Applications of Periodic and Quasi-Periodic Expansions to Options Pricing

8/3/2019 Applications of Periodic and Quasi-Periodic Expansions to Options Pricing

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Applications of Periodic and Quasi-Periodic Expansions toOptions Pricing

Dominique Bang1

1 Bank of America Merrill Lynch, [email protected]

Abstract. We present a new formulation of the call payoff as a convergent trigonometric series, where the corresponding frequencies are selected accord-ing to a minimum variance algorithm. When the characteristic function of the log-moneyness is available, this provides an efficient alternative to trans-

form methods for option valuation (such as Fourier Integral), reducing the

frequency spectrum from a continuum to a discrete set.

1 Introduction

A standard approach to pricing vanilla options is based on integral representations(such as the Fourier integral) of a call pay-off, requiring the characteristic functionof the log-moneyness to be known in a closed form, as is the case for many modelsof interest (e.g. Black&Scholes, Heston, Stein&Stein, Cox process, Levy processes

etc..). These methods generally rely on a numerical integration of the (inverse)Fourier Integral, which can be computationally intensive. Moreover, the relatedintegral being highly oscillatory and thus prone to inaccuracy, care needs to betaken in the way the integral is truncated and discretized. In particular, in thecase of Heston dynamics, a number of tricks can be used to improve the accuracyof the numerical integration while keeping reasonable the number of evaluations of the characteristic function as described comprehensively in [2]. Alternatively, aninteresting idea has been developped in Fang and Oosterlee [4] 1 where the call priceis approximately expanded in a Fourier Series using an approximate density basedon integral truncation; the authors demonstrate the efficiency of their approach

compared with Carr and Madan [3] brute force integration in the case of HestonDynamics. In this note, we develop a different approach, seeking for an exact seriesrepresentation of a call payoff in term of trigonometric functions. The idea is to find(recursively) the combination of trigonometric functions that is closest(in a sense tobe specified later) to the call price and then, as for the Fourier Integral, to apply theexpectation operator. In a sense, this approach amounts to performing a PrincipalComponents Analysis of the Call option payoff itself, using trigonometric functions(for which the expectation is known in a closed form via the characteristic functionof the log-moneyness). We demonstrate constructively the existence of such a series,show its convergence toward the call payoff in a strong sense (for the chosen norm)

1

Thanks to Leif Andersen for pointing out this reference and for constructive conversations.

1



and give an error estimate when the call price is computed using a truncation of the series. Finally, as an illustration, we apply this method to option pricing underHeston dynamics. Additionnally, we provide in passing a simple (approximate but

arbitrarily accurate) representation of a call option in terms of Fourier series, similarto Fang and Oosterlee’s cosine expansion, but with an improved rate of convergence.

2 Notations and General Framework

The payoff of a call struck at K maturing at T on the underlying S T is given by:

(S T − K )+ = K (ex − ex2 e−

|x|2 )

x∆= ln(

S T K

)

From contour integral theory, we have that:

φ(x)∆= e−

|x|2 =

1

π

+∞0

f λ(x)

λ2 + 14

dλ, f λ(x)∆= cos(λx)

yielding(S T − K )+ = S T − Ke

x2 φ(x) (1)

This decomposition formula shows how to recover the call payoff using a con-tinuum of frequencies in the Fourier space. Applying the expectation operator tothis equality yields a valuation formula for the call option, provided that the char-

acteristic function of the log-moneyness at maturity, χ(s) = E T (e(1

2+is)x), is knownin closed form. As pointed out above, this integral, being highly oscillatory, needsparticular care in the way it is discretized and/or truncated. More over, as a Fouriertransform of an unsmooth fonction, and per Heisenberg principle, the integrand de-cays slowly to zero, or, in other terms high frequencies are necessary to describe wellthe underlying pay-off function. The aim of this note is to develop an alternative tothis approach by providing a trigonometric series representation of φ(x) rather thanan integral, that is to project the function φ on a discrete set of frequencies ratherthan a continuum. More precisely, we aim to write

φ =

∞n=1

φnen

where φn is a scalar and en is a function such that en ∈ span

f ( pj)λj

1≤ j≤n

, where

the differentiation is performed w.r.t λ (the element ∂ p

∂λpf λ (x) corresponds to a

polynomial of degree p times a sin() or a cos() depending on the parity of p). Thefrequencies λ j are selected through a minimum variance algorithm (which, in ourcase, is formulated as a maximization of a normalized scalar product2 to be defined

2This approach is comparable to the one developped by Laskar [5] in the context of KAM theory.The latter proved to be very successful in areas as diverse as Celestial Mechanics and High Energy

Particles Dynamics.

