在擁有部分資訊且需付手續費之財務模型中之最佳投資策略

國

立

交

通

大

學

應用數學系

碩

士

論

文

在擁有部分資訊且需付手續費的財務模型中

之最佳投資策略

Optimal Trading Strategy with Transaction Cost

in a Partial Information Model

研 究 生：胡仲軒

指導教授：吳慶堂 副教授

在擁有部分資訊且需付手續費的財務模型中

之最佳投資策略

Optimal Trading Strategy with Transaction Cost

in a Partial Information Model

研 究 生：胡仲軒 Student：Chung-Hsuan Hu

指導教授：吳慶堂 Advisor：Ching-Tang Wu

國 立 交 通 大 學

應 用 數 學 系

碩 士 論 文

Submitted to Department of Applied Mathematics College of Science

National Chiao Tung University in partial Fulfillment of the Requirements

for the Degree of Master

in

Applied Mathematics

July 2008

Hsinchu, Taiwan, Republic of China

-i-在擁有部分資訊且需付手續費的財務模型中

之最佳投資策略

學生：胡仲軒 指導教授：吳慶堂

國立交通大學應用數學系碩士班

摘

要

本論文介紹當投資者在財務市場做交易的時候需要付一筆固定比率的手續

費時，投資人應該如何決定最佳投資策略。在這篇論文裡對於離散時間的財務模

型我們分別給風險中立、風險趨避兩種投資人一些結果。另一方面，本篇論文也

討論當投資人在市場裡只能觀察到部分資訊時，投資人又該如何決定最佳投資策

略。

-ii-Optimal Trading Strategy with Transaction

Cost in a Partial Information Model

Student：Chung-Hsuan Hu Advisors：Dr. Ching-Tang Wu

Department of Applied Mathematics

National Chiao Tung University

ABSTRACT

In this thesis, we study that when the investor needs to pay a constant proportional

transaction cost at each trading in a financial market how he (or) she decides the

optimal trading strategy. We give some main results for the risk-neutral and

risk-averse investors, respectively, in discrete financial model. And we also discuss

the optimal trading strategy for the investor when he (or she) can only observe partial

information in the financial market.

-iii-誌

謝

此篇論文能夠完成，首先，我最想感謝的是恩師也就是指導教授─吳慶堂教

授。自從大五那年旁聽了吳慶堂教授的高等微積分之後，便為我建立了穩固的數

學分析能力；因為有了老師的教導，讓我如願的考上交通大學的應用數學系研究

所，有機會當個研究生。之後，我懇求吳慶堂教授能夠擔任我的指導教授；於是，

展開了我的研究生活。在這段時間中，老師給了我許多的指導，好讓我對於在財

務數學的領域中有了更多的認識；此外，更要感謝指導教授不辭辛苦、不畏沉悶

花時間陪伴著我 meeting，聽著一遍又一遍老師已經熟透的內容，並適時的給予

我想法，加深我對於相關內容的認知。另外，我還想感謝陳育慈學姊，提供了我

許多關於財金方面的知識，讓我對於金融市場的現況有更多的了解；以及陳冠羽

學長給我數值模擬的建議，以完成數值結果。最後，還想感謝蔡明誠學長、蔡明

耀學長以及陳偉國學長，在我有課業問題時，能夠跟我討論，提供我許多的想法。

在口試期間，要感謝韓傳祥教授、王太和教授以及陳冠宇教授，費心的審閱

我的論文，並提供我許多的意見，我讓我的論文能夠更加完整，學生永銘在心。

最後，我要感謝我的家人，謝謝他們陪伴著我，給予我非常多的支持以及

鼓勵，好讓我能夠順利的完成學業；也要感謝我的朋友們能夠提供我休閑娛樂，

讓我有紓解壓力的管道。但願所有關心我的人，能夠和我一同分享完成此篇論文

的喜悅。

-iv-目

錄

中文提要

i

英文提要

ii

誌謝

iii

目錄

iv

一、

Introduction

1

二、

Optimal Strategy with Transaction Cost in Discrete

Model

3

2.1

Model setup

3

2.2

Trading Strategy with Transaction Cost and Backward

Inductions

5

2.3

Optimal Strategy with Transaction Cost under

Risk-Neutral Utility

7

2.4

Optimal Strategy with Transaction Cost Under

Risk-Averse Utility

9

2.5

Numerical Results

18

三、

Optimal Strategy with Transaction Cost in a Partial

Information Discrete Model

27

3.1

Model Setup

27

3.2

Optional Projection and The Gaussian Case

28

3.3

Examples for Risk-Neutral and Risk-Averse Utility

Functions

30

四、

Future Works

41

CHAPTER 1

Introduction

In a classic paper, Merton (1971) developed optimal portfolio and consumption rules for an investor managing a portfolio of risky assets whose prices evolve as geometric Brownian motions. In Merton’s model, it assumed that investors trade costless. However, investors in real capital markets face nontrivial transaction costs, so it is interested to discuss the effect on trading strategies when the assumption of frictionless is removed.

Magill and Constantinides (1976) got optimal trading policies which are more reasonable in continuous time theory formulated by Merton. Since they introduced transaction costs in the model, the investors traded at suitable disjoint intervals of time rather than trading at anytime. Davis and Norman (1990) investigated the optimal consumption and investment decisions with transaction costs equal to a fixed proportion of the amount transacted.

For the perspectives of the investors, we invest in some assets in discrete time, thus, the information that we observed from the market is collected in discrete time. Here we face a problem that if we only can observe the information which is collected in discrete time, what decisions will be the best? What trading strategies will make the maximal profit? Or under what decisions we will not be bankrupt in finite time? Moreover, what trading strategies will make the maximal profit if we only can observe the stock prices in discrete time model? Since in real capital market the drift term b and the noise term B in the equation (2.1) are not observable. This

problem has been studied widely, for example, Karatzas and Xue (1991), Lakner (1995, 1998), Bouchard and Pham (2003), and Xiong and Yang (2005). In such a situation, we call the model with “partial information”.

In this thesis, we assume that the stock price is governed by a simple discrete model similar to the Black-Scholes model with the interest rate 0. In Chapter 2, we assume all coefficients in the model are deterministic and the noise term is Gaussian. And we also assume an investor only need to pay constant proportional transaction cost when he (or she) sells some stocks. We discuss two different utilities, says risk-neutral and risk-averse, here. And for the risk-averse utility, we will find a “no trading” interval and give some numerical results. In Chapter 3, we consider the appreciation return of stock b as a random variable and assume b is Gaussian, then use the method (optional projection theorem) in Huang (2007) to rewrite the stock price model. Finally, we give a risk-averse utility example. In last chapter, we give some ideas for the future work.

CHAPTER 2

Optimal Strategy with Transaction Cost in Discrete Model

The basic problem for the investor in the financial mathematics is to reach the maximal profit via trading in the financial market. The trading strategy plays an important role for every investor in a financial market. The main question is how to find the best strategies in different cases. In this chapter we consider the case of the market model with one risky asset (stock) and one riskless asset (bond).

Let (Ω, F , P) be a complete probability space. 2.1. Model Setup

Let (Sn)n≥0 be stock prices in a financial market, where S0 ∈ R+is given, and we

assume that the interest rate is identical to 0. At time n, suppose that the investor can only observe the stock prices up to time n. Thus, the information the investor observe is Gn, the natural filtration generated by S0, S1, ..., Sn.

Assume that the stock prices follows the relation

(2.1) Sn+1− Sn= Sn[bn+ σn(Bn+1− Bn)], n ≥ 0,

where bn is the appreciation return of the stock, σn is volatility of the stock, and

(Bn) is a noise.

Assumption 1.

(1) All the coefficients bn and σn are assumed as deterministic.

(2) (Bn+1−Bn)n≥0is a Gaussian process with mean 0 and variance 1, and is (totally)

Under the Gaussian assumption it is not easy to know the distribution of Sn.

However, if we consider the stock return Xn+1 =

Sn+1− Sn

Sn

, then the stock price equation (2.1) will be rewritten as (2.2) Xn+1 = bn+ σn(Bn+1− Bn).

Remark 2.

(1) (Xn+1) is a Gaussian process with mean bn and variance σn2.

(2) {Xn+1, n ≥ 0} are totally independent for all n ≥ 0.

We denote G_n∗the natural filtration generated by {X0, X1, · · · , Xn} with X0 = S0.

