在期望效用理論與累積前景理論下最佳投資策略的比較

國

立

交

通

大

學

應用數學系

碩

士

論

文

在期望效用理論與累積前景理論下最佳投資策略的比較

Comparison of Optimization in the Sense of Expected Utility

Theory and Cumulative Prospect Theory

研 究 生：謝孟蓁

指導教授：吳慶堂 副教授

在期望效用理論與累積前景理論下最佳投資策略的比較

Comparison of Optimization in the Sense of Expected Utility

Theory and Cumulative Prospect Theory

研 究 生：謝孟蓁 Student：Meng-Chen Hsieh

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

A Thesis

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 2009

Hsinchu, Taiwan, Republic of China

i

在期望效用理論與累積前景理論下最佳投資策

略的比較

學生：謝孟蓁 指導教授：吳慶堂

國立交通大學應用數學系碩士班

摘 要

本論文的重點為比較在期望效用理論與累積前景理論下最佳投資策略的差異。而且

分成兩個部分來探討─不必考慮手續費的財務市場以及在賣股票後需繳交固定比率手

續費的財務市場。我們給一些在一個交易時間的財務模型中關於最佳投資策略以及最佳

避險策略結果，並且討論在這兩套不同理論下其結果的差異。

Comparison of Optimization in the Sense

of Expected Utility Theory

and Cumulative Prospect Theory

Student：Meng-Chen Hsieh Advisors : Dr. Ching-Tang Wu

Department of Applied Mathematics

National Chiao Tung University

ABSTRACT

In this thesis, our main point is that compare the optimal strategy in

the sense of expected utility theory and cumulative prospect theory.

Moreover, we suppose that there exist two kind of market model: A

model without transaction cost and a model with transaction cost. We

give some results about optimal trading strategy and optimal hedging

strategy in one period market model. And we discuss the difference of the

results in these two senses.

誌

謝

此篇論文能夠完成，首先，我最想感謝的是我的指導教授也就是我的恩師 ─吳慶堂教授。剛從台東大學來到交大這個新學校來唸研究所的我，對於研究所 的課業方向一無所知，在跟許多老師聊過之後，我決定懇求吳慶堂教授能夠指導 我；於是，我開始了我的研究生活。由於在財務數學這個領域有許多的先修知識 須要瞭解，才能夠繼續下去，因此，老師常不辭辛勞的陪伴我們 meeting，一一 在學識不足的地方給予我指導，讓我在財務數學的領域中有更多的認識。另外我 還想感謝陳育慈學姊，提供我許多在財經實務方面的知識，讓我對於金融市場的 現況有更多的了解；以及學姐在理論數學的豐厚知識，也常常在我遇到瓶頸的時 候能給予我方向，提供我許多想法。最後，我還想感謝胡仲軒同學，他總是在我 的身邊陪伴我，照顧我，讓我無後顧之憂的全心投入研究，不管是在學業或是在 生活上我都受益良多。 在口試期間，我要感謝韓傳祥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，讓我的論文能夠更加的完整，學生永銘在心。 最後我要感謝我的家人，謝謝他們陪伴著我，提供我非常多的支持和鼓勵， 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閒娛樂，讓我在 課後閒暇之餘能夠有紓解壓力的管道。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。 iii

iv

目

錄

中文提要

i

英文提要

ii

誌謝

iii

目錄

iv

一、

Introduction

1

二、

Theoretical Framework

5

2.1

Preference Relation and Expected Utility Theory

5

2.2

Prospect Theory

7

2.3

Cumulative Prospect Theory

13

三、

Comparison of Optimization in the Sense of Expected

Utility Theory and Cumulative Prospect Theory I:

A Model Without Transaction Cost

17

3.1

Optimal Trading Strategy in One Period Model

17

3.2

Reference Point Effect

27

3.3

Optimal Hedging Strategy in One Period Model

34

四、

Comparison of Optimization in the Sense of Expected

Utility Theory and Cumulative Prospect Theory II:

A Model With Transaction Cost

43

五、

Conclusion

59

Bibliography

61

CHAPTER 1

Introduction

Expected utility theory has been extensively used for investors to make a deci-sion under risk for several decades; however it still has some questions. This theory can not describe the behavior of investor precisely, such as Allais (1953) paradox. Kahneman and Tversky (1979) proposed a new theory, called prospect theory, to explain the major violation of expected utility theory in making choices between risky investments with at most two different nonzero outcomes. The most impor-tant views of this theory are as following: Firstly, a value function is concave for gains, convex for losses, and steeper for losses than for gains. Secondly, a nonlinear transformation of the probabilities, which overweights small probabilities and un-derweights high probabilities. Therefore, we can explain the behavior of individual decision making more adequately in the sense of prospect theory.

In the later, several authors, e.g., Quiggin (1982), Schmeidler (1989), Yarri (1987), Weymark (1981), proposed a new opinion, called the rank-dependent or the cumulative functional, which introduced a decision weighting function. Decision weighting function is different from probability weighting function. It transforms objective probabilities into entire cumulative probabilities rather than individual probabilities. In some literatures applying rank-dependent can describe the behav-ior of investors more precisely.

There still has some questions under prospect theory. For example, it only

works with prospects which have at most two different outcomes, and stochastic dominance would not still hold. Kahneman and Tversky (1992) proposed a new version of prospect theory that adopted the rank-dependent, which was called cu-mulative prospect theory. Cucu-mulative prospect theory is more general than prospect

theory because it allowed any number of outcomes of risky prospects and can be extended to a continuous case. One of the key point of this theory is that there exist two different weighting functions, one for probabilities of gains and the other for probabilities of losses. The other key point is that it transformed the entire cu-mulative probability rather than individual probability of each outcome separately. Moreover, Wakker and Tversky (1993) claimed that stochastic dominance has still acceptable under cumulative prospect theory. Hence cumulative prospect theory has nowadays become one of the most famous version for investors to make a decision under uncertainty.

Qingwei Liu, Yi Li and Shouyang Wang (2008) studied the influence of the reference point on a hedger’s decision based upon prospect theory and had a result about that how the prior outcomes affect decision making. They claimed that a hedger who takes the prior losses into consideration will become a speculator in the futures market. In this thesis we think about the same question in the sense of cumulative prospect theory and want to extend some conclusions to general cases.

In this paper, we interest in comparing the optimal strategies in two senses, ex-pected utility theory and cumulative prospect theory. Furthermore, we also discuss the optimal hedging strategies in these two senses. And we add transaction costs to the financial market.

This thesis proceeds as follows. In chapter 2, we introduce the preference order and expected utility theory. And we give some problems to explain why the expected utility theory can not describe the behavior of investors to make decision. Then we introduce the prospect theory and the most important elements of this theory, such as the properties of value function and probability weighting function. Finally, we introduce cumulative prospect theory.

In chapter 3, we will consider the fundamental problem in the financial math-ematics to seek out the optimal trading strategy. We consider one period market

1. INTRODUCTION 3

model with one riskless asset (bond) and one risky asset (stock). We want to find out the differences of optimal strategies in the sense of expected utility theory and cumulative prospect theory. First, we are interested in talking about that in what situation an investor is willing to buy the risky asset? The result is under expected utility theory if the mean of gains is greater than the mean of losses, an investor is willing to buy the risky asset. However, under cumulative prospect theory the condition is relative to the degree of sensitivity in facing losses. Second, we discuss reference point effect. If we take the prior losses into account and try to make for the prior loss, our attitudes toward risk become risk seeking. Finally, we compare the optimal hedging strategies in these two senses. Conclusion is that in the sense of expected utility theory, full hedging is the optimal hedging strategy; however, under cumulative prospect theory, the optimal strategy is more complicated. Moreover if the prior losses are sufficiently large, an investor will take vary large positions showing no consideration of risk.

In chapter 4, we add transaction costs to the financial market. Practically, in financial market we need to pay transaction costs when we sell the stock. In this situation, we have a result that there exists a ”no trading” interval. In other words, we will neither buy nor sell the stock in this interval. Besides, if we consider the value function given by

v(x) =        1 − exp(−θx) x ≥ 0 −λ(1 − exp(θx)) x < 0 ,

then we can get that the length of ”no trading” interval in the sense of cumulative prospect theory is longer than the length of ”no trading” interval in the sense of expected utility theory. This inference implies that investors become more conser-vative under cumulative prospect theory than under expected utility theory.

CHAPTER 2

Theoretical Framework

2.1. Preference Relation and Expected Utility Theory

Let X be a collection of all investments in the financial market. Suppose there are two alternative investments, X and Y , in the financial market. An arbitrage investor may prefer X to Y , or prefer Y to X. That is, there is a preference relation when we make a decision between two alternative investments. Here we define a preference relation on X .

