Επιχειρησιακή Έρευνα / Operational Research. An International Journal. Vol.2, No.2 (2003), pp.157-186 157

Valuation of Exotic Options Using Moments

Mauro D'Amico1, Gianluca Fusai2, Aldo Tagliani3 1 IMQ, Università L. Bocconi Viale Isonzo 25, Milano, Italia

Ε-mail: mauro.damico@uni-bocconi.it 2 SEMEQ, Università del Piemonte Orientale, Facoltà di Economia

Via Perrone 18, Novara, Italia Ε-mail: gianluca.fusai@eco.unipmn.it

3 Disa, Università di Trento Via Inama 1, 38100 Trento, Italia Ε-mail: ataglian@cs.unitn.it

Abstract

In this paper we discuss the problem of recovering a density from its moments. For theoretical reasons, we propose the use of fractional moments combined with the Maximum Entropy density. We then discuss the application to the pricing of exotic options. Keywords: Moment problem, Exotic options, Maximum Entropy principle 1. Introduction Aim of the present paper is to discuss the use of moments for recovering an unknown density and then for pricing exotic options. This problem occurs very often in finance. For example in pricing Asian options under the Geometric Brownian Motion (GBM) stochastic process, a common approximation is based on the use of a lognormal density that shares with the unknown density the first two moments [Levy (1992)].

However, it is well known that to recover a unknown distribution from its moments (the so called Stieltjes moment problem) arises both existence and determinacy questions. So approximating the density on the basis of algebraic moments is equivalent to assume that the moment problem is determinate. But in general this is very difficult to prove.

The question is then if we can find the characterizing moments, i.e. a sequence of expected values of the random variable under study that characterize in a unique way the unknown density. As shown in the paper the answer is positive: for example the so called fractional moments, i.e. the expected value of Xγ , with γ a real number, characterize in a unique way the unknown distribution.

Once the determinacy problem is solved, another problem arises. How to exploit the information content of the moments. For this reason a choice regard the approximating distribution has to be made. Restricting our information to the

158 Επιχειρησιακή Έρευνα / Operational Research. An International Journal. Vol.2, No.2 (2003), pp.157-186 moments content, the Maximum Entropy (ME) density arises as natural approximation. One of the main reason for this choice is that the ME density converges in L1 norm to the unknown density as the number of considered moments increases. L1 norm convergence entails convergence in distribution, so that expected values (the prices) may be accurately calculated. Moreover, we can also obtain a bound to the error in pricing put options. The bound becomes tighter as we increase the number of moments. In this way we obtain a simple criterion for choosing the optimal number of moments in constructing the approximating density.

The paper is organised as follows. In the first section we give some preliminary results on the moment problem. Then we discuss the Maximum Entropy approximation and the problem of approximating the option price. Finally, we consider three concrete applications to option pricing problems: the standard Black-Scholes model, the Asian option pricing problem in the case of GBM and Square Root processes for the underlying asset. We give in the Appendices the main proofs.

2. Some results on the moment problem

The general moment problem can be formalised in the following form [Stoyanov (1998)]. Let µ0=1, µ1, µ2, … be a sequence of real numbers and I be a fixed interval, I R1. Suppose there is at least one density function f(x), x7I, such that:

( ) , 0,1,2,...nnI

x f x dx nµ= =∫

If the density function is uniquely specified by {µn} we say that the moment problem is determinate, i.e. the distribution is uniquely determined by its moments, otherwise the moment problem is indeterminate. In the case I=[0, +∞) the moment problem is called the Stieltjes moment problem, while in the case I=(-∞, +∞) we speak of the Hamburger moment problem.

There is some initial reason to think that matching moments will give a good approximation. For example, Lindsay and Roeder (1997) show that if two distributions functions have the same first n moments, then they must cross each other at least n times. Moreover, Akhiezer (1965, page 66) shows that the possible error can be bounded, and establishes the relevant relationships of the difference between two distributions that share the same 2n moments. Let FX(x) be the unknown distribution function and let GX(x) a distribution that shares with FX(x) the same 2n moments. Then we have:

( ) ( ) ( )X X nF x G x xρ− ≤ (1) where

Επιχειρησιακή Έρευνα / Operational Research. An International Journal. Vol.2, No.2 (2003), pp.157-186 159

( )

11

1 1

1 2

1 11 1 ...

n

nn

nn

n n n

xx x

xx

µ µµ µ

ρµ µ µ

−

+

+

=

…… …… …

…

Unfortunately, Lindsay and Basak (1995) show that this bound results to be tight only in the tails of the distribution. However, in next paragraph we will see how to improve this bound making some restriction on the form of the approximation to the unknown distribution.

In Stoyanov (1997) we can find several sufficient conditions for the moment problem to be determinate or indeterminate. Criterion 1 (moment generating function condition): In Stieltjes moment problem an infinite sequence of moments determines a unique distribution if this distribution decays asymptotically as an exponential exp(-axλ), a>0, λ ≥1⁄2.

For example, this criterion is not satisfied in the case of the lognormal density, that is, there are multiple distributions with exactly the same moments [Stoyanov (1997), pagg. 102-4]. This distribution has indeed tails decaying slower than prescribed. Criterion 2 (Carleman condition). This is the most popular criterion and gives a sufficient condition for the determinacy of the moment problem. Indeed if:

( ) ( )1/ 22

1(Hamburger moment problem)n

nn

µ∞

−

=

= +∞∑ (2)

or

( ) ( )1/ 2

1(Stieltjes moment problem)n

nn

µ∞

−

=

= +∞∑ (3)

then the moment problem is determinate. Criterion 3 (Krein condition). This criterion gives a sufficient condition for the indeterminacy of the moment problem. If:

( )( )2

ln(Hamburger moment problem)

1f x

dxx

+∞

−∞

− < ∞+∫ (4)

or

160 Επιχειρησιακή Έρευνα / Operational Research. An International Journal. Vol.2, No.2 (2003), pp.157-186

( )

( )2

20

ln(Stieltjes moment problem)

1

f xdx

x

+∞

− < ∞+∫ (5)

then the moment problem is indeterminate. The problem in using the above criterions is that they require the knowledge of

the moment generating function or of the density function, or they give sufficient conditions only, so that if they are not satisfied nothing about the determinacy problem can be said. The consequence is that in general it results to be very difficult to prove the determinacy of the moment problem using algebraic moment.

A recent result by Lin (1992) provides an alternative way of solving the problem considering the fractional moments instead of the algebraic ones. Indeed the result gives a sufficient condition for which the determinacy is automatically verified. Criterion 4 (Lin condition). Lin has proved that given a sequence of positive and distinct numbers {ξj} j=0,…,∞ converging to some finite ξ∗ , with E[Xξ∗] < ∞, then the sequence of moments E[Xξj] characterizes uniquely the distribution of the r.v. X.

Using the result in Lin, the trouble of the Stieltjes moment problem determinacy is bypassed. 3. The Maximum Entropy Distribution In previous section we have discussed the problem of which moments to use for recovering a density. The Lin's result suggests the use of fractional moments. In this way using fractional moments we can characterize in a unique way the unknown density. This cannot be guaranteed using integer moments.

