A Diffusion Approximation for the Riskless Profit Under Selling of Discrete Time Call Options: Non-identically Distributed Jumps

IHS Economics Series Working Paper 164

January 2005

A Diffusion Approximation for the Riskless Profit Under Selling of Discrete Time Call Options: Non-

identically Distributed Jumps

Alexander V. Nagaev

Sergej Nagaev

Impressum Author(s):

Alexander V. Nagaev, Sergej Nagaev, Robert M. Kunst

A Diffusion Approximation for the Riskless Profit Under Selling of Discrete Time Call Options: Non-identically Distributed Jumps

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164 Reihe Ökonomie Economics Series

A Diffusion Approximation for the Riskless Profit Under Selling of Discrete Time Call Options

Non-identically Distributed Jumps

164 Reihe Ökonomie Economics Series

A Diffusion Approximation for the Riskless Profit Under Selling of Discrete Time Call Options

Non-identically Distributed Jumps

Alexander V. Nagaev, Sergei A. Nagaev, Robert M. Kunst January 2005

Institut für Höhere Studien (IHS), Wien

Institute for Advanced Studies, Vienna

Contact:

Alexander V. Nagaev

Faculty of Mathematics and Computer Sciences Nicolaus Copernicus University

Chopin str. 12/18 87-100 Torun, POLAND email: nagaev@mat.uni.torun.pl Sergei A. Nagaev

Department of Economics and Finance Institute for Advanced Studies Stumpergasse 56

1060 Vienna, AUSTRIA email: sergej@ihs.ac.at Robert M. Kunst

Department of Economics and Finance Institute for Advanced Studies Stumpergasse 56

1060 Vienna, AUSTRIA and

University of Vienna Department of Economics Brünner Straße 72 1210 Vienna, AUSTRIA email: robert.kunst@univie.ac.at

Founded in 1963 by two prominent Austrians living in exile – the sociologist Paul F. Lazarsfeld and the economist Oskar Morgenstern – with the financial support from the Ford Foundation, the Austrian Federal Ministry of Education and the City of Vienna, the Institute for Advanced Studies (IHS) is the first institution for postgraduate education and research in economics and the social sciences in Austria.

The Economics Series presents research done at the Department of Economics and Finance and aims to share “work in progress” in a timely way before formal publication. As usual, authors bear full responsibility for the content of their contributions.

Das Institut für Höhere Studien (IHS) wurde im Jahr 1963 von zwei prominenten Exilösterreichern – dem Soziologen Paul F. Lazarsfeld und dem Ökonomen Oskar Morgenstern – mit Hilfe der Ford- Stiftung, des Österreichischen Bundesministeriums für Unterricht und der Stadt Wien gegründet und ist somit die erste nachuniversitäre Lehr- und Forschungsstätte für die Sozial- und Wirtschafts- wissenschaften in Österreich. Die Reihe Ökonomie bietet Einblick in die Forschungsarbeit der Abteilung für Ökonomie und Finanzwirtschaft und verfolgt das Ziel, abteilungsinterne Diskussionsbeiträge einer breiteren fachinternen Öffentlichkeit zugänglich zu machen. Die inhaltliche Verantwortung für die veröffentlichten Beiträge liegt bei den Autoren und Autorinnen.

Abstract

A discrete time model of financial markets is considered. It is assumed that the relative jumps of the risky security price are independent non-identically distributed random variables. In the focus of attention is the expected non-risky profit of the investor that arises when the jumps of the stock price are bounded while the investor follows the upper hedge.

The considered discrete time model is approximated by a continuous time model that generalizes the classical geometrical Brownian motion.

Keywords

Asymptotic uniformity, local limit theorem, volatility

JEL Classification

G12, G11, G13

Comments

This article is prepared within the project "Risk-less profit of a writer under selling discrete time options"

Contents

1 Introduction 1

2 Basic results 6

3 "Local" profit of investor 10 4 Proof of Theorem 2.2 14 5 The limit value of the upper rational price 18 6 The local limit theorem and its applications 21

References 25

1 Introduction

Consider the simplest financial market in which securities of two types are circulating.