The following lemma tells us that X0, X1, · · · , Xn and S0, S1, · · · , Sn generate the

same filtration for all n.

Lemma 3. G_n∗ = Gn for all n ≥ 0. PROOF. Due to Sn+1 = SnXn+1+ Sn,

(1) When n = 0, G₀∗ = G0.

(2) When n = k, assume that G_k∗ = Gk.

(3) When n = k + 1, we have Xn+1 =

Sn+1− Sn

Sn

∈ Gk+1 and Sn+1= SnXn+1+ Sn ∈ Gk+1∗ .

By mathematical inductions, we have G_n∗ = Gn for all n ≥ 0.

Remark 4. Since G0 is the σ-field generated by S0 and S0 ∈ R+ is given, we

have G0 = {∅, Ω}. Then for any integrable random variable X, we have

Remark 5.

(1) By (2.2), Bn+1− Bn is independent of Gn∗ for all n ≥ 0.

(2) By Lemma 3, Bn+1− Bn is also independent of Gn for n ≥ 0.

2.2. Trading Strategy with Transaction Cost and Backward Inductions In this section we introduce the trading strategy with transaction cost in discrete time model, which is derived from the model specified in Kabanov (2002). Suppose that the random variables ξn+1 and ηn+1 describes the number of shares of assets

invested in stock and bond at time n (after the trading), respectively. Thus the wealth process at time n is given by

(2.3) Vn = ξn+1Sn+ ηn+1.

Moreover, if the initial endowment is given by x, then the initial wealth x = ξ0S0+ η0.

Remark 6. (ξ, η) is called a trading strategy if both of ξnand ηnare predictable

with respect to the filtration (Gn), i.e., ξnis the number of shares of the stock between

the n − 1 (after the trading) and the time n (before the trading). Thus, our wealth at time n is Vn (after the trading) defined by (2.3).

We assume that the investors need to pay the transaction cost when they sell stocks and we denote the transfers from the stock to the bond by L10_n (amount by money) at time n. Moreover, we consider a model with constant proportional transaction costs and the proportion is λ10∈ (0, 1). Thus, the wealth process (after

the trading) is given by Vn = ξn+1Sn+ ηn+1 = Vn−1+ ξn(Sn− Sn−1) − λ10L 10 n n ≥ 1, V0 = x − λ10L 10 0 . And, by (2.3) we have ηn+1 = ηn+ (ξn− ξn+1)Sn− λ10L 10 n.

Remark 7. In Kabanov (2002), it introduces the following model with transac-tion cost. Let V1

n denote the total value (amount by money) of the stock at time

n under transaction cost, V0

n denote the total value of the bond at time n under

transaction cost. And ξk represents the number of shares of the stock the investor

holds at time k − 1 (after the trading).

The portfolio value evolves according to the equation V_n1 = v1+ n X k=0 ξk(Sk− Sk−1) + n X k=0 L01_k − n X k=0 (1 + λ10)L10_k V_n0 = v0− n X k=0 L01_k + n X k=0 L10_k

where Lij_n represents transfers from the ith to the jth asset at time n under transac-tion costs, v0 _{and v}1 _{are initial endowments in the bond and the stock respectively.}

It is easy to verify that our model is equivalent to that introduced by Kabanov, since V_n1+ V_n0 = v0+ v1+ n X k=0 ξk(Sk− Sk−1) − n X k=0 λ10L10_k = Vn−1+ ξn(Sn− Sn−1) − λ10L 10 n.

Problem 8. For some future time n = N , in order to make the most profit, how do we invest at the beginning under the model in a small investor perspectives?

Due to the argument in game theory, we know that the optimal decision can be constructed by the backward induction, see, e.g., Fudenberg and Tirole (1991).

Definition 9 (Backward Induction). This is a mathematical technique for finding the optimal choice in each step in a game. The idea is to start by solving the optimal strategy of the last step, and then work backward to compute the optimal strategy before.

At time N − 1 our goal is to find the optimal strategy ξ_N∗ such that (2.4) max

ξN

E[U (VN −1+ ξN(SN − SN −1))|GN −1],

for given strategy (ξn, ηn)n≤N −1, i.e., we aim to solve (2.4) to get the optimal solution

ξ_N∗. Moreover, at time m ≤ N − 2 we choose the optimal ξ_m+1∗ satisfying max{E[U (Vm+ N X k=m+1 ξk(Sk− Sk−1) − N −1 X k=m+1 λ10L10_k )|Gm]},

for given strategy (ξn, ηn)n≤m and ξk= ξ∗k for k ≥ m + 2.

Remark 10. If the terminal time is 1, the optimal trading strategy solved by the backward induction is that solved by the same as the maximal expected utility.

2.3. Optimal Strategy with Transaction Cost Under Risk-Neutral Utility

In mathematical finance, utility function is a measure of the relative satisfaction gained by consuming different bundles of goods and services. In our model utility is a measure of the relative satisfaction gained by making different profits.

In this section we consider the risk-neutral utility function denoted by U (x) = x. First we give a one-period example under the risk-neutral utility function and show some results.

Example 11. If the terminal time is 1, by the maximal expected utility we want to compute

E[U (V0+ ξ1(S1− S0) − x)] = E[ξ1(S1− S0) − λ10L 10 0 ].

(1) If we buy some stocks at time 0 (ξ0 < ξ1) and ξ1 is G0-measurable, then

E[ξ1(S1 − S0) − λ10L 10

0 ] = E[ξ1(S1− S0)]

= ξ1S0E[b0+ σ0(B1− B0)].

Due to the Gaussian assumption we get

ξ1S0E[b0+ σ0(B1− B0)] = ξ1S0b0.

Hence,

(2.5) E[U (V0+ ξ1(S1− S0) − x)] = ξ1S0b0.

(2) If we sell some stocks at time 0 (ξ0 > ξ1), then

E[ξ1(S1− S0) − λ10L 10 0 ]

= E[ξ1S0[b0+ σ0(B1− B0)] − λ10S0(ξ0− ξ1)]

= −λ10S0(ξ0− ξ1) + ξ1S0b0+ E[ξ1S0σ0(B1− B0)].

Because of the Gaussian assumption, we get E[ξ1(S1− S0) − λ10L 10 0 ] = −λ 10 S0(ξ0 − ξ1) + ξ1S0b0 Hence, (2.6) E[U (V0+ ξ1(S1− S0) − x)] = −λ10S0(ξ0− ξ1) + ξ1S0b0 ≤ ξ1S0b0,

since ξ0 > ξ1. From (2.5), (2.6), in the case without any restrictions on the trading

strategy, the investors trade according the sign of b0. If b0 ≥ 0, the investor should

buy the stocks as much as possible. If b0+ λ10 < 0, the investors should make

short-sell as much as possible. In the case with constraint, for example, 0 ≤ ξ1 ≤ x/S0,

no short-sell and no loan, the optimal strategy is given by

ξ₁∗ =      x/S0, if b0 ≥ 0, 0, if b0+ λ10< 0 .

2.4. Optimal Strategy with Transaction Cost Under Risk-Averse Utility We consider the risk-averse utility function given by

U (x) = −1

θexp(−θx),

where θ > 0 is the absolute risk aversion and be a constant. First we show a one-period example below.

Example 12. If the terminal time is 1, by the maximal expected utility we have to compute E  −1 θexp(−θ(V0+ ξ1(S1− S0) − x))  = E  −1 θexp(−θ(−λ 10_L10 0 + ξ1(S1− S0)))  = E  −1 θexp(θλ 10_L10 0 − θξ1(S1− S0))  = −1 θE h exp(θλ10L10₀ − θξ1S0[b0+ σ0(B1− B0)]) i = −1 θ exp(−θξ1S0b0)E h exp(θλ10L10₀ − θξ1S0σ0(B1 − B0)) i . We have to discuss it in two cases.

(1) If we buy stocks at time 0, then −1 θexp(−θξ1S0b0)E h exp(θλ10L10₀ − θξ1S0σ0(B1− B0)) i = −1 θexp(−θξ1S0b0)E[exp(−θξ1S0σ0(B1− B0))]. Due to the Gaussian assumption, we get

−1 θ exp(−θξ1S0b0)E[exp(−θξ1S0σ0(B1− B0))] = − 1 θexp(−θξ1S0b0 + 1 2θ 2_ξ2 1S 2 0σ 2 0). Hence, E[−1 θ exp(−θ(V0+ ξ1(S1− S0) − x))] = − 1 θexp(−θξ1S0b0+ 1 2θ 2_ξ2 1S02σ02).