On the basis of this result we try to recover the true density using the maximum entropy approach. The maximum entropy principle, Jaynes (1978) and Golan et al. (1996), is the most popular one in choosing the approximating distribution and is well known and widely diffused in Statistics. In Finance has been already used for estimating the implied risk neutral distribution from the option prices [Buchen and Kelly (1996), Avellaneda (1998)]. Assuming the given moments as known information, the maximum entropy (ME) principle chooses, out of the distributions consistent with the given partial information, the one having maximum entropy, or equivalently the most uncertain, accomplishing a principle of scientific honesty. Moreover, using this approximating density, we can show that: a) for a given number of moments, the approximated distribution should be positive, b) for a given number of moments, a bound to the error made in pricing exotic

options is provided;

Επιχειρησιακή Έρευνα / Operational Research. An International Journal. Vol.2, No.2 (2003), pp.157-186 161 c) increasing the number of moments considered, the sequence of approximants is

converging to the true distribution in some norm. This result allows us to obtain a bound to the error in pricing exotic options.

We remark that property a) is usually not guaranteed when we approximate the true density with an Edgeworth or Laguerre series expansion. Property b) has never been discussed in the literature except in the present paper that gives a bound to the error in terms of the number of moments considered. Property c) is a delicate question when for example we use Edgeworth series. Indeed in practical work, the Edgeworth series converges for a small number of terms and then diverges, whilst the proposed ME approximation is converging to the true distribution in L1 norm.

If a finite set jE X ξ , j=0,..M of expected values is considered and the

Maximum Entropy principle is invoked then the following approximate probability density function ( ) ( )M

Xf x is obtained:

( ) ( ) ( )0

exp , 0,jM

MX j

jf x x xξλ

=

= − ∈ ∞

∑ (6)

to be supplemented by the constraints, for j=1,…,M:

( ) ( )0 0:j j M

j XE X x f x dxξ ξµ∞

= = ∫

allowing us to obtain the unknown coefficients λ0, …, λM. These coefficients can be found solving the minimization problem, compare [Kesavan (1992)]:

( )1

1 0,..., 1 1min ,..., ln exp j

M

M M

M j j jj j

x dxξ

λ λλ λ λ µ λ

+∞

= =

Γ = + −

∑ ∑∫ (7)

whilst λ0, the normalization constant, is obtained by (6) with j=0 ( 0 0ξ = ) imposing that the density integrates to 1.

The choice of the ME approximant is due mainly to the fact that the ME density ( ) ( )M

Xf x converges to the true density in entropy, Appendix A, i.e. if we define:

( ) ( ) ( ) ( ) ( )0

0

: lnM

M M MX X X j j

jH f f x f x dx λ µ

∞

=

= − = ∑∫

and

( ) ( )0

: lnX X XH f f x f x dx∞

= − ∫

162 Επιχειρησιακή Έρευνα / Operational Research. An International Journal. Vol.2, No.2 (2003), pp.157-186 then:

( )lim MX XM

H f H f→∞

=

holds. Let us introduce the Kullback-Leibler measure of directed divergence

( ) ( ) ( )( ) ( )0

, : lnM XX X X M

X

f xD f f f x dx

f x∞

= ∫

Since ( )Xf x and ( ) ( )MXf x share the same moments, then (Lemma C.1):

( ) ( ) [ ], M MX X X XD f f H f H f = −

holds. Moreover, from the inequality, [Kullback (1967)]:

( ) ( ) ( ) ( )( )2

0

1,2

M MX X X XD f f f x f x dx

∞ ≥ − ∫ ,

we have also convergence in the L1 norm:

( ) ( ) ( ) ( ) [ ]0

2M MX X X Xf x f x dx H f H f

∞ − ≤ − ∫

Then L1 norm convergence entails convergence in distribution. Then expected values (the prices) may be accurately computed by replacing ( )Xf x with ( ) ( )M

Xf x . Indeed, for each bounded function g(x), |g(x)| ≤ G, we have:

[ ] [ ] ( ) ( ) ( ) [ ] ( )( )0 0 02M MM

X X X XE g E g G f x f x dx G H f H f+∞

− ≤ − ≤ − ∫ (8)

Then (8) suggests which fractional moments have to be chosen:

{ }01

jM

jE X ξ

= such that ( )M

XH f is minimum

The estimate of the ME density should then be obtained through two nested minimization procedures:

1 10 0,..., ,..., 1 1

min min ln expj j

M M

M M

j jj j

E X x dxξ ξ

ξ ξ λ λλ λ

+∞

= =

+ − ∑ ∑∫ (9)

If, in addition to one of the previous expected values other requirements have to be taken into account, then such extra-requirements may be included in a priori

Επιχειρησιακή Έρευνα / Operational Research. An International Journal. Vol.2, No.2 (2003), pp.157-186 163

probability density w(x). Then the approximate ( )

( )_ M

Xf x is obtained by minimizing the Kullback-Leibler measure of directed divergence, [Kesavan (1992)]:

[ ] ( ) ( )( )0

min || : ln XX Xf

f xD f w f x dx

w x∞

= ∫ (10)

constrained by:

( ) 00j j

Xx f x dx E Xξ ξ∞ = ∫ (11)

with ξj positive quantities and j=0,…,M.

Then ( )

( )_ M

Xf x assumes the analytical form, [Kesavan (1992)]:

( )

( ) ( )_

0

exp , 0,j

M M

jXj

f w x x xξλ=

= − ∈ ∞

∑ (12)

Assuming positive real moments as a given information then the approximate ( )

( )_ M

Xf x converges in directed divergence to ( )Xf x (see Appendix A) in the sense that

( )_lim || ||

M

XXMD f w D f w

→∞

=

(13)

Since ( )Xf x and ( )

( )_ M

Xf x satisfy the same constraints, we have, [Kesavan (1992) page 161]:

( ) ( )_ _|| || ||

M M

X XX XD f f D f w D f w

+ =

(14)

so that (13) entails ( )

[ ]_

|| ||M

XXD f w D f w ≤

, whilst (14) brings to

( )( ) ( )

( )_ _

lim || 0M M

M MX XX XM

D f f H f H f→∞

= − =

(15)

164 Επιχειρησιακή Έρευνα / Operational Research. An International Journal. Vol.2, No.2 (2003), pp.157-186

Entropy-convergence (15) turns out to provide L1-norm convergence, by referring to the inequality [Kullback (1967)]:

( ) ( )

( )( )

( )2

_ _ _1||2

M M M

X X XX X XD f f H f H f f x f x dx

= − ≥ − ∫ (16)

3.1. How many moments? In our previous discussion we assumed that the first M common moments of the unknown distribution are preassigned. This may be justified in a theoretical work but in practice it is often necessary to decide how many moments M should be taken in a given situation. We know from Appendix A that the monotonic decreasing sequence {H[fX

(j)], j=1,2, …} converges to H[fX]. Now to have an estimate of the error for large values of j, we refer to Aitken ∆2-method as described below.