The price evolution of the securities of the first type is given by the equations b

= b

ρ

, k = 0, 1, 2, . . . ,

where b

> 0, ρ

≥ 1. The prices are registered at the equidistant moments of time t

= a + kh. With no loss of generality we put a = 0, h = 1, i.e. t

= k.

The price of the security of the second type at the moment k is represented as s

= s

ξ

· · · ξ

, k = 0, 1, 2, . . . ,

where the relative jumps ξ

are random.

The securities of the first type are riskless having the interest rates (ρ

− 1) · 100%.

Let us call them conventionally bonds. It is clear that possessing the securities of the second type is concerned with a risk of their devaluation. We call them conditionally stocks.

Taken together in certain amounts β and γ the securities of both types constitute a so-called portfolio (writer’s investment portfolio) whose worth at the time moment k is βb

+ γs

. Playing in the considered financial market consists of successive changing of the portfolio content at the moments k = 1, 2, . . . , n − 1. The successive pairs (β

, γ

), (β

, γ

), . . . , (β

, γ

) constitute a so-called strategy of the game or a trading strategy. Obviously, as a basis for choosing (β

, γ

) serves the evolution of the stock price up to this moment i. e. s

, s

, . . . , s

. In other words

β

= β

(s

, s

, . . . , s

), γ

= γ

(s

, s

, . . . , s

).

The player is called a writer (seller, investor).

A trading strategy is called self-financing if the changing of the portfolio content does not affect its value i.e.

β

b

+ γ

s

= β

b

+ γ

s

, k = 1, . . . , n − 1.

The final goal of the game is to meet the condition

x

= β

b

+ γ

s

≥ f(s

) (1.1) where f(s) is a so-called pay-off function of the simplest option of the European type having n as a maturity date. For more about the mathematical and substantial aspects of the option pricing theory see e.g. Shiryaev (1999).

Basic problems of the mathematical theory of options are the evaluation of the

so-called rational option price and a corresponding to it strategy leading to (1.1).

Both the problems are easily solved within the framework of the so-called binary model that is in the case where ρ = const ∈ (d, u) and ξ

take only two values d and u. In this case (see e.g. Ch. VI in Shiryaev (1999)),

x

= ρ

n

X

k=0

C

p

(1 − p

)

f(s

u

d

) (1.2) where

p

= ρ − d u − d .

It is worth emphasizing that (1.2) does not assume any restrictions on the measure which governs the evolution of the stock price (ξ

, . . . , ξ

). Furthermore, there exists the unique self-financing trading strategy (β

, γ

), (β

, γ

), . . . , (β

, γ

) leading to the equality

x

= β

b

+ γ

s

= f (s

). (1.3) The strategy is defined by the formulae

β

= uf

(s

d) − df

(s

u)

ρb

(u − d) (1.4)

and

γ

= f

(s

u) − f

(s

d)

s

(u − d) (1.5)

where

f

(s) = ρ

n−k

X

j=0

C

p

(1 − p

)

f(su

d

). (1.6) The successive values of the portfolio are

x

= f

(s

), k = 0, 1, . . . , n − 1.

In particular, x

= f

(s

) is the rational or fair price. The rational option price is the minimal initial capital x

which always allows the investor to meet contract terms under proper behavior. Note that any smaller initial capital never ensures the required pay-off.

Now, assume that the market model is binary but d and u are not constant. More precisely, assume that ξ

takes the values d

and u

.

Proposition 1.1 In order to guarantee the equality

x

= f(s

) (1.7)

the investor must have at the preceding moment the capital

x

= ρ

(p

f (s

u

) + (1 − p

)f(s

d

))

2

where

p

= ρ

− d

u

− d

.

Furthermore, x

must be distributed between bonds and stocks in the following way β

= u

f (s

d

) − d

f(s

u

)

ρ

b

(u

− d

) , γ

= f (s

u

) − f(s

d

) s

(u

− d

) .