Definition 1. A preference relation on X is a binary relation  with the fol-lowing two properties:

(1) (asymmetry) If X is better than Y , then Y would not be better than X. That is, if X  Y , then Y  X.

(2) (negative transitivity) If X is better than Y , and there exists investment, Z, in the financial market such that Y is better than Z, then Z would not be better than X. That is, X  Y , Y  Z and Z  X can not hold at the same time.

We also introduce weak preference and indifference relation. The weak preference order, denoted by %, is defined by

X % Y ⇐⇒ _{Y  X,}

and the indifference relation, denoted by ∼ is given by

X ∼ Y ⇐⇒ _{X % Y} and _{Y % X.}

Next, we state several axioms of the preference relation.

(1) (Comparability) For all X and Y are in X , one of X  Y , Y  X or X ∼ Y must be hold.

(2) (Transitivity) For all X, Y and Z are in X , if X  Y and Y  Z then X  Z. In other words, if X is better than Y , and Y is better than Z, then X must be better than Z.

(3) (Independence) For all X, Y and Z are in X and for all α ∈ (0, 1), if X  Y then αX + (1 − α)Z  αY + (1 − α)Z must be true.

(4) (Continuity) For all X, Y and Z are in X , if X  Y and Y  Z then there are α and β in (0, 1) such that αX + (1 − α)Z  Y and Y  βX + (1 − β)Z. We can reject or accept above axioms. However, if all of these axioms are ac-cepted, there exists a von Neumann-Morgenstern representation.

Definition 2. _{(1) If there exists a function U : X → R such that}

(2.1) X  Y ⇐⇒ U (X) > U (Y ),

we call it a numerical representation of preference relation . Moreover such function U is called a utility function.

(2) Let M be the collection of all probability distributions on a space (S, s), and there is u : S → R. A von Neumann-Morgenstern representation is a numerical representation of a preference relation , satisfying

U (µ) = Z

S

u(x)µ(dx),

for all µ ∈ M. Moreover such u is called the von Neumann-Morgenstern utility function.

In addition, if the above function u satisfies the conditions, strictly increasing, strictly concave and continuous on S, (2.1) can be written as

2.2. PROSPECT THEORY 7

called expected utility representation. Moreover, the conditions of u, strictly concave and strictly increasing, means that the preference relation  has the properties, risk averse and monotone, respectively.

Therefore when there are two alternative investments X, Y in X , and there exists expected utility representation of a preference relation , we will choose X rather than Y if the expected utility of X is greater than the expected utility of Y . In other words, we conclude that

X  Y ⇐⇒ E[u(X)] > E[u(Y )]

and

X % Y ⇐⇒ E[u(X)] ≥ E[u(Y )].

This standard of making decision is called expected utility theory. Under expected utility theory our main goal is to find out X ∈ X such that E[u(X)] reaches the maximal value.

2.2. Prospect Theory

Expected utility theory has been used popularly for investors to make decision under risk; however it still has some serious shortcomings. For instance, it can not explain the behavior of investor correctly. Many investors would not obey the principle of the expected utility theory, usually. And there exist many counterexam-ples, the best famous, was called Allais’ parodox, which was introduced by French economist Maurice Allais (1953). The example is given as following:

Problem 1: Choose between

A :            2500 with probability 0.33 2400 with probability 0.66 0 with probability 0.01 B : 2400 with certainty

Problem 2: Choose between C :      2500 with probability 0.33 0 with probability 0.67 D :      2400 with probability 0.34 0 with probability 0.66

The statistic shows that 82 percent of subjects chose B in Problem 1, and 83 percent of subjects chose C in Problem 2. The main result of this example is that in problem 1, investment B is better than investment A, and in problem 2, investment C is better than investment D. The first preference implies

0.33u(2500) + 0.66u(2400) + 0.01u(0) < u(2400).

Without loss of generality, suppose that u(0) = 0 and rearranging above inequality we obtain

0.33u(2500) < 0.34u(2400).

However, the second preference implies

0.33u(2500) > 0.34u(2400).

Thus we get a contradiction.

In order to modify the expected utility theory, Kahneman and Tversky (1979) proposed the prospect theory. They found out that investor’s attitude toward risk of gains is different from that of losses. Therefore they used value function, which is concave for gains and convex for losses, to instead of utility function that is concave everywhere. The implication of this transformation was that investors are not always risk aversion, they are risk seeking of losses. Moreover, in prospect theory it transfers probability into decision weight, called probability weighting function.

2.2. PROSPECT THEORY 9

However, decision weights are not probabilities, since it does not keep the axioms of probability measure.

Under prospect theory we use value function v, and probability weighting func-tion w to replace utility funcfunc-tion and probability measure in expected utility theory, respectively. And then consider a prospect (X, P ) = (x1, p1; x2, p2; ...; xm, pm) which

means that yield outcome xi with probability pi, where p1+ p2+ · · · + pm = 1. To

simplify the notation, we omit null outcomes and use (x, p) to indicate (x, p; 0, 1−p); furthermore, the riskless prospect with outcome x is denoted by (x). Then the value of this prospect is defined by

V (X, P ) = w(p1)v(x1) + w(p2)v(x2) + · · · + w(pm)v(xm) = m

X

i=1

w(pi)v(xi),

where v(0) = 0, w(0) = 0, and w(1) = 1. More precisely, if we consider a prospect (X, P ) = (x−m, p−m; x−m+1, p−m+1; ...; x−1, p−1; x0, p0; x1, p1; ...; xn, pn), with p−m +

· · · + pn = 1, the value of this prospect is defined by

V = V++ V−, where

V+(X, P ) = w+(p0)v+(x0) + w+(p1)v+(x1) + · · · + w+(pn)v+(xn)

and

V−(X, P ) = w−(p−m)v−(x−m) + w−(p−m+1)v−(x−m+1) + · · · + w−(p−1)v−(x−1).

Definition 3. For all prospects (X, P ), if (X, P ) is indifference of (c) for an investor, then for all k ∈ R+, (kX, P ) is indifference to (kc) for an investor. we call an investor exhibits preference homogeneity.

The main point of prospect theory is that value function is concave for gains and convex for losses. In other words, investors are risk-averse and risk-seeking when facing gains and losses, respectively.

In the rest of this section we introduce the value function and probability weight-ing function of prospect theory more precisely.

2.2.1. Value Function. The most important feature of prospect theory is that investors evaluate the value of prospects depend on the change of wealth rather than final wealth. Therefore there are two main viewpoints of value function: takes the current wealth as a reference point, and the size of change from that reference point. In other words, we separate value function into two parts, gains and losses, which above the reference point and below the reference point, respectively.

For many investors the difference in value between a gain of 100 and a gain of 200 becomes more attractive than the difference between a gain of 1000 and a gain of 1100. Hence Kahneman and Tversky (1979) applied this principle to the evaluation of monetary changes. They proposed that the value function is concave above the reference point and convex below it. That is, the value function is concave for gains and convex for losses.

Moreover, the characteristic of attitudes of investors toward the change of wealth is that losses loom larger than gains. In other words, for many investors the degree of miserable in losing a sum of money is greater than the degree of happy in gaining the same amount of money. This is because that most investors are not interesting in symmetric bets with the form (x, 0.5; −x, 0.5). Thus Kahneman and Tversky (1979) proposed that the value function for losses is steeper than that for gains.

Definition 4. Value function v satisfies v(0) = 0, strictly increasing and if |v(−x)| > v(x) for x > 0, then v has the property, called loss aversion.

In summary, we point out the properties of value function that Kahneman and Tversky (1979) proposed. First, value function is defined on deviations from the reference point. Second, it is concave for gains and convex for losses. Third, value

2.2. PROSPECT THEORY 11

value function that satisfies above properties is a S-shaped value function which is steepest at the neighborhood of reference point. Furthermore, Kahneman and Tversky suggested that the form of value function is

v(x) =        xα x ≥ 0 −λ(−x)β _{x < 0} .

The estimations of α, β and λ are given by α = β = 0.88 and λ = 2.25. In addition, if value function takes the form as above, preference homogeneity must be true for all prospects (X, P ). However, for the purpose of computing easily, we often consider the other useful value function given by

v(x) =        1 − exp(−θx) x ≥ 0 −λ(1 − exp(θx)) x < 0 .

It is easy to check that such function satisfies all properties of value function.

2.2.2. Probability Weighting Function. In this subsection we introduce the other major opinion of prospect theory, called probability weighting function.