Thus from the fundamental calculus, we maximize the conditional expectation to get the optimal ξ1 and from x = ξ0S0+ η0 we have

(2.7) ξ1 = b0 θS0σ02 > ξ0 = x − η0 S0 . (2) If we sell stocks at time 0, then

E  −1 θ exp(−θ(V0+ ξ1(S1 − S0) − x))  = −1 θexp(−θξ1S0b0)E[exp(θλ 10_S 0(ξ0− ξ1) − θξ1S0σ0(B1− B0))].

Due to the Gaussian assumption of B1− B0, we have

E[exp(θλ10S0(ξ0− ξ1) − θξ1S0σ0(B1− B0))] = exp(θλ10S0(ξ0− ξ1) + 1 2θ 2_ξ2 1S 2 0σ 2 0). Hence, E  −1 θexp(−θ(V0+ ξ1(S1− S0) − x))  = −1 θ exp  −θξ1S0b0+ θλ10S0(ξ0− ξ1) + 1 2θ 2 ξ₁2S₀2σ₀2  .

Thus from the fundamental calculus, we maximize the conditional expectation to get the optimal ξ1 and from x = ξ0S0+ η0 we have

(2.8) ξ1 = b0+ λ10 θS0σ02 < ξ0 = x − η0 S0 .

From (2.7), (2.8), we conclude that the investors will neither buy nor sell stocks when their initial wealth (amount by money) is in the interval

 b0 θσ2 0 + η0, b0+ λ10 θσ2 0 + η0  .

Moreover, if one wants to buy stocks he may choose the strategy ξ1 =

b0

θS0σ20

at time 0 to reach the maximum profit, and if one wants to sell stocks he may choose the strategy ξ1 =

b0+ λ10

θS0σ02

at time 0 to reach the maximum profit. Remark 13. Now the “no trading” interval is

 b0 θσ2 0 + η0, b0+ λ10 θσ2 0 + η0  .

And if one wants to buy (sell) stocks he may choose the strategy ξ1 =

b0

θS0σ02

(ξ1 =

b0 + λ10

θS0σ02

) at time 0. We find some phenomenons from this example.

(1) For fixed b0 and σ0, if the initial stock price S0 is too high, we will be

conservative for our strategy in the stock.

(2) For fixed σ0 and S0, if the appreciation rate of the stock be positive and

grows up, we invest the number of shares of the stock more.

(3) For fixed b0 and S0, if the volatility of the stock grows up, we are

conserva-tive when investing the stock.

(4) The length of the “no trading” interval is λ

10

θσ2 0

, if λ10 _{increases or σ}2 0

Example 14. If the terminal time is 2, by backward induction we figure out this situation in two steps.

Step1 : Compute the case from time 1 to time 2. E  −1 θexp(−θ{V1+ ξ2(S2− S1) − V0− ξ1(S1− S0)})     G1  = E  −1 θexp(−θ{ξ2(S2− S1) − λ 10_L10 1 })     G1  = E  −1 θexp(−θ{ξ2S1[b1+ σ1(B2− B1)] − λ 10_L10 1 })     G1  . (1) If we sell stocks at time 1, then

E  −1 θexp(−θ{ξ2S1[b1+ σ1(B2− B1)] − λ 10_L10 1 })     G1  = E  −1 θexp(−θξ2S1[b1+ σ1(B2− B1)] + θλ 10_(ξ 1− ξ2)S1)     G1  .

Since (B2− B1) is independent of G1 and is normally distributed, we have

E  −1 θ exp(−θξ2S1[b1+ σ1(B2− B1)] + θλ 10_(ξ 1− ξ2)S1)     G1  = −1 θexp  −θξ2S1b1+ θλ10(ξ1− ξ2)S1+ 1 2θ 2 ξ₂2S₁2σ2₁  . Hence, E  −1 θexp(−θ{V1+ ξ2(S2− S1) − V0− ξ1(S1− S0)})     G1  = −1 θ exp  −θξ2S1b1+ θλ10(ξ1− ξ2)S1+ 1 2θ 2_ξ2 2S 2 1σ 2 1  .

Thus from the fundamental calculus, we maximize the conditional expec-tation to get the optimal ξ2 and from the assumption ξ2 < ξ1 we have

ξ2 =

b1+ λ10

θS1σ12

(2) If we buy stocks at time 1, then using the similar argument in Example 1 we have ξ2 = b1 θS1σ12 > ξ1.

Thus, for given ξ1, the optimal solution is given by

(2.9) ξ₂ = b1+ λ 10 θS1σ21 I {ξ1>b1+λ10 θS1σ21 }+ b1 θS1σ12 I_{ξ 1< b1 θS1σ12 }+ ξ1I{ b1 θS1σ12 ≤ξ1≤b1+λ10 θS1σ21 }.

Step2 : Consider the trading period from time 0 to time 1, we replace the ξ2 by ξ2

which is given in (2.9) and compute the conditional expectation

E  −1 θ exp(−θ{V1+ ξ2(S2− S1) − x})     G0  = E  −1 θ exp(−θ{V0+ ξ1(S1− S0) − λ 10_L10 1 + ξ2(S2− S1) − x})     G0  = E  −1 θ exp(−θ{−λ 10_L10 0 − λ 10_L10 1 + ξ1(S1− S0) + ξ2(S2− S1)})     G0  .

Similar as the argument as the trading period from time 1 to time 2. we have to separate it into two cases.

(1) If we sell stocks at time 0, i.e., L10₀ = (ξ0− ξ1)S0, then we have

E  −1 θexp(−θ{−λ 10_L10 0 − λ 10_L10 1 + ξ1(S1 − S0) + ξ2(S2− S1)})     G0  = E " −1 θexp(−θ{−λ 10_(ξ 0− ξ1)S0− λ10L 10 1 (I_{ξ 1>b1+λ10 θS1σ21 }+ I{ξ1< b1 θS1σ21 }+ I{b1+λ10 θS1σ21 ≥ξ1≥ b1 θS1σ21 }) +ξ1(S1− S0) + ξ2(S2− S1)})   G0 i .

By some calculation, we get E  −1 θ exp  θλ10(ξ0− ξ1)S0− θξ1(S1− S0) + {θλ10(ξ1− b1+ λ10 θS1σ12 )S1 −θb1+ λ 10 θS1σ21 (S2− S1)}I_{S 1>b1+λ10 θξ1σ12 }+ {−θ b1 θS1σ12 (S2− S1)}I_{S 1< b1 θξ1σ12 } +{−θξ1(S2− S1)}I_{ b1 θξ1σ21 ≤S1≤b1+λ10 θξ1σ12 } !   G0 # = E[−1 θ exp(θλ 10 (ξ0 − ξ1)S0− θξ1(S1− S0) +{θλ10ξ1S1− (b1 + λ10)2 σ2 1 −b1+ λ 10 σ1 (B2− B1)}I_{S 1>b1+λ10 θξ1σ12 } +{−b 2 1 σ2 1 − b1 σ1 (B2− B1)}I_{S 1< b1 θξ1σ21 } +{−θξ1S1b1 − θξ1S1σ1(B2− B1)}I_{ b1 θξ1σ21 ≤S1≤b1+λ10 θξ1σ21 })|G0]

Using the tower property of the conditional expectation, we may rewrite the above equation in the following form

E[−1 θexp(θλ 10 (ξ0− ξ1)S0− θξ1(S1− S0) + {θλ10ξ1S1− (b1 + λ10)2 σ2 1 }I {S1>b1+λ10 θξ1σ21 } −b 2 1 σ2 1 I_{S 1< b1 θξ1σ12 }− θξ1S1b1I{ b1 θξ1σ21 ≤S1≤b1+λ10 θξ1σ21 })E[exp({− b1+ λ10 σ1 I {S1>b1+λ10 θξ1σ12 } −b1 σ1 I_{S 1< b1 θξ1σ21 }− θξ1S1σ1I{ b1 θξ1σ21 ≤S1≤b1+λ10 θξ1σ21 }}(B2 − B1))|G1]|G0].