2



later on) and can be seen as a PCA. Next, we introduce the framework in which thisanalysis takes place, which is naturally Hilbert space. More precisely, consider theweighted space H µ of integrable functions (w.r.t the weight µ) and the associated

scalar product:

g, h = g, hµ =

+∞0

g(x)h(x)µ(x)dx

|g| = |g|µ =

g, gµ(x) > 0 +∞

0

x pµ(x)dx < ∞, p ≥ 0

where the last technical condition is designed to ensure that for any λ ≥ 0, theelements f λ remains in H µ through differentiation of arbitrary order; we note thatthis condition is almost always satisfied in practice. We plan to build an orthonormalfamily (en)n∈N ∗ in H µ (using Gram-Schmidt construction) based on the free family(f λ)λ≥0, satisfying some optimal constraints and complete w.r.t φ. To gain furtherintuition about the forthcoming construction, we emphasize the following statement:

Lemma 1. let gλ denote a parametric family of non zero elements in H µ and ξdenote a fixed element in H µ. Then the two following problems are equivalent

• Find

λ,

ν

= arg min |ξ − νgλ|2 .

• Find λ = arg max ξ, gλ|gλ|2 , ν = ξ,gλ

|gλ|2.

Proof 1. From |ξ − νgλ|2 = 0 we must have ν = ξ,gλ|gλ|2 , so that

|ξ − νgλ|2 = |ξ|2 − 2 ν ξ, gλ + ν 2 |gλ|2

= |ξ|2 −

ξ,gλ|gλ|2

which shows that minimizing |ξ − νgλ|2

is equivalent to maximizing ξ,gλ|gλ| (the

lemma is also clear from a Lagrange Multipliers perspective).The first formulation amounts to finding the element gλ which captures the max-

imum variance of ξ, and can be essentially regarded as a Principal ComponentAnalysis. However, for practical computation, we opt for the second one, which iswell adapted to the iterative procedure we intend to use. From eq.1 we see thatthe success of any approach based on characteristic function relies on the ability torepresent φ in an efficient manner. The main idea of this paper is to apply the mini-mum variance (maximum scalar product) strategy to φ on a recursive basis: find the

first element f λ1 which maximizes φ, f λ

|f λ| , set e1 =f λ1

|f λ1|and φ1 = φ,

f λ1

|f λ1| and

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http://slidepdf.com/reader/full/applications-of-periodic-and-quasi-periodic-expansions-to-options-pricing 4/17

restart the same procedure with φ1 = φ−φ1e1 and so forth. In section 4, we providea rigorous formalization of this algorithm, stating the main result of this note, thatis the existence of (en)n∈N ∗ along with the convergence of the series

∞i=1 φiei to-

wards φ., but prior to that, in section 3 we give a self-contained result which allowsfor (approximate) call pricing in a very efficient and simple manner. This result isbased on Fourier Series expansion of a periodization of φ.

3 Periodic decomposition of a Quasi-call payoff

The following result3 gives a proxy for the call price using the characteristic functionχ(s) = E T (e(

1

2+is)x) of the log-moneyness.

Theorem 3.1 Let A be a real positive number. Consider the 2A− periodic function φA(x) such that φA(x) = φ(x) on

|x

| ≤A, and C A(F, K ) the (approximate)

call price by replacing φ with φA in eq .1. Then we have:

C A(F, K ) = F − K ∞l=0

clχ

lπ

A

cl = 4A1 − (−1)le−

A2

A2 + (2πl)2, c0 = 2

1 − e−A2

A

and the following error estimates hold:

•Error in prices:

|C (F, K ) − C A(F, K )| ≤ √F K

N (K exp−A) + 1 − N (K expA)

• Error in normal ( N ) and Black ( B) vol for an at-the-money option:

σN − σN A

≤

2π

T F


σB − σBA

≤

2π

T

N (K exp−A) + 1 − N (K expA) (≈)

where N () is the cumulative distribution function of S T . Proof: see Appendix A.Remark 3.i The idea of using Fourier Series rather than Integral has been firstexploited by Fang and Oosterlee in [4]. Though, the approach is slightly different intheir paper, as it is the integral representation of the density itself which is truncated,leading eventually to a cosine series. We note that their approach involves a total of three levels of approximations whereas ours only involves two(in the choice for thethreshold A and the truncation of the Fourier Series) with an improved decay rateof the Fourier coefficients. The latter being rescaled by χ (.) which usually vanishesexponentially fast, this shouldn t make too much of a difference in practice, except

3The idea to directly periodize φ originates from a conversation with Alex Lipton. We wish to

thank him in passing for his interest in this work.

4



when the distribution of the underlying is close to singular (e.g. when the optionmaturity is very small) and where the call is close to its intrinsic value. Also, ourFormula is simpler, and the error estimate for the choice of A quite transparent

(N (x) can be easily estimated for small/large x). The previous formula has beenmentioned in passing, as the core subject of this paper is to provide an exact seriesexpansion, rather than a proxy. However, its great simplicity compared to theforthcoming quasi-periodic expansion sometimes outweights the fact that it is onlyan imperfect representation of the call.

4 Quasi-Periodic decomposition of a call payoff

The next theorem is the main result of this note: it proves constructively the exis-tence of a countable trigonometric family (en)n

∈N ∗ satisfying some optimality criteria

and on which φ can be decomposed, in a Hilbert sense.Theorem 4.1 There exists an orthonormal family (en)n∈N ∗ ∈ H N

∗µ with asso-

ciated frequencies (λn ≥ 0)n∈N ∗ , indices ( pn ≥ 0)n≥1and scalars (φn ≥ 0)n∈N ∗such that:

φ =∞i=1

φiei

φi∆= φ, ei

ei ∈ span

f ( pj)λj 1≤ j≤i

where the convergence of the series should be understood as a strong convergence in H µ. The algorithm to compute (en)n∈N ∗ is performed recursively as follows: initial-ization for n = 1 according to step(1) and iteration of step(2).

1. find λ1 such that:

λ1∆= arg max

λ≥0Ω0(λ) where Ω0(λ)

∆=

φ,

f λ|f λ|2

(2)

set and store:

e1∆=

f λ1 +∞0

cos2(λ1x)µ(x)dx, φ1

∆= φ, e1 , p1

∆= 0 (3)

2. find λn+1 for n > 1 such that:

λn+1∆= arg max

λ≥0Ωn(λ) where Ωn(λ)

∆=

φ,

f λ −ni=1 f λ, ei ei

|f λ −ni=1 f λ, ei ei|

2

5



set and store:

en+1∆= lim

λ

→λ+n+1

f λ −

ni=1 f λ, ei ei

|f λ

−ni=1

f λ, ei

ei

|φn+1

∆= φ, en+1

pn+1∆= min

p such that f

( p)λn+1

−ni=1

f ( p)λn+1

, ei

ei = 0

Moreover (for completeness and practical interest) we have for any n ≥ 1 :

limλ→λn

Ωn(λ) = 0

Proof: see Appendix B. As for the existence of the family (en)n∈N ∗, see B.1, forthe convergence of the associated series toward φ, see B.2 and for the last statementΩn(λn) = 0, it comes directly from a second order Taylor expansion associated with

a maximality argument.Remark 4.i In (b) the limit is taken from above limλ→λ+n+1

. This choice is made

to resolve the intrinsic ambiguity on en+1 (which can be replaced by −en+1 withoutany disruption).Remark 4.ii The fact that Ωn(λn) = 0 is useful in practice, providing informa-tion about the zeros of Ωn(λn) when solving for λn+1(in particular, λn+1 = λn, andmoreover λn+1 should be fairly far from λn). In practice, one can observe a persis-tence of the zeros of Ωn() that is Ωn(λi) = 0 not only for i = n but also for i ≤ n(or at least it remains small). However, this is not necessarily the case for all n.Thus, a frequency λ j<n−1 may reappear when solving for arg max Ωn(λ). When this

