SometopicsinMathematicalFinance:Asianbasketoptionpricing,Optimalinvestmentstrategies Universit´eLibredeBruxelles

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Universit´e Libre de Bruxelles

Facult´e des Sciences D´epartment de Math´ematique

Some topics in Mathematical Finance:

Asian basket option pricing, Optimal investment strategies

Ibrahima DIALLO

Th`ese present´ee en vue de l’ obtention

du grade de Docteur en Sciences, orientation Sciences actuarielles

Promoteur: Griselda DEELSTRA

ULB, Janvier 2010

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Taking this opportunity I would like to thank all those people without whom the time I have been working on this thesis would not have been possible.

First of all my thanks go to Prof. Griselda Deelstra for all her very helpful advice as well as her comments and suggestions on various parts of this text. I am especially grateful to her for enabling me to attend to scientific meetings and to present my work there and for bringing me into contact with many distinguished researchers.

It is also a particular pleasure for me to thank Prof. Mich`ele Vanmaele for accepting to be the second referee. Her remarks and suggestions were very helpful for the content of Chapters 2 and 3.

Likewise, I would like to thank Dr. Roger Lord, Yves Demasure and Prof. Holger Kraft for some fruitful discussions.

Further, I wish to thank Prof. Reinhard and Prof. Patie for their comments and suggestions.

Also, I wish to thank Maude Gathy, Mme Patria Semeraro and Mme Jacqueline Botte- manne for their help.

I acknowledge the financial support of Fondation Universitaire David et Alice Van Burren.

Last but not least, I am grateful to my family and friends for their unconditional moral support and encouragement to follow my dreams and accomplish my goals.

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1 Introduction 1

2 Bounds for Asian Basket 5

2.1 Introduction . . . 5

2.2 The Model . . . 6

2.3 Bounds based on comonotonicity and conditioning . . . 7

2.3.1 Comonotonicity . . . 7

2.3.2 Comonotonic upper bound . . . 10

2.3.3 Comonotonic lower bound . . . 13

2.3.4 Bounds based on the Rogers and Shi approach . . . 17

2.3.5 Partially exact/comonotonic upper bound . . . 20

2.4 Non-comonotonic lower bound and UBRS approach . . . 23

2.5 Generalization of an upper bound based on the method of Thompson . . . . 25

2.6 Generalization of an upper bound based on the method of Lord . . . 30

2.7 Numerical results . . . 34

2.7.1 Asian Basket option . . . 35

2.7.2 Basket option . . . 39

2.8 Conclusion . . . 40

3 Moment Matching for Asian Basket 43 3.1 Introduction . . . 43

3.2 Splitting the price by conditioning . . . 44

3.3 Choice of conditioning random variable and integration bound . . . 45

3.4 Moment matching lognormal approximation . . . 48

3.5 Moment matching log-extended-skew-normal approximation . . . 49

3.5.1 Log-extended-skew-normal random variable . . . 50

3.5.2 Log-extended-skew-normal approximation of the underlying port- folio . . . 51

iii

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iv TABLE OF CONTENTS 3.5.3 Log-extended-skew-normal approximation after splitting and con-

ditioning . . . 52

3.6 Numerical results . . . 54

3.7 Conclusions . . . 57

4 Optimal Investment Strategies 59 4.1 Introduction . . . 59

4.2 The structure of the model . . . 61

4.2.1 The inflation . . . 61

4.2.2 The financial Market . . . 61

4.3 The portfolio processes . . . 71

4.4 The optimization problem . . . 72

4.5 Solution of the optimization problem . . . 74

4.6 Conclusion . . . 90

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Chapter 1 Introduction

This thesis presents the main results of my research in the field of computational finance and portfolios optimization. We focus on pricing Asian basket options and portfolio prob- lems in the presence of inflation with stochastic interest rates.

The fair price of a financial derivative can be expressed in terms of a risk-neutral ex- pectation of a random payoff. In some cases the expectation is explicitly computable, the Black & Scholes [6] formula for a call option on assets modeled by a geometric Brownian motion being a prime example. However, considering an exotic option of the type of an Asian, a basket or an Asian basket option, there exists no closed form expression for the price. Pricing Asian and basket options is an important subject of intensive research. The difficulty arises from the fact that the distribution of the sum of correlated lognormals has no closed form representation. Several approaches have been proposed in the literature. In the settings of Asian options, Kemna and Vorst [44] show that the value of an Asian option based upon arithmetic average is the solution of a second order partial differential equa- tion (PDE) and use Monte Carlo simulation to price and hedge such options. Although the Monte Carlo approach is a very flexible method for pricing Asian options which are path-dependent options, the numerical computation is very time consuming. Geman and Yor [33] derived a closed form expression for the value of a continuously sampled Asian option using the theory of Bessel processes and Laplace Transforms. Since their valuation requires the numerical evaluation of complex integrals, their approach also belongs to the class of numerical methods.