First, we introduce an example, based on Maurice Allais, that violate the ex-pected utility theory.

Problem 1: Choose between

A :      4000 with probability 0.8 0 with probability 0.2 B : 3000 with certainty

C :      4000 with probability 0.2 0 with probability 0.8 D :      3000 with probability 0.25 0 with probability 0.75

The data shows that 80 percent of subjects choose B in problem 1, and 65 percent of subjects choose C in problem 2. But in fact, we only reduce the probability by equal proportion. Hence we can obtain a result that reducing the probability from 1 to 0.25 makes greater influence than reducing the probability from 0.8 to 0.2. And we call this phenomenon common-ratio effect.

Thus Kahneman and Tversky (1979) explained the common-ratio effect by the method of a nonlinear transformation of probabilities, called ”probability weighting function”. Next, we illustrate the properties of probability weighting function as follows:

(1) First, the regressive property, explains the attitudes toward risk. For many investors are risk aversion for gains with large probability and for losses with small probability. In addition, they are risk seeking for gains with small probability and for losses with large probability. Moreover the transforma-tion of probabilities into probability weighting functransforma-tion is overweighting for small probabilities and underweighting for large probabilities.

(2) Second, changes in probabilities have greater influence on the boundary of probability interval. That is, increasing the probability from 0 to 0.1 have greater effect than increasing the probability form 0.5 to 0.6.

The probability weighting function satisfying above two properties is given by

w+(p) = p

γ

2.3. CUMULATIVE PROSPECT THEORY 13

and

w−(p) = p

δ

(pδ_{+ (1 − p)}δ₎1_δ,

where γ = 0.61 and δ = 0.69 for gains and losses, respectively.

Under prospect theory, probability weighting function for gains and for losses may be different. But if preference homogeneity and loss aversion both hold for all prospects, then the probability weighting function for gains and for losses are the same, i.e., w+ _{= w}−_.

Although prospect theory asserts that we replace utility function with value func-tion which is concave for gains and convex for losses and transform probability into probability weighting function which is nonlinear. There are still some drawbacks. For example, it only works with prospects that have at most two different nonzero outcomes, and stochastic dominance does not still hold. In order to modify the drawbacks of prospect theory Kahneman and Tversky (1992) proposed cumulative prospect theory.

2.3. Cumulative Prospect Theory

Though prospect theory explained the major violations of expected utility theory in decision making under risk, there still exist two problems. First, it does not always satisfies stochastic dominance. Second, it can not be extended to prospects with a large number of outcomes. In order to modify the drawback of prospect theory, Kahneman and Tversky (1992) proposed cumulative prospect theory. The most important element of this theory is that instead of transforming each probability separately, this model transforms the entire cumulative distribution function, called cumulative weighting function or weighting function for short. Further this theory applies the cumulative functional separately to gains and to losses.

We first rearrange the outcomes of each prospect in increasing order, such as

with x−m < x−m+1 < · · · < x−1 < x0 < x1 < · · · < xn, and p−m + · · · + pn = 1.

Without loss of generality, we suppose that x0 = 0 and take x0 as a reference

point. We interpret x−m, ..., x−1 as losses and x1, ..., xn as gains. Then cumulative

prospect theory asserts that there exist a strictly increasing value function, satisfying v(0) = 0, and probability weighting function w+(p) and w−(p) such that the value of this prospect is defined by

V (X, P ) = V+(X, P ) + V−(X, P ), where V+(X, P ) = π₀+v+(x0) + π1+v +_(x 1) + · · · + πn+v +_(x n) = n X i=0 π+_i v+(xi) and V−(X, P ) = π_−m− v−(x−m) + π−−m+1v − (x−m+1) + · · · + π−1− v − (x−1) = −1 X i=−m π−_i v−(xi).

The decision weights are defined by

π+_i =        w+_(p n) , i = n w+_(p i+ · · · + pn) − w+(pi+1+ · · · + pn) , 0 ≤ i ≤ n − 1 and π_i−=        w−(p−m) , i = −m w−(p−m+ · · · + pi) − w−(p−m+ · · · + pi−1) , 1 − m ≤ i ≤ −1 .

where w+ and w− are strictly increasing functions from the unit interval to itself satisfying w+_{(0) = w}−_{(0) = 0, and w}+_{(1) = w}−_{(1) = 1.}

Since we transform the entire cumulative distribution function rather than trans-form each probability separately, and consider the rank-dependent models, rear-ranging the outcomes in increasing order, cumulative prospect theory can extend the original version of prospect theory in several respects. First, it can work with prospects that have infinite outcomes and extend to continuous model. Second, it

2.3. CUMULATIVE PROSPECT THEORY 15

allows different decision weights for gains and losses. Furthermore in the version of cumulative prospect theory it no longer violets stochastic dominance. Therefore, cumulative prospect theory has nowadays become one of the most famous version for investors to make a decision under risk.

CHAPTER 3

Comparison of Optimization in the Sense of Expected

Utility Theory and Cumulative Prospect Theory I:

A Model without Transaction Cost

Since expected utility theory can not provide an adequate description of individ-ual choice, Kahneman and Tversky proposed cumulative prospect theory to explain the major violations of expected utility theory in choices between risky prospects.

In this chapter, we interest in seeking out optimal strategies, which make us to gain the maximal profit, in the sense of expected utility theory and cumulative prospect theory, respectively. Our main goal is to compare the optimal strategies in the sense of expected utility theory and cumulative prospect theory. Moreover, we also discuss the difference of optimal hedging strategies in these two senses.

3.1. Optimal Trading Strategy in One Period Model

A fundamental problem in the financial mathematics is to find out the opti-mal trading strategies which can reach the maxiopti-mal profit. In this section, we are interesting in comparing the optimal trading strategies under two different senses: expected utility theory and cumulative prospect theory.

In the beginning, we set up the market model as follow: Consider a one-period market model in which time points are denoted by 0 and 1, and the market model has one risky asset (stock) and one riskless asset (bond). In time 1 there exist two market states: ω1 and ω2, with probability p1and p2, respectively. The current bond

price B0 is 1, and the stock price S0 is s, and in time 1, the stock price denoted by 17

S1 is given by S1(ω1) = l and S1(ω2) = u where l < s < u.

Suppose that the interest rate at time 1 is 0, and initial wealth is x.

Let h = (h0, h1) be the trading strategy at time 0, where h0 and h1 means the

number of shares invested in bond and stock, respectively. And denote the optimal strategy by h∗ = (h∗₀, h∗₁).

Theorem 1. In the sense of expected utility theory, if the mean of gains is greater than the mean of losses, the optimal inveatment amount of stock is greater than 0, i.e., h∗₁ > 0 provided that p2(u − s) > p1(s − l). Moreover, the converse is

also true. That is, h∗₁ > 0 if and only if p2(u − s) > p1(s − l).

PROOF. Without loss of generality, suppose that the initial wealth is equal to 0.

In order to get the optimal trading strategy under expected utility theory, we must to find the strategy h = (h0, h1) such that f (h0, h1) = p1U (h0+ lh1) + p2U (h0+ uh1),

reaches the maximal value, subjected to h0 + sh1 = 0, and where U is a concave

function. Thus we can transform f (h0, h1) into

f (h1) = p1U ((l − s)h1) + p2(U (u − s)h1),

then by first order derivative, we have

f0(h1) = p2(u − s)U0((u − s)h1) − p1(s − l)U0((l − s)h1).

Thus, if f0(h1) = 0, p2(u − s) p1(s − l) = U 0_{((l − s)h} 1) U0_{((u − s)h} 1) . Since p2(u − s) > p1(s − l), we get U0((l − s)h1) U0_{((u − s)h} 1) > 1.

Hence h1 > 0 because of the property of concave function U . Moreover by second

order derivative we acquire

3.1. OPTIMAL TRADING STRATEGY IN ONE PERIOD MODEL 19

Therefore the optimal trading strategy is obtained as h∗₁ > 0.

Next, we want to show that if h∗₁ > 0 then p2(u − s) > p1(s − l). Suppose not,

if p2(u − s) ≤ p1(s − l) and h1 > 0, then

f0(h1) = p2(u − s)U0((u − s)h1) − p1(s − l)U0((l − s)h1) ≤ 0.

This implies that f (h1) is a nonincreasing function, thus h1 > 0 can not be the

optimal trading strategy which contradicts to h∗₁ > 0. Therefore if optimal trading strategy occurs when h∗₁ > 0 then p2(u − s) > p1(s − l). We complete the proof.

