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Diallo, I. (2010). Some topics in mathematical finance: Asian basket option pricing, Optimal investment strategies (Unpublished doctoral dissertation).

Université libre de Bruxelles, Faculté des Sciences – Mathématiques, Bruxelles.

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Université Libre de Bruxelles

Faculté des Sciences Départment de Mathématique

Some topics in Mathematical Finance:

Asian basket option pricing, Optimal investment strategies

Ibrahima DIALLO

Thèse présentée en vue de I’ obtention

du grade de Docteur en Sciences, orientation Sciences actuarielles

Promoteur: Griselda DEELSTRA

ULB, Janvier 20 lO

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Université Libre de Bruxelles

Faculté des Sciences Départment de Mathématique

Some topîcs in Mathematical Finance:

Asian basket option pricing, Optimal investment strategies

Ibrahima DIALLO

Thèse présentée en vue de 1’ obtention

du grade de Docteur en Sciences, orientation Sciences actuarielles

Promoteur: Griselda DEELSTRA

ULB, Janvier 2010

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Taking this opportunity 1 would like to thank ail those people without whoin the time I hâve been working on this thesis would not hâve been possible.

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

It is also a particular pleasure for me to thank Prof Michèle 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 fruitfül discussions.

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

Also, 1 wish to thank Maude Gathy, Mme Patria Semeraro and Mme Jacqueline Botte- manne for their help.

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

Last but not least, 1 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 intégration 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-nomial 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-nomial 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 présents tlie 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 dérivative can be expressed in ternis 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 géométrie 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 lias no closed form représentation. Several approaches hâve 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 équa­

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 Transfomis. Since their valuation requires the numerical évaluation of complex intégrais, 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 estimâtes 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 intégration over the lognormal distribution by an intégration over another distribution with the same moments of low order and such that this last intégration can be donc in closed form. Tumbull and Wakeman [71] use an Edgeworth sériés expansion to ap-

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proximate the density function of the arithmetic average of lognormally distributed random variables and they obtain closed forrn fonnulae for Asian options. Levy [52] also approxi- mates the unknown distribution of the arithmetic average of lognormal distributed random variables by a lognomial distribution. But whereas Levy détermines the parameters of the lognormal distribution in such a way that only mean and variance of true and lognormal distributions are matched, Tumbull 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 infinité sum of lognormally distributed random variables is reciprocal gamma distributed.

Recently, Albrecher and Predota [1] dérivé approximations for discrète arithmetic Asian options in a Négative Inverse Gaussian (NIG) Levy framework, one of which is obtained by using an Edgeworth sériés expansion to approximate the density function of the arith­

metic sum by a lognomial 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 dérivé a closed- fomi approximation for the value of the option, Rogers and Shi [63] introduce the condi- tional expectation approach. They start by conditioning the payofif 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 fines of the Rogers and Shi approach by using one spécifie standardized normally dis­

tributed conditioning variable in a Black & Scholes setting and dérivé 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 ail 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 dépendent 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 Camiona and Durrleman [ 12] derived lower and upper bounds for basket options using the linear programming approach and obtained a lower bound that perfomis well.

In Chapter 2, we concentrate upon the dérivation 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 dépendent 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 Cliapter is that we show liow to dérivé a lower and upper bound based on the Rogers and Shi approaeh [63] in the non-eomonotonic ease. Further, we generalize the method of Thompson [69] and of Lord [56] to the Asian basket ease. In Thompson’s approaeh we include an additional parameter which is optimized as in the optimization algorithm in [56],

Curran [16] introduced a method whieh eombines the ideas of [52] and the Rogers and Shi [63] approaeh for pricing Asian options. Whereas Rogers and Shi choose a general random variable, Curran carries out his calculation with the géométrie average of the rele­

vant asset priées as conditioning variable. By conditioning on the géométrie average he is able to split the price into two parts, one giving a closed fomi solution and the second being approximated by moment matching as in [52]. The main advantage of this décomposition is that it is possible to dérivé a closed form solution for the exact part which is responsible for about 99% ofthe 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 discrète arithmetic Asian basket options in a Black & Scholes framework. We generalize the approaeh of [16] and of [22] in several ways. We do this by looking at other condi­

tioning variables and in particular by using an approaeh 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 géométrie average, and by taking as condition­

ing variables the standardized logarithm of this géométrie 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 lognoimal 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 géométrie 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 approaeh is that of solving the Hamilton-Jacobi-Bellman équation of dynamic programming. This approaeh leads to a characterization of the value function as a solution of a partial differential équation. This équation 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 représentation theorem for which explicit

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solutions may be produced in vcry 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 Omstein-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 dérivé the nominal price of a zero- coupon bond by using the évolution PDF which can be solved by reducing the problem to the solution of three ordinary differential équations (ODE). To solve the corresponding control problems we apply a vérification 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 Computalional and Applied Mathematics, 218, 215-228.

Chapter 3: “Moment Matching Approximation of Asian basket option prices”, Journal of Computalional 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 discrète arithmetic Asian basket options in a Black & Scholes framework. An Asian basket option is an option whose payoff dépends on the average value of the prices of a portfolio (or basket) of assets (stoeks) at different dates.