occurs, there is an indeterminacy in limλ→λ+n+1

f λ−

ni=1

f λ,ei

ei

|f λ−ni=1f λ,eiei| , as both numerator

and denominator collapse, involving naturally the derivatives of the elements f λ (seeL’Hopital’s rule). The integer pn can be seen as a counter of the visits of the fre-quency λn before time n and also corresponds to the order of differentiation in f λnwhen λn = 0. For the special case λn = 0 the degree of differentiation is twice thenumber of visits n.Remark 4.iii We insist that, by construction, once en has been computed andstored, only one additional frequency is necessary to generate en+1. This meansthat the algorithm can be used on an adaptative basis with a low computationalcost (the numbers used to compute en can be stored and reused for en+1). This is of

prime importance in option pricing, where the expectation operator will be appliedto the elements en and thus, will lead to evaluations of the characteristic function,which is generally computationnally intensive. In our case, the evaluations of thecharacteristic function will be stored to be possibly reused when/if incrementing n.

5 Option Pricing

5.1 Valuation Formula and Error Estimate

We apply the previous approach to the pricing of a call option in a model for which

the characteristic function of the log-moneyness χ(λ) = E T (e(1

2+iλ)x) is known, as-

6



suming that (λ j) j≥1 , ( p j)

j≥1 , (e j) j≥1 and (φ j)

j≥1 have been found following theprocedure in 4 such that:

e− |x

|2 =

∞i=1

φiei, ei =

i j=1

C ij ∂

pj

f λ∂λ pj |λj

F ∆= E (S T )

for some constants C i,j. Also we introduce ρ (.), the density function of the log-

moneyness x = logF T K

, along with its symmetrization ρ(s)

∆= ρ(s) + ρ(−s). We

define the exact and approximate (truncated) price of the call option by

C (F, K )∆= E (S T − K )+ C n(F, K )

∆= F − KE

ex2

n

i=1φiei

and assume that the norm ||µ dominates the norm ||

ρ, that is: there exists C µ > 0

such that |f |ρ

< C µ |f |µ for any f ∈ H µ ⊂ H ρ.

Theorem 5.1 Under the previous assumptions, the following results hold:

• Option Valuation

For the exact price we have

C (F, K ) = F − K ∞i=1

φi

i j=1

C ij∂ pjχ(λ)

∂λ pj|λ=λj

and for the proxy:

C n(F, K ) = F − K

n j=1

Γn j

∂ pjχ(λ)

∂λ pj|λ=λj

Γn j

∆=

ni= j

C ijφi

• Error Estimate

We have the following bound for the truncation error:

|C (F, K ) − C n(F, K )| ≤ C µ√

F K

φ, φ −n

i=1φ2i

and for the normal implied vol error atm we have:

σN − σN n

≤ C µF √

2π

φ, φ −ni=1

φ2i

and for the log-normal vol atm approximately

σB − σB

n ≤ C µ

√2π

φ, φ −n

i=1φ2i

7



Proof: See appendix CRemark 5.i As for the standard Fourier approach, this formula allows a strip

of options for different strikes to be priced at once since

E

exp

(

1

2+ iz)x

= exp

−(

1

2+ iz)ln(K )

E

exp

(

1

2+ iz)ln(F T )

and the evaluations of the characteristic functions can be stored and reused for dif-ferent strikes.Remark 5.ii We emphasize that the computation of C n(F, K ) requires exactly nevaluations of the characteristic function χ (or its derivatives); moreover, as alreadystated above, one increment of n will result in a single new evaluation of χ, pro-vided the past evaluations have been stored, which makes an adaptative procedure

possible.Remark 5.iii The error estimate shows clearly the impact of the norm ||µ on thedeviation of the proxy from the exact price. Obviously C µ should be reasonable sothat the series converge sufficiently rapidly.