If a investor makes the decision under expected utility theory, he prefers risky asset to riskless asset only as the mean of gains is greater than the mean of losses. After that we take cumulative prospect theory into account, and assume that the probability weighting functions of gains and losses are the same with form

w+(p) = w−(p) = p

r

(pr_{+ (1 − p)}r₎1r

.

Theorem 2. In the version of cumulative prospect theory, if

(3.1) (w(p2)(u − s) − λw(p1)(s − l))(λw(p1) − w(p2)) > 0

holds, the optimal amount of stock is greater than 0. In fact, the converse is also true, i.e., h∗₁ > 0 if and only if (3.1) must be hold.

Remark 1. We can separate (3.1) into two cases:

(3.2) w(p2)(u − s) > λw(p1)(s − l), where λw(p1) w(p2) > 1 and (3.3) w(p2)(u − s) < λw(p1)(s − l), where λw(p1) w(p2) < 1.

PROOF. (Proof of theorem 2) Without loss of generality, suppose that the initial

acquire the optimal trading strategy, we need to find out the strategy h = (h0, h1)

such that

f (h1) = w(p1)v−((l − s)h1) + w(p2)v+((u − s)h1)

reaches the maximal value, where w is a probability weighting function and v−, v+

are the value functions of gains and losses, respectively. That is, v− is a convex function and v+ is a concave function. Use the property of value function, we can get

f (h1) = −λw(p1)v+((s − l)h1) + w(p2)v+((u − s)h1).

For simplicity, we denote v+_{= v. Then by first order derivative, we have}

f0(h1) = −λw(p1)(s − l)v0((s − l)h1) + w(p2)(u − s)v0((u − s)h1),

and f0(h1) = 0, which implies

(3.4) λw(p1)(s − l) w(p2)(u − s) = v 0_{((u − s)h} 1) v0_{((s − l)h} 1) . If (3.2) holds , v0((u − s)h1) v0_{((s − l)h} 1) < 1.

This indicates that (u − s)h1 > (s − l)h1. Moreover, due to u − s > s − l, we can

get h1 > 0. Use the same argument we have if (3.3) holds, h1 > 0. Finally, we can

check that

f00(h∗₁) = −λw(p1)(s − l)2v00((s − l)h∗1) + w(p2)(u − s)2v00((u − s)h∗1) < 0,

where h∗₁ satisfies the equation (3.4). Thus such h∗₁ is the optimal trading strategy. Conversely, since h∗₁ > 0 is the optimal strategy, h∗₁ must satisfies the equation

λw(p1)(s − l) w(p2)(u − s) = v 0_{((u − s)h}∗ 1) v0_{((s − l)h}∗ 1) .

3.1. OPTIMAL TRADING STRATEGY IN ONE PERIOD MODEL 21

Case 1: If u − s > s − l holds, v0((u − s)h∗₁) < v0((s − l)h∗₁) must be true. This implies

λw(p1)(s − l) < w(p2)(u − s).

Case 2: If u − s < s − l holds, v0((u − s)h∗₁) < v0((s − l)h∗₁) must be true. This implies

λw(p1)(s − l) > w(p2)(u − s).

We complete the proof.

In this theorem, we have a result that if a investor makes a decision in the sense of cumulative prospect theory, he is willing to buy the stock only when (3.1) holds. However, it is independent of the form of vale function.

Next, we give a one-period model example that was given as above. Our main goal is that find out the optimal strategies in the senses of expected utility theory and cumulative prospect theory. The results of this example support Theorem 1 and Theorem 2, stated before.

Example 1. Consider the market model, sat up as before in the beginning of

this section. For simplicity we denote it by b =    1 s    and D =    1 1 l u    where b

and D are called price vector and payoff matrix, respectively. Moreover, let p1 = p

and then p2 = 1 − p.

Suppose that the value function which an investor takes is given by

v(x) =        1 − exp(−θx) x ≥ 0 −λ(1 − exp(θx)) x < 0

Furthermore we assume that the probability weighting functions of gains and losses are the same, with form

w+(p) = w−(p) = p

r

In the sense of expected utility theory, we want to find out the strategy h = (h0, h1)

such that

f (h0, h1) = p(1 − exp(−θ(h0+ lh1))) + (1 − p)(1 − exp(−θ(h0+ uh1))),

to reach the maximal value that subjects to h0+ sh1 = 0. It is not difficult to get

that h1 = 1 θ(u − l)ln (1 − p)(u − s) p(s − l) .

By second order derivative test, we have

f00(h1) = −θ2(p(s − l)2exp(θh1(s − l)) + (1 − p)(u − s)2exp(−θh1(u − s))) < 0.

Therefore, the optimal trading strategy is

h∗ = (− s θ(u − l)ln (1 − p)(u − s) p(s − l) , 1 θ(u − l)ln (1 − p)(u − s) p(s − l) ).

And we can find out that if (1 − p)(u − s) > p(s − l) holds, the optimal trading strategy h∗₁ > 0, that is, an investor is willing to buy the stock if and only if the average of gains is greater than that of losses.

Next, we consider the same market model under cumulative prospect theory. Since in the version of cumulative prospect theory, it is defined over gains and losses relative to a specific reference point instead of final wealth. In this case, the optimal trading strategy is h = (h0, h1) such that

f (h1) = c(p)(1 − p)r(1 − exp(−θ(u − s)h1)) − λc(p)pr(1 − exp(θ(l − s)h1)),

where

c(p) = 1

(pr_{+ (1 − p)}r₎1_r,

to reach the maximal value that constrains to h0+sh1 = 0. By first order derivative,

we have

3.1. OPTIMAL TRADING STRATEGY IN ONE PERIOD MODEL 23

and if f0(h1) = 0 implies

ln (1 − p)r(u − s) − θ(u − s)h1 = ln λpr(s − l) − θ(s − l)h1.

Thus the optimal trading strategy is

h∗ = (− s θ(u + l − 2s)ln (1 − p)r(u − s) λpr_{(s − l)} , 1 θ(u + l − 2s)ln (1 − p)r(u − s) λpr_{(s − l)} ).

And we can find out that if

(3.5) ((1 − p)r(u − s) − λpr(s − l))(λpr− (1 − p)r_{) > 0}

holds, h∗₁ > 0. In this case, it is easy to check f00(h∗₁) < 0. Remark 2. If (3.5) holds, one of

(1 − p)r(u − s) − λpr(s − l) > 0 where λp r (1 − p)r > 1 or (1 − p)r(u − s) − λpr(s − l) < 0 where λp r (1 − p)r < 1 must be true.

According to this example, the results support the above two theorems. For investor who uses the version of expected utility theory to make a decision, he is willing to buy the risky asset if (1 − p)(u − s) > p(s − l) holds. In the sense of cumulative prospect theory, if (3.5) holds, the investor is willing to buy the risky asset more than riskless asset. Moreover, if we know the form of value function and probability weighting function that the investor takes, we can compute the optimal trading strategy. Then we want to find out the relation of optimal trading strategies in the sense of expected utility theory and cumulative prospect theory.

Remark 3. Suppose that we take the probability weighting function with the form given by

w+(p) = w−(p) = p

r

where r < 1, and assume that the sensitivity of losses is about 2.25, which proposed

by Kahneman and Tversky. If 1

3 < p < 1 2 holds, we have λw(p) w(1 − p) = λ( p 1 − p) r > 1.

In this situation, if w(1 − p)(u − s) > λw(p)(s − l) holds, (1 − p)(u − s) > p(s − l) must hold at the same time.

The main result is that if w(1 − p)(u − s) > λw(p)(s − l) where 1

3 < p < 1 2, an investor is willing to buy the risky asset in the sense of expected utility theory. That is, an investor is willing to buy the stock in these two senses.

Up to now we only consider the market model with two states space, then we take four states market into account. And give an example as following.

Example 2. Suppose the market model is similar to the above example, that the value function and probability weighting function are the same as above, respectively. But in this example we may assume that there are four states in time 1, called ω1,

ω2, ω3 and ω4, with probability p1, p2, p3 and p4, respectively. The stock price at

time 0 is S0 = s and price in time 1 is S1(ω1) = l1, S1(ω2) = l2, S1(ω3) = u1 and

S1(ω4) = u2, where l1 < l2 < s < u1 < u2.