Since (B2− B1) is independent of G1 and is normally-distributed, we get E[−1 θ exp({θλ 10_(ξ 0− ξ1)S0− θξ1(S1− S0) + θλ10ξ1S1− (b1+ λ10)2 2σ2 1 }I {S1>b1+λ10 θξ1σ12 } +{θλ10(ξ0− ξ1)S0− θξ1(S1− S0) − b2 1 2σ2 1 }I_{S 1< b1 θξ1σ12 } +{θλ10(ξ0− ξ1)S0− θξ1(S1− S0) − θξ1S1b1+ 1 2θ 2 ξ2₁S₁2σ2₁}I { b1 θξ1σ12 ≤S1≤b1+λ10 θξ1σ21 })] = −1 θ{ Z ∞ b1+λ10 θξ1σ21 exp(θλ10(ξ0 − ξ1)S0− θξ1(x − S0) + θλ10ξ1x − (b1+ λ10)2 2σ2 1 )f (x)dx + Z b1 θξ1σ12 −∞ exp(θλ10(ξ0− ξ1)S0− θξ1(x − S0) − b2 1 2σ2 1 )f (x)dx + Z b1+λ 10 θξ1σ12 b1 θξ1σ12 exp(θλ10(ξ0− ξ1)S0− θξ1(x − S0) − θξ1xb1+ 1 2θ 2_ξ2 1x 2_σ2 1)f (x)dx},

where f (x) is the probability density function of S1, i.e.,

(2.10) f (x) = √ 1 2πS0σ0 exp(−(x − S0− S0b0) 2 2S2 0σ20 ). Due to the first order condition with respect to ξ1, we have

Z ∞ b1+λ10 θξ1σ21 (−θλ10S0+ θλ10x − θ(x − S0)) exp(θλ10(ξ0− ξ1)S0+ θλ10ξ1x (2.11) − (b1+ λ 10₎2 2σ2 1 − θξ1(x − S0))f (x)dx + Z b1 θξ1σ21 −∞ (−θλ10S0− θ(x − S0)) exp(− b2 1 2σ2 1 + θλ10(ξ0− ξ1)S0− θξ1(x − S0))f (x)dx + Z b1+λ 10 θξ1σ21 b1 θξ1σ21 (−θb1x + θ2x2σ12ξ1− θλ10S0− θ(x − S0)) exp(−θξ1b1x + 1 2θ 2_ξ2 1x 2_σ2 1 + θλ10(ξ0 − ξ1)S0− θξ1(x − S0))f (x)dx = 0.

(2) If we buy stocks at time 0, i.e., L10₀ = 0, then we have E[−1 θexp(−θ{V1+ ξ2(S2− S1) − x})|G0] = E[−1 θexp(−θ{−λ 10_L10 1 (I_{ξ 1>b1+λ10 θS1σ12 }+ I{ξ1< b1 θS1σ21 }+ I{b1+λ10 θS1σ21 ≥ξ1≥ b1 θS1σ21 }) +ξ1(S1− S0) + ξ2(S2− S1)})|G0].

By some calculation, we get

E[−1 θexp(−θξ1(S1− S0) + {θλ 10_(ξ 1− b1+ λ10 θS1σ12 )S1− θ b1+ λ10 θS1σ12 (S2− S1)}I_{S 1>b1+λ10 θξ1σ21 } +{−θ b1 θS1σ21 (S2− S1)}I_{S1< b1 θξ1σ21 }+ {−θξ1(S2− S1)}I{ b1 θξ1σ21 ≤S1≤b1+λ10 θξ1σ21 })|G0].

A similar argument as in case (1), we have

E[−1 θ exp(−θξ1(S1− S0) + {θλ 10 ξ1S1− (b1+ λ10)2 σ2 1 }I {S1>b1+λ10 θξ1σ21 } −b 2 1 σ2 1 I_{S 1< b1 θξ1σ21 }− θξ1S1b1I{ b1 θξ1σ12 ≤S1≤b1+λ10 θξ1σ21 })E[exp({− b1+ λ10 σ1 I {S1>b1+λ10 θξ1σ21 } −b1 σ1 I_{S 1< b1 θξ1σ12 }− θξ1S1σ1I{ b1 θξ1σ12 ≤S1≤b1+λ10 θξ1σ12 }}(B2− B1))|G1]|G0].

Since (B2− B1) is independent of G1 and the Gaussian assumption we get E[−1 θexp({−θξ1(S1− S0) + θλ 10_ξ 1S1− (b1+ λ10)2 2σ2 1 }I {S1>b1+λ10 θξ1σ21 } +{−θξ1(S1− S0) − b2 1 2σ2 1 }I_{S 1< b1 θξ1σ21 } +{−θξ1(S1− S0) − θξ1S1b1 + 1 2θ 2 ξ₁2S₁2σ₁2}I { b1 θξ1σ21 ≤S1≤b1+λ10 θξ1σ21 })] = −1 θ{ Z ∞ b1+λ10 θξ1σ21 exp(−θξ1(x − S0) + θλ10ξ1x − (b1+ λ10)2 2σ2 1 )f (x)dx + Z b1 θξ1σ21 −∞ exp(−θξ1(x − S0) − b2 1 2σ2 1 )f (x)dx + Z b1+λ 10 θξ1σ12 b1 θξ1σ21 exp(−θξ1(x − S0) − θξ1xb1+ 1 2θ 2_ξ2 1x 2_σ2 1)f (x)dx},

where f (x) is the probability density function of S1, as in (2.10).

Due to the first order condition with respect to ξ1, we have

Z ∞ b1+λ10 θξ1σ12 (θλ10x − θ(x − S0)) exp(θλ10ξ1x − (b1+ λ10)2 2σ2₁ − θξ1(x − S0))f (x)dx (2.12) + Z b1 θξ1σ12 −∞ (−θ(x − S0)) exp(− b2₁ 2σ2 1 − θξ1(x − S0))f (x)dx + Z b1+λ 10 θξ1σ12 b1 θξ1σ12 (−θb1x + θ2x2σ12ξ1− θ(x − S0)) exp(−θξ1b1x + 1 2θ 2_ξ2 1x 2_σ2 1− θξ1(x − S0))f (x)dx = 0.

Conclusion : By the Backward Induction, we observe that when the investors sell stocks at time 0 (i.e., ξ1 < ξ0), the optimal trading strategy ξ1∗ (sell) at time 0

satisfies the equation (2.11); when the investors buy stocks at time 0 (i.e., ξ1 > ξ0),

the optimal trading strategy ξ₁∗ (buy) at time 0 satisfies the equation (2.12). And, the optimal ξ₂∗ at time 1 is chosen as in (2.9)

Remark 15. In Example 14, it is difficult to find the closed form for ξ1 in

equation (2.11) and (2.12). However, we can find some property of the equation (2.11) and (2.12). For example, considering the equation (2.12), if we let

F (ξ1) = Z ∞ b1+λ10 θξ1σ12 (θλ10x − θ(x − S0)) exp(θλ10ξ1x − (b1+ λ10)2 2σ2 1 − θξ1(x − S0))f (x)dx + Z b1 θξ1σ12 −∞ (−θ(x − S0)) exp(− b2₁ 2σ2 1 − θξ1(x − S0))f (x)dx + Z b1+λ 10 θξ1σ12 b1 θξ1σ12 (−θb1x + θ2x2σ12ξ1− θ(x − S0)) exp(−θξ1b1x + 1 2θ 2_ξ2 1x 2_σ2 1− θξ1(x − S0))f (x)dx, then F0(ξ1) = Z ∞ b1+λ10 θξ1σ21 (θλ10x − θ(x − S0))2exp(θλ10ξ1x − (b1+ λ10)2 2σ2 1 − θξ1(x − S0))f (x)dx + Z b1 θξ1σ21 −∞ (−θ(x − S0))2exp(− b2 1 2σ2 1 − θξ1(x − S0))f (x)dx + Z b1+λ 10 θξ1σ21 b1 θξ1σ21 (−θb1x + θ2x2σ12ξ1− θ(x − S0))2 exp(−θξ1b1x + 1 2θ 2_ξ2 1x 2_σ2 1 − θξ1(x − S0))f (x)dx.

So, F0(ξ1) ≥ 0 for all ξ1, which implies that either the equation (2.12) has a solution

or the optimal ξ₁∗ happens at the two end points. And a similar result for the equation (2.11). Then we can conclude that there is still a “no trading” interval in two period case.

2.5. Numerical Results

In this section, we give some numerical results for the equation (2.11), (2.12) and try to find the corresponding “no trading” intervals.