5.2 How to choose the truncation level n?

From the last section, it is clear that the error depends directly on the productC µ φ, φ −n

i=1 φ2i . Once µ has been chosen, the number of terms necessary to

achieve a given accuracy η can be easily inferred by setting

C µ√

F K φ, φ −

ni=1

φ2i < η

when C µ is known or at least is known to be reasonable. As an illustration, let usconsider a tractable example with BlackScholes dynamics:

x = σ√

TZ, Z ∼ N (0, 1)

ρ(s) = ρ(−s) =1

σ√

2πT exp(− s2

2σ2T )

where N (0, 1) denotes the standard normal distribution. For µ, we use an exponen-tial weight

µ(s) = ν exp(−νs)

A few calculations show that |f |ρ

< C µ |f |µ with the optimal choice ν = 1σ√ T

yielding C µ =

2eπ

≈ 1. 147 This means that if the variance of our process is

σ2T = 1, a good choice for our proxy distribution will be µ(s) = exp(−s). In thatcase, our experiments suggests that values n ≈ 15 − 20 produce very good results,not only when the terminal distribution of the underlying is log-normal, but also

for richer dynamics such that stochastic volatility, as long as the variance of the

8



process is reasonnably in line with the one implied by µ. When C µ is not knowna priori, another possible strategy is to evaluate the elements (ei)1≤i≤100 once andfor all, and then increment the number of elements used within this family until

convergence is achieved (e.g the results do not change by more than some givenquantity). This adaptative procedure is made possible as the evaluations of thecharacteristic functions are stored and reused. In the next section we examine thechoice for the weight µ.

5.3 Example of Weights

In this section we provides useful quantities related to the practical implementationwhen the weights are either gaussian or exponential. We recall that the scalarproduct in H µ is defined by:

g, hµ = +∞0

g(x)h(x)µ(x)dx

where, without loss of generality, we impose +∞0

µ(x)dx = 1.

5.3.1 Gaussian Weight

This case corresponds to Black&Scholes dynamics for the underlying, thus, for an at-the-money option, the log-moneyness can be assumed to be gaussian (and centeredfor simplicity). This results in the following table:

distribution Gaussian

µ (x) 2σ√ T √ 2π

exp(− x2

2σ2T )

f λ, f µ exp−λ2+µ2

2σ2T

cosh(λµσ2T )

|f λ|2 12

[1 + exp (−2λ2σ2T )]

φ, f λ [exp(m2)erfc(m)] , m =12− λi

σ

T 2

φ, φ 2N (−σ√

T ) exp(σ2T 2

)

Remark 5.ivThe Gaussian distribution is not necessary appropriate when the un-derlying’s distribution has fat tails, which is the case in stochastic volatility models.

5.3.2 Exponential Weight

The last remark suggests that we might consider a more slowly decaying density,and a good tractable candidate is the exponential density for which we will analyze

9



in more details.distribution Exponentialµ (x) ν exp(−νx)

f λ, f µ ν 2 ν 2+λ2+µ2

(ν 2+(λ+µ)2)(ν 2+(λ−µ)2)|f λ|2 ν 2+2λ2

ν 2+4λ2

φ, f λ ν (ν + 1

2)(ν + 1

2)2+λ2

φ, φ ν 1+ν

In this case, we can work out the first term in the expansion. A short calculationleads to:

Ω0(λ, ν ) =φ, f λ2

f λ

, f λ

=

ν (ν + 1

2)(ν + 1

2)2+λ2

2

ν 2+2λ2

ν 2+4λ2

λ1

√ 54

ν 4 + 8

25ν 2ν 2 + 32

25ν 3 − ν 2

|cos(λ1·)| 12

6ν +2

√(25ν 2+8+32ν )√

(25ν 2+8+32ν )−ν

φ18√ 2ν (1+2ν )

√(25ν 2+8+32ν )−ν

4+16ν +11ν 2+ν √

(25ν 2+8+32ν )

3ν +√

(25ν 2+8+32ν )

for ν = 1 this becomes:

λ1

√√ 65

−5

4 ≈ . 4374φ1

24√

8959+737√ 65

1777+303√ 65

≈ . 6942

e1 (x)cos

√ √65−54

x

√ 31+

√65√

65−1

≈ 1. 1299 cos (0. 4374 x)

The first element e1 capture φ21 ≈ . 48198 of the total variance φ, φ = 0.5. That is, in

relative terms approximatively 96.4% of the total variance. The 20 first frequencieshave been computed numerically:

Λ = 0.4374, 1.6489, 0.0000, 0.8941, 2.9970, 4.6829, 2.3117, 6.5990, 3.7550, 1.2114,

8.7081, 5.5708, 10.997, 7.5851, 13.455, 9.7852, 16.074, 18.836, 12.160, 0.0000 Φ =

69.42%, 10.61%, 5.202%, 4.034%, 3.963%, 1.910%, 1.077%, 1.056%, 0.800%, 0.663%0.645%, 0.474%, 0.423%, 0.306%, 0.293%, 0.212%, 0.211%, 0.158%, 0.153%, 0.138%

Note that the twentieth term is the first occurrence of a frequency revisit. Using thefirst three terms covers 99.2% of the variance etc.. Next, we apply this methodologyto Heston Dynamics.