In contrast with the PDE and Monte Carlo methods, analytical approximations can be very useful to fastly generate accurate estimates of the options value. For example, Jarrow and Rudd in [39] provide a general method based on Edgeworth expansions. Their idea is to replace the integration over the lognormal distribution by an integration over another distribution with the same moments of low order and such that this last integration can be done in closed form. Turnbull and Wakeman [71] use an Edgeworth series expansion to ap-

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proximate the density function of the arithmetic average of lognormally distributed random variables and they obtain closed form formulae for Asian options. Levy [52] also approxi- mates the unknown distribution of the arithmetic average of lognormal distributed random variables by a lognormal distribution. But whereas Levy determines the parameters of the lognormal distribution in such a way that only mean and variance of true and lognormal distributions are matched, Turnbull and Wakeman go beyond by additionally adjusting for higher moments in order to get a better fit with respect to skewness and kurtosis. Like in [71] and [52], Milevsky and Posner propose in [58] and [59] to approximate the true distribution of an arithmetic average by the Reciprocal Gamma distribution. They justify their choice by the fact that under suitable parameter restrictions the distribution of the infinite sum of lognormally distributed random variables is reciprocal gamma distributed.

Recently, Albrecher and Predota [1] derive approximations for discrete arithmetic Asian options in a Negative Inverse Gaussian (NIG) Levy framework, one of which is obtained by using an Edgeworth series expansion to approximate the density function of the arith- metic sum by a lognormal one.

Whereas the method based on Edgeworth expansions try to approximate the true dis- tribution of the arithmetic average by another distribution which allows to derive a closed- form approximation for the value of the option, Rogers and Shi [63] introduce the condi- tional expectation approach. They start by conditioning the payoff of an Asian option by a very general random variable that is normally distributed and apply Jensen’s inequality to obtain lower and upper bounds on Asian option prices. Nielsen and Sandmann [61] follow the lines of the Rogers and Shi approach by using one specific standardized normally dis- tributed conditioning variable in a Black & Scholes setting and derive an analytic solution for the lower bound. Bounds can themselves serve as an approximation if they are suffi- ciently tight. Thompson [69] derived an upper bound that sharpens those of Rogers and Shi. Lord [56] revised Thompson’s method and showed how to sharpen Thompson’s upper bound such that it was tighter than all upper bounds. In the setting of Asian and basket options Vanmaele et al. [74] and Deelstra et al. [22] used techniques based on comonotonic risks and derived upper and lower bounds for stop-loss premiums of sums of dependent random variables, as explained in Kaas et al. [41] and Dhaene et al. [26]. They improved the upper bound for Asian options that was based on the idea of Rogers and Shi. Recently Carmona and Durrleman [12] derived lower and upper bounds for basket options using the linear programming approach and obtained a lower bound that performs well.

In Chapter 2, we concentrate upon the derivation of bounds for Asian basket options in a Black & Scholes framework. This Chapter is based on the publication of Deelstra et al.

[23]. We start from methods used for basket options and Asian options. First we use the general approach for deriving upper and lower bounds for stop-loss premiums of sums of dependent random variables as in Kaas et al. [41] and Dhaene et al. [26]. We generalize the methods in Deelstra et al. [22] and Vanmaele et al. [74]. One of the main results of

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3 this Chapter is that we show how to derive a lower and upper bound based on the Rogers and Shi approach [63] in the non-comonotonic case. Further, we generalize the method of Thompson [69] and of Lord [56] to the Asian basket case. In Thompson’s approach we include an additional parameter which is optimized as in the optimization algorithm in [56].

Curran [16] introduced a method which combines the ideas of [52] and the Rogers and Shi [63] approach for pricing Asian options. Whereas Rogers and Shi choose a general random variable, Curran carries out his calculation with the geometric average of the rele- vant asset prices as conditioning variable. By conditioning on the geometric average he is able to split the price into two parts, one giving a closed form solution and the second being approximated by moment matching as in [52]. The main advantage of this decomposition is that it is possible to derive a closed form solution for the exact part which is responsible for about99%of the option value. Curran’s idea is used and again improved by Deelstra et al. [22] in the setting of basket options.

In Chapter 3, we propose some moment matching pricing methods for European-style discrete arithmetic Asian basket options in a Black & Scholes framework. We generalize the approach of [16] and of [22] in several ways. We do this by looking at other condi- tioning variables and in particular by using an approach based on the idea to rescale the underlying sum in the Asian basket payoff such that the new rescaled arithmetic average is larger than the corresponding new rescaled geometric average, and by taking as condition- ing variables the standardized logarithm of this geometric average. We create a framework that allows for a whole class of conditioning random variables which are normally dis- tributed. We moment match not only with a lognormal random variable but also with a log-extended-skew-normal random variable. We also show how to improve upper bounds of Chapter 2. The results of Chapter 3 are published in Deelstra et al. [24]

In Chapter 4, we consider an optimal investment problem of an agent who maximizes his expected utility of terminal wealth in the presence of inflation with a stochastic affine structure for the interest rates. Optimal investment problems were introduced by Merton [57] under the assumption that the risky asset follows a geometric Brownian motion with deterministic interest rates. By applying standard methods and results from stochastic con- trol theory, Merton derived closed form solutions for the value function and the optimal portfolio when the utility function is of a Constant Relative Risk Aversion (CRRA) type.