Proof. Let x and (β, γ) be respectively the investor capital and its distribution in the portfolio at moment n − 1. For the sake of simplicity we omit the subscript n − 1. So,

x = βb

+ γs

. The value of the potfolio at the moment n equals

x

= βb

+ γs

= βb

ρ + γs

ξ

. Taking into account the condistion (1.7) we obtain

f(s

ξ

) = βb

ρ

+ γs

ξ

or





f (s

u

) = βb

ρ

+ γs

u

f (s

d

) = βb

ρ

+ γs

d

. Solving the system we find out that

β

= u

f (s

d

) − d

f(s

u

)

ρ

b

(u

− d

) , γ

= f (s

u

) − f(s

d

)

s

(u

− d

) . (1.8) It is easily verified that this portfolio contains the capital

x = ρ

ρ

− d

u

− d

f (s

u

) + u

− ρ

u

− d

f (s

d

)

!

= ρ

(p

f(s

u

)+(1−p

)f(s

d

)).

The proposition is proved.

Set

f

(s) = ρ

(p

f(su

) + (1 − p

)f(sd

)).

From Proposition 1.1 it follows that

x

= f

(s

). (1.9)

So, we derived a pay-off function for a new European option with the maturity time n − 1. By the proposition, in order to guarantee (1.9) the investor must have at the moment n − 2 the capital

x

= ρ

(p

f

(s

u

) + (1 − p

)f

(s

d

))

or substituting the formula for f

x

= ρ

ρ

(p

p

f(s

u

u

) + p

(1 − p

)f (s

u

d

)+

(1 − p

)p

f (s

d

u

) + (1 − p

)(1 − p

)f(s

d

d

)) = f

(s

) where

f

(s) = ρ

ρ

(p

p

f (su

u

) + p

(1 − p

)f(su

d

)+

(1 − p

)p

f (sd

u

) + (1 − p

)(1 − p

)f (sd

d

)).

Obviously, in order to meet the contract obligations at the moment k the investor must have the capital

x

= f

(s

) (1.10)

where

f

(s) = ρ

· · · ρ

ik∈{0,1}^n−k

p(i

)f (sa(i

)), (1.11) while

i

= (i

, . . . , i

) and

a(i

) = u

d

· · · u

d

, p(i

) = p

(1 − p

)

· · · p

(1 − p

)

. This capital must be distributed in accordance with

β

= u

f

(s

d

) − d

f

(s

u

)

ρ

b

(u

− d

) , γ

= f

(s

u

) − f

(s

d

) s

(u

− d

) .

(1.12) In particular, the rational price is given by the formula

x

= ρ

· · · ρ

i0∈{0,1}ⁿ

p(i

)f(s

a(i

)) (1.13) and the initial portfolio must be of the form

β

= u

f

(s

d

) − d

f

(s

u

)

ρb

(u

− d

) , γ

= f

(s

u

) − f

(s

d

) s

(u

− d

) .

If ξ

, k = 1, 2, . . . , n, take more than two values then it is impossible to guarantee the desired relation (1.3) with probability 1. However, sometimes it is possible to guarantee (1.1). For example, this is possible when f (s) = f

(s) is convex. If f (s) is convex so are all the functions f

(s), k = 0, 1, . . . , n − 1. If, furthermore, ξ

∈ [d

, u

]

4

then the hedging capital sequence is evaluated by the same formulae (1.10), (1.11) and (1.2).

This fact was, first, proven in Tessitore and Zabczyk (1996) for the case of constant d and u by the methods of control theory (see also Zabczyk (1996) and Motoczy´ nski and Stettner (1998)). Later on in Shiryaev (1999) the rational price is derived as the solution of a extreme problem (see Theorem V.1c.1 ibidem).

Consider the sequence

¯

x

= f

(s

), k = 0, . . . , n − 1, (1.14) and let (β

, γ

) be defined as in (1.12).

Possessing after (k−1)−th step the capital x ¯

distributed in portfolio in accordance with (1.12) at the next step k the investor gains the capital

x

= β

b

+ γ

s

= u

− ξ

u

− d

f

(s

d

) + ξ

− d

u

− d

f

(s

u

).