10



5.4 An application to Heston Dynamics

In this section we assume that the underlying follows Heston dynamics, that is,using similar notations than in Lipton[1]:

dS tS t

= σ√

vtdW (S )t

dvt = κ (1 − vt) dt + ε√

vtdW (v)t , v0 = 1

ρdt = dW (S )

t dW (v)

t

From now on we consider an initial spot S 0 = 5% and the base case: σ = 40%,ε = 100%, ρ = 0, κ = 10% and T = 5y. The standard deviation in the base case is80% justifying the use of an exponential weight with a caracteristic constant ν = 1.In the tables in Appendix D, we vary individually the different model parameters

and compare a 20 terms quasi-periodic expansion with a benchmark brute forceintegration, reporting the errors between the two. The results suggest that a verygood accuracy is reached with 20 terms, for a large spectrum of model parameters.

6 Conclusion

The contribution of this paper is two folds: first we have slightly simplified and im-proved an original idea developped in [4] and second we have developped a new exactdecomposition of the call payoff in a polynomial-trigonometric series. Existence and

convergence results have been established in a rigorous framework, yielding a val-uation formula along with error estimates when truncation is applied for practicalpurpose. The expansion has been tested under Heston dynamics, demonstratingthe robustness and excellent accuracy of the approach for a number of terms aslow as 20. The method could, in principle, be applied to other dynamics wheneverthe caracteristic function of the log-spot is known in a closed form. We plan, in anear future, to apply this technique to close to singular distribution (such as thoseappearing for very short term maturity) known to pose challenging problems whenused in a Fourier Transform framework.

AknowledgementsI am indebtful to Cyril Grunspan for initial motivating conversations, to Yann Ticot for his insights through out this work and to Professor Peter Hawkes for useful suggestions. Also I want to thank my colleagues in Bank Of America Merrill Lynch,and more specially Leif Andersen, Alex Lipton and Henrik Rasmussen for useful comments and support. Last, i am grateful to prof. Cornelius W. Oosterlee and prof. Fang Fang for interesting feed back and discussion.

11



A Proof of Theorem 3.1

φA(x) being 2A− periodic, even, piecewise C 1 and C 0, can be decomposed into its

Fourier (cosinus) Series:

φA(x) =∞l=0

clcos

lπ

Ax

cl>0 = 4A1 − (−1)le−

A2

A2 + (2πl)2, c0 = 2

1 − e−A2

A

Thus, we have that:

C A(F, K ) = E T

S T − Ke

x2 φA(x)

= F − K ∞l=0

clE T ex2 cosl π

Ax

yielding

C A(F, K ) = F − K ∞l=0

clχ

lπ

A

By Cauchy-Schwartz,

|C (F, K ) − C A(F, K )| = K

E T

ex2 (φ(x) − φA(x))

≤ K E T [ex] E T (φ(x) − φA(x))2 ds

≤ K

E T [ex]

E T

1|x|>A

ds

=√

F K


At the money K = F , we have, in term of normal volatilities:

σN − σN A

≤ 2π

T F


and in term of log-normal volatility, approximately:

σB − σBA

≤ 2π

T


B Proof of Theorem 4.1

B.1 Existence result

In this section we show that the recursive construction described in 4 is valid at anyorder

12



Proof: We prove the result by induction. For n = 1, we have that

|f λ

|=

+∞

0

cos2(λx)µ(x)dx > 0

φ, f λ =

+∞0

cos(λx)φ(x)µ(x)dx

Ω0(λ)∆=

+∞0

cos(λx)φ(x)µ(x)dx2

+∞0

cos2(λx)µ(x)dx, λ ≥ 0

which shows that Ω1 is a positive continous function of λ. Moreover, using Lebesgue’sLemma, we see that:

limλ→∞ +∞0cos

2

(λx)µ(x)dx =1

2 +∞0µ(x)dx

limλ→∞

+∞0

cos(λx)φ(x)µ(x)dx = 0

so that:limλ→∞

Ω0(λ) = 0

Relying now on the continuity of Ω1 associated with a compacity argument, we canensure that there exist