However, the crucial point in his approach is that of solving the Hamilton-Jacobi-Bellman equation of dynamic programming. This approach leads to a characterization of the value function as a solution of a partial differential equation. This equation is highly nonlinear and in general the value function is not smooth.

With the growing application of stochastic calculus to finance, martingale methods were introduced by Karatzas et al. [42] or Cox and Huang [15]. Their main idea is to character- ize the optimal portfolio through the martingale representation theorem for which explicit

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solutions may be produced in very few cases. Recently many authors refine the approach of Merton to include either transaction costs, market imperfections of the model, labor in- come, inflation or a stochastic term structure for the interest rates. For example, Brennan and Xia [10] analyzed the portfolio problem of a finite-lived investor who can invest in stock or nominal bonds, when the interest rate and the expected rate of inflation follow correlated Ornstein-Uhlenbeck processes and the risk premia are constant.

In Chapter 4, we use the stochastic dynamic programming approach in order to extend Brennan and Xia’s unconstrained optimal portfolio strategies by investigating the case in which interest rates and inflation rates follow affine dynamics which combine the model of Cox et al. [14] and the model of Vasicek [75]. We first derive the nominal price of a zero- coupon bond by using the evolution PDE which can be solved by reducing the problem to the solution of three ordinary differential equations (ODE). To solve the corresponding control problems we apply a verification theorem without the usual Lipschitz assumption given in [48] or [45].

Publications and Papers

The Chapters of this thesis are related to the following list of reseach papers:

Chapter 2: “Bounds for Asian basket options”, Journal of Computational and Applied Mathematics, 218, 215-228.

Chapter 3: “Moment Matching Approximation of Asian basket option prices”, Journal of Computational and Applied Mathematics, doi: 10.1016/j.cam.2009.03.004.

Chapter 4: “Optimal investment strategies in the presence of inflation and stochastic interest rates”, Working Paper.

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Chapter 2

Bounds for Asian basket options

2.1 Introduction

In this chapter we propose pricing bounds for European-style discrete arithmetic Asian basket options in a Black & Scholes framework. An Asian basket option is an option whose payoff depends on the average value of the prices of a portfolio (or basket) of assets (stocks) at different dates.

Within a Black and Scholes [6] setting, no closed form solutions are available for Asian basket options involving the average of asset prices taken at different dates. Dahl and Benth value such options in [17] and [18] by quasi-Monte Carlo techniques and singular value decomposition. But as this approach is rather time-consuming, it would be ideal to have accurate analytical and easily computable bounds or approximations of this price.

In the setting of Asian options, an analytical lower and upper bound in the case of con- tinuous averaging is obtained by the methods of conditioning in Curran [16] and in Rogers and Shi [63]. Thompson [69] used a first order approximation to the arithmetic sum and derived an upper bound that sharpens those of Rogers and Shi. Lord [56] revised Thomp- son’s method and proposed a shift lognormal approximation to the sums and he included a supplementary parameter which is estimated by an optimization algorithm. In [61], Nielsen and Sandmann applied the Rogers and Shi approach to arithmetic Asian option pricing by using one specific standardized normally distributed conditioning variable and only in a Black & Scholes setting. Simon et al. [66] derived an easy computable upper bound for the price of an arithmetic Asian option based on the results of Dhaene et al. [25]. Dhaene et al.

[26] and [27] studied extensively convex upper and lower bounds for sums of lognormals, in particular of Asian options. Vanmaele et al. [74] used techniques based on comonotonic risks for deriving upper and lower bounds for stop-loss premia of sums of non-independent random variables, as explained in Kaas et al. [41] and the already mentioned [26] and [27].

Vanmaele et al. [74] improved the upper bound that was based on the idea of Rogers and Shi [63], and generalized the approach of Nielsen and Sandmann [61] to a general class of normally distributed conditioning variables. In Deelstra et al. [22] these methods for

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Asian options were generalized to the case of basket options. Here we concentrate upon the derivation of bounds for Asian basket options and we develop results obtained in [23].

We start with extending the methods of Deelstra et al. [22] and Vanmaele et al. [74] to the Asian basket case.

Besides, we are able to derive in the non-comonotonic case a simple analytical lower bound and an upper bound based on the Rogers and Shi [63] approach. Finally, we generalize the method of Thompson [69] and of Lord [56] to the Asian basket case. In Thompson’s ap- proach, we include an additional parameter which is optimized by using an optimization algorithm as in Lord [56]. Numerical results are included and based on several numerical tests, we give a conclusion which should help the reader to choose a precise bound accord- ing to the situation of moneyness and time-to-maturity that she is confronted with.

The chapter is organized as follows. In Section 2.2 we describe the dynamics of the assets and the price structure of a discrete arithmetic Asian basket option. In Section 2.3, we deal with procedures for obtaining lower and upper bounds for prices, by using the concept of comonotonicity as explained in [41], [26] and [27], along the lines of [74] and [22]. In Section 2.4, we derive a lower bound in a non-comonotonic situation, which is then used to obtain the upper bound in the Rogers and Shi approach. In Section 2.5, we generalize the upper bound based on the idea of Thompson [69] to discrete arithmetic Asian basket options. In Section 2.6, we improve the method in Section 2.5, and generalize the approach of Lord [56] to a discrete arithmetic Asian basket option. In Section 2.7, we discuss the quality of all these bounds in some numerical experiments and give a guideline of which bound to use in which situation.