(1) In the sense of expected utility theory, let

f (h0, h1) = p1(1 − exp(−θ(h0+ l1h1))) + p2(1 − exp(−θ(h0+ l2h1)))

+p3(1 − exp(−θ(h0 + u1h1))) + p4(1 − exp(−θ(h0+ u2h1)))

we want to find h = (h0, h1) such that f (h0, h1) reaches the maximum, which

sub-jects to h0+ sh1 = 0. Then we transfer f (h0, h1) into

3.1. OPTIMAL TRADING STRATEGY IN ONE PERIOD MODEL 25

By first order derivative we have

f0(h1) = −θseθsh1(p1e−θl1h1 + p2e−θl2h1 + p3e−θu1h1 + p4e−θu2h1)

+θeθsh1_(l

1p1e−θl1h1 + l2p2e−θl2h1 + u1p3e−θu1h1 + u2p4e−θu2h1)

= θeθsh1_((l

1− s)p1e−θl1h1+ (l2− s)p2e−θl2h1

+(u1− s)p3e−θu1h1 + (u2− s)p4e−θu2h1).

In order to obtain the extreme vale, f0(h1) = 0 must hold. In other words, if h∗1 is

optimal strategy, it needs to satisfy the equation

(u1− s)p3e−θu1h ∗ 1 _{+ (u} 2− s)p4e−θu2h ∗ 1 _{= (s − l} 1)p1e−θl1h ∗ 1 _{+ (s − l} 2)p2e−θl2h ∗ 1_.

Finally, we only need to check f00(h∗₁) < 0. Since

f00(h1) = θ2seθsh1(−θ)((l1− s)l1p1e−θl1h1 + (l2− s)l2p2e−θl2h1

+(u1− s)u1p3e−θu1h1 + (u2− s)u2p4e−θu2h1)

and l1 < l2 < s < u1 < u2, we have f00(h∗1) < 0. Therefore h ∗

1 which satisfies the

equation (u1− s)p3e−θu1h ∗ 1 + (u 2− s)p4e−θu2h ∗ 1 = (s − l 1)p1e−θl1h ∗ 1 + (s − l 2)p2e−θl2h ∗ 1

is the optimal trading strategy.

(2) Under cumulative prospect theory, we add the assumption, u1− s > s − l1, to

this market model. Our main goal is to find the strategy h = (h0, h1) such that

f (h1) = (w(p1))(−λ(1 − eθ(l1−s)h1)) + (w(p1+ p2) − w(p1))(−λ(1 − eθ(l2−s)h1))

+(w(p3+ p4) − w(p4))(1 − e−θ(u1−s)h1) + (w(p4))(1 − e−θ(l1−s)h1)

reaches the maximal value, subjected to h0+ sh1 = 0. For convenience we give some

p3, and w(p4) = p4. By first order derivative, we have

f0(h1) = λp1θ(l1− s)eθ(l1−s)h1 + λp2θ(l2− s)eθ(l2−s)h1

+p3θ(u1− s)e−θ(u1−s)h1 + p4θ(u2− s)e−θ(l1−s)h1.

Moreover, f0(h1) = 0 only if h1 satisfies the equation

p3(u1− s)e−θ(u1−s)h1+ p4(u2− s)e−θ(u2−s)h1

= λp1(s − l1)e−θ(s−l1)h1 + λp2(s − l2)e−θ(s−l2)h1.

At last, we need to check that if h∗₁ such that p3(u1− s)e−θ(u1−s)h ∗ 1 _{+ p} 4(u2− s)e−θ(u2−s)h ∗ 1 = λp1(s − l1)e−θ(s−l1)h ∗ 1 _{+ λp} 2(s − l2)e−θ(s−l2)h ∗ 1

holds, implies f00(h∗₁) < 0. By second order derivative, we have f00(h1) = θ2(λp1(s − l1)2e−θ(s−l1)h1 + λp2(s − l2)2e−θ(s−l2)h1

−p3(u1− s)2e−θ(u1−s)h1 − p4(u2− s)2e−θ(u2−s)h1).

Since u1− s > s − l1 implies s − l2 < s − l1 < u1− s < u2− s, we can get that

p3(u1 − s)2e−θ(u1−s)h1+ p4(u2− s)2e−θ(u2−s)h1

> λp1(s − l1)2e−θ(s−l1)h1 + λp2(s − l2)2e−θ(s−l2)h1,

i.e. f00(h∗₁) < 0. Therefore h∗₁ which satisfies the following equation p3(u1− s)e−θ(u1−s)h ∗ 1 + p 4(u2− s)e−θ(u2−s)h ∗ 1 = λp1(s − l1)e−θ(s−l1)h ∗ 1 + λp 2(s − l2)e−θ(s−l2)h ∗ 1

3.2. REFERENCE POINT EFFECT 27

The main point of this section is that when we are making decision under ex-pected utility theory, the optimal strategy is different from the optimal strategy in the sense of cumulative prospect theory. Under expected utility theory we only concern about the final wealth and it says that investors are risk aversion in choice between risky investments. On the other hand, in the sense of cumulative prospect theory the investor’s evaluation of risk depends on gains or losses relative to a refer-ence point; furthermore, the value function is concave for gains and convex for losses, and steeper for losses than for gains. In other words, investors are risk aversion for gains and risk seeking for losses. Further, investors are more sensitive for losses then for gains. Therefore the optimization strategy in the sense of cumulative prospect theory depends on the degree of sensitivity of losses.

3.2. Reference Point Effect

Since the investor evaluates the prospect in the sense of cumulative prospect theory depending on gains and losses relative to a reference point rather than on final wealth; moreover, the investor’s attitudes toward risk are different from gains and losses. Therefore the reference point would affect the investor’s trading strategy. Furthermore, the reference point is decided by investor’s subjective feeling, such as the past experience, and different from people to people. In this section we talk about the influence of reference point on trading strategy.

Consider a one-period market model in which time points are denoted by 0 and 1. Suppose there are two kinds of investors: A and B. Both of them take the same form of value function v(x) which is concave for gains and convex for losses, and x is the wealth change. Assume that before time 0, both A and B made wrong decisions and suffered a loss of w0. Furthermore suppose that both of them make decision

under cumulative prospect theory. However, they take different attitude toward this prior loss. Investor A only takes the current wealth X0 into account and takes X0

up for the prior loss with gains in time 1. Hence investor B takes X0+ w0 as the

reference point.

In such financial market model, there exists a simple lottery, denoted by (−x, p; y, 1− p), where x > 0, y > 0, and 0 < p < 1. This means that the lottery has probability p to lose x, and gain y with probability 1 − p.

Then the value of this lottery for investor A, who takes the current wealth as the reference point is

F = w(p)v(−x) + w(1 − p)v(y),

where w(p) is a probability weighting function. If investor A does not take any action, his wealth change is 0 and thus v(0) = 0.

For investor B who takes the prior loss into account, in other words the reference point, he takes, is X0+ w0. The value of this lottery for investor B is

F−w0 = w(p)v(−x − w0) + w(1 − p)v(y − w0) − v(−w0) if y − w0 ≥ 0,

or

F−w0 = w(p)v(−x − w0) + (1 − w(p))v(y − w0) − v(−w0) if y − w0 < 0,

where v(−w0) is the value when investor B does not take any action.

The main point of this section is to compare the optimal amount of lotteries that investor A and investor B are willing to hold, respectively.

Theorem 3. If the gain of lottery is greater than two times of the prior loss, the value of this lottery for investor B is greater than for investor A. That is, if y > 2w0

holds, we can get F−w0 > F .

PROOF. Since y > 2w0, we have

3.2. REFERENCE POINT EFFECT 29

And because of the property of probability weighting function

w(p) + w(1 − p) ≤ 1,

we can get

F−w0 ≥ w(p)v(−x − w0) + w(1 − p)v(y − w0) − (w(p) + w(1 − p))v(−w0)

= w(p)(v(−x − w0) − v(−w0)) + w(1 − p)(v(y − w0) − v(−w0)).

Without loss of generality, let v(0) = 0. If w0 ≤ x, we can get

v(−x − w0) − v(−w0) − v(−x) = v(−x − w0) − v(−x) − v(−w0) + v(0).

By mean value theorem, there exist c1, c2, where −x − w0 < c1 < −x and −w0 <

c2 < 0 such that

v(−x − w0) − v(−x) = (−w0)v0(c1)

and

v(−w0) − v(0) = (−w0)v0(c2).

Since w0 ≤ x implies c2 > c1 and the value function is convex for losses, i.e. v0 > 0

and v00 > 0, we can acquire

v(−x − w0) − v(−x) − v(−w0) + v(0) = w0(v0(c2) − v0(c1)) > 0.

Therefore we have the main result

v(−x − w0) − v(−w0) > v(−x).