If ξ

∈ [d

, u

], k = 1, . . . , n, then due to convexity of f

(s) we have δ

= x

− x ¯

= f

(s

d

) u

− ξ

u

− d

+ f

(s

u

) ξ

− d

u

− d

− f

(s

ξ

) ≥ 0. (1.15) If f

(s

ξ) is strictly convex in [d

, u

] then δ

= 0 if and only if ξ

= d

or ξ

= u

. Otherwise δ

> 0. Thus, if ξ

takes at least one value lying in (d

, u

) then a profit can arise. If the extreme values d

and u

belong to the support of the distribution of ξ

then x ¯

is the minimal capital that allows such a profit. It implies that

¯

x

= ρ

· · · ρ

i0∈{0,1}ⁿ

p(i

)f(s

a(i

)) (1.16) is the minimal starting capital that allows the investor to meet his contract obligations with probability 1 provided he follows the strategy determined by (1.4) and (1.5). This strategy forms the so-called upper hedge. It determines the sequence (¯ x

, x ¯

, . . . , x ¯

) of the hedging capitals. Here, x ¯

is called the upper rational price.

The investor may dispose of the so arisen profit in various ways. The simplest one is to withdraw from the game the superfluous quota δ

which to the maturity date acquires the value δ

ρ

· · · ρ

. So, the self-financing condition is fulfilled only in the part which bans any capital inflow.

Having withdrawn unnecessary quota one should follow the "binary" optimal strategy determined by (1.4) and (1.5). As a result to the maturity date the investor accumulates a riskless profit

∆

= δ

ρ

. . . ρ

+ δ

ρ

. . . ρ

+ · · · + δ

. (1.17) It should be emphasized that the upper hedge admits an arbitrage opportunity in the sense that the investor always meet his obligations, i.e.

P(x

≥ f (s

)) = 1,

and may have a riskless profit in the sense that P(∆

> 0) > 0.

It seems hopeless to find an acceptable formula for the expected value of riskless profit E∆

. So, the question arises how to approximate it. It is one of such approximations that is a basic goal of the paper.

It is worth emphasizing that a similar problem was studied in A. Nagaev and S.

Nagaev (2003) (see also S. Nagaev (2003)). In these papers the authors considered the simplest case where the random variables ξ

, k = 1, 2, . . . , n, were i.i.d. Here, if the pay-off function is not smooth then chaotic phenomena arise. The typical example of such a function is provided by the call option. Unfortunately, the considered models do not take into account such intrinsic property of the stock price evolution as volatility.

The basic goal of the present paper is to extend the main results of the latter work to the case where the stock price jumps are non-identically distributed.

The paper is organized as follows. In Section 2 the basic results concerning the expected value of the riskless profit under selling the call and put options are formulated.

The "local" profit in the case where the model converges to a geometrical Gaussian process with independent increments is studied in Section 3. In Section 4 the limit value for the expected value of the total riskless profit is established. The limit value for the upper rational price is given in Section 5. Auxiliary facts concerning limit theorems for sums of independent variables are given in Section 6.

2 Basic results

In what follows we consider the simplest case of the standard call and put options determined, respectively, by the pay-off functions

f (s) = (s − K )

, f (s) = (K − s)

. (2.18) Since the random variables ξ

, k = 1, 2, . . . , n, are not identically distributed it is convenient to build an approximation based on a small parameter. This parameter should be linked with the maturity time. Let ε be such that

ε → 0, nε → T, 0 < T < ∞.

Assume that

ξ

= ξ

= exp(h(kε)ε + η

ε

), k = 1, 2, . . . , n, (2.19) where independent random variables η

∈ [−y, x] and h(t) is a function continuous in [0, T ]. Further, x and y are positive constants. Obviously, ξ

∈ [d

, u

], where

u

= u

exp(h(kε)ε + xε

) d

= d

= exp(h(kε)ε − yε

), k = 1, 2, . . . , n, (2.20)

6

and

s

= s

= s

ξ

· · · ξ

, k = 1, 2, . . . , n. (2.21) Consider the random process

x

(t) =

k−1

X

j=1

h(jε)ε + ε

k−1

X

j=1

η

, k − 1

n ≤ t < k

n , k = 1, 2, . . . , n. (2.22) It is easily seen that the trajectories of the process belong to D[0, 1].

Definition 2.1 We say that the sequence of independent variables η

, η

, . . . , η

, satisfies Condition A if:

(A1)

Eη

= 0, k = 1, 2, . . . , n;

(A2) there exists a strictly positive continuous function σ(t) defined on [0, T ] such that Var η

= σ

(kε) + ω

, k = 1, 2, . . . , n,

where

lim

sup

1≤k≤n

|ω

| = 0;

(A3) [−y, x] is the minimal interval that contains the supports of all the distributions F

(u) = P(η

< u), k = 1, 2, . . . , n.