2.2 The Model

We consider a basket withn assets whose pricesSi(t), i = 1, . . . , n, are described, under the risk neutral measureQand withrsome risk-neutral interest rate, by

dSi(t) =rSi(t)dt+σiSi(t)dWi(t), (2.1) where{Wi(t), t >0}are standard Brownian motions associated with the price of asseti andσi(≥0)are the corresponding volatilities. Further, we assume that the different asset returns are instantaneously correlated in a constant way i.e.

cov(dWi, dWj) = ρijdt. (2.2)

Given the above dynamics, thei-th asset price at timetequals Si(t) =Si(0)e(r21σi2)t+σiWi(t).

An Asian basket option is a path-dependent multi-asset option whose payoff combines the payoff structure of an Asian option with that of a basket option. The price of a discrete

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2.3. Bounds based on comonotonicity and conditioning 7 arithmetic Asian basket call option with a fixed strikeK and maturityT onm averaging dates at current timet = 0is determined by

ABC(n, m, K, T) = erTEQ

 Xn

ℓ=1

a mX1

j=0

bjS(T −j)−K

!

+

 (2.3)

witha andbj positive coefficients, which both sum up to1, and with(x)+ = max{x,0}. ForT ≤m−1,the Asian basket call option is said to be in progress and forT > m−1, we call it forward starting. We consider forward starting Asian basket call options but the methods apply in general. The prices of Asian basket put options follow from the call op- tion prices by the call-put parity relation. Indeed, if the price of an Asian basket put option with a fixed strikeK, maturityT andm averaging dates is denoted by ABP(n, m, K, T), static arbitrage arguments lead to the following call-put parity relation:

ABC(n, m, K, T)−ABP(n, m, K, T) = Xn

ℓ=1 mX1

j=0

abjS(0)erj −erTK.

Asian basket options are suitable for hedging exposure as their payoff depend on an average of asset prices at different times and of different assets. Indeed, averaging has generally the effect of decreasing the variance, therefore making the option less expensive.

Moreover the Asian basket option takes the correlations between the assets in the basket into account. For example, it is extremely difficult to manipulate the payoff of this option because an investor has to influence the prices of n different assets at m different time points during the lifetime of the option. Asian basket options are especially important in the energy markets where most delivery contracts are priced on the basis of an average price over a certain period.

2.3 Bounds based on comonotonicity and conditioning

In this section we generalize the bounds of Deelstra et al. [22] and Vanmaele et al. [74] to the Asian basket case. In these papers the pricing of discrete arithmetic basket and Asian options are studied by using the notion of comonotonicity, as explained in Kaas et al. [41], Dhaene et al. [26] and [27]. They further improve the bounds by incorporating the ideas of Curran [16], Rogers and Shi [63] and Nielsen and Sandmann [61], and by looking for good conditioning variables.

2.3.1 Comonotonicity

We shortly recall from Dhaene et al. [25] and [26] and references therein the procedures for obtaining the comonotonic counterpart of the sum of dependent random variables.

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Definition 2.3.1 (Stop-loss Order) A random variable X is said to precede a random variableY in stop-loss order, written X ≤sl Y, if for all retentions d > 0, the stop-loss premium for randomXis smaller than that for randomY:

E(X−d)+ ≤E(Y −d)+.

Additionally, two random variables that have the same expectation leads to the so-called convex order.

Definition 2.3.2 (Convex Order) Consider two random variables X and Y. Then X is said to precedeY in the convex order sense, notationX ≤cx Y, if and only if

E(X) = E(Y)

E(X−d)+ ≤ E(Y −d)+, −∞< d < +∞.

The inverse of a cumulative distribution function (cdf) is usually defined as follows:

Definition 2.3.3 The inverse of the cumulative distribution functionFX(x) = P r[X ≤x]

of a random variableX is given by

FX1(p) = inf{x∈R|F(x)≥p}, p∈[0,1] (2.4) withinf∅= +∞by convention.

Next, we define comonotonicity of a random vector.

Definition 2.3.4 (Comonotonicity) A random vector(Y1, . . . , Yn)with marginal cdfsFYi(x) is said to be comonotonic if it has the same distribution as(FY11(U), . . . , FYn1(U)), withU a random variable which is uniformly distributed in the unit interval(0,1).

