e. constants formed from the rationals, using

and

. Provided that Schanuel's conjecture holds, such an algorithm was given by Richardson [28]. 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 [22].

2.5Limits and asymptotic scales

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 8. In Example 3, we have given a method for the explicit computation of an equivalent in

for any

. This both shows that

admits an effective limit map and that it admits

as an effective asymptotic scale. Similarly, Example 4 shows that the same holds for

, in which case the asymptotic scale becomes

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 9. Let

be an effective Hardy field and let

be as in Example 5. If

admits an effective limit map, then so does

, since

for all

. If

admits an effective asymptotic scale

, then

admits

as an effective asymptotic scale, with

for all

3Composition and functional inversion

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.

3.1First order functional inversion

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

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

for sufficiently large

. Since

is strictly increasing, it follows that

. In other words,

for sufficiently large

3.2First order right composition

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

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

. Post-composing with

, we again obtain

Let us finally assume that

. Then the above arguments prove that

. Consequently,

The above arguments 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

As to the general case, let

be such that

. By what precedes, we have

for all sufficiently large

. This shows that

3.3General composition

Lemma 12. Let and . Let and be such that and . Then for any with and , we have

for all sufficiently large

. Now Lemmas 10 and 11 imply

and similarly

. Consequently,

Proof. Let us first consider the case when

, so that

. For any

, we have

, whence

. Consequently,

Assume next that

, so that

. We again have

for all

, but this time, we rather obtain

, since

. Therefore,

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

Proof. The existence of

is ensured by Lemma 13. Since

is effective, we have an algorithm for doing the test

, which enables us to compute

. Setting

, our assumption that

ensures that

. The result now follows from Lemma 12.

Remark 15. In addition, by considering both cases

and

, it can be verified that

, that

implies

, and that

implies

3.4General functional inversion

A well-known way to solve functional equations of the form

is Newton's method [6]. We will now show that this method indeed yields a quadratic convergence in our setting.

Proof. Since

, we notice that

and

. Let

. For all sufficiently large

, we have

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

Theorem 17. Let . The algorithm invert is correct and terminates after at most iterations of the main loop.

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,

At the very start, we have

. At every iteration

, we have

. Lemma 16 therefore ensures that

doubles at least, whereas the algorithm terminates as soon as

. This happens after at most

iterations.

3.5Effective asymptotic expansions

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 18. Assume that and admit effective asymptotic expansions over . Then so does . If , then .

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 19. Assume that admits an effective asymptotic expansion over . Then so does and .

4Examples and applications to finance

4.1Lambert

function

The Lambert

function is defined to be the inverse function of

. Using our algorithm, we can compute the asymptotic expansion of the inverse function

. This also yields the asymptotic expansion of

for large

4.2Gaussian function

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 (7) gives itself an asymptotic expansion of a Gaussian Value-at-Risk

in terms of its confidence level

4.3Gaussian expected shortfall

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

4.4Incomplete Gamma function

where

is a Brownian motion and

denotes for instance a forward rate, the probability of absorption at zero before

-time is given by

Inverting (9) gives a confidence interval where the probability of absorption of

is lower than

, asymptotically, for

4.5Black–Scholes formula

By definition, a call option is a contract which gives to the owner the value

at a future

-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 “intrinsic value”

and lower than the spot value

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

For simplicity, we have assumed that the interest rate is

. If

are fixed, then it is easy to see that the function

is non-decreasing and one to one from

. 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 [10, 30], it is interesting to invert the Black–Scholes function

in (10). For instance, using techniques from 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 [4, 20]. 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 [14] that the asymptotic expansion of the “time value”

of the call price

, defined by

Formula (12) is nothing but another expression for Black-Scholes formula. Hence we get an asymptotic expansion for

in terms of

. Note that (12) is an asymptotic expansion of a call price in terms of

for large

. So, it gives also an asymptotic expansion of a call price when

is large i.e. small strike or large strike.

Notice also that there is another direct formula

when

, which gives an asymptotic expansion of

in terms of

At the limit when

, the first author previously obtained a similar result [14]. Setting this time

4.6Bachelier's formula

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

for

. So, using (8), 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 (11), we obtain an equivalence between normal volatility and lognormal volatility in the cases when

. For example, when

, the first terms of the expansion are given by

where

denotes the “moneyness”

and

. Note that at the money (

), we have a closed form formula [13]:

This is a refinement of a classical result

. In general, (15) shows that if

, then the slope of the volatility smile for the lognormal volatility is negative and this condition holds up to order 1 in

[13].

4.7Implied volatility of a local volatility process

By inverting (12), 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–Pleijel expansion - modulo conditions of regularity [2]. So, in general this allows to translate an asymptotic expansion of

for

in an asymptotic expansion of

As an example, let us consider a local volatility model

. Then Tanaka's formula reads:

with

and

is explictly given by induction [11, 37]. The asymptotic expansion (17) can be integrated by part and thus give an asymptotic expansion for

thanks to (16). Finally, by inverting (12), we get asymptotic expansion of the lognormal implied volatility.

4.8Constant elasticity of variance model

Let

be the probability density function to get state

-time starting from

at time

. There is a closed form formula for

in terms of the modified Bessel function

For short maturities

, the combination of (18) and (20) yields an asymptotic expansion of

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 4.5, we deduce an asymptotic expansion of the implied lognormal volatility of the CEV model at any order in

4.9Experimental Mathemagix implementation

We did an experimental implementation of our algorithm in the Mathemagix system [24]. Each of the above examples comes down to the computation of the functional inverse of a function

with an asymptotic expansion of the form

5Conclusion

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 [17]. More generally, it would be interesting to combine our approach with other algorithms for the calculation of heat kernel coefficients such as [1] to get automatic asymptotic expansion at any order of the implied volatility of option call prices for a large class of financial models [2].

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1Introduction

2Effective Hardy fields

2.1Hardy fields

2.2Basic properties

2.3Effective Hardy fields

2.4Adjunction of steep exponentials

2.5Limits and asymptotic scales

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

4Examples and applications to finance

4.1Lambert function

4.2Gaussian function

4.3Gaussian expected shortfall

4.4Incomplete Gamma function

4.5Black–Scholes formula

4.6Bachelier's formula

4.7Implied volatility of a local volatility process

4.8Constant elasticity of variance model

4.9Experimental Mathemagix implementation

5Conclusion

Bibliography