λ ≥ 0 such that

λ = arg maxλ≥0 φ,ξλ

|ξλ|2

we now let

λ1 = λe1 =

f λ1 +∞0

cos2(λ1x)µ(x)dx

and assume now a valid construction of (ei)1≤i≤n and (λi)1≤i≤n up to the rank n.We proceed as for the first term, considering

eλ =f λ −n

i=1 f λ, ei ei|f λ −n

i=1 f λ, ei ei|λ = λi, i = 1..n

which is well defined since |f λ|2 >n

i=1 f λ, ei2 as f λ /∈ Span(e1,..,en) when λ =λi, i = 1,..,n.

Let us consider now λ j ≥ 0, j ≤ n. We have trivially f λj −ni=1 f λj, ei ei = 0

so that we can consider

p = maxl≥

0/∀

0≤

k≤

l,∂ kf

∂λk |λj =

n

i=1 ∂ kf

∂λk |λj , ei ei

13



then, using Taylor expansion up to order p, we have that:

f λ−

n

i=1 f λ, ei

ei =

(λ − λ j) p+1

( p + 1)! ∂ p+1f

∂λ p+1 |

λj

−

n

i=1 ∂ p+1f

∂λ p+1 |

λj , ei ei + o(1)so that

eλ =(λ − λ j) p+1

|λ − λ j| p+1∂ p+1f ∂λp+1

|λj −ni=1

∂ p+1f ∂λp+1

|λj , ei

ei∂ p+1f ∂λp+1

|λj −ni=1

∂ p+1f ∂λp+1

|λj , ei

ei

is well defined and Ωn(λ) can be continously extended on λ ≥ 0. Similar argument

as previously shows that Ωn(∞) = 0 and thus λn+1∆= arg maxλ≥0 Ωn(λ) < ∞. Also,

in virtue of what preceed,

en+1 = limλ

→λ+n+1

f λ−n

i=1f λ,eiei

|f λ−

n

i=1f λ,ei

ei

|is well defined. Thus, we can perform this

construction at any order.

B.2 Convergence result

Φn =ni=1

φiei

(ei)1≤i being orthonormal, we have:

φ, φ

≥ Φn, Φn

=

n

i=1 φ2i

As an increasing bounded sequence, Φn, Φn is convergent with limit l ≥ 0 (thisalso yields φ∞ = 0) . Per

|Φn+ p − Φn|2 = Φn+ p − Φn, Φn+ p − Φn

=

n+ pi=n+1

φ2i ≤

∞i=n+1

φ2i

we see that Φn is a Cauchy sequence in H µ and therefore is convergent. Let us

denote its limit by Φ. We want to prove that Φ = φ. Let us assume momentarilythat this condition is violated, that is: ψ = φ − Φ = 0. (λ j)

j≥1 being a discrete

family, there exists λ such that:

[ψ, e∞]2 = η > 0

e∞ =f λ −∞

i=1

f λ, ei

ei f λ

2 −∞i=1

f λ, ei

2by orthogonality, we have that:

ψ, e∞ = φ, e∞14



and from φ∞ = 0, there exists N 1 such that ∀n ≥ N 1 we have φ2n < η

2. Also, we

have that:

limn→∞ en = e∞ en =f λ −n−1

i=1

f λ, ei

ei f λ

2 −n−1i=1

f λ, ei

2such that there exists N 2 > 0 verifying ∀n ≥ N 2, φ, eN 2 > η

2. Considering now

N = max(N 1,N 2), we see that we must have simultaneously φ, eN 2 > η2

and φ2N < η

2

which clearly violates the optimality of eN . Thus we must have ψ = φ−Φ = 0 whichcompletes the proof.