For every distribution unlike the maximum entropy distribution, the contents of information are shared by the whole sequence of fractional moments and a further addition of a moment causes an entropy decrease. The size of decrease is not the same for every moment. In fact for most of the distributions met in practice, we find that the first few moments generate a higher decrease than the latter in the list. Consequently it seems reasonable to expect that the sequence of points (j, H[fX

(j)]), j=1, 2,…∞ lie on a convex curve. Therefore the second order differences:

( ) ( ) ( ) ( )1 22 2 , 2j j j jX X X XD H f H f H f H f j− − = − + >

are positive. Next using Aitken's ∆2-procedure, we define a new accelerated sequence as:

( ) ( )( ) ( )( )

( )

21

2

j jX Xj jacc

X X jX

H f H fH f H f

D H f

− − = −

which converges faster to H[fX] than the initial sequence. Letting [ ] ( )Macc

X XH f H f ≈ , we obtain the following estimate of window or the error term:

( )( ) ( )( )

( )

21

2

M MX XM

X X MX

H f H fH f H f

D H f

− − − ≈

(17)

Επιχειρησιακή Έρευνα / Operational Research. An International Journal. Vol.2, No.2 (2003), pp.157-186 165 required in (17). As a result for any prefixed error bound, we can determine an optimal value of M. 3.2. Application to option pricing The results given in previous section allows us an accurate pricing of the put option on an asset of which we are able to compute the sequence of moments. Indeed, if:

( ) ( )0

rtXp e K x f x dx

∞ +−= −∫ (18)

and

( ) ( ) ( )0

MrtM Xp e K x f x dx

∞ +−= −∫ (19)

denote the true and the approximate price, we have:

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) [ ]( )

0 0

0

0

2

MrtM X X

MrtX X

MrtX X

MrtX X

p p e K x f x dx K x f x dx

e K x f x f x dx

e K f x f x dx

e K H f H f

∞ ∞+ +−

∞−

∞−

−

− = − − −

≤ − −

≤ −

≤ −

∫ ∫

∫∫

(20)

and so we obtain a bound for the error, although it depends on the unknown quantity H[fX]. But this quantity can be estimated resorting to the Aitken accelerating sequence and obtain the approximation in (17), so that the error bound may be obtained uniquely in terms of calculated quantities:

( ) ( )( )( ) ( ) ( )

21

1 22

2

M MX Xrt

M M M MX X X

H f H fp p e K

H f H f H f

−

−− −

− − ≤ − +

(21)

4. Numerical examples

In this section we apply the proposed technique to the following cases: a) Black-Scholes plain vanilla option pricing model;

166 Επιχειρησιακή Έρευνα / Operational Research. An International Journal. Vol.2, No.2 (2003), pp.157-186 b) Fixed strike Asian option for the lognormal process; c) Fixed strike Asian option for the square root process. The standard Black-Scholes model is considered in order to illustrate the importance of using fractional moments. Indeed, it is well known that the lognormal distribution is not uniquely determined by the infinite sequence of its algebraic moments. There exist other distributions with exactly the same sequence of moments [Stoyanov (1997), pages 102-4], provides some examples. However, in the lognormal case we can easily compute the fractional moments.

The other two examples are related to the pricing of Asian options, assuming two different processes for the underlying asset, i.e. lognormal and square root.

The pricing of Asian option in the lognormal case is particularly interesting because no analytical pricing formula is available. Moreover, given that there are multiple distributions with exactly the same moments as the lognormal, similarly one might expect that approximations based on a sequence of moments might either fail to converge, or converge to another distribution with the same moments also in the case of the arithmetic average. However, many approximations are based on integer moments. In this case as well we discuss how to compute fractional moments and how they are useful in providing accurate estimate of the option prices.

Finally in the third example we discuss the pricing of Asian options using a square root process. In this case we are able to compute the moment generating function and we can show how to compute fractional moments from it. In this case as well we can provide accurate option prices. We remark that in the literature no result on pricing Asian options using the square root process is available, so this example is of interest on its own. 4.1. Black Scholes model We assume as random variable X of interest the price St of an asset and we assume that St evolves according to the well-known Geometric Brownian Motion (GBM) process:

2

20

tr t W

tS S eσ σ

− +

= (22)

where r is the instantaneous risk free rate, σ is the instantaneous volatility and Wt is a Wiener process. Under the assumption of GBM, we know that:

22 21

2 20 0

tn r t n tn ntE S S e

σ σ

− + = (23)

Επιχειρησιακή Έρευνα / Operational Research. An International Journal. Vol.2, No.2 (2003), pp.157-186 167

Using this sequence of moments we can reconstruct the ME density and then compute the option price that can be compared with the analytical result given by the Black-Scholes formula. A proper choice of ξj is obtained considering the properties of (23). In the following we exploit only the information content of the expression in (23), without using the fact that St has a lognormal distribution. 4.1.1. Asymptotic behavior of ( )Sf s When n takes integer values, the power moments increase faster than moments corresponding to any function having asymptotic value exp(-ζsα), α, ζ>0. Indeed, being higher moments heavily dependent on the tail region, we have:

( )0

exp exp ln .nn

ns s ds ne

αµ ςςα

+∞ = − =

∫

Then the density underlying (23) decreases slower than ( )exp , 0sας α− ∀ >

4.1.2. Behavior in the neighboring of the origin Being (23) defined ∀n ∈ R then fS(s) admits a countable number of negative moments. It is reasonable to assume as maximum entropy approximating function which satisfies both requirements in the neighboring of the origin and in far in tails:

( ) ( )0

exp lnM

M jS j

jf s sλ

=

= −

∑

When M=2, then fS(s) coincides with a lognormal, the true density underlying (23). Nevertheless, being (23) the Mellin transform of the true density fS(s), we have:

( ) ( ) ( ) ( )000

0

ln lnj n

t j jS tj

n

d E Ss f s ds E S

dn∞

=

= =∫

On the other hand, djE0(Stn)/dnj|n=0, through a finite difference scheme may be

approximated by values ( )0 ,ktE Sξ ξk=k∆n where k=0, ±1, ±2,… and ∆n a proper step.

Equivalently, the first derivatives djE0(Stn)/dnj|n=0, may be approximated by positive

and negative fractional moments ( )0 ,ktE Sξ with ξk<0 if k<0. The approximating

density will be found through two steps:

168 Επιχειρησιακή Έρευνα / Operational Research. An International Journal. Vol.2, No.2 (2003), pp.157-186 i) Assuming ( )0 ,k

tE Sξ k=-N,...,1 as expected values, the ME technique provides an

approximating density ( ) ( ) ( )1exp kNS jk N

f s sξλ−

=−= −∑

constrained by:

( ) ( ) ( )00, ,...,1k kN

S ts f s ds E S k Nξ ξ∞ − = = −∫

Solving (7) we obtain explicitly fSt(-N)(s)

ii) In the second step we assume

( ) ( ) ( )NS Sw s f s−=

as weight function. Then considering the expected values ( )0k

tE Sξ , k=2,…,M through

minimization of direct divergence, we have the required approximant: ( )

( ) ( )_

2

exp expk k

M M M

S j jSk k N

f s w s s sξ ξλ λ= =−

= − = −

∑ ∑

When M→∞, ( )_

t

M

Sf converges in entropy and then in L1-norm to the true density

tSf , see Appendix A.