Within a Black and Scholes [6] setting, no closed fomi 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 décomposition. But as this approach is rather time-consuming, it would be idéal to hâve 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

tinuons 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 whieh 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 spécifie 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 partieular 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 dérivation 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 dérivé 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 précisé 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 discrète 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 Unes of [74] and [22]. In Section 2.4, we dérivé 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 discrète arithmetic Asian basket options. In Section 2.6, we improve the method in Section 2.5, and generalize the approaeh of Lord [56] to a discrète arithmetic Asian basket option. In Section 2.7, we discuss the quality of ail 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 with n assets whose prices Si{t), i = 1,..., ?r, are described, under the risk neutral measure Q and with r some risk-neutral interest rate, by

dS^[t) = vSr{t)dt. + a,Si{t)dWi{t)., (2.1) where {lT)(t), I. > 0} are standard Brownian motions associated with the price of asset i and a, (> 0) are the corresponding volatilities. Further, we assume that the different asset retums are instantaneously correlated in a constant way i.e.

cov{dWi,dWj) = pijdi. (2.2)

Given the above dynamics, the i-th asset price at time t equals S,{L) =

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 discrète

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2.3. Bounds based on comonotonicity and conditioning 7

arithmetic Asian basket call option with a fixed strike K and maturity T on m averaging dates at current time t — 0 \s determined by

with and bj positive coefficients, which both sum up to 1, and with (æ) + = maxjo;, 0}.

For T < rn — 1, the Asian basket call option is said to be in progress and for T > 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 front the call op­

tion prices by the call-put parity relation. Indeed, if the price of an Asian basket put option with a fixed strike K, maturity T and m averaging dates is denoted by ABP(n, m, K, T), static arbitrage arguments lead to the following call-put parity relation:

Asian basket options are suitable for hedging exposure as their payoff dépend 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 corrélations 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 vi 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 prieing of discrète 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)

ABC(7i, m, K, T) - ABP(n, m, K, T) =

EE

aebjSc(O)e-rj __

2.3.1 Comonotonicity

We shortly recall front Dhaene et al. [25] and [26] and référencés therein the procedures for obtaining the comonotonic eounterpart of the sum of dépendent random variables.

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Définition 2.3.1 (Stop-loss Order) A random variable X is said to précédé a randoni variable Y in stop-loss order. written X Y, if for ail retentions d > 0, the stop-loss premium for random X is smaller than that for random Y :

¥.{X -d) y <E(y-d)+.

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

Définition 2.3.2 (Convex Order) Consider two random variables X and Y. Then X is said to précédé Y in the convex order sense, notation X <„ Y, if and only if

E(.Y) = E(y)

E(A' — f/)+ < E(y — d)+, —oo < d < +0O.

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

Définition 2.3.3 The inverse of the cumulative distribution function F\-{x) = P7'[X < .t] ofa random variable X is given by

= inf {.T G K I F{x) > p} , p G [0,1] (2.4) with inf 0 = +00 by convention.

Next, we define comonotonicity of a random vector.

Définition 2.3.4 (Comonotonicity) A random vector (yj,..., y„ ) with marginal cdfs Fy^ (;r) is said to be comonotonic if it has the same distribution as (Ff^{U),.... Ff\U)), with U a random variable which is uniformly distributed in the unit interval (0,1).

Consider a random vector (yi,..., Yf. Its comonotonic counterpart {Y{,..., Yf) is a comonotonic random vector with the same marginal distribution:

(Y,=,.,,.y;:):= (F,T'F,T'(£/)). (2.5)

As proven in Dhaene et al. [25], the convex-largest sum of the components of the vector (y'i,..., ly) is obtained by the comonotonic sum = Y{' + ... -1- Yf, with

n

(2.6) (=1

The cdf of F^c{x) is defined by

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2.3. Bounds based on comonotonicity and conditioning 9

Let us now assume that we hâve some additional information available eonceming the stochastic nature of (Yi,..., V„). More precisely, we assume that there exists some random variable A with a given distribution function, such that we know the conditional cumulative distribution functions, given A = A, of the random Yj, for ail possible values of A. In fact Kaas et al. [41] define the improved comonotonic upper bound §“ as

S’' = + F,i(L0 + • • • + (2.8)

where is the notation for the random variable //((/, A), with the function /) defined by fi{u. A) = with U being independent of A. Given A = A the cdf of §“ can be deduced from (2.7);

Fs.<|a=a (-t) = sup e (0,1) I FyYxiv) <

The cdf of S“ then follows from

»=i

(2.9)

p—OO

Fs.(.x-)=/ Fs.\x^x{x)dE,{X). (2.10)

J H-oo

We recall from Kaas et al. [41] how to obtain a lower bound, in the sense of convex order, for the sum S = Yi + • • • + Y„ by conditioning on a random variable A. We remark that this idea can also be found in Rogers and Shi [63] for the continuons case for obtaining lower bounds for the price. Let us dénoté the conditional expectation by S^:

§^ = E[S| A]. (2.11)

Let us further assume that the random variable A is such that ail E[yi | A] are continuons and monotonie functions of A. For a non-increasing and continuons function of i\, the cdf of the lower bound follows from (2.7):

F^x) = sup I P 6 (0,1) I f^E[Y, I A =

l i=l

P)] < ^ Similarly, for a non-decreasing and continuons functions of A, we hâve

(2.12)

Fse.{x) = sup J P G (0,1) 1 I A = ^(p)] <

Z=1

(2.13) We now consider a normally distributed random variable A 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 = Xu where the ternis A',, are not mu- tually independent, but the multivariate distribution function of the random vector A = (A'i...., Afc) is not completely specified because one only knows the marginal distribu­

tion functions of the random variables Xi. As mentioned in Dhaene et al. [26], to be able to make decisions it may be helpfui to find the dependence structure for the random vec­

tor (A'i,..., A'fc) producing the least favourable aggregate daims S with given marginals.