Consider a random vector(Y1, . . . , Yn). Its comonotonic counterpart(Y1c, . . . , Ync)is a comonotonic random vector with the same marginal distribution:

(Y1c, . . . , Ync):= (Fd Y11(U), . . . , FYn1(U)). (2.5) As proven in Dhaene et al. [25], the convex-largest sum of the components of the vector (Y1, . . . , Yn)is obtained by the comonotonic sumSc =Y1c +. . .+Ync, with

Sc :=d Xn

i=1

FYi1(U). (2.6)

The cdf ofFSc(x)is defined by FSc(x) = sup

(

p∈(0,1)| Xn

i=1

FYi1(p)≤x )

. (2.7)

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2.3. Bounds based on comonotonicity and conditioning 9 Let us now assume that we have some additional information available concerning the stochastic nature of(Y1, . . . , Yn). More precisely, we assume that there exists some random variableΛwith a given distribution function, such that we know the conditional cumulative distribution functions, givenΛ =λ, of the randomYi, for all possible values of λ. In fact Kaas et al. [41] define the improved comonotonic upper boundSu as

Su =FY11|Λ(U) +FY21|Λ(U) +· · ·+FYn1|Λ(U), (2.8) whereFY1

i|Λ(U)is the notation for the random variablefi(U,Λ), with the functionfidefined byfi(u, λ) = FY1

i|Λ=λ(u), withU being independent ofΛ. GivenΛ =λthe cdf ofSu can be deduced from (2.7):

FSu|Λ=λ(x) = sup (

p∈(0,1)| Xn

i=1

FYi1|Λ=λ(p)≤x )

. (2.9)

The cdf ofSu then follows from

FSu(x) = Z −∞

+

FSu|Λ=λ(x)dFΛ(λ). (2.10) We recall from Kaas et al. [41] how to obtain a lower bound, in the sense of convex order, for the sumS=Y1+· · ·+Ynby conditioning on a random variableΛ.We remark that this idea can also be found in Rogers and Shi [63] for the continuous case for obtaining lower bounds for the price. Let us denote the conditional expectation byS:

S =E[S| Λ]. (2.11)

Let us further assume that the random variable Λ is such that all E[Yi|Λ] are continuous and monotonic functions ofΛ. For a non-increasing and continuous function ofΛ, the cdf of the lower boundSfollows from (2.7):

FS(x) = sup (

p∈(0,1)| Xn

i=1

E[Yi |Λ =FΛ1(1−p)]≤x )

. (2.12)

Similarly, for a non-decreasing and continuous functions ofΛ, we have

FS(x) = sup (

p∈(0,1)| Xn

i=1

E[Yi |Λ =FΛ1(p)]≤x )

. (2.13)

We now consider a normally distributed random variable Λ and we construct upper and lower bounds for the Asian basket option.

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2.3.2 Comonotonic upper bound

Let S be a random variable of the type S=Pk

i=1Xi, where the terms Xi are not mu- tually independent, but the multivariate distribution function of the random vector X = (X1, . . . , Xk) is not completely specified because one only knows the marginal distribu- tion functions of the random variablesXi.As mentioned in Dhaene et al. [26], to be able to make decisions it may be helpful to find the dependence structure for the random vec- tor(X1, . . . , Xk)producing the least favourable aggregate claimsSwith given marginals.

Therefore, given the marginal distributions of the terms in a random variableS=Pk i=1Xi, we shall look at the joint distribution with a smaller resp. larger sum, in the convex order sense. In short, the sumSis bounded below and above in convex order (cx)by sums of conditioning and comonotonic variables:

S cx ScxSu cxSc, which implies by definition of convex order that

Eh

S−d

+

i≤E

(S−d)+

≤E

(Su−d)+

≤E

(Sc−d)+ for alldinR+, whileE

S

=E[S] =E[Su] =E[Sc]. Remark that the double sumS=Pn

ℓ=1aPm1

j=0 bjS(T −j), showing up in equation (2.3), is a sum of lognormal distributed variables and can be written as

S= Xmn

i=1

Xi = Xmn

i=1

αieYi (2.14)

with

αi =ai

mb(i1) modmSi

m(0)e(r

1 2σ2

i

m)(T(i1) modm)

(2.15) and

Yii

mWi

m(T −(i−1) modm)∽N(0, σY2i2i

m(T −(i−1) modm)) (2.16) for all i = 1, . . . , mn, where N(0,1)is the standardized normal distribution, ⌈x⌉ is the smallest integer greater than or equal toxand

ymodm=y− ⌊y/m⌋m, where⌊y⌋denotes the greatest integer less than or equal toy.

As proven in Dhaene et al. [26], the comonotonic counterpartScof (2.14) is the random variable

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2.3. Bounds based on comonotonicity and conditioning 11

Sc = Xmn

i=1

FXi1(U) = Xmn

i=1

αieE(Yi)+sign(αiYiΦ−1(U) (2.17) whereUis a uniform(0,1)random variable and the usual inverse of a distribution function, which is the non-decreasing and continuous functionFX1(p),is defined by (2.4).

We shall look at the comonotonic boundsSc of (2.14) and we will derive the comono- tonic upper bound for the option price ABC(n, m, K, T), denoted by

CUB.

Proposition 2.3.1 Suppose the sum S is given by (2.14)-(2.16). Then the comonotonic upper boundScis given by

Sc = Xn

ℓ=1 mX1

j=0

abjS(0) exp

(r−1

2)(T −j) +σ

pT −jΦ1(U)

(2.18) whereΦ(·)is the standard normal cdf.