The above two steps are the correct way to proving ( )_

t

M

Sf converges in L1-norm. For practical purposes, we can consider simultaneously all the negative and positive fractional moments ( )0

ktE Sξ , k=-N,…,M and consider the corresponding

approximant:

( ) ( ) exp k

MM

S jk N

f s sξλ=−

= −

∑

Finally, we remark that in the lognormal case we have an explicit expression for the entropy. Indeed, given the expression for the lognormal density of St, we can compute analytically the entropy that results to be:

( ) ( ) ( )2 2

00

ln 21ln ln2 2 2S S

tf s f s ds S r t

πσ σ∞ − = + + + −

∫ (24)

so that in minimizing the potential function, once we multiply by (-1) the expression above, we have the minimum value we can achieve. In the lognormal case we can

Επιχειρησιακή Έρευνα / Operational Research. An International Journal. Vol.2, No.2 (2003), pp.157-186 169 then compute also the bound in (20) without using the accelerating Aitken's approximation:

In Table 1 we find the ME density using different sets of fractional moments, i.e. fractional moments of order ±1/2, ±3/4, ±1/4, ±1/8. In Table 1 we give the values of the optimal value of the potential function in (7) multiplied by (-1), the theoretical value of the Entropy as from formula (24), the bound to the option price (assuming K=1) and the effective difference between the true Black-Scholes value and the numerical approximation. The bound is obtained for the price of the put option, but using the put-call parity, it is valid for the call option price as well. Comparing the error bound value in equation (20), where we have used the true entropy value, and the effective difference between the true BS price and the numerical approximation, we can see that the bound is conservative: the effective difference is always much smaller. In Table 2 we report the call option prices for different volatility levels and using the same set of fractional moments. As we can see from the table, also for high volatility levels two fractional moments (±1/2) are sufficient to recover with great accuracy the lognormal density. This is also evidenced from the difficulty in the minimization of obtaining lower values for the potential value increasing the number of considered moments. With two moments, we obtain three digits accuracy in the option price. Finally, we remark that using the algebraic moments of power 1 and 2, we obtained very poor results.

Table 1. Entropies for Different Set of Fractional Moments and the Option Price Bound (r = 0.1, t = 1, S0 = 1, K = 1)

Fractional Moments

(±1/2) (±1/2; ±1/4)

(±1/2; ±1/4; ±1/8)

(±1/2; ±3/4; ±1/4; ±1/8)

Theoretical entropy

σ=0.1 0.78865 0.78865 0.78865 0.78865 0.78865 Bound in (20) 0.00047 0.00047 0.00048 0.00045 - Effective difference -0.00001 -0.00001 -0.00001 -0.00001 -

σ=0.3 -0.26998 -0.26998 -0.26998 -0.26998 -0.26997 Bound in (20) 0.00479 0.0044 0.00434 0.004 - Effective difference -0.00009 -0.00009 -0.00009 -0.00007 -

σ=0.5 -0.70094 -0.70085 -0.70085 -0.70081 -0.70079 Bound in (20) 0.01552 0.00975 0.00968 0.0058 - Effective difference -0.00004 -0.00038 -0.00036 -0.00021 -

170 Επιχειρησιακή Έρευνα / Operational Research. An International Journal. Vol.2, No.2 (2003), pp.157-186 Table 2. Plain Vanilla Option Prices under the Lognormal Process K is the Strike, BS is the

Price Obtained with the Black-Scholes Formula

r=0.1, t=1, S0=1, σ=0.1

K BS (±1/2) (±1/2; ±1/4) (±1/2; ±1/4; ±1/8)

(±1/2; ±3/4; ±1/4; ±1/8)

0.9 0.18631 0.18631 0.18631 0.18631 0.18631 0.95 0.14304 0.14304 0.14304 0.14304 0.14304 0.1 0.10308 0.10309 0.10309 0.10309 0.10309 0.105 0.06883 0.06884 0.06884 0.06884 0.06884 0.11 0.04217 0.04217 0.04217 0.04217 0.04217 r=0.1, t=1, S0=1, σ=0.3 0.9 0.2251 0.22519 0.22518 0.22518 0.22517 0.95 0.19474 0.19483 0.19483 0.19483 0.19481 0.1 0.16734 0.16743 0.16743 0.16743 0.16741 0.105 0.1429 0.14297 0.14297 0.14297 0.14296 0.11 0.12131 0.12136 0.12136 0.12137 0.12135 r=0.1, t=1, S0=1, σ=0.5 0.9 0.28644 0.28665 0.28679 0.28677 0.28665 0.95 0.2619 0.26204 0.26228 0.26226 0.26212 0.1 0.23927 0.23931 0.23964 0.23963 0.23948 0.105 0.21843 0.21837 0.2188 0.21879 0.21863 0.11 0.1993 0.19912 0.19964 0.19963 0.19947

4.2. Fixed Strike Asian option (lognormal process) As second example, we consider the pricing problem for an Asian option, again assuming that under the martingale measure the underlying asset price evolves according to the GBM process given in (22). The payoff of an Asian option depends on the average price over a fixed time interval. Assuming that the average is computed continuously in time in the period (0, t), we define:

0

1 t

t uA S dut

= ∫ (25)

Επιχειρησιακή Έρευνα / Operational Research. An International Journal. Vol.2, No.2 (2003), pp.157-186 171 and the payoff at time t of a fixed strike Asian put option is given by max[K – At, 0]. The pricing of this contract requires the computation of:

[ ] ( ) ( )0 0

Krt rtt Ae E K A e K x f x dx+− −− = −∫ (4.5)

where Af denotes the density of the arithmetic average At. Then the pricing problem is essentially related to the determination of this density.

Before showing how to use the ME approach in order to price this option, we summarize some known results in the literature. 1. In the case of the lognormal density there are multiple distributions with exactly

the same moments. This distribution has indeed tails decaying slower than exp(-axλ). As a result, in the case of the arithmetic average one might expect that approximations based on a sequence of integer moments might fail to converge, or converge to another distribution with the same moments. A difficult proof would no doubt be required and maybe a similar approximation could be numerically very unstable for more than a few moments.

2. Geman and Yor (1993) have provided the algebraic moments µn =E0[Atn], n=1,

2,… without proving whether they determine uniquely the distribution. Morevor, Geman and Yor prove that the sufficient Carleman's condition is not satisfied, but they use a criterion that has to be used for the so called Hamburger's moment problem, i.e. when the r.v. is defined over the entire real axis, whilst in the present case the r.v. is defined over the positive axis.

3. Although the determinacy problem in this case is unsolved, there are several approximations based on trying to recover the density of average using the information content of its algebraic moments. For example, Turnbull and Wakeman (1991) and Levy (1992) approximate the density of the average using a lognormal distribution with the same first two moments and this approximation results to be very common also among the practitioners. It is common to use also Edgeworth series expansion around a lognormal, although we can obtain a bimodal density with negative values. This approximation provides good results when σ√t<0.2.