Therefore, given the marginal distributions of the tenus in a random variable S = El, -E, we shall look at the joint distribution with a smaller resp. larger sum, in the convex order sense. In short, the sum S is bounded below and above in convex order (^c,t) by sums of conditioning and comonotonic variables;

s S" ^c,r S'", which implies by définition of convex order that

< E [(S - d)+] < E [(§“ - d)^] < E [(S*^ d),-]

for ail d in M+, while E [S^'] = E [S] = E [S“] = E [§"].

Remark that the double sum § = o,( Eli ~ showing up in équation (2.3), is a sum of lognomial distributed variables and can be written as

with

and

nin 7ïin

s = j2 -E =

i=l i=l

Qj mocl rrî‘5['_L'| (0)e

(r—)(T—(i—1) mocl nt)

(2.14)

(2.15)

Yi - — (i — 1) niodm) ^ A/’(0,cry-. ■ ali (T — (i — 1) modm,)) (2.16) for ail i = 1; ■ • •, mn, where A/"(0,1) is the standardized normal distribution, [.t] is the smallest integer greater than or equal to x and

ymodî/i = y — [y/rn\rn, where [y\ dénotés the greatest integer less than or equal to y.

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

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2.3. Bounds based on comonotonicity and conditioning 11

S" = F^-!{U) = Y (2.17)

i=\ i=l

where U is a unifomi (0,1) random variable and the usual inverse of a distribution function, which is the non-decreasing and continuons function F^\p), is defined by (2.4).

We shall look at the comonotonic bounds of (2.14) and we will dérivé the comono- tonic upper bound for the option price ABC(?r, m. K, T), denoted by CUB.

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

§" = J]

atb.jSt{Q)

e.xp

c=i j=a

{r- - (2.18)

where $(■) is the standard normal cdf.

Proof Froni (2.17) we write

mil

S'" :=

i=l

rjE(^"i)+sign(o;)<TVi^

By (2.16), we obtain

t=l

which is équivalent to (2.18). □

Theorem 2.3.1 Suppose the sum S is given by (2.14)-(2.I6). Then the comonotonic upper bound for the option price ABC[n, m, K, T) in (2.3) is determined by:

n in—\

CUB = EE acbjS({0)e

e=i j=o

(2.19) where the value F%<-{K) of the cumulative density function of the comonotonic sum can be foiind by solving

~ ^-\FsfK)) — e ^iT

K{l-Fs4K))

EE

m-lacbjS({Q)

exp

^=1. j=Q

{r-\o}){T^]) + <,,yT = K, (2.20)

with 4>(-) the standard normal cdf.

Proof Froni expressions for the cumulative distribution function and the stop-loss premiums of S‘^ the comonotonic upper bound for the price of a discrète Asian basket call

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option at current time t = 0 with strike K is detennined, by applying results in [27] or [22], as follovvs:

77171 ^2

ABC(n, rn, K,T) [sigri(aj)(Tv, — $~^(Fst;(A'))]

7=1

- e-^^A(l - Fs.(A)) (2.21)

where the cdf of the comonotonic sum F^r[K) can be found by solving

^.^.gE*‘‘(Ki)+sign(a;)<Tv,.

= K.

1=1

(

2

.

22

)

Froin (2.15) and (2.16) we find that the upper bound in (2.21 ) can be rewritten precisely as

(2.19) and the équation (2.22) becomes (2.20). □

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

n 777 — 1 71 /777—1

^ ae ^ b,S,(T - j) -K] < a, ( J] b,Se{T ~ j) ~ K,

^e=\ ,7=0 1=] \j=0

n Tii — 1

êEE cifbj {S({T — j) ~ F'cj)_

(=\ j=0 as well as

71 771—1 777—1 / 71

5]

a,

5^ - i) - /t < E ''d E - i) -

V /=1 J=0 / j=0 V 1=1 / +

n ni—1

sEE acbj (Se(T - j) - K(j)^ ,

f=l j=0

(2.23)

(2.24)

(2.25)

(2.26)

with

n m—1 77 m—1

Y. «.,K, = Y >>F<1 = E E “Ar'n = K (2-27)

f=l j=() t'=l .7=0

By a no-arbitrage argument we find that the time zéro price of such Asian basket option should satisfy the following two relations:

71 7n —1

ABC(77-, ?n, K, T) aeACeirn, Kc, T) < ^ J] a.cbje-^'^Ce{Kej, T - j) (2.28) f=l j=0

(’=!

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2.3. Bounds based on comonotonicity and conditioning 13

m —1 n 7n~]

ABC{n,m, K,T) < BC{n, Kj.T — j) < EE acbjC ’'^CfXKfj, T ~ j).

.7=0 e=l .7=0

(2.29) This means that the Asian basket call option can be superreplicated by a static' portfolio of vanilla call options Ce on the underlying assets Se in the basket and with different matu- rities and strikes. Also an average of Asian options ACe or a combination of basket options BC with different maturity dates form a superreplicating strategy. Since the weights a,e as well as bj sum to one, a possible choice for the strikes in the décompositions (2.28) is lie = li] = ll(j = 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 resuit 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 priées 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, for S = Xi is

= E [S I A]

where A is a normally distributed random variable. If | A] are ail non-decreasing functions of A or ail non-increasing functions of A, 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 where LBA dénotés ‘lower bound using the conditioning variable A’ and stands for [(S^ — /f)+]. The non-comonotonic situation for Asian basket options is solved in Section 2.4.