In Example 14, we consider a two period model and get a result that if we buy some stocks (ξ0 < ξ1), then the optimal strategy ξ1 must satisfy the equation (2.12)

and if we sell some stocks (ξ0 > ξ1), then the optimal strategy must satisfy the

equation (2.11).

Here we give some numerical results and observe how these parameters affects the “no trading” interval.

2.5.1. The relation between b0 and nontrading interval. First, we fix

S0 = 1, θ = 1, b1 = 1, λ10= 0.5, σ0 = 1, σ1 = 1, and discuss the relation between b0

and the “no trading” interval from (2.11), (2.12).

b0 1 2 3 4 5 6 7

ξ1 (buy) 0.66 1.79 3.62 5.36 6.99 8.53 10.02

ξ1 (sell) 1.36 2.88 4.43 5.93 7.39 8.82 10.22

nontrading interval 0.70 1.09 0.81 0.57 0.40 0.29 0.20

From the table of the relation between b0 and ξ1, we observe that ξ1 (buy) and

ξ1 (sell) are increasing when b0 becomes bigger. And from the Figure 2.1, the size

of the “no trading” interval may reach a maximum when b0 ∈ [1, 3].

In practice, the proportion of the transaction cost is usually 0.003, the appreci-ation return of the stock b and the volatility of the stock σ have values between 0.5 and 1.5. So, we give another data here.

We fix S0 = 1, θ = 1, b1 = 1, λ10 = 0.003, σ0 = 0.5, σ1 = 0.5, and discuss the

Figure 2.1. The relation between b0 and ξ1

b0 0.5 0.6 0.7 0.8 0.9 1.0 1.1

ξ1 (buy) 1.99 2.39 2.79 3.19 3.59 3.99 4.39

ξ1 (sell) 2.01 2.41 2.80 3.20 3.60 4.00 4.41

nontrading interval 0.02 0.02 0.01 0.01 0.01 0.01 0.02

From the table of the relation between b0 and ξ1, we observe that ξ1 (buy) and

ξ1 (sell) are increasing when b0 becomes bigger. From the Figure 2.2, the size of “no

Figure 2.2. The relation between b0 and ξ1

2.5.2. The relation between b1 and nontrading interval. Second, we fix

S0 = 1, θ = 1, b0 = 1, λ10= 0.5, σ0 = 1, σ1 = 1, and discuss the relation between b1

and the “no trading” interval.

b1 1 2 3 4 5 6 7

ξ1 (buy) 0.66 0.89 0.98 0.99 0.99 0.99 0.99

ξ1 (sell) 1.36 1.40 1.45 1.48 1.49 1.50 1.50

Figure 2.3. The relation between b1 and ξ1

From the table of the relation between b1 and ξ1, we observe that ξ1 (buy) and

ξ1 (sell) increase slowly when b1 becomes bigger. And from the Figure 2.3, the size

of the “no trading” interval has a very small change even b1 gets bigger.

In practice, the proportion of the transaction cost is usually 0.003, the appreci-ation return of the stock b and the volatility of the stock σ have values between 0.5 and 1.5. So, we give another data here.

We fix S0 = 1, θ = 1, b0 = 0.5, λ10 = 0.003, σ0 = 0.5, σ1 = 0.5, and discuss the

Figure 2.4. The relation between b1 and ξ1

b1 0.5 0.6 0.7 0.8 0.9 1.0 1.1

ξ1 (buy) 1.99 1.99 1.99 1.99 1.99 1.99 1.99

ξ1 (sell) 2.00 2.00 2.00 2.01 2.01 2.01 2.01

nontrading interval 0.01 0.01 0.01 0.02 0.02 0.02 0.02

From the table of the relation between b1 and ξ1, we observe that ξ1 (buy) and

ξ1 (sell) increase slowly when b1 becomes bigger. From the Figure 2.4, the size of

2.5.3. The relation between σ0 and nontrading interval. Third, we fix

S0 = 0.05, θ = 1, b1 = 1, λ10 = 0.5, b0 = 1, σ1 = 1, and discuss the relation between

σ0 and the “no trading” interval.

σ0 1 2 3 4 5 6 7

ξ1 (buy) 13.39 4.72 2.21 1.25 0.79 0.55 0.40

ξ1 (sell) 27.31 6.86 3.24 1.86 1.19 0.83 0.61

nontrading interval 13.92 2.14 1.03 0.61 0.40 0.28 0.21

From the table of the relation between σ0 and ξ1, we observe that ξ1 (buy) and

ξ1 (sell) decrease very quickly when σ0 becomes bigger. And from the Figure 2.5,

the size of the “no trading” interval decrease very rapidly when σ0 gets bigger.

In practice, the proportion of the transaction cost is usually 0.003, the appreci-ation return of the stock b and the volatility of the stock σ have values between 0.5 and 1.5. So, we give another data here.

We fix S0 = 1, θ = 1, b1 = 0.5, λ10 = 0.003, b0 = 0.5, σ1 = 0.5, and discuss the

relation between σ0 and the “no trading” interval.

σ0 0.5 0.6 0.7 0.8 0.9 1.0 1.1

ξ1 (buy) 1.994 1.386 1.019 0.780 0.617 0.499 0.413

ξ1 (sell) 2.006 1.394 1.025 0.785 0.620 0.502 0.415

nontrading interval 0.012 0.008 0.006 0.005 0.003 0.003 0.002

From the table of the relation between σ0 and ξ1, we observe that ξ1 (buy) and

ξ1 (sell) decrease very quickly when σ0 becomes bigger. From the Figure 2.6, the

CHAPTER 3

Optimal Strategy with Transaction Cost in a Partial

Information Discrete Model

In this chapter, we relax the restriction on the drift term bn and we use the

method introduced by Huang (2007).

Let (Ω, F , P) be a complete probability space. 3.1. Model Setup

Let (Sn)n≥0 be the stock prices in the market, where S0 ∈ R+is given. The stock

price follows the recursive relation

(3.1) Sn+1− Sn = Sn[bn+ σn(Bn+1− Bn)],

where bn is the appreciation return of the stock, σn is volatility of the stock, and

(Bn) is a noise.

Assumption 16.

(1) σn is assumed to be deterministic for all n and (bn)n≥0 is assumed as a sequence

of random variables.

(2) (Bn+1− Bn) is a Gaussian process with mean 0 and variance 1, and they are

(totally) independent for all n ≥ 0.

(3) The processes (Bn+1− Bn) and (bn) are independent.

At time n, we observe the stock prices up to time n, so we let (Gn)n≥0 be the

Consider the stock return

Xn+1 =

Sn+1− Sn

Sn

, then the model will be rewritten by

Xn+1 = bn+ σn(Bn+1− Bn).

Also recall that we have G_n∗ = Gn as in Chapter 2, for Gn∗ is the natural filtration

generated by X0, X1, · · · , Xn.

3.2. Optional Projection and The Gaussian Case

Denote by ˆb the optional projection, with respect to the filtration Gn, of the

process bn and we have

ˆ_b

n = E[bn|Gn].

Consider the Gn-measurable process

(3.2) Lˆn= Sn− S0− n−1 X k=0 ˆ_b kSk.

From some computation we have ( ˆLn) is a martingale with respect to (Gn).

Proposition 17 (Huang (2007), Proposition 28). Let D2_n= σ2_n+ E[bn− ˆbn]2

and assume that Dn is bounded away from 0 for n ≥ 0. There exists a martingale

process ( ˆBn)n≥0 with respect to the filtration (Gn) such that

(3.3) Lˆn= n−1

X

k=0

DkSk( ˆBk+1− ˆBk),

From (3.1),(3.2) and (3.3), our model is described by Sn+1− Sn= Sn[bn+ σn(Bn+1− Bn)]

= Sn[ˆbn+ Dn( ˆBn+1− ˆBn)].

Thus,

Dn( ˆBn+1− ˆBn) = (bn− ˆbn) + σn(Bn+1− Bn).

Assumption 18. We assume that bn is a Gaussian process and {bn} is

indepen-dent of {Bn+1− Bn}. Recall Xn+1 = Sn+1− Sn Sn = bn+ σn(Bn+1− Bn)

and Gn= Gn∗ for all n ≥ 0. Let

Ln= LX,n = {c0X0+ c1X1+ · · · + cnXn where cj ∈ R for 0 ≤ j ≤ n}.

Then we have the following two Lemma.