4. Recently, Dufresne (2000) has shown that the distribution of 1/At is completely determined by its moments. In this sense, it could be reasonable to recover the density of 1/At and then the density of At. Dufresne exploits the results for obtaining a representation of the option price using Laguerre series. However the determination of the moments of 1/At requires to solve numerically a recursive equation involving the numerical computation of derivatives, and this creates numerical problems in the calculations.

5. Fusai and Tagliani (2002) have shown that the distribution of ln(At) is determined by its moments. In order to compute these moments, a two step procedure is

172 Επιχειρησιακή Έρευνα / Operational Research. An International Journal. Vol.2, No.2 (2003), pp.157-186

required. Indeed Geman and Yor (1992) provide a Laplace transform with respect to time for µγ =E0[At

γ], γ∈R. The logarithmic moments require then the computation of the derivatives with respect to γ and a numerical Laplace inversion. The analytical form of the density of ln(At) is then approximated referring to a Maximum Entropy technique. This approximation turns out to be very accurate using only the first four moments.

Respect to the use of the logarithmic moments as in Fusai and Tagliani (2002), in this section we construct the Maximum Entropy density using the fractional moments. In order to compute these moments, we can use the results in Geman and Yor (1992). Defining:

( )2

0s

h W vshD e ds+= ∫ (26)

where v=2r/σ2-1, we have, by the scaling property of the BM:

224

4t t

A Dt σσ

= (27)

Geman and Yor (1992) give the Laplace transform of E0[Dhγ] with respect to h=σ2t/4,

for every γ∈R:

( )( ) ( )( )

0 00

1 12 2:

2 12 2

n h nh h h

n

v vn nE D e E D dh

v vn

λ

µ µ

µ µλ

∞ −

+ − Γ + Γ + Γ − = =

− + Γ Γ + +

∫ (28)

so we have that:

( )( ) ( )( )0 0n n n

h t h hE A h E D= (29)

Remark that E0[Dhn]is nothing but the Mellin transform of the density of Dh. The

fractional moments require then the numerical inversion of the Laplace transform in (29). This inversion can be done accurately using the Abate-Whitt algorithm as described in [Fusai and Tagliani (2002)].

In Table 3, using fractional moments of order (± 1/2; ±1/4; ±1/8), we give a) the optimal values of the potential function in (7) multiplied by (-1) and b) the bound to the option price error, equation (20), (assuming K=1) computed using the Aitken's extrapolation formula. In Table 4 we report the call option prices for different volatility levels and using the same set of fractional moments. In this case we do not have a exact option value and we compare our numerical approximations with the lower and upper bounds computed according to the procedure described in Thompson

Επιχειρησιακή Έρευνα / Operational Research. An International Journal. Vol.2, No.2 (2003), pp.157-186 173 (1999), with the numerical Laplace inversion of the Geman and Yor formula and with the ME approximation computed using the logarithmic moments. As in the lognormal case, comparing the error bound value and the lower and upper bounds, we can see that the error bound in equation (20) is conservative: the effective difference is always much smaller. However, respect to the lognormal case, also for low volatility levels, two moments are not enough: three-digits accuracy requires at least four fractional moments.

Table 3. Entropies for different set of fractional moments and the option price bound (r = 9%, t = 1 year, S0 =1, Κ=1)

Fractional Moments

σ (±1/2) (±1/2; ±1/4) (±1/2; ±1/4; ±1/8)

Bound in (20)

0.1 -1.23901 -1.37881 -1.37881 0.00314 0.3 -0.2983 -0.29921 -0.29921 0.00013 0.5 0.176657 0.173992 0.173991 1.3988*10

In Table 4 we report the call option prices for different volatility levels and using

the same set of fractional moments. As we can see from the table, also for high volatility levels two fractional moments (±1/2) are sufficient to recover with great accuracy the lognormal density: we obtain three digits accuracy in the option price. This is also evidenced from the impossibility of obtaining lower values for the potential value increasing the moments. Again, using algebraic moments of power 1, 2, 3 and 4, we obtained very poor results as well as troubles in the minimization of the potential function. LB and UB are the lower and upper bounds calculated according to Thompson (1999). CIL is the numerical inversion of the option price Laplace transform obtained in Geman and Yor. ME with log moments is the Maximum Entropy approximation computed using the first four logarithmic moments, as described in Fusai and Tagliani (2002).

Table 4. Asian option prices under the Lognormal Process

r = 9%, t = 1 year, S0 =1, σ=0.1

K LB (±1/2) (±1/2; ±1/4) (±1/2; ±1/4; ±1/8)

ME with log. moments CIL UB

90 13.38519 13.519 13.38526 13.38525 13.3851 13.38537 13.38603 95 8.91183 9.36109 8.91272 8.91269 8.91301 8.91185 8.91296

100 4.91508 5.81117 4.91705 4.917 4.91826 4.91512 4.91541 105 2.06993 3.14564 2.07127 2.07125 2.07137 2.07007 2.07038

174 Επιχειρησιακή Έρευνα / Operational Research. An International Journal. Vol.2, No.2 (2003), pp.157-186

110 0.63006 1.42686 0.62935 0.62936 0.62773 0.63027 0.63102 r = 9%, t = 1 year, S0 =1, σ=0.3

90 14.98279 15.03082 14.98358 14.983 14.98304 14.98396 14.99285 95 11.65475 11.6879 11.65548 11.65502 11.65486 11.65589 11.66128

100 8.82755 8.83421 8.82785 8.82765 8.82782 8.82876 8.83329 105 6.51635 6.49166 6.51643 6.51658 6.51712 6.51779 6.52257 110 4.69491 4.64097 4.69487 4.69537 4.69639 4.69671 4.70265

r = 9%, t = 1 year, S0 =1, σ=0.5 90 18.18295 18.23573 18.18811 18.18809 18.18746 18.18884 18.22077 95 15.43707 15.44218 15.44299 15.44298 15.44169 15.44272 15.47216

100 13.02253 12.97387 13.02906 13.02907 13.02769 13.02816 13.0568 105 10.92375 10.82069 10.93101 10.93104 10.92981 10.92963 10.9588 110 9.11795 8.96421 9.12586 9.12591 9.12515 9.12431 9.156 4.3. Fixed Strike Asian option (Square Root process) In this section we consider again the problem of pricing an Asian option prices, but assuming a different process for the underlying asset. In particular we assume that the process for the underlying asset is described by a square root process, i.e.:

dS = r S dt + σ ∉S dWt

The advantage of this process respect to the GBM hypothesis is that it allows for a smile effect that contradicts the hypothesis of lognormality.

In this case we can compute the moment generating function (mgf) of the average and then the moment problem is uniquely determined, by Criterion 1.