In order to dérivé the lower bound we need to State the following lemma:

Lemma 2.3.1 Let (A', Z) and (T', Z) be jointly normally distributed. Then we hâve (a)

E [A I Z] = E [A] -f —(Z - E [Z]) (b)

cov {X, r I Z) = cov (A', Y) var (Z)

cov {X, Y) cov (y, Z) var (Z)

' When exercising an option at a maturity T — j with j G {1,.. ., t7i — 1), one has in addition to invest the payoff in the risk free money-account.

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rcj

var[X \ Z] = var(X) — cov {X, zy var{Z)

Proof See e.g. Beisser [8],

Next we State the convex lower bounds and we will dérivé the eomonotonic lower bound LBA.

Proposition 2.3.2 Suppose the sum § is given hy (2.I4)-(2.J6) and A is a normally dis- îributed conditioning variable siich that — j), A) are bivariate normally distributed for ail i and j and the corrélation coefficients

cov{W({T - i)p\)

~

C'A fT - J

----

are different from zéro for ail £ and j. Then the convex lower bound is given by

n rn — \ ' 1 .

E X] “A‘5r(0) exp (r - -rij(yj){T - j) + vej(Je^/T -

f=i j=o L

with f/ = $ ('AxIM

<y,\

(2.30)

(2.31)

Proof By the définition of we hâve [S I A]

n m—\

n m— 1

= E E ‘'■ihSiW exp ( (r - f - ]) ] E** |exp {a,Wi{T - ;)) | A|

= EE af6j5f(0)e^''“5'^f

f.= \ j=0

(2.32)

From lemma 2.3.1 we hâve

[aeWfT - j) \ A] = A - EQ [A]

G^

and

var [atWfT - j) \ A] = aj{T - j) - tJjg] (T - j) .

Substitution of these expressions in (2.32) leads to (2.31). □ Theorem 2.3.2 Suppose the sum § is given by (2.14)-(2.16) and A is a normally distributed conditioning variable such that {Wf{T — j), A) are bivariate normally distributed for ail £ and j and the corrélation coefficients j of (2.30) hâve the saine sign, when not zéro, for

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2.3. Bounds based on comonotonicity and conditioning 15

ail d and j. Then the comonotonic lower boiind for the option price ABC{n, m, K, T) in (2.3) is given by

n m~l

LB./\ =515^ a.ebjSe{0)e-^^^

e=i j=o

- [-sign{n,)^-\Fs,{K))]

sign{rij) [•rtjOt\/T - j - 0> \F^t{K)Ÿj

(2.33) where the value F^fK) of the cdfofthe comonotonic siim solves

n 717 — 1

EE

a.cbjSe{0) exp

t'=i j=()

( ^22

i){T - j) + rtjatVf^<l>-\F^fK)) = K.

(2.34)

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

[(§^ - a:).,..] =

f

du.

\f=l j=0 /

The corrélations rp j hâve the same sign. Then the function defined by

(2.35)

n VI—l

/w = EE

e=\ j=o

(2.36)

is continuons and monotone taking positive and négative values. Therefore there exists a unique Fÿe{K) such that f{Fgr{K)) = 0, or equivalently, (2.34) holds.

Let us assume that rpj < 0 for ail i and j. Then we hâve S'’ > K which is équivalent to U < F^t{K). Therefore the lower bound is equal to

LB./\ = e-rT

rFf(K) ( “ ___

*^0 V /J—1 ^_n

‘60 - K du

J=i j=o

or equivalently, LBA

n 771—1

--"EE

apb,Sc{0)e^''--AiF"l(^-P

c=^ j=u

-FAX)

'U

--'EE

aebjSc{0)e^'"

t:=i j=o I

- e-^''^'KFsf{K)

pu<y^dT^i'’ip{v)dv - e~'''^'KFsc{K)

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where (•) is standard normal density function. A change of variable leads to

n m—L

LBA =

e=] j=o

n m — 1

= E E («î>-’(FsKA')) - n,a,y/T^) - K Fsc{K).

r=i j=ü

Similarly, for ail T(^j > 0, we hâve

n 7/1—1

= E E

«A5r(0)e-^^<î> -

$-'(fV(/^)))

-e-’'^7v'(1 - F^e{K)).

t=\ j=0

To judge the quality of the stochastie lower bound [§ | A], 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 |A= A] should be minimized. In other words, to get the best lower bound, A and S 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 of §. Vanduffel et al. [72] propose a conditioning variable A such that the first order approximation of the variance of is maximized. We can take A = FAI, FA2, or FA3, such that for /c = 1,2,3 :

n m— 1

=

E E

- j) (2.37)

r=i j=o with

(2.38) For ail these choices of A, the corrélation coefficients vtj, which enter the lower bound, are easy to calculate. Their expressions contain the instantaneous corrélations p(j (2.2), which influence the sign of the ryj. Indeed when \ is given by (2.37) then the linearity property of covariance leads to

^p{x)dx - KFse{K

cov{W,{T - j).,FAk)

11 in—[

= E E aebnSe{0)5,{C,p)aeCov{W,{T - j), (T - p))

C=l p=0 n ni — l

= EE aebpSe.{0)5k{ê, p)aepu min (T - j, T - p)

f=l p=0

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2.3. Bounds based on comonotonicity and conditioning 17

and

'^FAA,- = CO'' (FAA:, FAk)

Tl in—l n 777 — ]

= <^'A’ibjbi,Sci0)Si{0)dk{(!: j)4(À pWeOiPti min (T - j, T - p) .