In particular, condition (A2) implies that for all sufficiently small ε we have Var η

> 1

2 min

0≤t≤T

σ

(t) > 0.

If η

, η

, . . . , η

, satisfy Condition A then the Lindeberg condition holds and, therefore, by the Central Limit Theorem the finite dimensional distributions of the process x

(t) converge to those of the process

x(t) =

tT

Z

0

h(u)du + y(tT ), 0 ≤ t ≤ 1,

where y(t) is the Gaussian process such that

y(0) = 0, Ey(t) ≡ 0, Ey(s)y(t) = B(t, s) =

min(s,t)

Z

0

σ

(u)du.

It is easily seen that y(t) has independent increments.

Actually, the process x

(t) weakly converges to x(t) in D[0, 1]. However, dealing with the expected value of the total profit ∆

it suffices to have the weaker convergence.

Consider the sums

ζ

=

k

X

j=1

η

, k = 1, . . . , n.

It is evident that as k → ∞

Var ζ

= b

(1 + o(1)).

where

b

=

k

X

j=1

σ

(jε).

As in Nagaev and S. Nagaev (2003), the following form of the local limit theorem plays a crucial role. There exists ε

such that as k → ∞

b

P(z ≤ ζ

< z + h) = hϕ(z/b

) + o(1) (2.23) uniformly in z ∈ IR

, ε ∈ [0, ε

] and h, 0 < h

≤ h ≤ h

< ∞. Here, ϕ(z) is the density of the standard normal law.

The most general, though not very convenient, condition guaranteeing (2.23) is the following: for all sufficiently small ε

sup

1≤k≤n

sup

0<δ≤|t|≤∆<∞

|Ee

| = ρ(δ, ∆) < 1. (2.24) In Section 6 we discuss more convenient conditions stated in terms of the distribution functions F

(u).

In addition to (2.20) and (2.19) assume that

ρ

= ρ

= exp(α(kε)ε) (2.25) where α(t) ≥ 0 is continuous in [0, T ]. Let ∆

= ∆

be determined by (1.15) and (1.17) with u

, d

, ρ

, ξ

and s

replaced, respectively, u

, d

, ρ

, ξ

and s

.

Define for t ∈ [0, 1]

ψ(t, z) = x + y

q

xyT (1 − t) ϕ











ln K − z −

tT

α(u)du +

xy(1 − t)T

q

xy(1 − t)T











(2.26)

8

and for t ∈ [0, T ]

I(t) =

Eψ(t, x(t/T ) + ln s

) =

√

B(t,t)+xy(T−t)

ϕ









ln(K/s0)−

t

R

0

h(u)du−

T

R

t

α(u)du+¹₂(T−t)xy

√

B(t,t)+xy(T−t)









.

(2.27)

The following theorem contains the basic result of the present paper.

Theorem 2.2 Assume that the sequence η

, j = 1, 2, . . . , n, satisfies Condition A. If (2.23) is also fulfilled, then as ε → 0, nε → T

E∆

= K 2

ZT

0

(xy − σ

(t))I(t)dt + o(1)

where K is the strike price from (2.18).

It is worth reminding that if the random variables η

, η

, . . . are independent then σ

= σ.

Note that Var η

≤ xy, k = 1, 2, . . . , n. This implies that sup

0≤t≤T

σ

(t) ≤ xy. So, the limit value of the sequence E∆

is non-negative. It should be emphasized that this limit value depends on x and y through xy. Furthermore, in the case of the call option, the upper rational price corresponding to x and y as n → ∞ converges to (see (5.47))

¯

x

→ c(xy) =

s

Φ









ln(s0/K)+

T

R

0

α(t)dt+T xy/2

√T xy









− K exp −

0

α(t)dt

!

Φ









ln(s0/K)+

T

R

0

α(t)dt−T xy/2

√T xy









.

As to the lower rational price given by the formula

x

= ρ

· · · ρ

(s

ρ

· · · ρ

− K )

it converges to

c(0) = s





1 − K s

· exp





−

T

Z

0

α(t)dt









+

.

So, the interval of the rational prices converges as n → ∞ to (c(0), c(xy)).