The moment generating function of Yt = ∩0tSudu can be obtained by the

Feynman-Kac equation and is given by:

( ) ( ) 00 ,0 0, ;

tuS du t Sv S t E e e

µ ϑ µµ− − ∫= =

where:

( ) ( )( )( ) ( )

2 exp 1,

expt

tr r tµ λ

ϑ µλ λ λ

−=

+ + − (30)

and

Επιχειρησιακή Έρευνα / Operational Research. An International Journal. Vol.2, No.2 (2003), pp.157-186 175

2 22rλ µσ= + (31)

In order to compute the fractional moments from the moment generating function (mgf), we exploit the fact that [see Cressie (1986)]: • the inverse moments are given by:

( )( ) ( )1

00 0

1 1 , ; , 0Y

t

f yE dy v S t d

Y yα

α α µ µ µ αα

∞ ∞ − = = > Γ

∫ ∫

where Γ(n) is the gamma function, • the positive fractional moments are given by:

( ) ( ) ( )( )

( )' 1 0

0 0

1 , ;,

'

m m

t Y m

v S tE Y y f y dy dα α α µ

µ µα µ

∞ ∞ −− ∂= =

Γ ∂∫ ∫

α=m-α′ >0, m=[α]+1, where [α] denotes the .greatest integer ≤α, so that 0<α′ <1. When fractional moments ( )0 ,k

tE Y ξ are used, the choice of ξk can be made

studying the analytical properties of v(S0, t, µ): a) v(S0, t, µ) has convergence abscissa µ0 = (r/σ)2/2. Then the underlying density

fY(y) has asymptotic decay exp(-µ0 y); b) for µ →∞, v(S0, t, µ)≈ ( )2

0exp 2 / Sµ σ− . From the tables of Laplace transform

the function ( )20exp 2 / Sµ σ− admits as original function

2

0 02 1 2exp4

S Syσ σ

−

Combining the results of step a) and b), we may assume as a priori density:

( )2

2023

1 2 1exp4 2Y

A S rw y yyy σ σ

= − −

where A≡A(r, σ, S0) is a normalizing constant. ( )Yw y takes into account both the asymptotic decay and the behavior in the neighboring of the origin. Then a proper choice of ξk leads to assume 0<ξk<1, according to asymptotic decay of ( )Yf y . By

minimizing the directed divergence, under the constraints ( )0 ,ktE Y ξ 0<ξk<1,

k=0, …,M, we have the approximating density:

176 Επιχειρησιακή Έρευνα / Operational Research. An International Journal. Vol.2, No.2 (2003), pp.157-186

( ) ( ) ( )0

exp k

MM

Y Y kk

f y w y yξλ=

= −

∑

which converges in L1 norm to the underlying density fY, when M→∞, see Appendix A. As in previous Example 1, ( )M

Yf is stable, by locating the nodes ξk in different subintervals of (0,1).

Comparison between call prices CME and CEX obtained by ( )MYf and by numerical

inversion of the mgf are illustrated in Tables 5 and 6. The numerical inversion of the mgf is obtained by Laplace transform inversion [Fusai (2001)]. The set of nodes ξk are chosen as (1; ¾), (1; ¾; ½), (1; ¾; ½; 3/8). The choice of the moment of order 1 is due to the asymptotic properties of the density function. Regard the choice of the other fractional positive moments, we remark that several tentatives have shown that different choices of ξk lead to similar results. In Table 5 we report the numerical entropies computed using 2, 3, and 4 moments. With just the first two fractional moments, we obtain a three-digits accuracy also for high volatility levels. CIL in Table 6 is obtained by inverting the Laplace transform using the Abate-Whitt algorithm as detailed in Fusai (2001).

Table 5. Entropies for different set of fractional moments and the option price bound (r = 10%, t = 1 year, S0 = 1)

Fractional Moments

σ (1; ¾) (1; ¾; ½) (1; ¾; ½; 3/8)Bound in (20)

0.1 -1.38587 -1.38589 -1.386 n.a. 0.2 -0.69342 -0.69376 -0.69376 0 0.3 -0.29173 -0.292 -0.29209 0.0088 0.4 -0.00984 -0.01086 -0.011 0.00583 0.5 0.205526 0.204353 0.203525 0.05704

Table 6: Asian option prices in the Square Root Process

r = 10%, t= 1 year, S0 = 1, σ = 0.1

K (1; ¾) (1; ¾; ½) (1; ¾; ½; 3/8) CIL 0.9 0.13734 0.13735 0.13735 0.13734

0.95 0.09292 0.09298 0.09298 0.092941 0.05245 0.05262 0.05262 0.05258

1.05 0.02239 0.02264 0.02264 0.02261.1 0.00656 0.00676 0.00676 0.00687

Επιχειρησιακή Έρευνα / Operational Research. An International Journal. Vol.2, No.2 (2003), pp.157-186 177

r = 10%, t= 1 year, S0 = 1, σ = 0.3 0.9 0.15432 0.15402 0.15397 0.15384

0.95 0.12036 0.12014 0.1201 0.120011 0.09086 0.09077 0.09076 0.09075

1.05 0.06624 0.06632 0.06633 0.06641.1 0.04655 0.04678 0.04682 0.04696

r = 10%, t= 1 year, S0 = 1, σ = 0.5 0.9 0.18753 0.18721 0.18677 0.18691

0.95 0.15842 0.15823 0.15805 0.158211 0.13228 0.13226 0.13236 0.13253

1.05 0.10917 0.10931 0.1097 0.109871.1 0.08903 0.08933 0.09 0.09016

5. Conclusions In the present paper we have considered the problem of recovering an unknown density from its moments. In particular we have discussed the reason for using fractional moments combined with a Maximum Entropy approximation and then we have presented some applications to option pricing. References Abate, J. and Whitt, W. (1992). The Fourier-series method for inverting transforms of

probability distributions, Queueing Systems Theory Appl. vol. 10, 5-88. Akhiezer, N.I. (1965). The Classical Moment Problem and some Related Questions in

Analysis. Oliver & Boyd, London. Avellaneda, M. (1998). Minimum Entropy Calibration of Asset Pricing Models.

Journal of Theoretical and Applied Finance vol. 1(4), 447-472. Buchen, P.W. and Kelly, M. (1996). The Maximum Entropy Distribution of an Asset

Inferred from Option Prices, Journal of Financial and Quantitative Analysis vol. 31(1), 143-159.

Cressie, N. (1986). The Moment Generating Function has its Moments. Journal of Statistical Planning and Inference vol. 13, 337-344.

Dufresne, D. (1989). Weak Convergence of Random Growth Processes with Applications to Insurance. Insurance: Math. Econonom. vol. 8, 187-201.

178 Επιχειρησιακή Έρευνα / Operational Research. An International Journal. Vol.2, No.2 (2003), pp.157-186 Dufresne, D. (2000). Laguerre Series for Asian and Other Options. Mathematical

Finance vol. 10(4), 407-428. Fusai, G. (2001). Applications of the Laplace Transform for Evaluating Occupation

Time Options and other Derivatives. PhD. Thesis, University of Warwick. Fusai, G. and Tagliani, A. (2002). An Accurate Valuation of Asian Options Using

Moments, International Journal of Theoretical and Applied Finance vol. 5(2), 147-169

Geman, H. and Yor, M. (1992). Quelques Relations Entre Processus de Bessel, Options Asiatique et Fonctions Confluentes Hypergéométriques. C.R. Acad. Sci. Paris vol. 314(1), 471-474.