C=\ j=0 i=l p=0

Nielsen and Sandmann [61] suggest to look at the géométrie average G which in the Asian basket case is defined by

n TO-l n /in—\ ,

G=n n =n ( n '

t=\ j=0 ^=1 \j=Q

and to consider its standardized logarithm as conditioning variable

G/l = In G — E [In G]

<7i,iG

ELi E"Lo^ aebjaeWeiT - j) C’’lnG

(2.39) where

n n 771— 1 777 — 1

-- = EEEE o.ca.ihjhk(Tt(Tipci min (T - j, T - k).

e=i i=i j=o /c=o

When .'\ is given by (2.39) the covariance follows from

n rn—\

cov{Wi{T - j), GA) = - J-.T - p) ■

f=i p=ü

Only when the (non-zero) corrélation coefficients hâve the same sign for ail ( 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 variable Y and Z:

0 < E [E(y+ I Z) - E(y I Z)+] < ^Ev/var(y | Z).

In this case, we obtain

0 < Eû [(S - K)_^_ I A] - (§<' - A')+] < -E^ [\/var(§ |A)1,

Thus, we find as upper bound for the Asian basket option

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ABC(n, ?n, K, T) < etT 1

[(S^-/0+] +-E^[\/var(§|A) (2.40)

According to an idea of Nielsen and Sandmann [61], we détermine d,\ G M for each of the four different A’s (2.37) and (2.39) such that A > implies that S >K\

n r?i—1

f=i j=ü for A; = 1, 2, 3 and

dvAk — A - (1+ (r- A?) {T-3) - ln(4(f.j))) (2.41)

n m —1

1 / \ / 1

dcA = ---; XI ~ d) {r -

\ ^=1 j=0

Using the same reasoning as in Deelstra et al. [22], it follows from (2.40) that

(2.42)

ABC(n, ni, K, T) < [(S^ - A')+] + (var(S | A = A)5 f/,{X)dX

^ •/ — CO

= [(§' - KU] + [(var(S |A))^ l^A<rf,

< e-''^^'E^ [(S^ - A')+]

(e® [var(S |A)l,A<a,,] ’ E® [l,„<a,,] ’)

(2.43)

where the Hôlder inequality lias been applied and /a is the normal density function of A. In the following theorem we give upper bounds which are denoted by UBRSA with A being a conditioning variable:

Theorem 2.3.3 Let § be given by (2.14)-(2.I6) and Kbea normal/y distributed condition­

ing variable such that {W({T — j), A) are bivariate normally distributed for ail C and j.

Further, suppose that there exists a d>\ G K such that A > d\ implies that § >K. Then an upper bound to the option price ABC{n, ni, K, T) in (2.3) is

UBRSA =e“"^E^ [(S^ - A")+] + {$ (fZ*J}5 {71 n m—Im—1

EEEE

aeu.kbjbpSe{0)Sk{0) e=i k=\ j=ü p=o

^ ^r{2T-j-p) (çOfCTkptk _ ^rrjr^.^j.accrj,. \/(T-j}(r-p)

(2.44)

x<I) - rtjOifT - j - rk,pOk\/T - p

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2.3. Bounds based on comonotonicity and conditioning 19

with d*^ = r(j and Tk.p lhe corrélation coefficients (2.30) and p(k the ins tan ta­

nçons corrélations (2.2).

A])' l{.A<./,v} ■ (2.45) Proof We remark that

E'® [var(S 1A)1{a<^,}] = E® [E^ [S^l A] 1{a<^,j] - E^

The first term of the right-hand side of (2.45) can be rewritten as

E^ [E« [§2|A]

n n m—1 m—1

a,a,bffi,¥P^ [5, {T - j) {T - p) \ A= A]

t-\ fc=l j=0 ;)=0 n 11 m~irn—\

e=\ /c=i j=o p=o

X e -J

J —OCi

where we use the transformation V = <I> ^2—j g^d where

4-,„ = var(a,lV,(T~j) + a,W,(T-p}) (2.46)

= a^(T - j) + al{T - p) + 2aiOkPik min(T - j,T -p), and

rtjMp =

cov{g(W({T - j) + ükWkjT - p), A)

^Cj,kp^ A

(^e\/T - J ^ ^ cTfex/T-p _

~t t.j I ^ fc.p'

^tj.kp

(2.47)

Using the well-known formula

e''‘*’“‘^"V'A(A)dA = eT$(d;_6),

d\

(Ia - E^ [A]

CTa

(2.48) we can rewrite:

E^ [E^ [§2|A]

n n m— 1m— 1

= E E E E

e=l k=l ,7=0 ;;=0

X (d\ - rcj^kpOtj.kp)

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or equivalently, [S->]

71 n rn—Im—1

= E E E E

{dl - vc^,k,,<yejMp) ■

(=[ k=\ j=Q p=0

(2.49) The second terni of the right-hand side of (2.45) equals:

Ev (E« |S |A|)" 1,

/ (E«[S|A = A])"a(AMA

J —CO

/d\ ( ™.-l \ 2

EE ) /A(A)f/A

\e=i j=o /

71 77. 777—1 771—1

f=l /c=l J=ü p=0 c/a

n n 777— 1 777 — 1 _________________

=

E E E E

o.ea.kbjb,,S(i0)Sk{0)e^'^-^'-^-^’^+^^^^^^^ (2.50)

C=\ k—V j=0 p=U

X $ - r(jŒ(x/T - :j - rk^pUk\/T - ?j) .