3 "Local" profit of investor

Let us denote by c any positive constant whose concrete value is of no importance.

Under such a convention we have e.g. c + c = c, c

= c etc. By θ we denote any variable taking values in [−1, 1].

Denote

p

= ρ

− d

u

− d

, λ

= ξ

− d

u

− d

.

From (1.6) it follows that the discounted "local" profit of the investor takes the form

∆

= δ

ρ

· · · ρ

=

ik∈{0,1}^n−k

p(i

)(λ

f (s

u

a(i

)) + (1 − λ

)f(s

d

a(i

))−

−f(s

ξ

a(i

)),

(3.28) where p(i

) and a(i

) are as in (1.11). For the time being we suppress the dependence of λ

, d, u, ξ

and s

on ε.

Let, first, f (s) = (s − K )

. Consider i

such that s

d

a(i

) > K. Then

λ

f (s

u

a(i

))+(1−λ

)f(s

d

a(i

))−f (s

ξ

a(i

)) = s

(λ

u

+(1−λ

)d

−ξ

)a(i

) = 0.

If s

d

a(i

) ≤ K then

0 = f (s

u

a(i

)) ≥ f (s

ξ

a(i

)) ≥ f (s

d

a(i

)).

It is worth reminding that d

≤ ξ

≤ u

. Thus,

∆

= δ

ρ

· · · ρ

=

(ik:sk−1dka(ik)≤K<sk−1dka(ik)

p(i

)(λ

(s

u

a(i

) − K)

+

(1 − λ

)(s

d

a(i

) − K )

− (s

ξ

a(i

) − K )

).

Denote

|i

| = i

+ · · · + i

. Define for d

≤ z ≤ u

¯

r

(z, Z ) = (n − k)p + ε

R





Z −

n

X

j=k

h(jε)ε





− R

w where

R = x + y, p = y

x + y , ln z = h(kε)ε + wε

. Let

r

(z) = ¯ r

(z, ln(K/s

)).

The following lemma plays an important role.

10

Lemma 3.1 Let i

and z satisfy the equation s

za(i

) = K

where ln z = wε

+h(kε)ε, −y ≤ w ≤ x. If 0 < x

≤ min(x, y) ≤ max(x, y) ≤ x

< ∞ Then

|i

| = r

(z) = [r

(d

)].

Proof. From the equation

s

za(i

) = K it follows that

ln(K/s

) = i

ln u

d

+ · · · + i

ln u

d

+ ln d

+ · · · + ln d

+ ln z.

According to (2.20)

ln u

d

= Rε

and, therefore,

ln(K/s

) = ε

R(i

+ · · · + i

) − (n − k)yε

+ wε

+

n

X

j=k

h(jε)ε.

or

ln(K/s

) = ε

R|i

| − (n − k)yε

+ wε

+

n

X

j=k

h(jε)ε.

So, |i

| = r

(z). If z varies within [d

, u

], then w stays in [−y, x]. It is easily seen that r

(d

) − r

(u

) = 1. It implies that

#{j : r

(u

) < j ≤ r

(d

)} = 1. (3.29) So, r

(z) = [r

(d

)]. The lemma is proved.

From the lemma it follows that

∆

= δ

ρ

· · · ρ

=

(ik:|i_k|=[rn−k(dk)])

p(i

)(λ

(s

u

a(i

) − K)

+

(1 − λ

)(s

d

a(i

) − K)

− (s

ξ

a(i

) − K )

).

Let z

= exp(w

ε

+ h(kε)ε) ∈ [d

, u

] be determined by the equality r

(z

) = [r

(d

)]. It is easily seen that

p − {r

(d

)} = − w

R . (3.30)

Furthermore,

s

u

a(i

) = K exp((x − w

)ε

) and

s

d

a(i

) = K exp(−(y + w

)ε

).

Further,

∆

=

(ik:|i_k|=[rn−k(dk)], sk−1ξka(ik)>K)

p(i

)(λ

(s

u

a(i

) − K ) − (s

ξ

a(i

) − K))+

λ

(ik:|i_k|=[rn−k(dk)]), sk−1ξka(ik)≤K)

p(i

)s

u

a(i

− K) = ∆

+ ∆

.