Lemma 19 (Huang (2007), Lemma 30). Let ˇb = E[bn|Ln]. Then there exists a

sequence of coefficients ˇcj for 0 ≤ j ≤ n such that

ˇ_b

n = ˇc0X0+ ˇc1X1+ · · · + ˇcnXn.

Lemma 20 (Huang (2007), Lemma 31). ˆbn = E[bn|Gn] = E[bn|Ln].

Recall

Dn( ˆBn+1− ˆBn) = (bn− ˆbn) + σn(Bn+1− Bn),

Proposition 21 (Huang (2007), Proposition 32). ( ˆBn+1− ˆBn) is a Gaussian

process with mean 0 and variance 1 and they are independent with each other for n ≥ 0.

From the Huang’s work above, we can rewrite our model as follows: Sn+1− Sn = Sn[ˆbn+ Dn( ˆBn+1− ˆBn)]

and recall our wealth process (after the trading)

Vn= ξn+1Sn+ ηn+1= Vn−1+ ξn(Sn− Sn−1) − λ10L 10

n ∀n ≥ 0

with the initial wealth x = ξ0S0+ η0 > 0. We will show the neutral and

risk-averse utility result. Our decision rule is also the backward induction. We first give an example for the Gaussian assumption with the utility

U (x) = x, and then give an example for the utility

U (x) = −1

θexp(−θx),

where θ > 0 is the absolute risk aversion and be a constant.

3.3. Examples for Risk-Neutral and Risk-Averse Utility Functions Example 22. If the terminal time is 1, by the maximal expected utility we want to compute

E[U (V0+ ξ1(S1− S0) − x)] = E[ξ1(S1− S0) − λ10L 10 0 ].

(1) If we buy some stocks at time 0 (ξ0 < ξ1) and ξ1 is G0-measurable, then E[ξ1(S1− S0) − λ10L 10 0 ] = E[ξ1(S1− S0)] = ξ1S0E[ˆb0+ D0( ˆB1− ˆB0)]. By Proposition 21, we get ξ1S0E[ˆb0+ D0( ˆB1− ˆB0)] = ξ1S0ˆb0. Hence, (3.4) E[U (V0+ ξ1(S1− S0) − x)] = ξ1S0ˆb0.

(2) If we sell some stocks at time 0 (ξ0 > ξ1), then

E[ξ1(S1− S0) − λ10L 10 0 ] = E[ξ1S0[ˆb0+ D0( ˆB1− ˆB0)] − λ10S0(ξ0− ξ1)] = −λ10S0(ξ0− ξ1) + ξ1S0ˆb0+ E[ξ1S0D0( ˆB1− ˆB0)]. By Proposition 21, we get E[ξ1(S1− S0) − λ10L 10 0 ] = −λ 10_S 0(ξ0 − ξ1) + ξ1S0ˆb0 Hence, (3.5) E[U (V0+ ξ1(S1− S0) − x)] = −λ10S0(ξ0− ξ1) + ξ1S0ˆb0 ≤ ξ1S0ˆb0,

since ξ0 > ξ1. From (3.4), (3.5), in the case without any restrictions on the trading

strategy, the investors trade according the sign of ˆb0. If ˆb0 ≥ 0, the investor should

buy the stocks as much as possible. If ˆb0+ λ10 < 0, the investors should make

no short-sell and no loan, the optimal strategy is given by ξ₁∗ =      x/S0, if ˆb0 ≥ 0, 0, if ˆb0+ λ10< 0 .

Example 23. If the terminal time is 1, by the maximal expected utility we have to compute E[−1 θ exp(−θ(V0+ ξ1(S1 − S0) − x))] =E[−1 θ exp(−θ(−λ 10_L10 0 + ξ1(S1− S0)))] =E[−1 θ exp(θλ 10 L10₀ − θξ1(S1− S0))] = −1 θE[exp(θλ 10_L10 0 − θξ1S0[ˆb0+ D0( ˆB1− ˆB0)])] = −1 θexp(−θξ1S0 ˆ_b 0)E[exp(θλ10L 10 0 − θξ1S0D0( ˆB1− ˆB0))].

(1) If we buy stocks at time 0, then −1 θexp(−θξ1S0 ˆ b0)E[exp(θλ10L 10 0 − θξ1S0D0( ˆB1− ˆB0))] = −1 θexp(−θξ1S0 ˆ b0)E[exp(−θξ1S0D0( ˆB1 − ˆB0))].

Since the Gaussian assumption, we get −1 θexp(−θξ1S0 ˆ_b 0)E[exp(−θξ1S0D0( ˆB1− ˆB0))] = − 1 θexp(−θξ1S0 ˆ_b 0+ 1 2θ 2_ξ2 1S02D20). Hence, E[−1 θexp(−θ(V0+ ξ1(S1− S0) − x))] = − 1 θ exp(−θξ1S0 ˆ_b 0+ 1 2θ 2_ξ2 1S 2 0D 2 0).

Thus from the fundamental calculus, we maximize the conditional expectation to get the optimal ξ1 and from x = ξ0S0+ η0 we have

(3.6) ξ1 = ˆ_b 0 θS0D20 > ξ0 = x − η0 S0 .

(2) If we sell stocks at time 0, then E[−1 θexp(−θ(V0 + ξ1(S1− S0) − x))] = −1 θexp(−θξ1S0 ˆ_b 0)E[exp(θλ10S0(ξ0− ξ1) − θξ1S0D0( ˆB1− ˆB0))].

Since the Gaussian assumption, we get

E[exp(θλ10S0(ξ0− ξ1) − θξ1S0D0( ˆB1− ˆB0))] = exp(θλ10S0(ξ0− ξ1) + 1 2θ 2_ξ2 1S 2 0D 2 0). Hence, E[−1 θexp(−θ(V0+ξ1(S1−S0)−x))] = − 1 θ exp(−θξ1S0 ˆ_b 0+θλ10S0(ξ0−ξ1)+ 1 2θ 2_ξ2 1S 2 0D 2 0).

Thus from the fundamental calculus, we maximize the conditional expectation to get the optimal ξ1 and from x = ξ0S0+ η0 we have

(3.7) ξ1 = ˆ_b 0+ λ10 θS0D20 < ξ0 = x − η0 S0 .

From (3.6), (3.7), we conclude that the investors will neither buy nor sell stocks when their initial wealth (amount by money) is in the interval

" ˆ b0 θD2 0 + η0, ˆ_{b0+ λ}10 θD2 0 + η0 # . Moreover, if one wants to buy stocks he may choose the strategy ξ1 =

ˆ_b

0

θS0D20

at time 0 to reach the maximum profit, and if one wants to sell stocks he may choose the strategy ξ1 =

ˆ

b0+ λ10

θS0D02

at time 0 to reach the maximum profit. Remark 24. Now the “no trading” interval is

" ˆ_b 0 θD2 0 + η0, ˆ b0+ λ10 θD2 0 + η0 # .

And if one wants to buy (sell) stocks he may choose the strategy ξ1 =

ˆ b0 θS0D02 (ξ1 = ˆ_b 0 + λ10 θS0D20

(1) For fixed ˆb0 and D0. If the initial stock price S0 is too high, we will be

conservative for our strategy in the stock.

(2) For fixed D0 and S0, if ˆbn is positive and grows up, we invest the number

of shares of the stock more.

(3) For fixed ˆb0 and S0, if the volatility of the stock grows up, we are

conserva-tive when investing the stock.

(4) The length of the “no trading” interval is λ

10

θD2 0

, if λ10 _{increases or D}2 0

de-creases, the “no trading” interval becomes larger.

Example 25. If the terminal time is 2, by backward induction we figure out this situation in two steps.

Step1 : Compute the one-period case

E[−1 θexp(−θ{V1+ ξ2(S2− S1) − V0− ξ1(S1− S0)})|G1] = E[−1 θexp(−θ{ξ2(S2− S1) − λ 10 L10₁ })|G1] = E[−1 θexp(−θ{ξ2S1[ˆb1 + D1( ˆB2− ˆB1)] − λ 10 L10₁ })|G1].

(1) If we sell stocks at time 1, then

E[−1 θexp(−θ{ξ2S1[ˆb1+ D1( ˆB2− ˆB1)] − λ 10_L10 1 })|G1] = E[−1 θexp(−θξ2S1[ˆb1+ D1( ˆB2− ˆB1)] + θλ 10_(ξ 1− ξ2)S1)|G1].