Proof From (2.17) we write Sc :=

Xmn i=1

αieE(Yi)+sign(αiYiΦ−1(U). By (2.16), we obtain

Sc :=

Xmn i=1

αieσYiΦ−1(U),

which is equivalent to (2.18). 2

Theorem 2.3.1 Suppose the sumSis given by (2.14)-(2.16). Then the comonotonic upper bound for the option priceABC(n, m,K, T)in (2.3) is determined by:

CUB = Xn

ℓ=1 mX1

j=0

abjS(0)erjΦh σ

pT −j −Φ1(FSc(K))i

−erTK(1−FSc(K)), (2.19) where the valueFSc(K)of the cumulative density function of the comonotonic sumSc can be found by solving

Xn ℓ=1

mX1 j=0

abjS(0) exp

(r−1

2)(T −j) +σ

pT −jΦ1(FSc(K))

=K, (2.20)

withΦ(·)the standard normal cdf.

Proof From expressions for the cumulative distribution function and the stop-loss premiums ofSc the comonotonic upper bound for the price of a discrete Asian basket call

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option at current timet= 0with strikeKis determined, by applying results in [27] or [22], as follows:

ABC(n, m, K, T)≤erT Xmn

i=1

αieEQ(Yi)+

σ2 Yi 2 Φ

sign(αiYi−Φ1(FSc(K))

−erTK(1−FSc(K)) (2.21)

where the cdf of the comonotonic sumFSc(K)can be found by solving Xmn

i=1

αieEQ(Yi)+sign(αiYiΦ−1(FSc(K))=K. (2.22) From (2.15) and (2.16) we find that the upper bound in (2.21) can be rewritten precisely as

(2.19) and the equation (2.22) becomes (2.20). 2

Interpretation of the comonotonic upper bound The payoff of the Asian basket call option satisfies

Xn ℓ=1

a

mX1 j=0

bjS(T −j)−K

!

+

≤ Xn

l=1

a

mX1 j=0

bjS(T −j)−K

!

+

(2.23)

≤ Xn

ℓ=1 mX1

j=0

abj(S(T −j)−Kℓj)+ (2.24) as well as

Xn l=1

a mX1

j=0

bjS(T −j)−K

!

+

mX1 j=0

bj

Xn l=1

aS(T −j)−Kj

!

+

(2.25)

≤ Xn

ℓ=1 mX1

j=0

abj(S(T −j)−Kℓj)+, (2.26)

with Xn

ℓ=1

aK =

mX1 j=0

bjKj = Xn

ℓ=1 mX1

j=0

abjKℓj =K. (2.27) By a no-arbitrage argument we find that the time zero price of such Asian basket option should satisfy the following two relations:

ABC(n, m, K, T)≤ Xn

ℓ=1

aAC(m, K, T)≤ Xn

ℓ=1 m1

X

j=0

abjerjC(Kℓj, T −j) (2.28)

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2.3. Bounds based on comonotonicity and conditioning 13

ABC(n, m, K, T)≤

mX1 j=0

bjerjBC(n, Kj, T −j)≤ Xn

ℓ=1 m1

X

j=0

abjerjC(Kℓj, T −j).

(2.29) This means that the Asian basket call option can be superreplicated by a static1portfolio of vanilla call optionsCon the underlying assetsS in the basket and with different matu- rities and strikes. Also an average of Asian optionsACor a combination of basket options BC with different maturity dates form a superreplicating strategy. Since the weights a

as well asbj sum to one, a possible choice for the strikes in the decompositions (2.28) is K = Kj = Kℓj =K. However this will not provide optimal superreplicating strategies.

In [66] and [2] it was noted that in the Asian option case the comonotonic upper bound can be interpreted as the price of an optimal static superreplicating strategy consisting of vanilla options. Hobson et al. [38] obtained a similar result for a basket option in a model free framework, while Chen et al. [13] extended this to a more general class of exotic op- tions.

Since prices for basket options can be simulated very fast, the expression (2.29) as a com- bination of basket options with different maturity dates might be useful.

2.3.3 Comonotonic lower bound

A lower bound, in the sense of convex order, forS=Pmn i=1Xiis S =E[S|Λ]

where Λ is a normally distributed random variable. If E[Xi |Λ] are all non-decreasing functions ofΛor all non-increasing functions ofΛ,S is a sum of comonotonic variables and the reasoning of Dhaene et al. [26] and [27] for the stop-loss premium leads to Theo- rem 2.3.2 below whereLBΛdenotes ‘lower bound using the conditioning variableΛ’ and stands forerTEQ

(S−K)+

. The non-comonotonic situation for Asian basket options is solved in Section 2.4.

In order to derive the lower bound we need to state the following lemma:

Lemma 2.3.1 Let(X, Z)and(Y, Z)be jointly normally distributed. Then we have (a)

E[X |Z] =E[X] + cov(X, Z)

var(Z) (Z−E[Z]) (b)

cov(X, Y |Z) = cov(X, Y)− cov(X, Y)cov(Y, Z) var(Z)

1When exercising an option at a maturity T −j withj ∈ {1, . . . , m−1}, one has in addition to invest the payoff in the risk free money-account.

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(c)

var[X |Z] =var(X)− cov(X, Z)2 var(Z)

Proof See e.g. Beisser [8]. 2

Next we state the convex lower bounds S and we will derive the comonotonic lower boundLBΛ.