Geman, H. and Yor, M. (1993). Bessel Processes, Asian Options and Perpetuities. Mathematical Finance vol. 3(4), 349-375.

Golan, A., Judge, G. and Miller, D. (1996). Maximum Entropy Econometrics: Robust Estimation with Limited Data, John Wiley & Sons.

Jaynes, E.T. (1978). Where Do we Stand on Maximum Entropy, in The Maximum Entropy Formalism, (R.D. Levine and M. Tribus, eds.). MIT Press Cambridge MA, 15-118.

Kesavan, H.K. and Kapur, J.N. (1992). Entropy Optimization Principles with Applications. Academic Press.

Kullback, S. (1967). A lower bound for discrimination information in terms of variation. IEEE Trans. on Information Theory IT-13, 126-127.

Levy, E. (1992). Pricing European Average Rate Currency Options. Journal of International Money and Finance vol. 11, 474-491.

Lin, G.D. (1992). Characterizations of Distributions via Moments. Sankhja: The Indian Journal of Statistics vol. 54, Series A, 128-132.

Lindsay, B.G. and Basak, P. (1995). Moments Determine the Tail of a Distribution (But not Much Else). Technical report 95-7, Center for likelihood Studies, Dept. of Statistics, The Pennsylvania State University, University Park, PA.

Lindsay, B.G. and Roeder, K. (1997). Moment-based Oscillation Properties of Mixture Models. Ann. Statist. vol. 25, 378-386.

Shohat, J.A. and Tamarkin, J.D. (1943). The Problem of Moments. AMS Mathematical Survey, 1, Providence RI.

Stoyanov, J. (1997). Counterexamples in Probability (2nd edition). Wiley Series in Probability and Statistics.

Tagliani, A. (2001). Recovering a Probability Density Function from its Mellin Transform, Applied Mathematics and Computation vol. 118, 151-159.

Thompson, G.W.P. (1999). Topics in Mathematical Finance. PhD. Thesis, Queens' College, January.

Turnbull, S. and Wakeman, L. (1991). A Quick Algorithm for Pricing European Average Options. Journal of Financial and Quantitative Analysis vol. 26, 377-89.

Επιχειρησιακή Έρευνα / Operational Research. An International Journal. Vol.2, No.2 (2003), pp.157-186 179 Wilmott, P., Dewynne, J.N. and Howison, S. (1993). Option Pricing: Mathematical

Models and Computation. Oxford Financial Press. A. Directed divergence convergence A.1. Some background Let's consider a probability density w(t) and a sequence of equispaced points αj=α* j/M, j=0,…, M, for a some finite α*>0 and:

( )( )

( )_

0: ,j j

M

j XE X x f x dxα αµ∞

= = ∫ (A.1)

with ( )

( )_

0

exp j

M M

jXj

f w x xαλ=

= −

∑

If we set * Mx tα= , from (A.1) we have for j=0,…,M, that

( )*

0 * *01

exp ln 1 ln ,

jj

M Mj j

jj

E X

M Mx w x x x dx

α

α

µ

λ λα α

∞

=

≡

= − − − − −

∑∫ (A.2)

which is a reduced Stieltjes moment problem for each fixed M value and a determinate Stieltjes moment problem when M→∞, see Theorem 1 in Lin (1992).

The following symmetric definite positive Hankel matrices are considered

00 1

0 0, 2 21 2

2

,...,M

M

M M

µ µµ µ

µµ µ

µ µ

∆ = ∆ = ∆ =

………

1 11 2

1 1, 3 2 12 3

1 2 1

,...,M

M

M M

µ µµ µ

µµ µ

µ µ

+ +

+

+ +

∆ = ∆ = ∆ =

………

(A.3)

whose (i, j) -th entry, i, j=0,1…, holds:

( ) ( ) ( )0

ji Mi j Xt t w t f t dtααµ

∞

+ = ∫

180 Επιχειρησιακή Έρευνα / Operational Research. An International Journal. Vol.2, No.2 (2003), pp.157-186

As a consequence the solvability of the Stieltjes moment problem (A.2) entails |∆i|>0, ∀i, [Shohat and Tamarkin (1943)]. Stieltjes moment problem is determined and the underlying distribution has a continuous distribution function. Then the maximal mass ρ(t) which can be concentrated at any real point t is equal to zero ([Shohat and Tamarkin (1943)], Corollary 2.8). In particular at t = 0, we have:

( ) ( )( )

0

0 20 0

2 1

1 2

0 lim : lim

i

ii iii ii

i i

µ µ

µ µρ µ µ

µ µ

µ µ

−

→∞ →∞+

+

= = = −

………………

(A.4)

where ( )0iρ indicates the largest mass that can be concentrated at a given point t=0 by

any solution of a reduced moment problem of order ≥ i and ( )0

iµ − indicates the minimum value of µ0 once the first 2i moments have been assigned.

Let us fix {µ0, …, µi-1, µi+1, …, µM,} while only µi, i=0,…varies continuously. From (A.1)-(A.3) we have

0

2 1

/

/

i

M i

M i

d de

d d

λ µ

λ µ+

∆ = −

(A.5)

where ei+1 is the canonical unit vector in RM+1, from which

00 0

2 1

/0,

/

iM M i

M ii i i i i

M i

d dd d d d de id d d d d

d d

λ µλ λ λ λ λµ µ µ µ µ

λ µ+

∆ = − = − > ∀

… … (A.6)

We are ready to prove the following theorem. A.2. Directed divergence convergence Let us define

[ ] ( ) ( )( )0

|| : ln XX X

f xD f w f x dx

w x∞

= ∫

Επιχειρησιακή Έρευνα / Operational Research. An International Journal. Vol.2, No.2 (2003), pp.157-186 181 and

( ) ( ) ( )( ) ( )

( )0|| : ln

MM M X

X Xf x

D f w f x dxw x

∞ = ∫

Theorem A.1 If j=0, …,M, α*>0, then

( )lim || ||MX XM

D f w D f w→∞

= (A.7)

Proof. From (A.1) we have:

Let's consider the first statement of (A.4). When only 2i varies continuously, taking into account (A.1), (A.3) and (A.5) we have:

( )0 0

00 0

|| 1M

M jX j

j

dd D f wd d

λµ λ λ

µ µ=

= − − = − + ∑

( )( )

2 1

21 20

20 0 2 0 0

1|| 0

M

M M MX M

M

d dD f wd d

µ µ

µ µλµ µ µ µ

+

+−

= − = = > ∆ −

………

Thus ( )( )||MXD f w is a differentiable convex function of µ0. When ( )

0 0Mµ µ −→

then ( )( )||MXD f w →∞, whilst at µ0 it holds ( )( )||M

XD f w < D(fX||w), being ( )MXf the

minimum directed divergence density once assigned (µ0, …, µM). Besides, when M→∞ then ( )

0 0Mµ µ− → . So the theorem is proved.