Since ( ˆB2− ˆB1) is independent of G1 and the Gaussian assumption we have

E[−1 θ exp(−θξ2S1[ˆb1+ D1( ˆB2− ˆB1)] + θλ 10_(ξ 1− ξ2)S1)|G1] = −1 θ exp(−θξ2S1 ˆ_{b1+ θλ}10_(ξ 1− ξ2)S1+ 1 2θ 2_ξ2 2S 2 1D 2 1).

Hence, E[−1 θexp(−θ{V1+ ξ2(S2− S1) − V0− ξ1(S1− S0)})|G1] = −1 θexp(−θξ2S1 ˆ b1+ θλ10(ξ1− ξ2)S1+ 1 2θ 2_ξ2 2S 2 1D 2 1).

Thus from the fundamental calculus, we maximize the conditional expec-tation to get the optimal ξ2 and from the assumption ξ2 < ξ1 we have

ξ2 =

ˆ_{b1+ λ}10

θS1D12

< ξ1.

(2) If we buy stocks at time 1, then using the similar argument in Example 1 we have ξ2 = ˆ b1 θS1D12 > ξ1. Step2 : Choose ξ₂ = ˆb1+ λ 10 θS1D12 I {ξ1> ˆ b1+λ10 θS1D12 }+ ˆ_b1 θS1D12 I {ξ1< ˆ_b1 θS1D21 }+ ξ1I{ ˆb1 θS1D21 ≤ξ1≤ ˆ_b1+λ10 θS1D21 }.

We replace the ξ2 by ξ2 and compute

E[−1 θ exp(−θ{V1+ ξ2(S2− S1) − x})|G0] = E[−1 θ exp(−θ{V0+ ξ1(S1− S0) − λ 10_L10 1 + ξ2(S2− S1) − x})|G0] = E[−1 θ exp(−θ{−λ 10_L10 0 − λ 10_L10 1 + ξ1(S1− S0) + ξ2(S2− S1)})|G0].

(1) If we sell stocks at time 0, i.e., L10₀ = (ξ0− ξ1)S0, then we have

E[−1 θ exp(−θ{−λ 10_L10 0 − λ 10_L10 1 + ξ1(S1− S0) + ξ2(S2− S1)})|G0] = E[−1 θ exp(−θ{−λ 10 (ξ0− ξ1)S0− λ10L 10 1 (I_{ξ 1> ˆ b1+λ10 θS1D21 }+ I{ξ1< ˆ b1 θS1D21 }+ I{ˆb1+λ10 θS1D21 ≥ξ1≥ ˆ b1 θS1D21 }) +ξ1(S1− S0) + ξ2(S2− S1)})|G0].

By some calculation, we get E[−1 θexp(θλ 10 (ξ0− ξ1)S0− θξ1(S1 − S0) + {θλ10(ξ1− ˆ b1+ λ10 θS1D12 )S1 −θˆb1+ λ 10 θS1D12 (S2− S1)}I_{S 1> ˆ_b1+λ10 θξ1D21 }+ {−θ ˆ b1 θS1D12 (S2− S1)}I_{S 1< ˆ b1 θξ1D21 } +{−θξ1(S2− S1)}I_{ ˆ_b1 θξ1D12 ≤S1≤ ˆ b1+λ10 θξ1D21 })|G0] = E[−1 θexp(θλ 10_(ξ 0− ξ1)S0− θξ1(S1 − S0) +{θλ10ξ1S1− (ˆb1+ λ10)2 D2 1 −ˆb1+ λ 10 D1 ( ˆB2− ˆB1)}I_{S 1> ˆ_b1+λ10 θξ1D12 } +{− ˆ_b2 1 D2 1 − ˆb1 D1 ( ˆB2− ˆB1)}I_{S 1< ˆ_b1 θξ1D21 } +{−θξ1S1ˆb1− θξ1S1D1( ˆB2− ˆB1)}I_{ ˆ_b1 θξ1D21 ≤S1≤ ˆ_b1+λ10 θξ1D12 })|G0]

Using the conditional expectation property, we have

E[−1 θexp(θλ 10_(ξ 0− ξ1)S0− θξ1(S1− S0) + {θλ10ξ1S1− (ˆb1 + λ10)2 D2 1 }I {S1> ˆ b1+λ10 θξ1D21 } −ˆb 2 1 D2 1 I {S1< ˆ b1 θξ1D12 }− θξ1S1 ˆ_b1I { ˆb1 θξ1D12 ≤S1≤ ˆ b1+λ10 θξ1D21 })E[exp({− ˆ_b 1 + λ10 D1 I {S1> ˆ b1+λ10 θξ1D21 } −ˆb1 D1 I {S1< ˆ_b1 θξ1D21 }− θξ1S1D1I{ ˆb1 θξ1D21 ≤S1≤ ˆ_b1+λ10 θξ1D12 }}( ˆB2− ˆB1))|G1]|G0].

Since ( ˆB2− ˆB1) is independent of G1 and the Gaussian assumption we get E[−1 θ exp({θλ 10 (ξ0− ξ1)S0− θξ1(S1− S0) + θλ10ξ1S1− (ˆb1+ λ10)2 2D2 1 }I {S1> ˆ b1+λ10 θξ1D21 } +{θλ10(ξ0− ξ1)S0− θξ1(S1− S0) − ˆ b2₁ 2D2 1 }I {S1< ˆ b1 θξ1D12 } +{θλ10(ξ0− ξ1)S0− θξ1(S1− S0) − θξ1S1ˆb1+ 1 2θ 2 ξ2₁S₁2D₁2}I { ˆb1 θξ1D12 ≤S1≤ ˆ b1+λ10 θξ1D21 })] = −1 θ{ Z ∞ ˆ_b1+λ10 θξ1D21 exp(θλ10(ξ0 − ξ1)S0− θξ1(x − S0) + θλ10ξ1x − (ˆb1+ λ10)2 2D2 1 )f (x)dx + Z ˆ_b1 θξ1D21 −∞ exp(θλ10(ξ0 − ξ1)S0− θξ1(x − S0) − ˆ_b2 1 2D2 1 )f (x)dx + Z ˆ b1+λ10 θξ1D21 ˆ_b1 θξ1D21 exp(θλ10(ξ0− ξ1)S0− θξ1(x − S0) − θξ1ˆb1x + 1 2θ 2_ξ2 1x2D12)f (x)dx},

where f (x) is the probability density function of S1, i.e.,

(3.8) f (x) = √ 1 2πS0D0 exp(−(x − S0− S0 ˆ_b0)2 2S2 0D02 ).

Due to the first order condition with respect to ξ1, we have

Z ∞ ˆ b1+λ10 θξ1D21 (−θλ10S0 + θλ10x − θ(x − S0)) exp(θλ10(ξ0− ξ1)S0+ θλ10ξ1x (3.9) −(ˆb1+ λ 10₎2 2D2 1 − θξ1(x − S0))f (x)dx + Z ˆ_b1 θξ1D21 −∞ (−θλ10S0− θ(x − S0)) exp(− ˆ_b2 1 2D2 1 + θλ10(ξ0− ξ1)S0− θξ1(x − S0))f (x)dx + Z ˆ b1+λ10 θξ1D21 ˆ b1 θξ1D21 (−θˆb1x + θ2x2D21ξ1− θλ10S0− θ(x − S0)) exp(−θξ1ˆb1x + 1 2θ 2_ξ2 1x 2_D2 1 + θλ10(ξ0− ξ1)S0 − θξ1(x − S0))f (x)dx = 0,

where f (x) is given in (3.8).

(2) If we buy stocks at time 0, i.e., L10₀ = 0, then we have

E[−1 θ exp(−θ{V1+ ξ2(S2 − S1) − x})|G0] = E[−1 θ exp(−θ{−λ 10 L10₁ (I {ξ1> ˆ_b1+λ10 θS1D21 }+ I{ξ1< ˆ b1 θS1D21 }+ I{ˆb1+λ10 θS1D12 ≥ξ1≥ ˆ b1 θS1D12 }) +ξ1(S1− S0) + ξ2(S2 − S1)})|G0].

By some calculation, we get

E[−1 θexp(−θξ1(S1− S0) + {θλ 10_(ξ 1− ˆ_b 1+ λ10 θS1D21 )S1− θ ˆ_b 1+ λ10 θS1D12 (S2− S1)}I_{S 1> ˆ b1+λ10 θξ1D21 } +{−θ ˆ_b1 θS1D12 (S2− S1)}I_{S 1< ˆ b1 θξ1D21 }+ {−θξ1(S2− S1)}I{ ˆb1 θξ1D21 ≤S1≤ ˆ b1+λ10 θξ1D12 })|G0].