If w0 > x, we have

Buy the same way, we can find c∗₁ and c∗₂, where −x − w0 < c∗1 < −w0 and −x <

c∗₂ < 0, such that

v(−x − w0) − v(−w0) = (−x)v0(c∗1)

and

v(−x) − v(0) = (−x)v0(c∗₂)

Since w0 > x and the value function is convex for losses, we obtain

v(−x − w0) − v(−w0) > v(−x).

Next use the similar argument and loss aversion property, −v(−w0) ≥ v(w0), we

have

v(y − w0) − v(−w0) − v(y) = v(y − w0) − v(y) − v(−w0)

≥ v(y − w0) − v(y) + v(w0) = v(y − w0) − v(y) + v(w0) − v(0)

= v0(c3)(−w0) + v0(c4)(−w0) = w0(v0(c4) − v0(c3)),

where y − w0 < c3 < y and 0 < c4 < w0. Owing to y > 2w0 we have c4 < c3, and

besides v is concave for gains. Then we can get

v(y − w0) − v(−w0) > v(y).

Therefore

F−w0 = w(p)v(−x − w0) + w(1 − p)v(y − w0) − v(−w0)

≥ w(p)v(−x) + w(1 − p)v(y) = F.

We complete the proof.

The main point of this theorem is that if the gain of the lottery is large enough, the value of this lottery for investor B who does not want to accept the reality of the prior loss is greater than for investor A who accepts the reality and takes the current wealth as the reference point.

3.2. REFERENCE POINT EFFECT 31

Then we interest in seeking out the difference of optimal trading strategies for investor A and investor B that takes different reference point into consideration.

Suppose that h∗ > 0 is the optimal amount of risky assets that investor A is willing to buy, and h∗_−w0 is the optimal amount of risky assets that investor B is willing to hold. Our main goal is to compare the number of h∗ and h∗_−w0. Before this we first give a lemma which describes one property of h∗_−w

0. The statement is

as following.

Lemma 1. If the optimal amount of risky asset that investor B is willing to hold is greater than 0, i.e. h∗_−w

0 > 0, −w0+ yh

∗

−w0 ≥ 0 must hold.

PROOF. Suppose that −w0 + yh∗−w0 < 0 holds, and we define ∆ > 0 to be a

small unit of asset such that yh∗_−w0 + ∆y − w0 < 0. Then we compare the profit of

portfolio h∗_−w

0 − ∆ with h

∗

−w0. Since h

∗

−w0 is the optimal strategy, we have

w(p)v(−xh∗_−w0 + ∆x − w0) + (1 − w(p))v(yh∗−w0 − ∆y − w0)

≤ w(p)v(−xh∗_−w0 − w0) + (1 − w(p))v(yh∗−w0 − w0)

By rearrangement, we get

(1 − w(p))(v(yh∗_−w0− w0) − v(yh∗−w0 − ∆y − w0))

≥ w(p)(v(−xh∗_−w0 + ∆x − w0) − v(−xh∗−w0 − w0)).

Because yh∗_−w

0 + ∆y − w0 < 0 and v(x) is a convex function defined on x < 0, by

the property of convex function and mean value theorem we have

v(yh∗_−w

0 − w0) − v(yh

∗

−w0 − ∆y − w0)

≤ v(yh∗−w0 + ∆y − w0) − v(yh

∗

and

v(−xh∗_−w0 + ∆x − w0) − v(−xh∗−w0 − w0)

≥ v(−xh∗_−w0 − w0) − v(−xh∗−w0 − ∆x − w0).

So combining the above equations, we have

(1 − w(p))(v(yh∗_−w0+ ∆y − w0) − v(yh∗−w0 − w0))

≥ (1 − w(p))(v(yh∗_−w 0− w0) − v(yh ∗ −w0 − ∆y − w0)) ≥ w(p)(v(−xh∗_−w 0 + ∆x − w0) − v(−xh ∗ −w0 − w0)) ≥ w(p)(v(−xh∗_−w 0 − w0) − v(−xh ∗ −w0 − ∆x − w0)). Therefore we get w(p)v(−xh∗_−w 0 − ∆x − w0) + (1 − w(p))v(yh ∗ −w0 + ∆y − w0) ≥ w(p)v(−xh∗ −w0 − w0) + (1 − w(p))v(yh ∗ −w0 − w0).

That is, investor B will prefer (h∗_−w

0+ ∆) to h

∗

−w0, which contradict to the fact that

h∗_−w0 is the optimal amount of risky asset that investor B is willing to hold.

Then we can get the main point of this section, and use the above lemma to complete the proof of the following theorem.

Theorem 4. If investor B is willing to buy a certain number of security, the optimal amount of risky asset that investor B is willing to hold is no less than that investor A is willing to hold, i.e., if h∗−w0 > 0, then h

∗

−w0 ≥ h

∗

.

PROOF. Suppose that h∗−w0 < h

∗ _{hold. Since h}∗

−w0 > 0, by the above lemma we

get −w0+ yh∗−w0 ≥ 0, and implies −w0+ yh

∗ _{≥ 0. Because h}∗ _{is optimal amount of}

risky security that investor A is willing to hold, we have the following equation:

w(p)v(−xh∗) + w(1 − p)v(yh∗) ≥ w(p)v(−xh∗−w0) + w(1 − p)v(yh

∗ −w0).

3.2. REFERENCE POINT EFFECT 33

Rearranging the above equation we can get

w(1 − p)(v(yh∗) − v(yh∗_−w0)) ≥ w(p)(v(−xh∗_−w0) − v(−xh∗)). If yh∗− w0 ≥ yh∗−w0 holds, we have v(yh∗_−w 0) − v(yh ∗ −w0 − w0) > v(yh ∗ ) − v(yh∗− w0),

because of the property of value function of gains. This implies

(3.6) v(yh∗) − v(yh∗_−w0) < v(yh∗− w0) − v(yh∗−w0 − w0).

If yh∗− w0 < yh∗−w0 holds, by mean value theorem and concave function v, we have

v(yh∗) − v(yh∗_−w

0) < v(yh

∗_{− w}

0) − v(yh∗−w0 − w0).

Due to the similar argument and the property of value function of losses, we can obtain (3.7) v(−xh∗_−w 0) − v(−xh ∗_{) > v(−xh}∗ −w0 − w0) − v(−xh ∗_{− w} 0).

By combining equation (3.6) and equation (3.7), we obtain

w(1 − p)(v(yh∗− w0) − v(yh∗−w0 − w0)) ≥ w(1 − p)(v(yh∗) − v(yh∗_−w0)) ≥ w(p)(v(−xh∗_−w0) − v(−xh∗)) ≥ w(p)(v(−xh∗_−w 0 − w0) − v(−xh ∗_{− w} 0))

Therefore we can get

w(p)v(−xh∗− w0) + w(1 − p)v(yh∗− w0)

≥ w(p)v(−xh∗_−w0 − w0) + w(1 − p)v(yh∗−w0 − w0),

which contradicts to the optimal amount of risky asset, h∗_−w

0, which investor B is

The main point of this theorem is that if the investor who takes the prior loss into account, his attitude toward risk become risk seeking. In other words, if the reference point that investor B takes is greater than that investor A takes, investor B tends to increase the amount of risky asset which he holds.

In this section we know that the attitude of an investor toward risk may become risk seeking, if he takes the prior loss into account and wants to make up for the prior loss. That is, if an investor who does not accept the prior loss and takes X0 + w0

as the reference point, risk seeking preference would play a dominant role when he evaluates the lottery (−x, p; y, 1 − p).

3.3. Optimal Hedging Strategy in One Period Model

In this section, we compare the hedger’s optimal strategy in the future market in the sense of expected utility theory and cumulative prospect theory. Furthermore, we suppose that at time 0 the hedger made wrong decisions and suffered a loss of w0. Under expected utility theory the investor evaluates the prospect depending on

the final wealth. But in the sense of cumulative prospect theory the investor may not accept the prior loss and then takes w0 as reference point.

First, we set up the market model as following: Consider a one period model in which time points are denoted by 0 and 1. At time 1, there are two market states, denoted by w1 and w2, with probabilities p and 1 − p, respectively. The current spot

price is S0 = s, and at time 1, the spot price is given by S1(w1) = u and S1(w2) = l,

where l < s < u. Suppose the futures market is unbiased and there exists no basis risk at time 1, thus the futures prices in time 0 and 1 are F0 = E(S1) and F1 = S1,

respectively. Moreover, we suppose that the form of value function which the hedger takes is given by v(x) =        1 − exp(−θx) x ≥ 0 −λ(1 − exp(θx)) x < 0

3.3. OPTIMAL HEDGING STRATEGY IN ONE PERIOD MODEL 35

where θ > 0 and λ > 1.