Proposition 2.3.2 Suppose the sum S is given by (2.14)-(2.16) and Λ is a normally dis- tributed conditioning variable such that(W(T −j),Λ)are bivariate normally distributed for allandj and the correlation coefficients

rℓ,j = cov(W(T −j),Λ) σΛ

T −j (2.30)

are different from zero for allandj. Then the convex lower boundSis given by

S = Xn

ℓ=1 mX1

j=0

abjS(0) exp

(r− 1

2r2ℓ,jσ2)(T −j) +rℓ,jσ

pT −jΦ1(U)

, (2.31)

withU = Φ

ΛEQ[Λ]

σΛ

.

Proof By the definition ofSwe have S = EQ[S|Λ]

= Xn

ℓ=1 m1

X

j=0

abjS(0) exp

(r−1

2)(T −j)

EQ[exp (σW(T −j))|Λ]

= Xn

ℓ=1 m1

X

j=0

abjS(0)e(r21σ2)(Tj)+EQW(Tj)|Λ]+12var[σW(Tj)|Λ]. (2.32)

From lemma 2.3.1 we have

EQW(T −j)|Λ] =rℓ,jσ

pT −jΛ−EQ[Λ]

σΛ

and

var[σW(T −j)|Λ] =σ2(T −j)−rℓ,j2 σ2(T −j).

Substitution of these expressions in (2.32) leads to (2.31). 2 Theorem 2.3.2 Suppose the sumSis given by (2.14)-(2.16) andΛis a normally distributed conditioning variable such that(W(T −j),Λ)are bivariate normally distributed for allandj and the correlation coefficientsrℓ,j of (2.30) have the same sign, when not zero, for

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2.3. Bounds based on comonotonicity and conditioning 15 alland j. Then the comonotonic lower bound for the option price ABC(n, m, K, T)in (2.3) is given by

LBΛ = Xn

ℓ=1 mX1

j=0

abjS(0)erjΦh

sign(rℓ,j) rℓ,jσ

pT −j−Φ1(FS(K))i

−erT

sign(rℓ,j1(FS(K))

(2.33) where the valueFS(K)of the cdf of the comonotonic sumS solves

Xn ℓ=1

mX1 j=0

abjS(0) exp

(r− 1

2rℓ,j2 σ2)(T −j) +rℓ,jσ

pT −jΦ1(FS(K))

=K.

(2.34)

Proof In view of (2.31) we can rewrite the lower bound, omitting the discount factor, as

EQ

(S−K)+

= Z 1

0

Xn ℓ=1

m1

X

j=0

abjS(0)e(r12rℓ,j2 σ2)(Tj)+rℓ,jσT−1(u)−K

!

+

du.

(2.35) The correlationsrℓ,j have the same sign. Then the function defined by

f(u) = Xn

ℓ=1 mX1

j=0

abjS(0)e(r12r2ℓ,jσ2)(Tj)+rℓ,jσT−1(u)−K (2.36) is continuous and monotone taking positive and negative values. Therefore there exists a uniqueFS(K) such thatf(FS(K)) = 0, or equivalently, (2.34) holds.

Let us assume thatrℓ,j < 0for all ℓ andj.Then we have S ≥ K which is equivalent to U ≤FS(K). Therefore the lower bound is equal to

LBΛ =erT

Z FS(K) 0

Xn ℓ=1

m1

X

j=0

abjS(0)e(r12rℓ,j2 σ2)(Tj)+rℓ,jσT−1(u)−K

! du

or equivalently, LBΛ

=erT Xn

ℓ=1 m1

X

j=0

abjS(0)e(r21rℓ,j2 σ2)(Tj)

Z FS(K) 0

erℓ,jσT−1(u)du−erTKFS(K)

=erT Xn

ℓ=1 m1

X

j=0

abjS(0)e(r21rℓ,j2 σ2)(Tj)

Z Φ−1(FS(K))

−∞

erℓ,jσTjvϕ(v)dv−erTKFS(K)

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whereϕ(·)is standard normal density function. A change of variable leads to LBΛ = erT

Xn ℓ=1

mX1 j=0

abjS(0)er(Tj)

Z Φ−1(FS(K))rℓ,jσ Tj

−∞

ϕ(x)dx−erTKFS(K)

= Xn

ℓ=1 mX1

j=0

abjS(0)erjΦ

Φ1(FS(K))−rℓ,jσ

pT −j

−erTKFS(K).

Similarly, for allrℓ,j >0,we have LBΛ =

Xn ℓ=1

mX1 j=0

abjS(0)erjΦ

rℓ,jσp

T −j−Φ1(FS(K))

−erTK(1−FS(K)). 2 To judge the quality of the stochastic lower bound EQ[S|Λ], we might look at its variance. To maximize it, i.e. to make it as close as possible to var[S], the average value of var[S|Λ=λ] should be minimized. In other words, to get the best lower bound,Λ andS should be as alike as possible.