As a corollary of the previous theorem, when a priori density w(x) is not defined

then ( ) ( )0

exp jM

MX j

jf x xξλ

=

= −

∑ and (A.7) is replaced by

( )lim MX XM

H f H f→∞

=

i.e. ( )MXf converges in entropy to fX(t).

( )

0

||M

MX j j

jD f w λ µ

=

= − ∑ (A.8)

182 Επιχειρησιακή Έρευνα / Operational Research. An International Journal. Vol.2, No.2 (2003), pp.157-186 B. Stability Analysis B.1. Relative error estimate Let ( )M

Xf a ME density obtained by fractional moments

( )( )

( )_

0: j j

M

j XE X x f x dxξ ξµ∞

= = ∫ . We consider two sets of moments {µ0, …, µM} and

{µ0+∆µ0,…, µM +∆µM} with | ∆µi|<<1 to which respectively correspond the density functions ( )M

Xf having λj, j=0, …, M, ( )MXg having λj,+∆λj j=0, …, M, Let us

introduce the relative error:

( )( ) ( ) ( ) ( )

( ) ( ),

M MX X

M j MX

g x f xx

f xε µ

−∆ = (B.1)

Taking into account (A.5) and by Taylor expansion we have:

( ),M jxε µ∆ = 0

exp 1jM

jj

xξλ=

− ∆ −

∑ (B.2)

= 0 0

exp 1jM M

ji

j i i

xξ λµ

µ= =

∆ − ∆ − ∆ ∑ ∑ (B.3)

≈ 0 0

jM M

ji

j i i

xξ λµ

µ= =

∆− ∆

∆∑ ∑

=

1

0 0 1

2

1 2

0 11

M

M

M

M M M M

x xξ ξ

µ µ µ µ

µ µ µ µ+

∆−

∆

∆

……

…

B.2. Entropy estimate

Επιχειρησιακή Έρευνα / Operational Research. An International Journal. Vol.2, No.2 (2003), pp.157-186 183 Let us consider the entropies H[f(M)] corresponding to {µ0, …, µM} and H[g(M)]corresponding to {µ0+∆µ0,…, µM,+∆µM}. Taking into account (A.8) and (A.5) and from Taylor expansion

∆H = H[g(M)] - H[f(M)] (B.4)

0

0 00

0 2

2

01

2

M

MM

j jj M

M M M

µ µµ µ µ

λ µ µ

µ µ µ=

∆ ∆∆

= ∆ − ∆ +∆

∆

∑

……

…

Setting ∆µj/µj=constant ∀j, we have:

( ) ( )2 00 02

H H Hµµ µ µ µ µ∆ = ∆ − − ∆ ≈ ∆ − (B.5)

so that the computation of entropy is stable. B.3. Expected values Let Π(x)=(K-ax)+, α>0, be a bounded function. Let us set pM the option price computed using M moments and pM,∆µ the option price computed using the moments µj+∆µj. Taking into account (B.1), we have:

|∆pM| = |pM - pM,∆µ|

= ( ) ( ) ( ) ( ) ( ) ( )0 0

M MX Xx f x dx x g x dx

∞ ∞Π − Π∫ ∫

≤ ( ) ( ) ( ) ( )0

, MM j Xx x f x dxε µ

∞Π ∆∫

≤ ( ) ( ) ( )0

, MM j XK x f x dxε µ

∞∆∫ (B.6)

If we assume ∆µj/µj=∆µ=constant ∀j, from (B.2) and (B.4) we have:

|∆pM|≤ K |∆µ| (B.7)

184 Επιχειρησιακή Έρευνα / Operational Research. An International Journal. Vol.2, No.2 (2003), pp.157-186 which proves the stability of the expected value (the option price in our case) evaluation. C. Appendix C From Kesavan (1992), we know that the Shannon-entropy maximizing distribution

( )MXF , which has fractional moments ( ): j

j E X αµ = , has the following density

function

( ) ( ) ( )0

exp , 0,jM

MX j

jf x x xαλ

=

= − ∈ ∞

∑ (C.1)

(here λ0, …, λM are Lagrange's multipliers) which must be supplemented by the condition that its first M moments coincide with µj, i.e. for j=0,…, M:

( ) ( )0

j Mj Xx f x dxαµ

∞= ∫

Its Shannon entropy ( )( )MXH f is given as:

( ) ( ) ( ) ( ) ( )0

0

lnM

M M MX X X j j

jH f f x f x dx λ µ

∞

=

= − = ∑∫

Given two probability densities fX and ( )MXf , there are two well-known measures

of the distance between them. Namely the divergence measure:

( ) ( ) ( ) ( )( ) ( )0

, lnM M XX X X M

X

f xI f f f x dx

f x+∞

= ∫ ,

and the variation measure:

( ) ( ) ( ) ( )0

,M MX X X XV f f f x f x dx

+∞ = − ∫

Lemma C.1 We have: ( ) ( ),M M

X X X XI f f H f H f = −

where H[fX] is entropy of the target density fX.

Επιχειρησιακή Έρευνα / Operational Research. An International Journal. Vol.2, No.2 (2003), pp.157-186 185 Proof. Using the expression for ( )M

Xf and the fact that both densities have the same first moments, we can rewrite:

( ) ( ) ( )( ) ( )

[ ] ( )

[ ]

( ) [ ]

0

00

0

, lnM XX X X M

X

Mj

X j Xj

M

X j jj

MX X

f xI f f f x dx

f x

H f x f x dx

H f

H f H f

λ

λ µ

+∞

+∞

=

=

=

= − +

= − +

= −

∫

∑ ∫

∑ (C.2)

In literature several lower bounds for the divergence measure (also called discrimination information) I based on V are available. We shall however use the following Kullback's bound.

Lemma C.2 We have:

2 4

2 36V VI ≥ + (C.3)

Proof. See Kullback (1976). Then the main result on the absolute difference between the two distributions may

be obtained.

Theorem C.1 We have:

( ) ( ) ( ) ( )( )1/ 21/ 243 1 1

9M M

X X X XF x F x H f H f − ≤ − + + −

Proof. From the inequality (C.3) we first obtain an upper bound for V in terms of I:

1/ 21/ 243 1 1

9V I

≤ − + +

(C.4)

Then we use the result to have a bound on the absolute difference between FX and ( )M

XF as:

186 Επιχειρησιακή Έρευνα / Operational Research. An International Journal. Vol.2, No.2 (2003), pp.157-186

Let G ∈F|F be any other member, then from a property of the divergence measure, we know [ ] ( ) ( ), , ,M M

X X X XI g f I f f H f H f ≥ = − showing that the bound in (C.5)

holds for all members of F.

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )

( )( )

0

0

0

1/ 21/ 243 1 19

xM MX X X X

x MX X

MX X

MX X

F x F x f u f u du

f u f u du

f u f u du

H f H f

+∞

− = −

≤ −

≤ −

≤ − + + −

∫

∫∫

(C.5)