Since E"® [l{A<rf\}] = ^(<^a) and substituting (2.49) and (2.50) into (2.43), we get the Lipper boiind (2.44) for the Asian basket option price. □ Remark that if the corrélation coefficients V(j hâve the saine sign, when not zéro, for ail i and j, then e~''^E'® [(§^ — K)+] equals the comonotonic lower bound LBA of Theorem 2.3.2. The explicit expression of e~'’^E‘® [(S^ — I^)+] in the non-comonotonic situation will be derived in Section 2.4. Therefore, it is one of the merits of this chapter, that it shows that even in a non-comonotonic situation the upper bound based on Rogers and Shi UBRSA can be obtained.

2.3.5 Partially exact/comonotonic upper bound

The so-called partially exact/comonotonic upper bound, denoted by PECUBA with A be- ing a conditioning variable, consists of an exact part of the option price and some improved comonotonic upper bound for the remaining part, and can be derived as in [74].

Let us first look at the improved comonotonic upper bound §“ (2.8) of (2.14) before

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2.3. Bounds based on comonotonicity and conditioning 21

deriving the PECUBA.

Proposition 2.3.3 Suppose the sitm S is given by (2.14)-(2.I6) and A is a normally dis- trihuted conditioning variable with zéro mean siich fhat {W^{T — 7), A) are bivariate nor- nialty distrihutedfor ail (: and j with the corrélation coefficients given by (2.30). Then the improved comonotonic iipper boimd S" is given by

n ni—\

f=i j=()

where U and V are miitually independent uniform (0,1) random variables.

Proof Use (2.14) and apply Theorem 1 in Dhaene et al. [27].

Theorem 2.3.4 Suppose the sum § is given by (2.14)-(2.16) and A is a normally distributed conditioning variable with zéro mean such that {WfT — ,7), A) are bivariate normally dis­

tributed for ail ( and j with the corrélation coefficients given by (2.30). Then the improved comonotonic upper boundfor the option price ABC[n, m, K, T) in (2.3) is given by

ri ni— l

ICUB = (2.52)

C=1 J=0

X* - #-■ (F5.|„.j (A-)))

K (1 - Fs.|a.j (A)) where Fsi'|a=a (F’) solves

(>-E^[Al)+<Tf (F'Ma^,\(/0) ^

t=l j=0

(2.53)

Proof Use (2.14) and apply results in Kaas et al. [41].

Theorem 2.3.5 Let § be given by (2.14)-(2.16) and A be a normally distributed con­

ditioning variable satisfying the assumptions of Theorem 2.3.3. Then the partially ex- act/comonotonic upper bound to the option price ABC{n, m, K,T) in (2.3) has the fol-

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lowing expression:

Il 77/— 1

PECUBA = 5] 51 atb,SeXQ)e-^^^^ ~ <v) " e-''^K{l - $«,)) r^id^)

i=\ j=o

77 7/1—1

+

EE

a(bjSe{0)e ■''' 2■?)

£=1 j=0

g''f,jCT£v'T=7<i' '(l')

X ^ [ae\/{T - j)(l - ri^) - $ ^(Es..|v=,;(A')) ) dv - Ke-'"^ I - / Fs-.\v=Àl<)dv ) ,

Hd\)

(2.54)

where E = 4> ( Fs>in/=.„(A') solves (^A

n m— 1

EE

(2.55)

£=1 j=0

Proof Recall that there exist d\ such that A > d,\ implies that § >K. Using this fact, we Write:

E® [(S-A”)^] = E® [E® [(S-7i)+ I A]] (2.56)

= E« [E® [(S-A)^ I A] 1(A>„,|1 + E® [E® [(S-A)^. | A] . The first term of the right hand side of (2.56) equals

E^î [E^' [(S-/0+ I A] 1{.A>.,}] = [E® [(S-/^) I A] 1{.a><,}]

Using the expression of given in (2.31), we write:

E«[E«[(S-/0., I A]1{a>,u}] (2-57)

7?. rn — 1

=

EE

ar6,5f(0)eA^-A$ /T - - d\) - K{\ - ^{d\)).

e=i j=o

In the second term of (2.56) we replace S by S“ of (2.51 ), in order to obtain its upper bound, and therefore

E^ [E^ [(S-K)^ I A] 1|A<,,}] < E^ [E^ [(S''-A-)+] 1{a<6/,}] ■

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2.4. Non-comonotonic lower bound and UBRS approach 23

This can be rewritten as

[E^ [(S’'-A')+] 1(A«,}] = / [(S"-A-)+ I A = A] h{X)dX

~oo

/ [(§"-A')+ I K = ü] dv,

JO

si„ce V = 4. (and d* = By (2.52) and (2.53) we obtain:

! [E« [(S»-/0 J 1(A<40]

” "i-l ___

/

a

EE

/î_T ■—(1

^=1 j=Ü

- iv'

X * ■ ’i,) (r - 3) - <t'~'(Fs.iv.JK))) <fo

<5K)- Fs..\v=v{K)dv]..

where Fg„\v^^{K) follows from

(2.58)

71 m— 1 e.=\ j=o

which is precisely (2.55).

Substituting (2.57) and (2.58) into (2.56) we get after discounting upper bound (2.54). □

2.4 Non-comonotonic lower bound and UBRS approach

In this section, we consider the case where not ail of (2.30) hâve the same sign. Then, will not be a comonotonic sum of random variables, making the detemiination of the lower bound more complicated since it does not follow from the comonotonicity literature.