A first idea to choose conditioning variables is based on [41] and [26] and consists in looking at first order approximations ofS. Vanduffel et al. [72] propose a conditioning variableΛ such that the first order approximation of the variance ofS is maximized. We can takeΛ =FA1, FA2,or FA3,such that fork = 1,2,3 :

FAk = Xn

ℓ=1 m1

X

j=0

abjS(0)δk(ℓ, j)σW(T −j) (2.37) with

δ1(ℓ, j) =e(r12σ2)(Tj), δ2(ℓ, j) = 1, δ3(ℓ, j) = er(Tj). (2.38) For all these choices ofΛ, the correlation coefficientsrℓ,j, which enter the lower bound, are easy to calculate. Their expressions contain the instantaneous correlationsρℓj (2.2), which influence the sign of therℓ,j. Indeed whenΛis given by (2.37) then the linearity property of covariance leads to

cov(Wi(T −j),FAk)

= Xn

ℓ=1 m1

X

p=0

abpS(0)δk(ℓ, p)σcov(Wi(T −j), W(T −p))

= Xn

ℓ=1 m1

X

p=0

abpS(0)δk(ℓ, p)σρiℓmin (T −j, T −p)

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2.3. Bounds based on comonotonicity and conditioning 17 and

σ2FAk = cov(FAk,FAk)

= Xn

ℓ=1 mX1

j=0

Xn i=1

m1

X

p=0

aaibjbpS(0)Si(0)δk(ℓ, j)δk(i, p)σσiρℓimin (T −j, T −p).

Nielsen and Sandmann [61] suggest to look at the geometric averageGwhich in the Asian basket case is defined by

G= Yn ℓ=1

mY1 j=0

S(T −j)abj = Yn ℓ=1

mY1 j=0

S(0)e(r12σ2)(Tj)+σW(Tj)bj

!a

and to consider its standardized logarithm as conditioning variable GA= lnG−E[lnG]

σlnG

= Pn

ℓ=1

Pm1

j=0 abjσW(T −j) σlnG

(2.39) where

σ2lnG = Xn

ℓ=1

Xn i=1

m1

X

j=0 mX1

k=0

aaibjbkσσiρℓimin (T −j, T −k). WhenΛis given by (2.39) the covariance follows from

cov(Wi(T −j),GA) = Xn

ℓ=1 m1

X

p=0

abpσρiℓmin (T −j, T −p).

Only when the (non-zero) correlation coefficients rℓ,j have the same sign for allℓ and j the comonotonic lower bound may be applied. Otherwise one has to employ the newly developed non-comonotonic lower bound of Section 2.4.

2.3.4 Bounds based on the Rogers and Shi approach

Rogers and Shi [63] derived an upper bound based on the lower bound starting from the following general inequality for any random variableY andZ:

0≤E[E(Y+ |Z)−E(Y |Z)+]≤ 1 2Ep

var(Y |Z).

In this case, we obtain 0≤EQ

EQ[(S−K)+ |Λ]−(S−K)+

≤ 1

2EQhp

var(S|Λ)i . Thus, we find as upper bound for the Asian basket option

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ABC(n, m, K, T)≤erT

EQ

(S−K)+

+1

2EQhp

var(S|Λ)i

. (2.40)

According to an idea of Nielsen and Sandmann [61], we determinedΛ∈Rfor each of the four differentΛ’s (2.37) and (2.39) such thatΛ≥dΛimplies thatS≥K:

dFAk =K− Xn

ℓ=1 mX1

j=0

abjS(0)δk(ℓ, j)

1 +

r− 1 2σ2

(T −j)−ln (δk(ℓ, j))

(2.41) fork = 1,2,3and

dGA = 1 σlnG

lnK− Xn

ℓ=1 mX1

j=0

abj

lnS(0) + (T −j) (r− 1 2σ2)

!

. (2.42)

Using the same reasoning as in Deelstra et al. [22], it follows from (2.40) that ABC(n, m, K, T) ≤ erTEQ

(S−K)+

+ 1 2erT

Z dΛ

−∞

(var(S|Λ =λ)12 fΛ(λ)dλ

= erTEQ

(S−K)+

+ 1

2erTEQh

(var(S|Λ))12 1

{Λ<dΛ}

i

≤ erTEQ

(S−K)+

(2.43)

+1 2erT

EQ

var(S|Λ)1

{Λ<dΛ}

12 EQ

1{Λ<dΛ}

12

where the H¨older inequality has been applied andfΛis the normal density function ofΛ. In the following theorem we give upper bounds which are denoted byUBRSΛwithΛbeing a conditioning variable:

Theorem 2.3.3 LetSbe given by (2.14)-(2.16) andΛbe a normally distributed condition- ing variable such that (W(T −j),Λ) are bivariate normally distributed for allandj. Further, suppose that there exists adΛ∈Rsuch thatΛ≥dΛimplies thatS≥K. Then an upper bound to the option price ABC(n, m, K, T)in (2.3) is

UBRSΛ =erTEQ

(S−K)+

+ 1

2erT {Φ (dΛ)}12 (2.44)

× ( n

X

ℓ=1

Xn k=1

mX1 j=0

m1

X

p=0

aakbjbpS(0)Sk(0)

×er(2Tjp)

eσσkρℓkmin(Tj,Tp)−erℓ,jrk,pσσk

(Tj)(Tp)

×Φ

dΛ−rℓ,jσ

pT −j−rk,pσk

pT −po12 ,

Figure

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