(3.31) Since λ

u − ξ

= −d(1 − λ

) we conclude that

∆

= K(1 − λ

)(1 − exp(−(y + w

)ε

))

(ik: |ik|=[rn−k(dk)], sk−1ξka(ik)>K)

p(i

) (3.32) while

∆

= Kλ

(exp((x − w

)ε

) − 1)

(ik:|i_k|=[rn−k(dk)]), sk−1ξka(ik)≤K)

p(i

). (3.33) The inequality s

ξ

a(i

) > K is equivalent to η

> w

. Taking into account (2.20)S and (2.19) we conclude that

λ

= R

(η

+ y) + O(ε

), 1 − λ

= R

(x − η

) + O(ε

).

Then we obtain

∆

= Kε

(σ

+ O(ε

))π

(3.34) where

π

=

|i_k|=[r_n−k(dk)]

p(i

) and

σ

=







(x − η

){r

(d

)} if η

> R({r

(d

)} − p) (η

+ y)(1 − {r

(d

)}) otherwise.

(3.35) Curiously, if f(s) = (K − s)

then the asymptotic formula (3.34) remains valid. In order to verify this one should slightly modify the calculations leading to (3.34).

Lemma 3.2 Let ψ(t, z) be defined as in (2.26). Under the conditions of Theorem 2.2 π

= ε

ψ(kn

, ln s

) + o(ε

)

uniformly in k, k ≤ (1 − δ)n.

12

Proof. Consider independent variables ζ

, k = 1, . . . , n such that P(ζ

= 1) = p

, P(ζ

= 0) = 1 − p

. It is evident that

π

= P(ζ

+ · · · + ζ

= j ) where j = [r

(d

)].

Taking into account (2.20) we obtain

u

− d

= (x + y)ε

+ x

− y

2 ε + O(ε

) while

ρ

− d

= yε

+ (α(kε) − h(kε) − y

/2)ε + O(ε

).

Therefore,

p

= p + α(kε) − h(kε) − xy/2

R ε

+ O(ε).

Denote

a

= p

+ · + p

, b

= p

(1 − p

) + · + p

(1 − p

).

Obviously,

a

= (n − k)p + ε

R





n

X

j=k+1

α(jε)ε −

n

X

j=k+1

h(jε)ε − (n − k)xyε 2





+ O(1) and

b

(n − k)p(1 − p) + O(ε−1/2).

If n − k → ∞ then by (5.44) we obtain π

= 1

b

ϕ r

(d

) − a

b

!

+ o(b

).

By Lemma 3.1

r

(d

) − a

= ε

R





ln(K/s

) −

n

X

j=k+1

α(jε)ε





+ (n − k)ε

xy

2R + O(1)

and, therefore, r

(d

) − a

b

= 1

q

(n − k)εxy





ln(K/s

) −

n

X

j=k+1

α(jε)ε





+

q

(n − k)εxy

2 + O(ε

).

Since nε → T r

(d

) − a

b

= 1

q

T (1 − kn

)xy





ln(K/s

) −

T

Z

kn⁻¹T

α(u)du)





+

q

T (1 − kn

)xy

2 +o(1).

(3.36) Since

b

= ε

R

(n − k)εxy it remains to recall (2.26). The lemma is proved.

From Lemma 3.2 and (3.34) it follows that

∆

= Kεψ(kn

, ln s

)σ

+ o(ε) (3.37) uniformly in k, k ≤ (1 − δ)n. This is the starting point for the proof of Th. 2.2.

4 Proof of Theorem 2.2

Represent the total profit ∆

as

∆

=

1≤k<δn

∆

+

δn≤k≤(1−δ)n

∆

+

(1−δ)n≤k≤n

∆

= ∆

+ ∆

+ ∆

(4.38) and estimate the expectations E∆

, E∆

and E∆

one after another.

According to (3.37) we have E∆

= Kε

δn≤k≤(1−δ)n

Eψ(kn

, ln s

)σ

+ o(1)

or in view of (2.19)

E∆

= KεEψ(kn

, ε

(η

+ · · · + η

) +

k−1

X

j=0

h(jε)ε + ln s

)σ

. Consider

A(u, v) = (x − v)uχ(u, v) + (v + y)(1 − u)(1 − χ(u, v)), (u, v) ∈ [0, 1] × [−y, x], (4.39) where

χ(u, v) =







1 if R(u − p) < v ≤ x, 0 ≤ u ≤ 1 0 if − y < v ≤ R(u − p), 0 ≤ u ≤ 1 In view of (3.35) we have

σ

= A({r

(d

)}, η

).