Using the conditional expectation property, we have

E[−1 θ exp(−θξ1(S1− S0) + {θλ 10_ξ 1S1− (ˆb1+ λ10)2 D2 1 }I {S1> ˆ b1+λ10 θξ1D12 } −ˆb 2 1 D2 1 I {S1< ˆ b1 θξ1D21 } − θξ1S1 ˆ_b1I { ˆb1 θξ1D21 ≤S1≤ ˆ b1+λ10 θξ1D21 })E[exp({− ˆ_b 1+ λ10 D1 I {S1> ˆ b1+λ10 θξ1D21 } −ˆb1 D1 I {S1< ˆ b1 θξ1D21 }− θξ1S1D1I{ ˆb1 θξ1D21 ≤S1≤ ˆ_b1+λ10 θξ1D12 }}( ˆB2 − ˆB1))|G1]|G0].

Since ( ˆB2− ˆB1) is independent of G1 and the Gaussian assumption we get E[−1 θ exp({−θξ1(S1− S0) + θλ 10_ξ 1S1− (ˆb1+ λ10)2 2D2 1 }I {S1> ˆ b1+λ10 θξ1D12 } +{−θξ1(S1− S0) − ˆ b2 1 2D2 1 }I {S1< ˆ b1 θξ1D12 } +{−θξ1(S1− S0) − θξ1S1ˆb1+ 1 2θ 2_ξ2 1S12D21}I_{ ˆ_b1 θξ1D12 ≤S1≤ ˆ b1+λ10 θξ1D21 })] = −1 θ{ Z ∞ ˆ b1+λ10 θξ1D12 exp(−θξ1(x − S0) + θλ10ξ1x − (ˆb1+ λ10)2 2D2 1 )f (x)dx + Z ˆ b1 θξ1D21 −∞ exp(−θξ1(x − S0) − ˆ_b2 1 2D2 1 )f (x)dx + Z ˆ_b1+λ10 θξ1D12 ˆ b1 θξ1D21 exp(−θξ1(x − S0) − θξ1ˆb1x + 1 2θ 2_ξ2 1x 2_D2 1)f (x)dx},

where f (x) is the probability density function of S1, as in (3.8).

Due to the first order condition with respect to ξ1, we have

Z ∞ ˆ b1+λ10 θξ1D21 (θλ10x − θ(x − S0)) exp(θλ10ξ1x − (ˆb1+ λ10)2 2D2 1 − θξ1(x − S0))f (x)dx (3.10) + Z ˆ b1 θξ1D12 −∞ (−θ(x − S0)) exp(− ˆ_b2 1 2D2 1 − θξ1(x − S0))f (x)dx + Z ˆ b1+λ10 θξ1D21 ˆ b1 θξ1D12 (−θˆb1x + θ2x2D21ξ1− θ(x − S0)) exp(−θξ1ˆb1x + 1 2θ 2 ξ₁2x2D2₁− θξ1(x − S0))f (x)dx = 0, where f (x) is given in (3.8).

CHAPTER 4

Future Works

In above chapters, we consider the discrete model with the transaction costs, and we find that “no trading” intervals and the optimal strategies corresponding to selling assets and buying assets respectively. But many results are still open. We will illustrate as follows :

(a) First, if the investor is a “large” investor, i.e., the amount of the investment would influence the stock price, how can the investors reach their optimal strategies?

(b) Second, in the discrete model, if the drift term bn is random and bn is

“not” independent of the noise term, can we find the “no trading” intervals and the optimal strategies corresponding to selling assets and buying assets respectively?

(c) In our model, the stock price is driven by the recursive relation Sn+1− Sn = Sn[bn+ σn(Bn+1− Bn)],

so the stock price may be negative. However, it will become confused if we do not assume that Sn is positive for all n. Because we will have the

conclusion that we sell the stocks at time 0 when b0 > θσ02(x − η0) − λ10

and buy the stocks at time 0 when b0 < θσ02(x − η0). So if we introduce

another stock price model in which the stock price Sn is positive for all n,

what strategy will be suggested? Simply, we can give the model setup as follows :

Let (Ω, F , P) be a complete probability space. Let (Sn)n≥0 be the stock

prices in the market and is driven by Sn+1 = inSn,

where in is a nonnegative random variable and un := ln in is normally

distributed, and S0 ∈ R+ is given. Hence similar problems will be discussed

as follows :

(1) First, if our market is frictionless (no transaction cost), then for a given risk-neutral or risk-averse utility, what strategy will be find under the backward induction in this model?

(2) Second, if there is transaction cost in this model, What is the result? Example 26. Here we give a one period model result for (1). If the terminal time is 1 and the utility function is U (x) = ln x, by backward induction we have to compute

E[U (V0+ ξ1(S1− S0))|G0] =E[ln{x + ξ1(S1− S0)}|G0]

=E[ln{x + ξ1S0(exp(u0) − 1)}|G0].

Since G0 = {φ, Ω} and assume that u0 is normally distributed with mean 0

variance 1, then we get

E[ln{x + ξ1S0(exp(u0) − 1)}|G0] =√1 2π Z ∞ −∞ ln{x + ξ1S0(exp(y) − 1)} exp(− y2 2)dy. Due to the first order condition with respect to ξ1, we have

Z ∞ −∞ S0(exp(y) − 1) x + ξ1S0(exp(y) − 1) exp(−y 2 2)dy = 0

By the change of variable, we get Z ∞ 0 S0(exp(y) − 1) x exp(y) − ξ1S0(exp(y) − 1) exp(−y 2 2 )dy = Z ∞ 0 S0(exp(y) − 1) x + ξ1S0(exp(y) − 1) exp(−y 2 2 )dy. Then we choose the optimal ξ1 =

x 2S0

.

This example gives a result for one period case in the frictionless market model. But how about the two period case or even for any finite terminal time N ? And how is the result for the same problem in the model with transaction cost?

(d) If there are more than one risky assets in our model, what is the optimal trading strategy?

(e) Finally, extending our discrete model to continuous time model and solve the similar problems by using other mathematical tools.

Bibliography

[1] B. Bouchard, B and H. Pham. Wealth-path dependent utility maximization in incomplete

markets. Finance and Stochastics. Vol. 8, Iss. 4, 579-603, 2004.

[2] M. H. A. Davis and A. R. Norman. Portfolio Selection with Transaction Costs. Math. Oper.

Res. 15. 676-713, 1990.

[3] F. Delbaen, Y. M. Kabanov and E. Valkeila. Hedging under Transaction Costs in Currency

Markets : A Discrete-Time Model. Mathematical Finance. Vol. 12, No. 1, 45-61, 2002.

[4] H. F¨ollmer and D. Sondermann. Hedging of Non-Redundant Contingent Claims. Contributions

to Mathematical Economics. 205-223, 1986.

[5] D. Fudenberg and J. Tirole. Game theory, First Edition. MIT Press. 1991.

[6] C.-C. Huang. Optimal Hedging Strategy with Partial Information in Discrete Financial Model.

Master Thesis in National University of Kaohsiung, 2007.

[7] I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus, Second Edition.

Springer. 1988.

[8] I. Karatzas and S. E. Shreve. Methods of Mathematical Finance, First Edition. Springer. 1998.

[9] I. Karatzas and X.-X. Xue. A note on utility maximization under partial observations.

Math-ematical Finance 1. 57-70, 1991.

[10] D. Lamberton and B. Lapeyre. Introduction to Stochastic Calculus Applied to Finance, First

Edition. Springer. 1996.

[11] P. Lakner. Utility maximization with partial information. Stochastic Processes and their

Ap-plications 56. 247-273, 1995.

[12] P. Lakner. Optimal trading strategy for an investor : the case of partial information. Stochastic

Processes and their Applications 76. 77-97, 1998.

[13] M. J. P. Magill and G. M. Constantinides. Portfolio Selection with Transaction Costs. J.

[14] R. C. Merton. Optimum Consumption and Portfolio Rules in a Continuous-Time Todel. J.

Econ. Theory 3. 373-413, 1971.

[15] B. Øksendal. Stochastic Differential Equations : An introduction with Applications, Sixth

Edition. Springer. 2005.

[16] J. Xiong and Z. Yang. Optimal investment strategy under saving/borrowing rates spread with