To détermine a lower bound, we follow the approach suggested in [56] for basket options.

From (2.35) we know that the lower bound can be rewritten as [(§^' - I<)+]

/ n 711,-1 '

dv.

,f=i j=o with V = 4>

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Let us consideralso the function / in (2.36). Notice that the function f{v) is no longer a monotone function of u (as in the comonotonic situation) when not ail rej hâve the same sign. The dérivative f'{v) with respect to v equals

7tl - l

Av) = aehjS({{))re,jat\/f' jepv 2 f'J crj)iT-j) + rt_jCrf^T-j>î’ ' (v) e=i j=o

where ip{-) is the standard nonrial density function. Obviously, the above denominator is strictly positive for v G (0,1). The nunierator, which we will dénoté by K{v), is a non- decreasing function of v since its dérivative with respect to v is positive. Moreover, this numerator has the following limits;

lim/T(i,>) = —oo and limA’(r;) = +oo.

i’—0 U—>1

Therefore, there exists a unique v* such that — 0 and consequently /'('//) = 0.

Since moreover

lim/('ü) = +00 and lirn/(u) = +cx),

î;—>0’ «r

we conclude that f{v) is either positive upon whole the interval [0,1], or has a strictly négative minimum /(u*). Hence, in the latter case, f{v) stays positive before a certain value c/ai ê]0,1[, is then négative until a value dj^.^ G]dAi, 1[ but has then again positive values on the interval 1]. Therefore, the following theorem can easily be proved:

Theorem 2.4.1 Lel § be given by (2.14)-(2.16) and let A be a normally distributed condi- lioning variable such that {W({T — j), A) are bivariate normally distributed for ail l and J. Suppose that not ail r^ j of(2.30) hâve the same sign and consider the function f intro- duced in (2.36). The non-comonotonic lower boundfor the option price ABC{n, ni, K, T) in (2.3) is such that

et) if f{v) > 0 for ail v, then

Il 777 I

LBA = aebjSe{0)e-’'^ - Ker^'^', (2.59)

f=i j=o

b) ifj{v*) < 0, with V* the solution off'{v) = 0, then

71 m—1

LBA=EE“AS<(0)e'"‘I’(rf;,

e=[ j=o

77 777—1

e=i j=o

(2.60)

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

where, for i = 1,2, df. = c/a, < dénoté the two solutions of the following équation in x:

m -1

^^aebjSfO)e = K. (2.61)

(=T. J=0

Proof Case of f{v) > 0 for ail v is trivial.

Caseof /(v*) < 0: A < c/ai or A > imply that and d\^ < A < implies

§^</\. □ Remarks

1. This lower bound can be used in the Rogers and Shi approach, so the upper bound UBRSA can also be derived in the non-comonotonic situation.

2. As a basket option is a spécial case of an Asian basket option with m — 1, the reasoning above and fomiula (2.60) (with rn = 1) remains valid for basket options in the cases where S'^ is not a comonotonic sum, providing a much simpler lower bound than in [22]. No optimization algorithm is needed.

3. The approach in this section is general and can also be used in other settings in which sums of non-comonotonic random variables show up with corrélations with mixed signs. In [73], Vanduffel et al. deal with cash flows with mixed signs and obtain a resuit with a similar taste.

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

In his paper [69], Thompson used intuition and simple optimization to dérivé an upper bound which tightened Rogers and Shi’s upper bound considerably for continuously sam- pled Asian options. His reasoning is based upon a first order approximation and is therefore referred to as FA. In his Ph.D. thesis [70], Thompson already suggested the idea of adding a supplementary parameter but he did not work it out. Thompson’s approximation is only justified when OfWfT — j) has a small variance (i.e. when a]{T — j) is small).

In case of Asian options. Lord [56] approximates the arithmetic sum by a shifted lognormal variable and then adds according to the ideas of Thompson a supplementary parameter. In this section, the method of Thompson [69] will be generalized to the Asian basket case by taking into account a supplementary parameter.

Let

.[({T - 3) = dt(T - j) + a

77. 771—1

afvVfT - j) -

EE

afkafV^iT - k)

i=l k=0

(2.62)

(34)

be a random function where — j) and a are detenninistic functions with

V ni—l

e.=i j=o

(2.63)

Then the price of an Asian basket option (apart from the discount factor) can be written and boLinded as:

Tl 777 — 1

\e=i j=o )

(

^ J] {a,h,St [T - j) - Kaeb.MT - j)) n rn — 1 e=\ j=o

n 771—1

s E’* E E (“''’J®' f®" - - ><vh,h(T J.=1 j=0

n m—1

= E E [(*5^ (T - ,7) - Kfe{T - j) ), ] . r=i ,y=o

-rj

(2.64)

An upper bound is obtained by minimizing the right hand side of (2.64) over the set of detenninistic functions — j) and a such that condition (2.63) is satisfied. By means of the Lagrangian method we detennine the optimal value of /if (T — j) and W. Let

71 m—i

L(A, {/if (T - .?) - - ^'))4-] (2-65)

e=i j=o

(

Tl 777—1 \

Y (khji^e{T - ,y) - 1

j

c=i j=o

J

be the Lagrangian, and consider its first order dérivative with respect to [ii({T — j)} :

c>L(A,{/if(r-j)},?r) r.- i x < .o

--- citbjE'^ [-A • l{Sf(T-j)-A7f(T-j)>o}J - (2-66) Equating (2.66) to zéro we obtain the following condition:

Q[5f(T-j)-A7f(r-j)>0] = -^ for any £ and T - (2.67) Defining

Yt{T-j) = Se{T-j)-âJ<

n 771—1 creWeiT - j)

- E E

i=l k=0

(2.68)

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2.5. Generalization of an upper bound based on the method of Thompson 27

condition (2.67) equals

Q [Y,{T -

j)

> Kfj,{T - ,7)1 =

- A

(2.69) where A is merely a négative constant. Because this cannot be re-arranged to détermine a relation between /j£(T - j) and A, we will approximate the random variable Yi{T - j) in order to solve the minimization problem approximately.