Suppose the position that the hedger holds in the futures market is h. The wealth change in time 1 is X1 = S1− s + h(F1− F0), i.e.,

X1 =       

u − s + h(1 − p)(u − l) = wu with probability p

l − s + h(−p)(u − l) = wl with probability 1-p

Since wu− wl = (u − l) + h(u − l) holds, we have

       wu ≤ wl if h ≤ −1 wu ≥ wl if h ≥ −1 .

Moreover, we can obtain E(S1) = pwu+ (1 − p)wl.

In the beginning, we talk about the hedger’s optimal strategy h∗ in the sense of expected utility theory.

Theorem 5. Under expected utility theory, full hedging is the optimal trading strategy for a hedger. That is, h = −1 is the utility maximizing point.

PROOF. In the sense of utility theory, our main goal is to find out h∗ such that

p(1 − exp(−θwu)) + (1 − p)(1 − exp(−θwl)), denoted by f (h), reaches maximum. By

first order derivative, we get

f0(h) = (−p)(−θ)(1 − p)(u − l) exp(−θwu) − (1 − p)(−θ)(−p)(u − l) exp(−θwl)

= p(1 − p)θ(u − l)(exp(−θwu) − exp(−θwl))

In order to let f0(h) = 0 hold, we have wu = wl. This implies that h must be −1.

Moreover, by second order derivative we can get

f00(h) = p(1 − p)θ(u − l)(−θ(1 − p)(u − l) exp(−θwu) − θp(u − l) exp(−θwl))

Therefore, h = −1 is the optimal strategy for the hedger in the sense of expected utility theory.

When the investor evaluates the value of prospects in the sense of expected utility theory, he would accept the prior loss and only cares about the final wealth. Besides, his attitude toward risk is risk aversion. Hence, under expected utility theory the optimal strategy for the hedger is full hedging.

Unlike conventional expected utility theory, cumulative prospect theory replaced the utility function with the value function, v(x). Moreover, it used decision weight-ing function, π(p), instead of probability measure. In the followweight-ing we discover that under cumulative prospect theory the optimal strategy for the hedger is much more complicated.

Theorem 6. Under cumulative prospect theory, the optimal strategy for the hedger is as following: (1) In the case h ≤ −1, (i) if E(S1) > w0, h∗ =        −1 − 1 θ(u−l) ln pw(1−p) (1−p)(1−w(1−p)) when w(1 − p) ≥ (1 − p) −1 when w(1 − p) < (1 − p) (ii) if E(S1) ≤ w0, h∗ =          l − w0 p(u − l) when θ(−2w0+ wu+ wl) + ln λ(1 − p)w(p) pw(1 − p) ≥ 0 −∞ when θ(−2w0+ wu+ wl) + ln λ(1 − p)w(p) pw(1 − p) < 0 . (2) In the case h ≥ −1, (i) if E(S1) > w0, h∗ =        −1 + 1 θ(u−l)ln (1−p)w(p) p(1−w(p)) when w(p) ≥ p −1 when w(p) < p

3.3. OPTIMAL HEDGING STRATEGY IN ONE PERIOD MODEL 37 (ii) if E(S1) ≤ w0, h∗ =          w0− u (1 − p)(u − l) when θ(−2w0+ wu+ wl) + ln λpw(1 − p) (1 − p)w(p) ≥ o ∞ when θ(−2wo+ wu+ wl) + ln λpw(1 − p) (1 − p)w(p) < 0 .

PROOF. Suppose that h ≤ −1 and given that E(S1) ≤ w0. When h ∈ {h |

wl− w0 ≤ 0},

F−w0 = w(p)(−λ(1 − exp(θ(wu− w0)))) + (1 − w(p))(−λ(1 − exp(θ(wl− w0))))

+λ(1 − exp(−θw0))

= −λ(w(p) + w(p) exp(θ(wu− w0)) + (1 − w(p))) + λ((1 − w(p)) exp(θ(wl− w0)))

+λ(1 − exp(−θw0))

By first order derivative, we have

dF−w0

dh = λw(p)θ(1 − p)(u − l) exp(θ(wu− w0)) + λ(1 − w(p))θ(−p)(u − l) exp(θ(wl− w0)) = λθ(u − l) exp(−θw0)((1 − p)w(p) exp(θwu) − p(1 − w(p)) exp(θwl)).

Moreover, in order to guarantee dF−w0

dh = 0, h must satisfies the following equation

(1 − p)w(p) exp(θ(u + h(1 − p)(u − l))) = p(1 − w(p)) exp(θ(l − hp(u − l))).

And this implies

(3.8) h = −1 − 1

θ(u − l)ln

(1 − p)w(p) p(1 − w(p)).

However, by second order derivative, we can obtain

d2F−w0

dh2 = λθ(u−l) exp(−θw0)((1−p)

2_{w(p)θ(u−l) exp(θw}

u)+p2(1−w(p))θ(u−l) exp(θwl)) ≥ 0.

Thus (3.8) is the utility minimizing point. Moreover, we can get that the maximum of F−w0 in the set {h | wl− w0 ≤ 0} is reached on the boundary. When h ∈ {h |

wl− w0 ≥ 0},

F−w0 = w(p)(−λ(1 − exp(θ(wu− w0)))) + w(1 − p)(1 − exp(−θ(wl− w0)))

+λ(1 − exp(−θw0))

= −λw(p)(1 − exp(θ(wu− w0))) + w(1 − p)(1 − exp(−θ(wl− w0)))

+λ(1 − exp(−θw0)).

By first order derivative, we get

dF−w0

dh = θ(u − l)(λ(1 − p)w(p) exp(θ(wu− w0)) − pw(1 − p) exp(−θ(wl− w0))).

Case 1: When

(3.9) θ(−2w0+ wu+ wl) + ln

λ(1 − p)w(p)

pw(1 − p)) ≥ 0,

dF−w0

dh ≥ 0, that is, F−w0 is increasing corresponding to h. Thus the maximum of

F−w0 in the set {h | wl− w0 ≥ 0} is reached on the boundary wl− w0 = 0. In other

words, the optimal hedging strategy is

h∗ = l − w0 p(u − l). Case 2: When (3.10) θ(−2w0+ wu+ w0) + ln λ(1 − p)w(p) pw(1 − p) < 0,

F−w0 is decreasing corresponding to h. Thus the maximum of F−w0 in the set

3.3. OPTIMAL HEDGING STRATEGY IN ONE PERIOD MODEL 39

Given that E(S1) ≥ w0. When h ∈ {h | wu− w0 ≥ 0},

F−w0 = w(1 − p)(1 − exp(−θ(wl− w0))) + (1 − w(1 − p))(1 − exp(−θ(wu− w0)))

+λ(1 − exp(−θw0))

= −w(1 − p) exp(−θ(wl− w0)) + 1 − exp(−θ(wl− w0)) + w(1 − p) exp(−θ(wu− w0))

+λ(1 − exp(−θw0)).

By first order derivative, we acquire dF−w0

dh = θ(u−l)(−pw(1−p) exp(−θ(wl−w0))+(1−p)(1−w(p)) exp(−θ(wu−w0))).

In order to ensure dF−w0

dh = 0, h must satisfies the following equation

pw(1 − p) exp(−θ(l − hp(u − l))) = (1 − p)(1 − w(1 − p)) exp(−θ(u + h(1 − p)(u − l))).

This equation implies

h = −1 − 1

θ(u − l)ln

pw(1 − p)

(1 − p)(1 − w(1 − p)).

In order to ensure such h ≤ −1, we must add a condition w(1 − p) ≥ (1 − p). Moreover, by second order derivative we have

d2_F −w0

dh2 = −θ(u − l)(p

2_{w(1 − p)θ(u − l) exp(−θ(w}

l− w0)))

−θ(u − l)((1 − p)2_{(1 − w(1 − p))θ(u − l) exp(−θ(w}

u− w0)))

≤ 0.

Therefore the optimal strategy for the hedger is

(3.11) h∗ = −1 − 1

θ(u − l)ln

pw(1 − p) (1 − p)(1 − w(1 − p))

as w(1 − p) ≥ (1 − p). However, if w(1 − p) < (1 − p) hold, optimal hedging strategy is reached on the boundary h ≤ −1. In other words, optimal hedging strategy is

Suppose h ≥ −1 and given that E(S1) ≤ w0. When h ∈ {h | wu− w0 ≤ 0},

F−w0 = w(1 − p)(−λ(1 − exp(θ(wl− w0)))) + (1 − w(1 − p))(−λ(1 − exp(θ(wu− w0))))

+λ(1 − exp(−θw0)).