14

It is evident that χ(u, v) admits a monotone ε−approximation by means of χ

(u, v) and χ

(u, v) where

χ

(u, v) =























v−R(u−p)

ε⁰

+ 1 if R(u − p) − ε

≤ v ≤ R(u − p), 0 ≤ u ≤ 1 0 if − y ≤ v ≤ R(u − p) − ε

, 0 ≤ u ≤ 1 1 if R(u − p) ≤ v ≤ x, 0 ≤ u ≤ 1 and

χ

(u, v) =























v−R(u−p)

ε⁰

if R(u − p) ≤ v ≤ R(u − p) + ε

, 0 ≤ u ≤ 1 0 if − y ≤ v ≤ R(u − p), 0 ≤ u ≤ 1 1 if R(u − p) + ε

≤ v ≤ x, 0 ≤ u ≤ 1.

Obviously, χ

(u, v) are continuous in [0, 1] × [−y, x] and χ

(u, v) ≤ χ(u, v) ≤ χ

(u, v).

Furthermore, 0 ≤

Z

[0,1]×[−y,x]

(χ

(u, v) − χ

(u, v))dudF

(v) ≤

Z

U_ε0

dudF

(v ) ≤ (2ε

/R) (4.40) where

U

= ((u, v) : u ∈ (0, 1), −y < v < x, |v − R(u − p)| ≤ ε

).

Therefore,

Eψ(kn

, ε

(η

+ · · · + η

) +

k−1

P

j=0

h(jε)ε + ln s

)A

({r

(d

)}, η

) ≤

Eψ(kn

, ε

(η

+ · · · + η

) +

k−1

P

j=0

h(jε)ε + ln s

)σ

=

Eψ(kn

, ε

(η

+ · · · + η

) +

j=0

h(jε)ε + ln s

)A({r

(d

)}, η

) ≤

Eψ(kn

, ε

(η

+ · · · + η

) +

k−1

P

j=0

h(jε)ε + ln s

)A

({r

(d

)}, η

) where

A

(u, v) = (x − y)uχ

(u, v) + (y + x)(1 − u)(1 − χ

(u, v)).

Obviously, the family ψ(t, z), δ ≤ t ≤ 1 − δ, is contained in the class G defined in Section 6. So, we may apply Corollary 6.4.

By the corollary

Eψ(kn

, ε

(η

+ · · · + η

) +

j=0

h(jε)ε + ln s

)A

({r

(d

)}, η

) =

Eψ(kn

, ν

B(kε, kε) +

kε

R

0

h(u)du + ln s

)

[0,1]×[−y,x]

A

(u, v)dudF

(v) + o(1) uniformly in k, δ ≤ kn

≤ 1 − δ. Here ν has the standard (0, 1)−normal distribution and F

is the distribution function of η

.

In view of (4.40)

Z

[0,1]×[−y,x]

A

(u, v)dudF

(v ) =

Z

[0,1]×[−y,x]

A(u, v)dudF

(v) + 2θε

.

It is easily verified that

Z

[0,1]×[−y,x]

A(u, v)dudF

(v) = 1

2(x + y) (xy − Var η

) . Since ε

is arbitrary we obtain

Eψ kn

, ν

B(kε, kε) +

0

h(u)du + ln s

!

σ

=

1

2(x+y)

(xy − Var η

)Eψ kn

, ν

B(kε, kε) +

kε

R

0

h(u)du + ln s

!

+ o(1) uniformly in k, δ ≤ kn

≤ 1 − δ.

Thus, E∆

= Kε

2R

X

δn≤k≤(1−δ)n

(xy − σ

(kε))Eψ





kn

, ν

B(kε, kε) +

Zkε

0

h(u)du + ln s





+o(1) or

E∆

= KT 2R

1−δ

Z

δ

(xy − σ

(tT ))Eψ





t, ν

B(tT, tT ) +

tT

Z

0

h(u)du + ln s





.

16