C Proof of Theorem 5.1

For the valuation formulae, we have that:

E

ex2 ei

=i

j=1

C ijE

ex2

∂ pjf λ∂λ pj

|λj

=i

j=1

C ij

∂ pj

∂λ pjE

ex2 eiλx

|λj

=i

j=1

C ij ∂ pjχ(λ)

∂λ pj|λ=λj

yielding:

C (F, K ) = F − K ∞i=1

φi

i j=1

C ij

∂ pjχ(λ)

∂λ pj|λ=λj

C n(F, K ) = F − K

n

j=1Γn j

∂ pjχ(λ)

∂λ pj|λ=λj

, Γn

j =n

i= jC ijφi

Error estimates are a simple consequence of Cauchy-Schwartz inequality.

D Numerical results and tables

In the next tables we detail the deviations from the exact values(computed viaFourier Integral and an adequate number of discretizations points) in term of impliedvolatilities for different set of parameters.

15



Table 1: Varying atm Volatilitystrike-σ 10% 20% 40% 50% 70% 100%1% -1.46E-02 -2.15E-04 6.55E-06 -5.17E-07 3.03E-05 2.36E-043% 5.62E-04 1.09E-04 3.30E-05 2.31E-05 2.39E-05 1.07E-045% -3.87E-04 -3.19E-05 1.36E-05 1.47E-05 2.10E-05 9.64E-057% -4.63E-05 3.45E-05 2.46E-05 1.96E-05 2.25E-05 1.01E-04

9% 2.21E-03 1.58E-04 3.54E-05 2.39E-05 2.44E-05 1.11E-04

Table 2: Varying correlationstrike-ρ -0.9 -0.5 0 0.5 0.91% -1.97E-05 1.46E-04 6.55E-06 -3.32E-05 1.58E-033% 6.89E-05 3.79E-05 3.30E-05 2.05E-05 -3.35E-055% 2.91E-05 1.99E-05 1.36E-05 2.05E-05 6.59E-057% 8.22E-06 1.10E-05 2.46E-05 3.38E-05 2.22E-059% 4.72E-05 3.48E-05 3.54E-05 2.80E-05 2.43E-05

Table 3: Varying vol of var

strike- 1% 10% 50% 100% 300% 500%1% 8.74E-05 8.22E-05 1.37E-05 -6.55E-06 4.74E-06 -7.56E-053% -2.80E-05 -2.78E-05 -2.64E-05 -3.30E-05 -1.10E-04 -1.44E-045% -2.45E-05 -2.44E-05 -2.23E-05 -1.36E-05 1.18E-04 5.29E-047% -2.60E-05 -2.59E-05 -2.44E-05 -2.46E-05 -1.72E-05 1.38E-059% -2.90E-05 -2.88E-05 -2.73E-05 -3.54E-05 -2.01E-04 -5.33E-04

Table 4: Varying time to maturitystrike-T 6M 1Y 2Y 5Y 10Y 20Y1% -1.34E-01 3.54E-03 2.88E-04 6.55E-06 1.59E-05 5.33E-05

3% 9.53E-04 2.29E-04 8.98E-05 3.30E-05 1.82E-05 2.74E-055% -8.20E-05 -1.81E-06 1.55E-05 1.36E-05 1.27E-05 2.45E-057% 2.16E-04 9.57E-05 5.31E-05 2.46E-05 1.58E-05 2.59E-059% 1.61E-03 3.06E-04 1.06E-04 3.54E-05 1.88E-05 2.83E-05

16



References

[1] Andersen L. and Andreasen J.(2002), Volatile Volatilities, Risk Magazine.

[2] Andersen L. and Piterbarg V.(2010), Interest Rates Modeling, Vol. 1, Chapt. 9,Atlantic Financial Press.

[3] Carr P.P. and Madan D.B.(1999), Option valuation using the fast Fourier transform. J. Comp. Finance, 2:61-73.

[4] Fang F. and Oosterlee C.W.(2008), A novel pricing method for european optionsbased on Fourier-Cosine series expansions, SIAM SISC.

[5] J. Laskar(2001), Frequency Map Analysis and quasi-periodic decompositions, pro-ceedings of the school ”Hamiltonian systems and Fourier analysis”, Porquerolles.

[6] Lipton A.(2002), The vol Smile Probem, Risk Magazine.

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Documents

Applications of Periodic and Quasi-Periodic Expansions to Options Pricing