Thompson suggests to use the first order approximation exp{cr£Wc{T — j)) ss 1 + acWc{T — j) which is valid for small ae\/T — j. This leads to the following first order approximations for S( {T - j) and Y(,{T - j) respectively:

Yf^^-j) = Sr{T-j)-âK

SP {T - :j) = (1 + aeW,{T - j)) (2.70)

îi m “1

(7emT - j) - k) . (2.71 )

i=l k=0

Denoting — j) by analogy to nt(T - j), we conclude that

Q [yriT - ,7) > /Wif (T - j)] = - A

must be independent of ^ and T-j. Using the fact that Yp‘^{T - j) is normally distributed, we deduce that

/x7if-'(T-j)-S,(0)e(--H)(^-^) v/var(y/’-'(r-j))

,.F.4

+ (2.72)

where is the cumulative distribution function of a normally distributed random variable, and hence

/xf(r - ,7) = A (2.73)

where

var(y/-^(r-j)) (2.74)

= aj (T - j)

n m—\

+ 2âK ^ ^ ^ c^bkaeaipa rnin(r - ,7, T - k)

i=l k=0 n rr/. —1 11 rn—l

+ ô-K^ EEEE cpahbkbpa,ahpzh miii(T - k, T - p).

i=l k=0 h=l p=0

(36)

The constraint (2.63) for /iPCT — j) implies that

_ A' - Et. E;r,‘

Etal E”7,' yvar(y™(r^j))

(2.75)

Theorem 2.5.1 g/ve/? by (2.14)-(2.I6) and a > 0, then upper bounds based on Thompson ’s method for the option price ABC{n, m, K, T) in (2.3) are given by

n m—1

ABC{nyrn, K,T) < e~’~^

EE

aebj

e=\ j=n

(T - :i, X, a)$

V d,{T-:uâ)

J

+d((T - j, a)(p (- jyx,a)\

d,{T-j.,â) J p{x)dx (2.76) with ip the standard normal density function, and with c^^{T — j, x, a) and d?({T — j, a) the conditional mean and variance:

cf\T - J, X, W) = _ âK —~ (2.77)

Il TTL 1 . / rjn 1 rfi -\ \

--- : V" V" h l'iiinr - r - 7) \ +at,XsJT - J - 2^ 2^Oih<Xi-

i=l k=:>

and

(n m —1 n ni—l

EEEE

aibkaiibj,a^(jf,p,ii rniri(T - A:, T - p)

z=l A.'=0 /i=l p=0

(Eili EfcL"o' aibi,a,p,t min(r ~ ky T ~ j)Ÿ\

- - - - ~j - - - - j (2.78)

where /if ''(T - j) is given by (2.73).

Proof As the upper bound in (2.64) holds for any function p.^(T — j) satisfying the above constraint, it also holds for the approximately optimal function /if''(T - j) :

ABC(n, m, K, T) Tl m—1 /!=I j=0

n m— 1

+atWf{T - j) -

EE a,6,a,H/;(T -

k)

i=\ k=0

Se {T - j) - O K kde^(J-j)

a (2.79)

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2.5. Generalization of an upper bound based on the method of Thompson 29

It is well-known that (T - j) - âl<

71 in— 1

a + aeWe{T - j) - k) (2.80)

i=l k=0

conditioned on Wf:{T — j) = x\JT — j is normally distributed. Define cp''^{T — j, x, rr) and as the corresponding conditional mean and variance:

- j, X, if) - al<, f4'\T - :i) ^ Æ---:- + acx y/T - J a

n 77/.-» 1

EE CHh;,aiE^ hv,{T - k) \ W({T - j) = xy/T - j i=l k=0

- üK —- + atxy/r^j

n m—1

; min(T - k, T - j) ^ o t \

“ )

î=i k=Q V ./ y

and

71 777—1

4(T - ,7,â) = ^ a,ha/V/T - k) | {T - j)

,7=1 A:=0

n m — \ n 77?--1

= ÏÏ^K-

EEEE

aihaiJ)j^a,(7,,pik min(T - k. T - p)

\ i=\ k={) h=l p=0

(EZi EL~o‘ (kh.T.p2cmm{T - k, T - j))“\

T-:i (2.82)

Then the upper bound (2.79) can be expressed as ABC(n, m, K, T)

71 771 — 1 P 0^1

^ E E - 3:^)Z) A '^{x)dx., (2.83)

» . . J ~ i>r\ L

£=1 j=()

where ip(-) dénotés the density of the standard normal distribution and Z has a A/’(0,1) distribution. Using the well-known relation

El(c+rfZ)J =c4.(£)+<i.pg),

the upper bound (2.83) leads to (2.76). □

Recall that the Lagrangian is defined by (2.65). Its dérivative with respect to a however

Figure

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