By first order derivative, we obtain

dF−w0

dh = λθ(u − l) exp(−θw0)((1 − p)(1 − w(1 − p)) exp(θwu) − pw(1 − p) exp(θwl)), and dF−w0 dh = 0 only if (3.12) h = −1 + 1 θ(u − l)ln pw(1 − p) (1 − p)(1 − w(1 − p))

holds. In addition, by second order derivative we get

d2F−w0

dh2 = λθ

2_{(u − l)}2_{((1 − p)}2_{(1 − w(1 − p)) exp(θw}

u) + p2w(1 − p) exp(θwl)) ≥ 0.

Hence, (3.12) is utility minimizing point. So we can get that the maximum of F−w0

in the set {h | wu− w0 ≤ 0} is reached on the boundary, that is, wu− w0 = 0, which

implies that the optimal strategy for the hedger is

h∗ = w0− u

(1 − p)(u − l).

When h ∈ {h | wu− w0 ≥ 0},

F−w0 = w(1 − p)(−λ(1 − exp(θ(wl− w0)))) + w(p)(1 − exp(−θ(wu − w0)))

+λ(1 − exp(−θw0)).

By first order derivative, we have

dF−w0

3.3. OPTIMAL HEDGING STRATEGY IN ONE PERIOD MODEL 41

Case 1. When

(3.13) θ(−2w0+ wu + wl) + ln

λpw(1 − p) (1 − p)w(p) ≥ 0,

we know that F−w0 is a decreasing function corresponding to h. Therefore, the

maximum of F−w0 in the set {h | wu − w0 ≥ 0} is reached on the boundary, which

means that

h∗ = w0− u

(1 − p)(u − l)

is the optimal strategy for the hedger in the futures market. Case 2. When

(3.14) θ(−2w0+ wu+ wl) + ln

λpw(1 − p) (1 − p)w(p) < 0,

we get that F−w0 is an increasing function corresponding to h. Hence the maximum

of F−w0 in the set {h | wu− w0 ≥ 0} is h

∗ _{= ∞.}

Given that E(S1) ≥ w0 and h ∈ {h | wl− w0 ≥ 0}, then we have

F−w0 = w(p)(1 − exp(−θ(wu− w0))) + (1 − w(p))(1 − exp(−θ(wl− w0)))

+λ(1 − exp(−θw0)).

By first order derivative, we obtain that dF−w0

dh = 0 implies

h = −1 + 1

θ(u − l)ln

(1 − p)w(p) p(1 − w(p)).

In order to ensure such h ≥ −1, w(p) ≥ p must hold. Then by second order derivative, we get d2F−w0 dh2 = −θ 2_(u−l)2_exp(θw 0)((1−p)2w(p) exp(−θwu)+p2(1−w(p)) exp(−θwl)) ≤ 0. Therefore (3.15) h∗ = −1 + 1 θ(u − l)ln (1 − p)w(p) p(1 − w(p))

is the optimal strategy for the hedger in the futures market as w(p) ≥ p. However, if w(p) < p holds, optimal hedging strategy is h∗ = −1.

According to this theorem, we obtain that under cumulative prospect theory there exists three cases. If the prior losses w0 are not sufficiently large, w0 < ES1,

the optimal strategy for the hedger is h∗ given in (3.11) in the case h ≤ −1 and w(1 − p) ≥ (1 − p), and in the case h ≥ −1 and w(p) ≥ p the optimal strategy h∗ of the form (3.15). If the losses before time 0 satisfy (3.9) or (3.13) , the investor will change his hedging strategy. If the prior losses are sufficiently large, and (3.10) or (3.14) holds, the investor will go crazy, and take large positions showing no consideration of risks.

Corollary 1. Optimal strategy for the hedger in the sense of expected utility theory is full hedging. Besides, under expected utility theory the optimal hedging strategy does not depend on the prior losses. On the other hand, in the view of cumulative prospect theory the optimal strategy for the hedger is more complicated and is related to the losses before time 0.

CHAPTER 4

Comparison of Optimization in the Sense of Expected

Utility Theory and Cumulative Prospect Theory II:

A Model with Transaction Cost

In this chapter we introduce trading strategy with transaction costs in one pe-riod market model, which is derived from the model specified by Kabanov (2002). Suppose that the financial market is one period model, and our portfolio is (h0, h1),

which means that the number of shares of assets invested in bond and stock, respec-tively. Moreover, we assume that the initial wealth is x0 = h0+ sh1, composed of

bond and stock.

Suppose that the investor need to pay the transaction costs when they sell the stock, and consider the model with constant proportional transaction costs, denoted by λ. Therefore, we find out that if we buy the stock at time 0, the value of the portfolio at time 0 after trading is v0 = h0+ sh1, and in this situation h1 > h1 must

hold. If we sell the stock at time 0, the value of the portfolio at time 0 after trading is v₀∗ = h0+ sh1− λs(h1− h1), and in this situation h1 < h1 must hold.

In the following of this section we assume that in this market model our portfolio values are only affected by the asset price fluctuation and transaction costs. Thus we get v0 = x0 and v∗0 = x0− λs(h1 − h1). Our main task is to find out optimal

trading strategy with transaction costs in the sense of expected utility theory and cumulative prospect theory, respectively.

Example 3. (Constant absolute risk aversion CARA)

Under expected utility theory, we consider the risk-averse utility function given by

U (x) = 1 − exp(−θx),

where θ > 0 is absolute risk aversion and be a constant. Moreover, the market model is sat up as the first section in chapter 3.

(1)If we buy the stock at time 0, then the final wealth is

w =        wl = x0+ h1(l − s) with probability p wu = x0+ h1(u − s) with probability 1 − p.

Under expected utility theory, we have to maximize the function f (h1), defined by

f (h1) = pU (x0+ h1(l − s)) + (1 − p)U (x0+ h1(u − s)),

to get the optimal strategy h∗₁. Using first order derivative, we have

f0(h1) = pθ(l − s) exp(−θ(x0+ h1(l − s))) + (1 − p)θ(u − s) exp(−θ(x0+ h1(u − s))),

and let f0(h1) = 0 we get

(4.1) h1 =

1 θ(u − l)ln

(1 − p)(u − s)

p(s − l) .

In order to guarantee h1 > 0, we should add the condition, (1 − p)(u − s) > p(s − l),

to this market model. Moreover, by second order derivative, we get

f00(h1) = −θ2(p(s − l)2exp(−θwl) + (1 − p)(u − s)2exp(−θwu)) ≤ 0.

Therefore we can get that if we buy the stock at time 0, the optimal trading strategy is

h∗₁ = 1

θ(u − l)ln

(1 − p)(u − s) p(s − l) > h1.

4. COMPARISON OF OPTIMIZATION IN THE SENSE OF EUT AND CPT II 45

(2) If we sell the stock at time 0, then the final wealth is

w =        wl = x0+ h1(l − s) − λs(h1− h1) with probability p wu = x0+ h1(u − s) − λs(h1− h1) with probability 1 − p

Suppose that l − s + λs ≤ 0. Under expected utility theory, we have to maximize the function f (h1), defined by

f (h1) = pU (x0+ h1(l − s) − λs(h1− h1)) + (1 − p)U (x0+ h1(u − s) − λs(h1− h1)),

to acquire the optimal strategy h∗∗₁ . Using first order derivative, we have f0(h1) = 0

(4.2) h1 =

1 θ(u − l)ln

(1 − p)(u − s + λs)

p(s − l − λs) .

Moreover, by second order derivative, we get

f00(h1) = −θ2(p(s − l − λs)2exp(−θwl) + (1 − p)(u − s + λs)2exp(−θwu)) ≤ 0.

Therefore we can get that if we sell the stock at time 0, the optimal trading strategy is

h∗∗₁ = 1

θ(u − l)ln

(1 − p)(u − s + λs) p(s − l − λs) < h1.

From (1), (2), we conclude that if an investor who wants to buy the stock at time 0, he may choose the strategy h∗₁ = (4.1) at time 0 to reach the maximum profit, and if an investor who wants to sell the stock at time 0 he may choose the strategy h∗∗₁ = (4.2) at time 0 to reach the maximum profit.

Remark 4. From above example, we have a result that if h1 > h∗1 , the investor

will not buy the stock at time 0, and if h1 < h∗∗1 , the investor is not willing to sell

the stock at time 0. Moreover, because of (1 − p)(u − s) > p(s − l) and

(1 − p)(u − s + λs) p(s − l − λs) >

(1 − p)(u − s)